Arash Sangari

Hasti Mirkia

Amir H Assadi

Perception of Motion and Architectural Form: Computational Relationships between Optical Flow and Perspective

Keywords: Visual perception, Vanishing point, Optical flow, MSTd, Eye movement, Architectural form

Perceptual geometry refers to the interdisciplinary research whose objectives focuses on study of geometry from the perspective of visual perception, and in turn, applies such geometric findings to the ecological study of vision. Perceptual geometry attempts to answer fundamental questions in perception of form and representation of space through synthesis of cognitive and biological theories of visual perception with geometric theories of the physical world. Perception of form, space and motion are among fundamental problems in vision science. In cognitive and computational models of human perception, the theories for modeling motion are treated separately from models for perception of form.

Introduction

Modeling visual perception of form and motion has been studied extensively with many success stories. Primate brain's perception of motion from optical flow is attributed to dynamics in networks of the visual areas V1 through MSTd [START_REF] Newsome | Deciding about motion: linking perception to action[END_REF]. In contrast, cortical processes required for perception of spatial form (e.g. as in the interiors of buildings) are still largely unknown. Computational models and theories of motion perception have not been studied in the context of perception of architectural forms; that is, shape perception and formation of the percept of planes in perspective are studied separately.

Below, we provide the theory and a model for perception of interior space in architecture. We propose the existence of neuroanatomical loci that play a pivotal role in perception of perspective, and demonstrate a mechanism for visual perception of interior spaces based on eye movement. We conclude that learning to estimate motion from optical flow is similar to learning to estimate a single vanishing point. Both perceptual tasks, therefore, potentially employ the same local circuit in the MSTd. To enhance our theory, we show that an ANN with sufficient accuracy in predicting motion from optical flow learns the tasks of detecting vanishing points more efficiently. In contrast, training a generic ANN with random initialization will not reach the corresponding level of accuracy under the same protocols. Our simulation suggests a neurobiological hypothesis for "sharing of the neuronal pathways" for detection of vanishing points and estimation of direction of motion from optical flow. More precisely, the flow of information from the retina through primary and higher areas of the human visual cortex employs plasticity of the cortical circuits in MSTd that are known to compute motion from optical flow. The "MSTd Hypothesis for Perception of Perspective" proposes that the vanishing point detecting neurons are in MSTd.

In view of the emerging advances due to understanding the role of eye movement in mental health research, we plan to develop a predictive theory for human affective response to variation of spatial features of interior spaces. This article provides the computational model and tools for such a theory.

Geometry Perception

The study of 3D perception from 2D images is the main issue of vision science in both biological and computer systems. Depending on the projection method used for mapping 3D space to a 2D plane, different images may be resulted. Among these projection methods, perspective projection is known as an accurate mapping for reconstructing geometrical features of real objects. One important concept in perspective projection is the "vanishing point". The vanishing point of a straight line under perspective projection is that point in the image beyond which the projection of the straight line cannot extend. Therefore, the vanishing points correspond to the "points at infinity" in the receding direction of the parallel straight lines in space.

Computationally, the vanishing point in interior spaces with a single vanishing point can be found by finding the point where most of the salient straight lines in the image intersect with each other. The salient straight lines can be found using processes such as edge detection and line extraction. The mechanism of edge detection in the brain is related to the lateral inhibition in the receptive field of the visual sensory system [START_REF] Palmer | Vision Science: Photons to Phenomenology[END_REF]. A sample of how edge detection occurs in an interior space is shown in Fig. 1-b.

The Gestalt principle of continuity shows how the brain can fill in the gaps between the dots in the image and extract the major lines. The Hough transform can be used to extract the slope and the constant term of lines in the image, based on black and white result of edge detection algorithms [START_REF] Assadi | Learning Gestalt of Surfaces in Natural Scenes, Neural Networks for Signal Processing IX Proceedings of the[END_REF]. The line drawings made from the interior spaces of Fig. 1-b carry pieces of information that the brain needs to estimate the orientation of surfaces and compute the perception of the interior space in perspective. By extending these lines and finding the intersection points, we can computationally find the vanishing point (Fig. 1-c).

In real word experience, additional information such as texture, color and cues provide much more useful information for richer perceptual experience of interior spaces [START_REF] Assadi | A Learning Theoretic Approach to Perceptual Geometry in Natural Scenes[END_REF][START_REF] Assadi | Geometry of the Perceptual Space[END_REF]. However, this paper focuses on simpler geometric objects as illustrated in the line drawings below.

Motion Field Estimation

Structure determination algorithms such as shape from shading are often used to model perception of 3D forms from 2D images. However, their accuracy is correlated to the complexity of the image. As it is known in neuroscience, the brain takes advantage of motion in the image plane to estimate 3D form. Similarly, additional temporal dimension could be used to design algorithms with increased accuracy in modeling 3D shape perception.

The relative motion between image plane and scene can be estimated from successive image frame gradients in time and space [START_REF] Palmer | Vision Science: Photons to Phenomenology[END_REF]. If the sequence of image frames is denoted by a 3D array ( , , ) I x y t , the projection of the relative velocity vector onto the image plane is:

0 x y I I I V V x y t ∂ ∂ ∂ + + = ∂ ∂ ∂ (1)
This equation for each pixel of the image in each frame yields a pair of relative velocity components:

( )

( , ) ( , ), ( , ) x y V m n V m n V m n = (2)
The latter pairs form a complex-valued matrix as in:

( , ) ( , ) ( , ) , 1 x y V m n V m n iV m n i = + = - (3)
Since this matrix shows the flow of image data (in pixels) motion in an image plane, in computer vision and vision science this matrix usually is called an optical flow matrix.

Eye Movement

Our long-term objective is to develop a unified dynamic framework for modeling perception of geometric forms and motions. The key biological observation is the dynamic nature of vision. Saccades are major players in visual perception of the physical world [START_REF] Martinez-Conde | The role of fixational eye movements in visual perception[END_REF]. Further, cortical neurons have receptive fields ([4] pp. 41-43), roughly, a cone-shaped region of space measured in terms of the visual angle. It is believed that the relatively limited light rays passing through that cone stimulate neurons to eventually invoke mechanisms of visual attention. The concepts of the receptive field and visual attention lead to a number of significant hypotheses concerning neuronal mechanisms for perception of regularity in the viewed images. Among them, one could hypothesize that there is a biologically realistic computational model of 'intermediate-level vision' that elucidates the process of neuronal detection of visual regularity in the presence of repetitive light patterns on the retina. This mechanism is a hybrid of hierarchical receptor field theory and visual attention, in the form of a network with multiple feedbacks.

Accordingly, a series of spike trains that encode a temporal sequence of short intervals T j preprocess the visual stimuli. The collection of T j undergo computational loops that implement an adaptive process of forming correlations among T j by improving the length of the interval T j and the choices of the subsequences (in reference to saccadic movements that need not be in a uniform order). The results of such recursive process is an adaptively computed output, which is dominated by stable and coherent patterns of dynamics that form the inputs to the higher level visual processing centers. Repetition of highly correlated spike trains result in arrival of even more highly correlated spike trains to the higher level visual centers, which in terms of Bayesian models increase the likelihood of the recognition of that common pattern, thus bringing visual attention mechanisms to take over and pursue the process of perception, cognition and interpretation/action. One may well speculate that a sequence of saccades repeatedly brings to the fovea the implicit information in the scene that translates to repetition of highly correlated spike trains in specific intervals.. We refer to the mechanism thus formulated as the Adaptive Eye Movement Hypothesis (AEMH.) Perceptual geometry investigates physiological evidence to establish the low-level computations of visual, tactile motor and auditory processes that contribute to the ability to perceive regularity in structure [START_REF] Assadi | A Learning Theoretic Approach to Perceptual Geometry in Natural Scenes[END_REF] [START_REF] Assadi | Learning Gestalt of Surfaces in Natural Scenes, Neural Networks for Signal Processing IX Proceedings of the[END_REF]. For example, perceptual geometry predicts that the human perception of form and spatial organization of planes in architectural interior space could be modeled via the computational elements that perform the local computations for reconstruction of self-motion from optical flow. Eye movement is the key ingredient in models of vision that refer to AEMH. In the context of this paper, AEMH refers to the ubiquitous and continual presence of eye movements of several sorts (the tiny, almost instantaneous jitters of the eyeballs) that provide the impressions of frames of a movie, even when we fixate and seemingly try to maintain our eyes, head and neck motion-less. Suffice it to say that according to AMEH, the perception of natural surfaces, bounding planes and other geometric elements that contribute to perception of architectural form arises from integration of the flow of sensory processes and subsequent neuronal network information processing of a temporal sequence of "single frames."

It is certainly true that saccades and other eye movements capture images that could change orientation and direction of fovea; thus, how such discontinuous current of information from scene geometry is a puzzle that could give rise to a cohesive and continuous Gestalt of the surfaces delimiting the spatial geometry. In a nutshell, the processes in the neuronal network need not follow the temporal order of saccades or other eye movements. Rather, short-distance feedback loops, dynamics of local attractors in coordinated sub-collections of local circuits, and the phenomena that are collectively referred to as "visual attention" could maintain just enough records of the "seemingly single image frames," in such a way that the processing could sustain an orderly organization of the pieces of the percept.

Therefore, our first attempt focuses on modeling adaptive processes that simulate saccades (Latimer, 1996). Below, we consider a simplified computational model that detects the Gestalt of lines in the formation of vanishing points, as in perspective drawings. This model is robust with respect to noise, partial occlusion and some irregularity in the unavailability of directly observable edges. Moreover, it lends itself to generalization to visual attention and theories of active vision. Fig. 3. Eye movement creates a sequence of snapshots (Alfred L, 1967 [START_REF] Yarbus | Eye Movements and Vision[END_REF]). In our model the tree of eye movement creates a flow of visual information comparable to the frames of a movie.

