domain: info stat

text:
Zhaoxin Chen

Eric Anquetil
email: eric.anquetil@irisa.fr

Harold Mouchére

Christian Viard-Gaudin
email: christian.viard-gaudin@univ-nantes.fr

Recognize multi-touch gestures by graph modeling and matching

Extract the features for a multi-touch gesture is difficult due to the complex temporal and motion relations between multiple trajectories. In this paper we present a new generic graph model to quantify the shape, temporal and motion information from multi-touch gesture. To make a comparison between graph, we also propose a specific graph matching method based on graph edit distance. Results prove that our graph model can be fruitfully used for multi-touch gesture pattern recognition purpose with the classifier of graph embedding and SVM.

Introduction

Due to the recent prevalence of multi-touch devices, multi-touch gesture recognition has gained a large interest in the last decade. It is possible to consider two different application frameworks for multi-touch gestures: gestures as direct manipulation or indirect command. The former study is usually based on low-level touch events (e.g., touch-down, touch-move, touch-up) and has been extensively applied in image or map view application (e.g., pinch or spread two fingers to scale an image). The latter is much similar to the character recognition as the recognition procedure is conducted after the entire gesture being performed and triggers a corresponding operation (e.g., copy, paste, etc.). To the best our knowledge, most research in the latter are based on mono-touch gesture. Our work focuses on the difficulties on multi-touch gesture recognition and tries to make it possible for indirect command.

Unlike mono-touch gesture recognition which tracks the movement of a single point of input, multi-touch gesture often tracks many points of contact in parallel as they appear, move and disappear. The recognition for multi-touch gestures is challenging because of the complex chronological relation between the fingers' trajectories. Consider a two-stroke gesture under a multi-touch input context. Although the shape of the strokes is fixed, the writing order could vary from in-sequence to full-synchronizing as explained in Figure 1. In the work (Hao Lu & Yang Li, 2012), the authors present a GestureCode system recognizing multi-touch gestures with state machines. They achieved around 75% accuracy on a 15-gesture database. In (Kenrick Kin et al., 2012), the Proton++ describes the multi-touch gestures as regular expressions of touch event symbols. GeForMt [START_REF] Kammer | Towards a formalization of multi-touch gestures[END_REF] uses context-free grammars to model multi-touch gestures.

In our prior work [START_REF] Chen | A graph modeling strategy for multi-touch gesture recognition[END_REF], we use graph to characterizes the stroke shape information on nodes and stroke spatial and temporal information on edges. In most of the cases, this approach could well recognize the multi-touch gestures which are alike in shape but different in stroke order. However, it only uses some discrete attributes as many previous works (Kenrick Kin et al., 2012) to indicate the state or shape of the strokes. In this paper, we propose an improvement of our prior work that quantifies these relations with numeric features to characterize the stroke shapes, spatial relationships, motion relationships and temporal relationships in the gesture.

Represent a multi-touch gesture by graph

Extracting global features has been proven to be efficient for handwriting character, symbol, icon or gesture recognition where the discrimination is mainly made according to the shape of the sample (Don [START_REF] Willems | Iconic and multi-stroke gesture recognition[END_REF]. For the multi-touch gesture case, since each stroke of gesture is no longer only written in sequence, the temporal relation as well as the relative motion relation of strokes should also be characterized. Therefore, in our graph model, we propose to use the node to describe the shape of each stroke and edge to indicate their temporal and motion relation. In the following paragraphs, the graph will be denoted by , , where denotes a finite set of nodes, denotes a set of edges.

Preprocessing and sub-stroke representation

In general, an on-line handwriting data is a set of consecutive points which records the coordinate ( , ), time ( ) for each input event. It is unwise to use raw coordinate point as node because the complexity of graph matching in the following procedure is exponential in the number of nodes. We consider to segment each raw strokes into a sequence of straight line sub-strokes , … , using polygonal approximation. The Ramer-Douglas-Peucker algorithm (Urs [START_REF] Ramer | An iterative procedure for the polygonal approximation of plane curves[END_REF] is used to find a polygonal chain with few segments that serves as an accurate approximation of the stroke. Each sub-stroke is then represented by a node in the graph with an attribute of 4 components, , , , , where is the position (unit square bounding box coordinates) of the center of the sub-stroke, is its angle relative to the horizontal, length of the sub-stroke, contains start and end times. Each feature is normalized to [0,1]. Thus, assume a multi-touch gesture with n strokes, the node set V is represented as , … , , … , , … , , … , , [1] where is the subset which contains the nodes belonging to the same stroke .

is the node which represents the ith sub-stroke of the jth stroke. Once the node set V is obtained, a link edge is built from to which indicates the written sequence of a stroke. Since the direction attribute has been contained in node (as angle), the edge only indicates the concatenating relation between nodes.

Representation for temporal and motion relation between strokes

Differ from the mono-touch handwritten gesture, the multiple strokes in a multi-touch gesture are not performed stroke after stroke. The relative temporal relation between each two strokes could be in-sequence, semisynchronizing or full-synchronizing. To quantify these relations, we define a fuzzy relative temporal relation function , as expressed by equation [2] and illustrated in figure 1, where is the time of overlap detected in the semi-synchronizing case, and are the time duration for the two strokes. In our graph, this relation function is implemented on each pair of nodes (sub-strokes) which do not belong to same stroke. When value of the relation function is larger than a threshold as , , a synchronization edge is built between the corresponding two nodes.

In-sequence Semi-synchronizing Full-synchronizing , 0 min 1, , 1 in-seq semi-sync full-sync [2]
We now proceed to the modeling of motion information. The synchronization