Contribution to Value Creation

Advances in computational sciences and engineering have opened many possibilities for development of robotics and automation of routine tasks. For example, robot motion planning provides myriad possibilities for transportation and delivery robots inside hospitals and other public or private places. Meanwhile, much remains toward the development of robots with higher levels of learning or proper response to human needs. In this paper, we have demonstrated that perception of architectural form (a challenging problem for robots) could be developed with the same types of computations that are needed in estimation of motion from optical flow (a problem which has made great advances.) In a broader context, the follow-up of this research will also contribute to better understanding of the human response to the architectural form, thus enabling the architects and engineers to design habitats better suited for populations with specific needs or of specific genre of affective and cognitive abilities. In summary, this research offers many possibilities through applications to industrial and scietal value creation.

Method

Below, we outline a unified framework for perception of geometry and motion in interior spaces. We designed two 3D interior spaces in a virtual reality environment. We used a PCA for reducing the dimension of successive frames of this simulation and used some of the most informative components in training an ANN (Optiflonet) for estimation of direction of movement on an image plane. Then we demonstrated that this neural network with a slightly different configuration can extract vanishing point of still images.

Construction of Simulation of Training Sets for Neural Network

In our interior model, we have simulated the sensory information by architectural rendering software: 3D Max. In order to test complex kinetics, virtual reality is used in rendering simple interior spaces (Fig. 4); we compared extraction of lines from simulated scenes versus extraction of lines in natural scenes.

Learning Motion Heading Direction from Optical Flow

When an observer moves in an interior space, optical flow provides a robust mechanism to estimate the motion and heading direction. The optical flow vectors indicate the distribution of stimulated local circuits that detect motion and heading direction.

Optical flow vectors, (the arrows in the Fig. 5) can provide the input for training our neural network Optiflonet. Validation and testing of neural networks are performed using 100*100 arrays with sparse sampling of the optical flows. As illustrated in Fig. 5, for simple interior spaces, optical flow vector can be approximated by a sparse matrix. Therefore, we use a principal component analysis for extracting the most informative components of optical flow vectors in all frames. We use the first 20 principal components of image frames in order to reduce the required number neurons in our neural network while avoiding reconstruction errors (Fig. 6 shows the relationship between number of principal components and reconstruction error in our case). Fig. 7 shows the whole structure of the Optiflonet system to estimate direction of motion on an image plane. The architecture of neural network that we used is also presented in Fig. 8. To test our theory we designed a neural network that learns to extract vanishing points from perspectives called Perspectinet. This network incorporates a simplified model of eye movement, where the scene is sampled as a flow of consecutive snapshots of interior spaces. The subsequent layer of Perspectinet receives dynamic information from the scene.

Design of Optiflonet is based on the Fig. 9. The optical flow from movement on the retina provides the input to the next layers of visual sensation and visual perception of motion.

The Proposed Model for Estimation of Information from Perspective

We trained Optiflonet using examples of motion in an interior space where each frame is a perspective projection of the interior space. The observer motion provides different projections on the retina, and consequently, differing perspectives with a possible shift of the vanishing point. The perspective could be tested in two ways. First, by turning the dynamic scene (perspective images) into movie that illustrates the effect of eye movement, the subsequent blocks will have layers of network and connection weights that are inherited from training the Optiflonet. Our problem is how to introduce and measure the performance of the Perspectinet when test images are from interior space without any training on interior space forms. In other words, the local circuits of Perspectinet in the visual motion processing layers are used as local neural circuits that are employed for computations in movies of eye movement snapshots from interior spaces forms.

There is a second method for feasibility study of the "MSTd Hypothesis". Namely, we train two neural networks and compare their performance and accuracy. The first one is the Perspectinet, and the second ANN is a copy of Perspectinet whose initial synaptic weights are determined from training as local circuits of Optiflonet. These two ANNs are, now, trained, validated, and tested for accuracy using the exact same input data and the same protocols for stopping time. The test for validity of MSTd Hypothesis is the superior performance of the second ANN within the same training time and exposure to stimuli.

Results

In order to train Optiflonet we used back propagation method on our training data set. The training data set contained the 150 frames from a simulation of walking through a corridor like that in Fig. 4, right. We used another set of 150 frames for validating and testing of our neural network. Fig. 10 shows the convergence of mean square error during the training, validating and testing.

Fig. 11 shows the histogram of output errors in these procedures. Based on this histogram we can find that the norm of output error of Optiflonet is highly concentrated at about zero pixel errors in detection of motion in the frames. Fig. 12 also shows the error in estimation of motion heading in the horizontal and vertical axes of each frame separately. Since in training and test data we used simulated walking, the kinetics of movement (velocity and orientation of camera) through interior space was the stochastic variable.

Conclusion

Our contribution proposes a computational model that provides feasibility results for the proposal that perception of form is closely related to perception of motion in the context of natural scenes. We use architectural indoor space as natural scenes that are sampled by eye movement and spatial organization of planes are detected through reconstruction of parallel lines from vanishing points. We demonstrate that the same kind of local computations that must process the stream of such images in a scene are equivalent to the similar local computations for perception of motion from optical flow. These results and the well-established results on cortical neurophysiology of motion naturally suggest that the brain area known as MSTd contains the local circuits for perception of architectural form from reconstruction of vanishing points. Applications include new models for human-like robot motion planning in indoor spaces.

Fig. 1 .

 1 Fig. 1. Two steps in detection of vanishing point: edge detection of local circuits in area V1 and the thalamic feedback system provide the information about the lines as above. We have simulated the extraction of lines using Hough transform. (a) Left: an image from the Wisconsin Institute for Discovery (WID). (b) Center: binary output of edge detection. (c) Right: overlay of the Hough transform output onto the original image.

2

 2

Fig. 2 .

 2 Fig. 2. This figure shows estimation of motion field (optical flow) from a movie. Far left: Lewis Hamilton in his Nürburgring 2011 qualifying lap. Next, gray scale image of the same frame with optical flow vectors. Third from left: heat map showing the magnitude of optical flow vectors (blue is zero and red is the largest magnitude). Right: The phase heat map.

Fig. 4 .

 4 Fig. 4. (Left top) Examples for training, validating and testing an ANN to extract vanishing points. Fig. 5. (Left bottom) Optical flow vectors in simulated interior space movement.Fig. 6. (Right) Reconstruction accuracy vs. the number of principal components, e.g. 20 principal components has reconstruction accuracy of about 95%.

Fig. 7 .

 7 Fig. 7. (Top) Flowchart of Optiflonet to estimate direction of motion. Fig. 8. (Middle) Shows the architecture of Optiflonet, with two layers, and 100 neurons in the hidden layer.Fig. 9. (Bottom) Flowchart of Perspectinet for extraction of vanishing point.

Fig. 9 .

 9 Fig. 7. (Top) Flowchart of Optiflonet to estimate direction of motion. Fig. 8. (Middle) Shows the architecture of Optiflonet, with two layers, and 100 neurons in the hidden layer.Fig. 9. (Bottom) Flowchart of Perspectinet for extraction of vanishing point.

  Fig. 13 presents a time series of camera velocity and orientation during recording frame data during training. The orientation of the camera is shown as 3 components of the unit vector parallel to the camera orientation. Finally, error in detection of location of the vanishing point in different frames is illustrated in two histograms in Fig. 14. The left histogram shows the vertical error while the right histogram shows the horizontal error. The maximum error along the horizontal axis is less than 2 degrees and less than about 13 degrees along the horizontal axis.

  Fig. 13. (Left) Time series of camera movement (right plots) and orientation (left plots) during the recording data for neural network training. Fig. 14. (Right) Histogram of output error during the testing of the Perspectinet neural network. (Top) error in estimation of the vertical component of the vanishing point in the image. (Bottom) error in estimation of the horizontal component of the vanishing point in the image. The unit of estimation error in these histograms is the pixel. Since the field of view of each frame is comprised of 100 pixels, covering 45 degrees, each pixel would cover a 0.45 degree of field of view.


Luís M R Oliveira

A J Marques Cardoso

Power Transformers Differential Protection Using the p-q Power Theory

Keywords: Transformers, differential protection, p-q power theory

This paper describes the application of the p-q power theory to the differential protection of power transformers. The information provided by the harmonic content of the differential active and reactive power components is used to detect winding insulation failures and to distinguish them from magnetizing inrush current transients. A variety of test cases is presented in the paper, demonstrating the effectiveness of the protection strategy.

Introduction

The vast majority of transformer differential relaying schemes use the second harmonic restraint method to discriminate between energization transients and internal faults. Harmonic restraint is based on the fact that the inrush current typically contains a large second-harmonic component, which is used to inhibit the relay operation during the transformer energization. However, inrush current parameters (peak, 2 nd harmonic and duration) of today's power transformers differ from those of older designs due to the use of higher grain oriented core steels, the step-lap core joint type, and higher rated design core induction values [START_REF] Girgis | Characteristics of Inrush Current of Present Designs of Power Transformers[END_REF]. In modern power transformers, the 2 nd harmonic during inrush may be very low, jeopardizing relay security [START_REF] Kasztenny | An Improved Transformer Inrush Restraint Algorithm Increases Security While Maintaining Fault Response Performance[END_REF]. Moreover, the 2 nd harmonic component may also be generated during internal faults in the transformer, with relevant magnitudes.

Researchers and designers have made remarkable progress in the development of protection strategies since microprocessor-based relays were adopted for the differential protection of power transformers. A wide variety of protection schemes have been proposed, many of them combining different methods, such as the improvement of the classical harmonic restraint schemes [START_REF] Kasztenny | An Improved Transformer Inrush Restraint Algorithm Increases Security While Maintaining Fault Response Performance[END_REF], [START_REF] Guzmán | A Current-Based Solution for Transformer Differential Protection -Part I: Problem Statement[END_REF], the introduction of new trip and blocking signals [START_REF] Kasztenny | An Improved Transformer Inrush Restraint Algorithm Increases Security While Maintaining Fault Response Performance[END_REF], [START_REF] Gajić | Sensitive Turn-to-Turn Fault Protection for Power Transformers[END_REF][START_REF] Baoming | An Equivalent Instantaneous Inductance-Based Technique for Discrimination Between Inrush Current and Internal Faults in Power Transformers[END_REF][START_REF] Oliveira | An EPVA-Based Strategy for Power Transformers Differential Protection[END_REF][START_REF] Yabe | Power Differential Method for Discrimination Between Fault and Magnetizing Inrush Current in Transformers[END_REF], the use of wave-shape recognition methods [START_REF] Guzmán | A Current-Based Solution for Transformer Differential Protection -Part I: Problem Statement[END_REF], and the application of advanced digital signal processing tools, fuzzy logic and artificial intelligence techniques [START_REF] Tripathy | Advances in Transformer Protection: a Review[END_REF].

Many of these strategies have proved to be successful to some extent, but very few have been incorporated in commercially available relays, where the 2 nd harmonic restraint method is still widely used, regardless of its limitations. This paper describes the application of the p-q power theory for the discrimination between inrush currents and internal faults in transformers differential protection. The so-called Power Signature Analysis, which has been previously used to diagnose AC motors faults [START_REF] Drif | The Use of the Instantaneous-Reactive-Power Signature Analysis for Rotor-Cage-Fault Diagnostics in Three-Phase Induction Motors[END_REF], is applied to the difference between the instantaneous powers at all the transformer's terminals. The information provided by the harmonic content of the resultant differential p-q power signals is used for the discrimination between inrush transients and interturn short-circuits. With this approach, the traditional 2 nd harmonic restraint limitations are avoided.

Contribution to Value Creation

In today's digital era, an affordable and reliable electricity supply is essential to the economic and social progress. Simultaneously, the development of deregulation in power systems leads to a higher requirement on power quality. In the area of relay protection this means that a faster and sensitive protection is needed, while undesirable operation of the protection system is almost unacceptable [START_REF] Lotfi-Fard | Improved Overcurrent Protection Using Symmetrical Components[END_REF]. It is therefore of crucial importance the development of new transformer protection schemes, which can improve the sensitivity of differential relays to detect internal faults and which can also provide a solution to the second harmonic restraint drawbacks. This will be of great value to improve the overall grid reliability, which is the key to the social and industrial development.

Simulated Test System: Power and Current Transformer Models

A three-phase, two winding, three leg, 10.3 kVA, 230/132 V transformer model was previously developed for winding faults studies and it was used for the evaluation of the proposed protection strategy. The non-linear model is based on the combination of both magnetic and electric lumped-parameters equivalents circuits, which allows the modeling and simulation of the transformer in its natural electromagnetic environment. The faults are introduced in the model by dividing the affected winding into two parts, which represent the healthy and the faulty subwindings, Fig. 1(a). The current in the shorted turns is adjusted by varying the value of the fault impedance. A detailed description of the model implementation and/or validation, under inrush and fault conditions, can be found in [START_REF] Oliveira | An EPVA-Based Strategy for Power Transformers Differential Protection[END_REF], [START_REF] Oliveira | A Permeance-Based Transformer Model and its Application to Winding Interturn Arcing Fault Studies[END_REF], [START_REF] Oliveira | Power Transformers Behavior Under the Occurrence of Inrush Currents and Turn-To-Turn Winding Insulation Faults[END_REF].

The influence of the current transformers (CT) core saturation on the functionality of the protective system is also considered in the simulation study: the current inputs of the differential relay are computed by using a non-linear CT model. The CT model is based on the same principles used to simulate the behavior of the power transformer. The diagram of the simulated test system is shown in Fig. 1(b).

i x i R i R i S i T i 1 g v 3 g v 2 g v 1 L i 2 L i 3 L i r i s i t i RS v ST v TR v

Protection Strategy Based on the p-q Power Theory

The active power differential method was firstly proposed in [START_REF] Yabe | Power Differential Method for Discrimination Between Fault and Magnetizing Inrush Current in Transformers[END_REF] for the discrimination between inrush transients and internal faults in power transformers. The differential power is computed as the difference between the instantaneous powers at all the transformer's terminals. In [START_REF] Yabe | Power Differential Method for Discrimination Between Fault and Magnetizing Inrush Current in Transformers[END_REF], the average of the instantaneous differential power is responsible for both the operation and the blocking signals of the relay. The discrimination criterion is based on the following basic principle: the average power is almost zero for energizing, but an internal fault consumes large power. However, the direct application of the average power as a decision signal for the discrimination of inrush/fault conditions can lead to misleading results, due to the oscillations of the average active power, as reported in [START_REF] Yabe | Power Differential Method for Discrimination Between Fault and Magnetizing Inrush Current in Transformers[END_REF].

The same basic principle mentioned before is particularly suitable for the application of the p-q theory to the power differential method, where the active power component is directly related to the internal fault within the transformer and the reactive power component is associated with the inrush transient.

Differential Power Components Computation

A two-winding transformer with a DYn5 winding connection [Fig. 1(b)] is considered here, but the method is valid for any multi-winding transformer. For simplicity, the differential power components are computed using phase quantities. The d-q components of the voltage and current, of high and low voltage sides of the transformer, are computed by applying the Park's transformation:

[ ]

1 1 T T D Q RS ST TR v v T v v v   = ⋅   , (1)
[ ]

2 2 T T D Q rn sn tn v v T v v v   = ⋅   , (2)
[ ]

1 1 T T D Q R S T i i T i i i   = ⋅   , (3)
[ ]

2 2 T T D Q r s t i i T i i i   = ⋅   , ( 4
)
where T is the Park's transformation matrix:

2 3 1 6 1 6 0 1 2 1 2 T - - -   =     . ( 5
)
Then, the primary-and secondary-side Park's power components (k = 1 and k = 2, respectively) are computed:

k Dk Dk Qk Qk p v i v i = ⋅ + ⋅ , 1, 2 k = , ( 6
) k Qk Dk Dk Qk q v i v i = ⋅ - ⋅ , 1, 2 k = , (7)
and finally the differential active and reactive power components are:

(load dependent) 1 2 d p p p = -, (8)
(load dependent) 1 2 d q q q = -. ( 9
)
The sensitivity of the method can be further enhanced by compensating for the internal active and reactive load-dependent power components. The winding copper losses can be subtracted in ( 8):

( ) ( ) 2 2 2 2 1 2 1 1 1 2 2 2 d D Q D Q p p p R i i R i i = --⋅ + -⋅ + , ( 10
)
where R 1 and R 2 are the primary-and secondary-side winding resistances, respectively. Similarly, the reactive power supplied to the short-circuit reactance can be deducted in ( 9):

( ) ( )

1 2 1 1 1 1 1 2 2 2 2 2 d Q D D Q Q D D Q q q q L di dt i di dt i L di dt i di dt i = --⋅ ⋅ - ⋅ -⋅ ⋅ - ⋅ , (11)
where L 1 and L 2 are the primary-and secondary-side leakage inductances, respectively.

Differential Power Signature During an Internal Fault

2 presents the results for the case of a winding fault with 6 shorted turns in the primary winding of phase R. The current in the fault impedance is limited to 5 times the rated current of the affected winding. The transformer operates at full load, with 0.8 power factor.

With these conditions the fundamental component of the differential current associated with the faulty phase increases about 7 times when the short-circuit occurs. This differential signal reaches up to 20% of the transformer winding rated current, which is in the range of the minimum differential protection pickup (e.g. 15%-20%) typically required for the operation of the relay. Therefore, it can be assumed that this fault would be the smallest interturn insulation defect that can be detected by the traditional protection systems. It should be noted, however, that a higher value of the minimum protection pickup setting is often used, in order to deal with the ratio and phase mismatches and avoid false trips of the protection system. In this case the turnto-turn fault may not be detectable initially and the winding protection is uncertain.

Fig. 2(a) presents the evolution of the instantaneous differential active and reactive powers during the turn-to-turn winding fault occurrence. After the fault inception there is an increase in the magnitude of both instantaneous power components. They have an oscillatory component at twice the supply frequency (f supply = 50 Hz).

A further insight into the characteristics of the differential power components can be obtained by performing the frequency analysis of the power signals. 100) components can be used for fault detection. However, as explained in the next subsection, it is not possible to discriminate between internal faults and inrush transients by using the 100 Hz component of the differential active power.

The evolution of the relevant harmonic components of the differential reactive power is presented in Fig. 2(c). Only the 100 Hz component is affected by the fault occurrence, but some limitations also exist in using this parameter as a fault/inrush discriminator. This will be addressed later.

The DC value of the active power is substantially affected by the presence of the fault. The average value of the active power increases about 8 times as a result of the winding fault occurrence, which results in only a slightly improvement of the fault detection sensitivity, as compared to the fundamental component of the line currents involved with the affected phase (which increases 7 times for the same type of fault, as mentioned before). However, the differential power method is not affected by the ratio mismatch due to on-load tap-changer, and enhanced fault detection sensitivity is thus achieved. Therefore, the value of p d(DC) is a reliable indicator of the presence of a fault, and can be used as a trip signal of the relay.

To detect this low-level turn-to-turn fault the trip signal should be activated when p d(DC) exceeds a threshold of 800 W. This is about 8 times the normal operating value (which is equal to the no-load losses). This allows the power differential relay to accommodate the measurement errors without nuisance tripping and still providing reliable fault detection. be stored in the windings. Comparatively, the instantaneous active power is much smaller. Moreover, its DC value (p d(DC) ) has a reduced variation and never reaches the trip threshold value (800 W), as seen in Fig. 3(b).

Differential Power Signature during an Inrush Transient

The instantaneous reactive power has an asymmetric waveform, with a frequency of 50 Hz (and not 100 Hz, as in the internal fault case). Accordingly, the q d(DC) and the q d(50) components have high values during the inrush transient, Fig. 3(c).

Figs. 3(d) to 3(f) present the same curves as in Figs. 3(a) to 3(c), but now considering the current transformers under saturated conditions. The DC component of the inrush currents drives the CT's into saturation, resulting in distorted secondaryside CT's current waveforms, with minor peak values, faster decay and slightly more symmetrical and bipolar waveforms, as compared to the non-saturated CTs case [START_REF] Oliveira | An EPVA-Based Strategy for Power Transformers Differential Protection[END_REF]. This results in smaller values of q d and p d , Fig. 3(d). The influence of the CT saturation on the harmonic content of the p-q power components during the inrush transient can be summarized as follows:

The p d(DC) take larger values, but never reaches the trip threshold, Fig. 3(e).

The behaviour of the p d( 50) is irregular and very dependent upon the CT's saturation conditions. The q d(DC) and the q d(50) components have smaller values than in the nonsaturated CT's case, Fig. 3(f). Even so, at the inrush start these values are at least 20 times higher than in the case of an internal fault, Fig. 2(c). The DC value and the 50 Hz harmonic component of the reactive power takes high values at the beginning of the energization transient, falling then slowly, with the same rate as the inrush currents, to the steady-state value. These two components are not affected by the interturn fault conditions, as seen before, and can be used to block the relay operation and avoid false trips.

The p d(100) and the q d(100) harmonic components are largely affected by both fault and inrush conditions, not allowing to discriminate between these two situations.

Inrush Blocking Signal

The transformer energization can be detected by using the ratio between the signals that characterize the inrush transient (q d(DC) and q d(50) ) and the trip indicator (p d(DC) ).

Accordingly, the proposed blocking signal is defined as:

( )

( ) (50) ( ) blocking signal d DC d d DC q q p = + . ( 12
)
The relay operation will be inhibited if the blocking signal is above a predefined threshold value. For the present case this threshold value can be assumed to be 10. A safety margin is introduced between the threshold and the normal operation status, to avoid blocking the relay trip signal under the simultaneous occurrence of inrush currents and internal faults. It is considered here that the threshold is about 8 times the normal operating value.

In the case of an inrush event, although the trip indicator never reaches the 800 W threshold, the blocking signal increases promptly and the relay is inhibited during the first cycles of the transient, as seen in Fig. 4(a). When a fault occurs the blocking signal has the opposite behavior, i.e., decreases to a value which is 10 times smaller than the one obtained under normal operating conditions, Fig. 4(b).

Fig. 5 presents the evolution of the trip indicator and the blocking signal for the case of a simultaneous occurrence of a transformer energization and a winding turn-to-turn fault (with the same severity as in subsection 4.2). It can be seen that the threshold value of the blocking signal is not reached and the relay, which is rapidly triggered by the large value of p d(DC) , is not blocked.

Other simulation studies, carried out under different inrush and fault conditions, lead to similar conclusions to the ones presented before.

Conclusions

This work explores the possibility of applying the Park's power components to solve the inrush/fault discrimination dilemma in power transformers differential protection.

The results indicate that the proposed p-q power-based strategy provides reliable inrush stabilization and enhanced sensitivity for internal faults. With this approach, the traditional 2 nd harmonic restraint shortcomings are avoided. Additionally, the error introduced by the current mismatch due to the tap-changer is minimized, and increased sensitivity is obtained in the differential protection system.

  b

Fig. 1 .

 1 Fig. 1. (a) Equivalent circuit for a winding fault; (b) diagram of the test system.

Fig. 2

 2 (b) presents the relevant harmonic content of the differential active power during the fault transient. There is an increase in the DC and the 100 Hz components of p d (p d(DC) and the p d(100) , respectively) as a consequence of the fault occurrence. After the short transient at the time of the fault introduction, the 50 Hz component, p d(50) , is negligible. It seems that the p d(DC) and the p d(

Fig. 3 (

 3 Fig. 3(a) presents the waveforms of the instantaneous active and reactive powers during a transformer energization transient. It is considered in this case that the CT's reproduces correctly the inrush currents. The reactive component is largely affected during the energization of the transformer, in order to supply the magnetic energy to

Fig. 2 .

 2 Fig. 2. Internal fault occurrence: (a) instantaneous waveforms of p d and q d . DC, 50 Hz and 100 Hz harmonic components of: (c) p d and (d) q d .

Fig. 3 .

 3 Fig. 3. Transformer energization transient. (a), (d): Instantaneous waveforms of p d and q d . (b), (e): Harmonic components of p d . (c), (f): Harmonic components of q d . Transformer operation under no-load conditions. Energization at t = 20 ms.

Fig. 4 .

 4 Fig. 4. Trip indicator and blocking signal, during: (a) energization (same conditions as in Fig. 3); (b) interturn fault occurrence (same conditions as in Fig. 2).

Fig. 5 .

 5 Fig. 5. Trip indicator and blocking signal, during a faulted transformer energization transient.


Oleksandr Husev

Andrei Blinov

Dmitri Vinnikov

Experimental Results of Parallel Active Filter Implementation in Nonideal Power Grid

Keywords: parallel active power filter, power grid, control system, efficiency

This paper is devoted to the implementation of a parallel active power filter prototype in a nonideal power grid. The control algorithm is described and realized. Simulation and experimental results proved the theoretical prediction. Analytical conditions of the usable implementation are shown. Some guidelines are presented in the conclusion.

Introduction

Enhanced quality requirements for the mains current supplied by the power network require devices compensating the reactive power in both the fundamental frequency and higher harmonics. The principal advantage of active filters (AF) is in their ability to compensate a wide spectrum of higher harmonics.

The design of all active filters includes a storage element, a converter and a charger [START_REF] Akagi | Trends in active power line conditioners[END_REF]- [START_REF] Cadaval | Active power line conditioner based on two parallel converters topology[END_REF]. Fig. 1 represents implementation of a parallel active power filter (PAPF) in a nonideal power grid. Active filters exist in a large variety of topologies [START_REF] Akagi | Trends in active power line conditioners[END_REF]- [START_REF] Dixon | Some improvements in 81-Level Inverters for Traction Applications[END_REF]. They allow compensation of a wide spectrum of the load current, although they are not free of shortcomings -namely, a limited maximum amplitude of reactive power fluctuations and the complexity of implementation. The latter issue becomes more urgent of increasing power quality demands for power energy consumers. Energy efficiency of old power distribution and consumption systems is at least three times lower than in European requirements.

Numerous articles are devoted to the shunt active filters that contain nothing but theoretical description [START_REF] Bhattacharya | Hybrid solutions for improving passive filter performance in high power application[END_REF], [START_REF] Husev | Neuro-fuzzy Control System for Active Filter with Load Adaptation[END_REF]. At the same time this problem should be solved taking into account real time power grid conditions. Old power distribution equipment cannot be replaced immediately, but the efficiency should be increased.

This paper covers practical implementation of a PAPF in a nonideal power grid. The power grid in the point common connection (PCC) is represented by a threephase power transformer with nonideal parameters. Focus is on the behavior of a PAPF in such PCCs.

Theoretical, simulation and experimental results are shown. Some guidelines are presented in the conclusion

Contribution to Value Creation

The value creation topic is strictly connected with power quality supplied system. The PAPF implementation poses new and interesting practical problems concerning power quality and their mutual influences. The quality of the delivered power often depends on the power load interfaces. Due to their nonlinear nature, these loads inject harmonics current into the power system and cause voltage harmonics distortion. In order to keep the quality of the power delivery system it is important to eliminate harmonics current and estimate influences of PAPF on the voltage quality. The presented work is a contribution for the power quality problem and thus for the increasing value creation of the general system

Control System

The control system is based on the power balance method [START_REF] Strzelecki | Power Quality Converters with Minimum Number of Current Sensor Requirement[END_REF]. Fig. 2 presents a block diagram of a control system for a PAPF. The control system consists of a measurement block, a reference current definition block and a switch control block. The main idea of the power balance method is to maintain the DC voltage level across the capacitors. The input values of the regulator enter the instantaneous value of the voltage across the capacitors (V DC1 , V DC2 ) and the reference voltage V REF , equal to the necessary values of the voltage. In order to eliminate voltage pulsations across the capacitors, only the constant component of the voltage is used. Further, an error on the voltage will be given to the PI regulator whose output determines the amplitude of the reference current. The form of the reference current is determined by the supply voltage, as a result, the current is inphase to the voltage. Further, the reference current will be given to the switch control block. The switch control block is built according to the direct current control method. Thus, transfer of the necessary active power from the grid is ensured. In the steady state mode the value of an error is equal to zero, values from the output of the integrator are constant. In this case the power consumed by the load is equal to the power that is consumed from the generator.

Measurement Block

The main advantage of the proposed algorithm is in the reduced number of current sensors. Taking into account the EMC problem that usually is present in the real system, the reduced number of measurement values increases the stability of the control system. For example, control systems based on the PQ theory are less feasible because they demand more measurement elements

Simulation Results

A simplified equivalent circuit of a power grid for one phase is shown in Fig. 3a. The induction voltage regulator is substituted by the voltage source e a , the internal resistances R 0 and the internal leakage inductance L 0 . The transformer is substituted by the equivalent circuit with primary and secondary internal resistances R 1 and R 2 , primary and secondary internal leakage inductances L 1 and L 2 and magnetizing inductance L M . We assume that these parameters are equal in all phases.

A PARF is represented taking into account the internal inductance resistance R a and transistor parameters. Nonlinear load is represented by a passive rectifier with a capacitor output filter that is one of the worst cases for a power grid.

Equivalent circuits for switching transistors are shown in Figs. 3b (T 1 is opened) and 3c (T 2 is opened).

From the analysis of equivalent circuits it is evident that active filter commutation disturbs the output transformer voltage shape. We can write:

, 2 ) ( 2 2 DC C a C C L S V dt di L R R i dt di L e ± ⋅ + + + ⋅ = (1)
, 2

DC C a C S V dt di L R i V ± ⋅ + ⋅ = (2)
. As it can be seen from Eq. ( 4), relative voltage distortion depends on the DC capacitor voltage and the ratio between the leakage inductance of the transformer and the inductance of the compensator. The lower the leakage inductance the lower the distortion is.

S S V V e + ∆ = (
DC C a C C L C C L S V dt di L R R i dt di L R i dt di L e V ± ⋅ + + + ⋅ ⋅ + ⋅ = ∆ . ( 4
In order to verify the presented control algorithm and the analytical prediction the MATLAB simulations were performed.

The passive components of the compensator and the minimum frequency of the commutation of power switches with the direct control method were calculated according to [START_REF] Ivanets | Method of choice elements of active line conditioner[END_REF]:

, 6 . 0 98 50 2 125 1 . 0 380 6 1 4 4 1 mF V V K Q C m DC m = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = π ω γ (5) , 2 . 2 10 2 . 1 98 125 4 ' 1 mH I V V L сm m DC = ⋅ - = - = (6)
where: It should be noted that an additional RLC passive filter was used in the measurement system to obtain the reference voltage shape.

The first simulation diagrams shown in Fig. 4 correspond to the idle power grid mode (Fig. 4a) and the nominal power mode without a PAPF (Fig. 4b). During the idle mode, primary and secondary voltage shapes are purely sinusoidal. After the load connection, voltage shapes are greatly disturbed. It shows the influence of the nonlinear load on the power quality in the power grid with nonideal parameters. The total consumed power is P 1 = 1300 W, on the secondary side is P 2 = 1050 W. As a result, the efficiency of the chosen transformer is about 80%. Voltage and current THD on the secondary side are about 19.5% and 47.3%, correspondingly.

The second simulation diagrams in Fig. 5 shows power grid conditions with a PAPF. The reference phase voltage is shown in Fig. 5a. The compensator current is represented in Fig. 5b. The primary and secondary power grid currents are shown in Fig. 5e. The reference voltage and source current THD on the secondary side are about 28.4% and 18.7%, respectively. We can see that after the inclusion of the PAPF, the current shape is improved but the voltage shape is impaired. It is proved by the secondary and primary winding voltage (Figs. 5b and5d). The DC link voltage is shown in Fig. 5f that proves the steady state mode. The total consuming power on the primary side is increased to P 1 = 1950 W, on the secondary side to P 2 = 1450 W. The output consuming power is about 1340 W. As a result, PAPF efficiency is about 92%. It should be noted that modeling losses parameters of a PAPF correspond to the worst real IGBT transistors that can be used in such systems. The frequency of commutation is 10 kHz. Simulation parameters and results are summarized and shown in Table 1.

V S1 = 133 V L L0 = 5 mH L L1 = 2.2 mH L L2 = 1.8 mH R 0 = 3 Ohm R 1 = R 2 = 1.5 Ohm V DC = V DC1 +V DC2 = 250 V L = 2.2 mH С 1 = С 2 =0.56 mF R A = 0.5 Ohm F=10 kHz R F = 300 Ohm L F = 0.5 mH С F =1 µF P 1 =

Experimental Results

A laboratory prototype is shown in Fig. 6. It consists of a transformer (Fig. 6b), a nonlinear load (three-phase rectifier, capacitor, resistor) and a PAPF. The PAPF is separately shown in Fig. 6a and consists of an inverter, optically isolated drivers for IGBTs, inductors, capacitors, measurement and control systems.

An inverter is built on BSM75GB120DLC 1200V/75A IGBT transistors. Hall effect current sensors and optically isolated voltage sensors represent the measurement system. The digital control system is based on the digital signal processor TMS320F28335. An induction regulator is used as a primary voltage generator. A serial buck transformer was used like a secondary voltage source. Leakage inductance and internal resistance of the secondary winding are 1.8 mH and 1.5 Ohms, respectively. The nominal transformer power is 2.5 kW, turns ratio 1.35.

Experiments with the PAPF were performed according to the simulation parameters (Fig. 7). Control was realized by the direct current control method at 10 kHz switching frequency.

It is evident that the voltage and the current are not of purely sinusoidal shape. It is explained by nonideal power grid parameters. The consuming primary power without a PAPF was equal to P 1 =3•115 V•3.9 A=1345 W. The consuming power in the load was P OUT = 1010 W.

Transformer efficiency was about 75 %. In the steady state mode with a PAPF the rms phase voltage V S2 and current I S2 were equal to 83 V and 5.3 A, respectively. The output DC voltage V OUT was equal to 195 V and the load current I OUT to 6.1 A. As a result, the output consuming power increased to P OUT = 1190 W. Taking into account the total input consuming power P 1 = 1720 W and the secondary winding consuming power P 2 = 3•83•5.2=1295 W, it can be concluded that transformer efficiency was about 75 %. The PAPF efficiency was about 91 % that is close to the simulation results.

Conclusions

This paper presents a PAPF implementation prototype in a nonideal power grid. The power balance based control algorithm is described and realized. Voltage distortion as the main problem in similar power grids was analytically estimated. Simulation and experimental results proved the theoretical prediction. Analytical conditions of the usable implementation were presented. A PAPF cannot be an effective solution in power grids where leakage inductance is close to the active filter inductance. In such systems a PAPF improved current quality and total power quality slightly. Total consuming power had increased (up to 30 %), the reactive power did not decrease significantly. At the same time voltage THD level increased. In order to improve voltage quality an additional serial active power filter is required

Fig. 1 .

 1 Fig.1. A nonideal power grid with a PAPF.

Fig. 2 .

 2 Fig. 2. Control system structure of a PAPF.

3 )

 3 Assuming that the voltage source e S has a pure sinusoidal shape we can express the relative voltage distortion

Fig. 3 .

 3 Fig. 3. Equivalent circuits of a nonideal power grid.

mQ

  -amplitude of the pulsation of the reactive power in the three-phase network; γ -the relative duration of the pulsation of the reactive power; ω -network frequency; m V -amplitude value of the phase voltage; DC V -value of voltage on the capacitor; K -ripple factor of voltage on the capacity; сm I ' -maximum value of the rate of current changing.

Fig. 4 .

 4 Fig. 4. Simulation diagrams without a PAPF.

Fig. 5 .

 5 Fig. 5. Simulation diagrams with a PAPF.

Fig. 6 .

 6 Fig. 6. Laboratory prototype of a PAPF

Fig. 7 .

 7 Fig. 7. Experimental results of a PAPF.

Table 1 .

 1 Simulation parameters and results.

		Power Grid	PAPF	RLC Filter	Simulation Results
		Parameters	Parameters	Parameters	Without PAPF With PAPF
	ω	2 ⋅ = π	50	Hz


António Roque
email: antonio.roque@estsetubal.ips.pt

Sónia Pinto
email: soniafp@ist.utl.pt

João Santana
email: jsantana@ist.utl.pt

Duarte Sousa
email: duarte.sousa@ist.utl.pt

Elmano Margato
email: efmargato@isel.ipl.pt

José Maia
email: jose.maia@estsetubal.ips.pt

Design of Current Power Sources for a FFC NMR Apparatus: A Comparison

Keywords: current power source, fast field cycling nuclear magnetic resonance, fast transients

The power supply of a Fast Field Cycling Nuclear Magnetic Resonance apparatus is typically a current power source, with characteristics that are not fulfilled by most of the commercially available power supplies. This current source is used to supply a specially designed magnet, and should be able to drive a cycling current with a slew rate lower than 3 A/ms. In this paper, two solutions for this current source are designed and discussed. The first uses two power supplies: the main voltage source and an auxiliary power supply to guarantee the current transitions from the low level to the high level. In the second solution the auxiliary power supply is replaced by a capacitor. To guarantee the required current slew rates, the possibility of using resistors to dissipate the energy stored in the magnet is also discussed.

Introduction

Fast Field Cycling (FFC) Nuclear Magnetic Resonance (NMR) is an experimental and powerful technique used to study the molecular dynamics of different types of compounds [START_REF] Noack | NMR Field-Cyclying Spectroscopy: Principles and Applications[END_REF], [START_REF] Seitter | Magnetic Resonance: Relaxometers. Encyclopedia of Spectroscopy and Spectrometry[END_REF]. A FFC NMR apparatus includes different modules, as represented in Fig. 1. In the last decades, the performance of this equipment has improved, incorporating the most up-to-date technological advances, which have resulted in: the optimization of the magnet, providing a magnetic induction with very high homogeneity (∆B/B lower than 10 -4 ) [1] - [START_REF] Anoardo | Fast-Field-Cycling NMR: Applications and Instrumentation[END_REF]; higher efficiency and higher performance of the current source; higher portability, reducing the size of the equipment. In this work, two solutions for the current source are described and compared, fulfilling the requirements of the FFC NMR equipment.

Contribution to Value Creation

FFC NMR is a powerful technique that owes its advances to the use of new power semiconductors, new materials, new optimization techniques and computational tools.

Fig. 1. Main parts of a FFC NMR apparatus

During the last decades the use of FFC relaxometers has been mainly confined to academic specialized laboratories in part due to the fact that these equipments were not commercially available. This has been changing recently and the use of FFC relaxometry has extended to other areas of research, not only in academy but also in industry. This work contributes to value creation, as the development of FFC NMR technology, allowing higher efficiency and performance of the current source, as well as its higher portability and low size will be a factor to spread the usage of this technology in industrial quality control and product development laboratories.

FFC NMR Power Supply

The power supply of the FFC NMR apparatus is basically a current source that cycles with tunable response times in the ms range. The power supply should allow fast switching of the magnetic field between different values with current slew rates adjusted to less than 3 A/ms [START_REF] Redfield | Precision High Speed Current Regulators for Occasionally Switched Inductive Loads[END_REF] - [START_REF] Sousa | Desktop Fast-Field Cycling Nuclear Magnetic Resonance Relaxometer[END_REF].

The typical behavior of the magnet current dynamics is represented in Fig. 2, where four different operation modes can be identified: A: current transition from a low level (I low ) to a high level (I high + ∆I/2); B: current high level or low level (with negative time derivative), bounded by a predefined error window (∆I);

C: current high level or low level (with positive time derivative), bounded by a predefined error window (∆I); D: current transition from a high level (I high ) to a low level (I low ).

To achieve these requirements with an adequate efficiency, the FFC current sources are generically designed using two power supplies [START_REF] Noack | NMR Field-Cyclying Spectroscopy: Principles and Applications[END_REF], [START_REF] Anoardo | Fast-Field-Cycling NMR: Applications and Instrumentation[END_REF]: a power supply driving the steady state current (main power supply); and a fast-response power supply driving the magnet current during the step up current transients (auxiliary power supply). Several solutions have been developed for each one of these power supplies [START_REF] Anoardo | Fast-Field-Cycling NMR: Applications and Instrumentation[END_REF], and therefore, to the global topology of the current source.

Fig. 2. Typical current waveform of a FFC NMR

In this paper, two possible topologies fulfilling the requirements of this application are presented and compared. For both solutions, the main power supply is based on the topology of a chopper converter. However, the first topology requires an additional dc voltage power supply (based on a full bridge rectifier) to drive the current step up, and includes a damping resistor to guarantee the required current slew rate when the current goes down. In the second topology, a capacitor is used instead of the additional dc power supply, storing energy when the current goes down, and using it to supply the magnet when the current steps up.

Current Source with Auxiliary Power Supply and Damping Resistor

The topology of the FFC NMR current source with an auxiliary power supply and a damping resistor is represented in Fig. 2, where S 1 and S 2 represent semiconductors with turn-off capability (as MOSFET); D 1 and D 2 are diodes; U 0 represents the voltage of the main power supply; U aux is the voltage of the auxiliary power supply (high voltage power supply based on a full bridge rectifier); R 1 is the damping resistor; L is the self-inductance of the magnet and R is its equivalent resistance.

As the magnet is immersed in liquid nitrogen and cools very slowly, it is difficult to guarantee a constant temperature at the copper coils, which will highly affect the resistance value. As a result, the magnet resistance R (Fig. 3) will not be constant. It has been experimentally measured for different temperature values and the results show that the equivalent resistance is nearly 8 times higher at ambient temperature (T a =298º K and R Ta ≈ 3.0 Ω) than at liquid nitrogen temperature (T N =77º K and R TN ≈ 0.4 Ω). These values will be further considered in the simulations.

With the topology of Fig. 3 it is possible to cycle the magnet current, controlling the states of S 1 and S 2 . The desired current dynamics is achieved using a high voltage auxiliary power supply U aux to step up the current (Fig. 4a) and a damping resistor to step down the current (Fig. 4b). The current high level and low level are obtained using only the main power supply U 0 (Fig. 4c) and the damping resistor R 1 (Fig. 4b).

Defining γ k as (1) and considering Fig. 3, the possible values of γ 1 and γ 2 are presented in Table 1. )( )

2 1 0 1 , k OFF is S if ON is S if k k k =    = γ (1)
magnet magnet aux magnet i L R i L R L U L U dt di - γ - γ - - γ + γ = 1 2 1 2 0 1 1 1 (2)
Equation ( 2) will be further used in the current controller design.

Current Source with Capacitor

The topology using a capacitor is represented in Fig. 5, where S 1 and S 2 are semiconductors with turn-off capability (as MOSFET); D 1 and D 2 are diodes; U 0 is the main power supply voltage; C is the storage capacitor; L is the self-inductance of the magnet and R is its equivalent resistance (variable with temperature). In this topology, the auxiliary power supply of Fig. 3 is replaced by a capacitor, and the damping resistor is not used. The capacitor is then used to store energy when the current goes down (Fig. 6d), and further used to supply the magnet when the current steps up (Fig. 6a). To guarantee the current high level and low level, only the main power supply (Fig. 6c) and free wheeling diode D 1 are used (Fig. 6b).

From (1), Fig. 5 and Fig. 6, the possible values of γ 1 and γ 2 are as presented in Tab. 2.

Table 2. States of γ 1 and γ 2 for each operating mode of the topology with capacitor

Topology with capacitor

A B C D γ 1 1 0 1 0 γ 2 1 1 0 0
From (1) and Fig. 5, it is possible to establish the equations representing the dynamics of the magnet current (3) and the capacitor voltage [START_REF] Redfield | Precision High Speed Current Regulators for Occasionally Switched Inductive Loads[END_REF].

From ( 4), ( 1) and Fig. 6, the capacitor voltage will change only when both semiconductors (S 1 and S 2 ) are ON (mode A) or when both are OFF (mode D). If only one semiconductor is ON (mode B or C), the capacitor charge will remain unchanged.

magnet c magnet i L R L U L U dt di - γ + γ = 2 0 1 (3) ( ) magnet c i C dt dU 2 1 1 1 γ - γ - = (4)
This solution seems clearly more efficient and less expensive than the previous. However it requires a well designed capacitor and a more complex control system.

Current Source Controller

An important requirement of a FFC NMR apparatus is to generate a magnetic induction field with very high homogeneity (magnetic induction field ripple ∆B/B lower than 10 -4 ) that is directly related to the current ripple. Also, it should guarantee adequate current damping transients. To achieve these requirements, the current control method plays an important role.

For both topologies, a sliding mode controller can be successfully implemented, using hysteretic controllers [START_REF] Sousa | Desktop Fast-Field Cycling Nuclear Magnetic Resonance Relaxometer[END_REF], [START_REF] Bühler | Réglage par Mode de Glissement[END_REF]. To guarantee the current high level (I high ) and low level (I low ), operation modes B and C (Fig. 4, Fig. 6), both presenting first order dynamics, are used. Then, the control of the magnet current is ensured by the convergence of the state trajectories to the sliding surface (5) [START_REF] Silva | Advanced Control of Switching Power Converters, Power Electronics Handbook[END_REF], according to the current error, e i [START_REF] Constantin | Fast Field-Cycling Nuclear Magnetic Resonance Spectrometer[END_REF], where i is the magnet current, i ref is the reference current and k is a constant (k>0).

( )

magnet ref _ magnet i e i i k e k S i - = = (5)
To control the magnet current, condition (6) must be guaranteed [START_REF] Silva | Advanced Control of Switching Power Converters, Power Electronics Handbook[END_REF]:

0 < i i e .

e S S (6)

The adequate operation mode is chosen from the sliding surface result (5), guaranteeing condition (6): Then, operation mode B should be chosen for both topologies;

• If 0 0 < ⇒ < i e e S i
• If 0 0 > ⇒ > i e e S i
and the magnet current should increase to guarantee 0 < i e . S [START_REF] Sousa | Novel Pulsed Switched Power Supply for a Fast Field Cycling NMR Spectrometer[END_REF]. Then, operation mode C should be chosen for both topologies.

Sizing of the Current Power Sources

To guarantee the adequate operation of both topologies it is necessary to design the damping resistor and the voltage of the auxiliary power supply (topology with damping resistor) and the storage capacitor values (topology with capacitor).

To design the auxiliary power supply it is assumed that the current switches from I low to (I high + ∆I/2) (mode A) with a slew rate of 2 ms. In this operation mode the current dynamics ( 6) is obtained from (2), considering γ 1 =γ 2 =0:

magnet aux magnet i L R L U L U dt di - + = 0 (6)
Solving ( 6), considering Fig. 1 and assuming steady-state operation, the auxiliary power supply voltage may be calculated from ( 7):

( ) 0 1 2 U e I I R e I U ) R / L ( T high ) R / L ( T low aux A A - - ∆ + - = - - (7)
Considering T A = 2ms, L = 270 mH, R = 3 Ω (at ambient temperature), U 0 = 24 V, I low = 0.1 A, I high = 5 A and ∆I = 4 mA, the voltage of the auxiliary power supply is around U aux =650V. To design the damping resistor it is considered the current transient from t e to t f (Fig. 2), where T D =t f -t e =2 ms. In this operation mode (D) the current dynamics ( 8) is obtained from (2), considering γ 1 =γ 2 =0:

magnet magnet magnet i L R i L R dt di - - = 1 (8)
The damping resistor ( 9) is calculated solving (8):

R I I ln T L R high low D -         - ≈ 1 (9)
From ( 9), assuming T D =2ms, L=270mH, R=3Ω (at ambient temperature), I low =0.1A, I high =5A and ∆I=4mA, the value obtained for the damping resistor is R 1 ≈530Ω. The storage capacitor is designed considering the operating mode D of the topology with capacitor (Fig. 6). During this transient, it is assumed that the energy stored in the coil is transferred to the storage capacitor, from t=t e to t=t f (Fig. 2) and at t=t e the capacitor is discharged.

U C I I L c low high 2 2 2 2 1 2 1 ≈       - (10)
From (10), assuming U C = 800 V, the estimated capacity value is 10.7 µF.

Simulation Results

Both topologies were simulated using Matlab/Simulink and considering a variable damping resistor. The main results are presented in Fig. 7 (topology with damping resistor) and Fig. 8 (topology with capacitor). To evaluate the performance of the two power supplies under severe operation conditions, the magnet resistance value is simulated as a sinusoidal waveform, assuming an amplitude of 50% of its rated value, and a frequency of 8 Hz.

The simulation results of Fig. 7 show that the current cycles, as required by the application, even for severe changes in the magnet resistance. In steady state operation the magnet voltage switches from 24V (U 0 ) to -2625V (R 1 ×I high ) or -52.5V (R 1 ×I low ), and during the transient from I low to I high , the magnet voltage is nearly 670V. The simulation results of Fig. 8 show that the current cycles are as required by the application, even when the magnet resistance changes. Also, this solution requires an initial charge of the capacitor, which does not affect the FFC NMR experiments. In steady state operation the magnet voltage switches between 24V (U 0 ) and a minimum of -15V (in the case of the maximum current) and during the transients from I low to I high or from I high to I low , it is possible to observe that the magnet voltage equals the storage capacitor voltage.

Conclusions

In this paper the design and sizing of two topologies for a FFC NMR magnet current source were presented and evaluated. From the simulation results both topologies fulfilled the requirements of the application, allowing the magnet current cycling, as expected. The magnet current ripple is bounded, according to the specifications, and even a 50% change of the magnet resistance (around its nominal value) did not affect the current controller performances.

Comparing both solutions, the topology with damping resistor is expected to be more expensive due to the additional auxiliary power supply and to the damping resistor. Furthermore, the global efficiency of this solution is also expected to be lower due to the energy losses in the damping resistor. Moreover, the voltage applied to the magnet is higher for a longer period (reaches 2kV), which will increase the insulation demand of this setup. In the topology with capacitor a more complex control system is needed and an initial charge of the storage capacitor is required.

Fig. 3 .Fig. 4 . 1 .From ( 1 )

 3411 Fig. 3. Topology of the current source with damping resistor

(

Fig. 5 .Fig. 6 .

 56 Fig. 5. Topology of the FFC NMR current source with capacitor

Fig. 7 .

 7 Fig. 7. Steady-state magnet voltage and magnet current for the topology with damping resistor considering a variable magnet resistance

Acknowledgements. This work was partially supported by the Portuguese Government, FCT project PEst-OE/EEI/UI4064/2011, CIEEE -"Center for Innovation in Electrical and Energy Engineering" of IST/TULisbon and POSC -"Programa Operacional da Sociedade do Conhecimento.


Luís Encarnação

José Fernando Silva

Sónia F Pinto
email: soniafp@ist.utl.pt

Luis M Redondo
email: lmredondo@deea.isel.pt

Grid Integration of Offshore Wind Farms using Modular Marx Multilevel Converters

Keywords: modular multilevel converter, offshore wind farm, dc capacitor voltage equalization

This paper proposes the use of a Modular Marx Multilevel Converter, as a solution for energy integration between an offshore Wind Farm and the power grid network. The Marx modular multilevel converter is based on the Marx generator, and solves two typical problems in this type of multilevel topologies: modularity and dc capacitor voltage balancing. This paper details the strategy for dc capacitor voltage equalization. The dynamic models of the converter and power grid are presented in order to design the converter ac output voltages and the dc capacitor voltage controller. The average current control is presented and used for power flow control, harmonics and reactive power compensation. Simulation results are presented in order to show the effectiveness of the proposed M 3 C topology.

Introduction

The multilevel conversion of electric energy is appropriate for high levels of voltage and current, and therefore suitable for power and energy systems applications. The technology for the energy transmission system between wind farm and onshore grid can be done using ac connection (High Voltage Alternating Current -HVAC) or dc connection (High Voltage Direct Current -HVDC) [START_REF] Adam | Comparison between Two VSC-HVDC Transmission Systems Technologies: modular and Neutral Point Clamped Multilevel Converter[END_REF]. This paper uses dc connection (HVDC -VSC voltage source converter) and proposes a three-phase modular multilevel Marx converter (3M 3 C), with five levels for each phase leg, to connect the dc wind farm to the inshore power electrical network. The modular multilevel converter is based on the Marx generator (M 3 C) [START_REF] Encarnação | A New Modular Marx Derived Multilevel Converter[END_REF][START_REF] Lesnicar | An Innovative Modular Multilevel Converter Topology Suitable for a Wide Power Range[END_REF][START_REF] Redondo | Repetitive High-Voltage Solid-State Marx Modulator Design for Various Load Conditions[END_REF] and solves two typical problems with known multilevel converter topologies [START_REF] Franquelo | The age of multilevel converters arrives[END_REF][START_REF] Encarnação | Reactive Power Compensation Using Sliding-Mode Controlled Three-Phase Multilevel Converters[END_REF], such as the Neutral-Point Clamped (NPC) and flying capacitor (FC). These topologies are non modular being difficult to balance the dc capacitor voltages, for topologies with levels higher than three. The five-level multilevel converter leg topology uses 8 basic cells for each arm (4 cells for each half arm). The three-phase 5 level multilevel converter uses 24 basic cells (3×8), and each basic cell has a capacitor which will be charged with U Ci =U dc /(n-1), where n represents the number of levels (in this case n=5, implying U Ci =U dc /4).

Section 2 presents the technological contribution of this paper to value creation. In Section 3, the 3M 3 C with five levels for each phase leg is presented. The dynamic models of the converter and power grid are presented in Section 4, in order to obtain the converter ac output voltages for a linear three-phase inductive load, and the dc capacitor voltage controller. The average current control [START_REF] Hagiwara | PWM Control and Experiment of Modular Multilevel Converters[END_REF][START_REF] Silva | Control Methods for Switching Power Converters[END_REF] is presented and used for power flow control, harmonics and reactive power compensation. A sigma-delta modulator is applied to a M 3 C leg, and a sliding-mode approach [START_REF] Encarnação | Sliding Condition Based Sliding Mode Modulators for Multilevel Power Converters[END_REF] is used to enforce the converter ac output voltages. The strategy for dc capacitor voltage equalization is presented with a detailed implementation of the algorithm. Simulation results are presented in Section 5 to show the effectiveness of the proposed solution.

Contribution to Value Creation

This paper proposes a solution for energy integration between an offshore Wind Farm and the power grid network, using a three-phase M 3 C with five levels for each converter arm. The solution presented enables cleaner electrical energy because it uses controllers to reactive power and harmonic compensation. This work will contribute to value creation as it also allows a better use of renewable energy and the development of new technologies to enhance economic activity, creating more labor, more jobs while showing the potential to reduce; 1) the price per kilowatt, 2) the costs of imported fossil fuels and 3) the carbon dioxide emissions.

Three-Phase Modular Multilevel Marx Converter 3M C

Three-phase modular multilevel Marx converter (3M 3 C) with five levels for each leg is described. The proposed modular converter is based on Marx generators and uses M 3 C topology (Fig. 1a). Its operation is described in detail in [START_REF] Encarnação | A New Modular Marx Derived Multilevel Converter[END_REF]. This topology has three bi-directional switches (S k D k , S ka D ka and S k k+1 D k k+1 ) and a capacitor denoted C k that will be charged with U Ck =U dc /(n-1), where n=5, implying U Ck =U dc /4, and k represents the number of M 3 C cells or modules. The strategy to equalize all C k capacitors will be explained in Section 4. [START_REF] Redondo | Repetitive High-Voltage Solid-State Marx Modulator Design for Various Load Conditions[END_REF].

With the basic M 3 C cell, n levels MCs can be obtained, using n-1 basic cells for the upper arm, and n-1 cells for the bottom arm. The five-level MC leg topology uses four basic cells (Fig. 1b) for each half arm, with a total of 8 cells for each converter arm (or phase). A three-phase 3M 3 C with five levels for each phase uses twenty four basic cells (Fig. 1b). To understand the operating principles of three-phase M 3 C, consider the five-level leg in which, to verify Kirchhoff's laws, (n-1) basic cells must present the voltage U dc /(n-1). Table 1 show all possible combinations with the basic cells (bottom and top arm) to have 4 basic cells turned off (S Ka on) and 4 turned on (S K on) and the corresponding voltage levels on each arm.

Table 1 shows the five voltage levels of voltage U Load and the number of vectors for each level (there are 36 redundant vectors in level 3).

Modeling and Controller Design of M 3 C Connection to Inshore Network

The dynamic models of the dc offshore fed M 3 C to the inshore ac network are obtained in this section, together with the average current control used for power flow, harmonic and reactive power compensation. The strategy to equalize all the capacitors voltages is presented with a detailed implementation of the Vector Decision algorithm. The ac side model ( Fig. 2) in system coordinates (123) can be obtained using the Kirchhoff voltage and current laws, considering R 1 =R 2 =R 3 and R 11 =R 22 =R 33 and L 1 =L 2 =L 3 and L 11 =L 22 =L 33 then R eq =R 1 +R 11 and L eq =L 1 +L 11 :

1 1 1 1 2 2 2 2 3 3 3 3 S e q eq S e q eq S e q eq di U R i L e dt di U R i L e dt di U R i L e dt  = + +    = + +    = + +   (1)
The system model in dq0 coordinates (2) can be obtained applying Park's transformation to [START_REF] Adam | Comparison between Two VSC-HVDC Transmission Systems Technologies: modular and Neutral Point Clamped Multilevel Converter[END_REF]. The average current control method uses this state-space model (2) for power flow control, harmonics and reactive power compensation.

( ) ( )

1 1 e q d d eq q d d eq eq q e q q eq d q q eq eq R

di i L i u e dt L L di R i L i u e dt L L  = - + ω + -     = - + -ω + -   (2)

Linear Control of the U dc Voltage

This section presents the dc capacitor voltage controller (Fig. 1b). The linear control of the U dc voltage provides the i d ref component which in turn will be used to enforce the converter ac output voltages references. The controller is designed from the converter model (dc side) in dq0 coordinates (Fig. 3). The transfer function results in a 2 nd order system (4) with a zero located in -k I /k P .

( ) ( )

P d I d eq eq dc P d I d dc ref eq eq k k s C C U s k k U s s s C C γ γ + = γ γ + + (4)
In order to obtain a fast response with reduced overshoot, the zero of the controller should obey

I P n k k - ≤ -
C k k C k C k C ξω  γ  ξω = =   γ   ⇒   γ ω   ω = =   γ   (5)
Using the M 3 C converter parameters, 100kV 2 mF ; 0.5 200kV

d eq d dc u C U = γ = = =
the PI parameters are k I =98.7 and k P =0.9.

Linear Control of i d and i q Currents

This section details the average current control of i d and i q currents. From the state-space model (2) in dq0 coordinates, making

d eq q d d
H L i u e = ω +and q eq d q q

H L i u e = -ω +the linear model ( 6) is obtained. To obtain the control value H d we use a PI controller where I dref (set-point) is obtained from the output of the U dc voltage controller. To obtain H q another PI controller is used, with the set-point I qref =0 to ensure reactive power compensation and harmonics mitigation.

e q d d d eq eq q e q q q eq eq R di

H i dt L L di R H i dt L L  = - +     = - +   (6)
Neglecting R eq the following block diagram was obtained (Fig. 5): The resulting 2 nd order system with a zero enables the calculation of k P2 and k I2 :

I n eq k L k L  ξ ω =     ω =   ⇒ 2 2 2 2 2 2 2 P n eq I n eq k L k L = ξ ω    = ω   (7)
The PI controllers are also designed for a damping factor 2 2 / 2 ξ = and ω n2 =2π×25 rad.s -1 . Considering L eq =L 1 +L 11 =32mH ( Fig. 2), the PI parameters are k I2 =789.6 and k P2 =7.1. Equations of H d and H q can be solved for u d and u q to obtain the ac output voltages (Fig. 6).

Fig. 6. Block diagram with average current control method to obtain the converter ac output voltages

Using inverse Park transformation three sinusoidal voltages references can be obtained from u d and u q . Then these voltages are fed to three sigma-delta modulators to obtain the three converter ac output voltages in phase coordinates (Fig. 7). Each sigma-delta modulator uses the sliding-mode stability condition to ensure the desired voltage applied to the load by increasing or decreasing the chosen level. The modulator uses five levels, u mk (±½ U dc =100kV, ±1/4 U dc =50kV and 0V) to obtain the voltage output U 1 .

Hd

DC Capacitor Voltage Balancing

To balance all the capacitor voltages a Vector Decision algorithm was developed. This algorithm does not need data from the capacitor voltage measurements.

For three-level MC leg topology the algorithm is not necessary to balance the capacitors voltages, because the S k k+1 semiconductor of each basic cell (Fig. 1a) guarantees voltage balancing. However, for M 3 C with leg levels higher than three, a different strategy is needed. The idea of the Vector decision algorithm is to equally select the n-1 vectors of each intermediate level (2, 3 and 4), where n represents the number of levels. To obtain the first (-U dc /2) and the fifth voltage level (+U dc /2), only one vector is available. Therefore, the control of each converter leg will use 14 vectors (1+4+4+4+1) (Table 2). The Vector decision algorithm must use different cells (turned on) in the same level to perform capacitor voltage balancing. Since in the intermediate levels there are (3×4) redundant vectors a counter for each leg was devised as a function of the respective level (Table 2). The redundant vectors produce the same output voltage in the converter but different effects in the capacitors (charge or discharge). Table 2 shows the redundant vector choice (a, b, c, d) in states -1, 0, 1 to guarantee capacitor voltage balance.

The algorithm assigns a different vector for each leg as defined in table 2. The decision vector is chosen from the output status C k and the output of the sigma delta modulator, γ k value. Each C k output depends on the status of three internal counters, one for each intermediate level (C 10 ⇔γ -1 , C 20 ⇔γ 0 , C 30 ⇔γ 1 ). This internals counters, reset to one after three increments, repeating the same sequential assignment for each cycle.

Table 2. Vector decision Table

γ k C k -2 -1 0 1 2 0 V1 V5 1 V2a V3a V4a 2 V2b V3b V4b 3 V2c V3c V4c 4 V2d V3d V4d

Simulation Results

The grid integration of an offshore wind farm using a modular Marx multilevel

[-2;2] [-1;1]
Vector Decision In order to verify the effectiveness of the proposed strategy power variations were simulated from offshore wind park (begin with 500A, then change to 700A at 0.4s, and after to 800A at 0.6s and finally a step current to 1000A at 0.8s. Fig. 9 shows the 8 capacitor voltages in arm 1 balanced within approximately ±5% of their working voltage (50kV).

Fig. 10a shows the linear control of the U dc voltage (top left waveform) and the 2 nd waveform presents the I d control. The 1 st waveform (top right) of Fig. 10b shows the H d component and the 2 nd waveform presents H q component. Fig. 11a presents input and output power figures to show the high efficiency of this power converter (≈98%). Reactive power compensation is observed in Fig. 11b, in which the phase angle between current (in line 1) and voltage (Us×0.01) (in phase 1) is zero. Fig. 12a show the three-phase converter ac output voltages (u m1 , u m2 and u m3 ). Fig. 12b presents converter ac output voltage in phase 1 and sinusoidal output voltage reference U S1ref .

Conclusions

This paper has presented modular multilevel converter for connecting an offshore wind park to the inshore ac power grid network (HVDC-VSC). The three-phase modular multilevel Marx converter (3M 3 C) has five levels in each arm.

The multilevel converter was built with modular MC cells that can be used to build multilevel converter with high number of levels. The capacitor voltage equalization is obtained with a strategy (Vector Decision algorithm) that does not need to measure the capacitor voltages. The dc link capacitor voltage control was designed using a linear controller. Sigma-delta modulators were used to obtain the converter ac output voltages, and average current control was used for power flow control, harmonics and reactive power compensation. Presented simulations have shown the effectiveness of the proposed strategy

Fig. 1 .

 1 Fig. 1. Modular Multilevel Marx Converter topology: a) Structure of the basic cell; b) Three-phase M 3 C with five levels for each phase leg (Ph→phase SM→sub-module cell)

Fig. 2 .

 2 Fig. 2. Electrical diagram of offshore dc fed M 3 C to the inshore ac network

  Considering

Fig. 3 .

 3 Fig. 3. Equivalent model of the converter DC side Neglecting the pole associated to a small time delay T d in the current controlled current source γ d I d , the block diagram for the U dc control is shown in Fig. 4.

Fig. 4 .

 4 Fig. 4. Block diagram of the linear control of the U dc voltage

Fig. 5 .

 5 Fig. 5. Block diagram of the linear control of I d

Fig. 7 .

 7 Fig. 7. Block diagram of sigma-delta modulator to obtain the converter ac output voltages

  using MATLAB/Simulink environment in accordance with Fig.8. These simulations illustrates the three-phase M 3 C operating as a multilevel inverter to deliver energy (3 sinusoidal output voltages with reference amplitude equal to 100kV) from the offshore wind park (bipolar connection) to the power grid. The main circuit parameters (Fig.1, Fig.2and Fig.8) are U dc =200kV, C A =C B =4mF, 24 Capacitor cell C 1-24 =400µF, R 1 =0.33Ω, L 1 =30mH, R 11 =0.1Ω, L 11 =2mH.

Fig. 8 .

 8 Fig. 8. Block diagram of the dc fed M 3 C Connection to Inshore ac Network

8 )Fig. 9 .

 89 Fig. 9. Simulation results showing balanced capacitor voltages in arm 1 of 3M 3 C

Fig. 11 .Fig. 12 . 1 e 1

 111211 Fig. 11. Simulation results a) Output and input powers b) Current and voltage×0.01 in phase1

Table 1 .

 1 Voltage

	Level	U Load	Number of ON Cells Upper | Bottom	Number of Vectors
	1	-U dc /2	0	4	1×1=1
	2	-U dc /4	1	3	4×4=16
	3	0	2	2	6×6=36
	4	+U dc /4	3	1	4×4=16
	5	+U dc /2	4	0	1×1=1

levels and number of vectors for a Five-Level M 3 C leg

  ξω , therefore the C(s) controller is designed for a damping

	factor	ξ =	2 / 2	and ω n =2π×25 rad.s -1 (	n ω	=	2 / π	d T	), the following relations hold:
				2	n		P d		P	2	n eq
							eq			d
				n	2	I d eq		I	n	2	d	eq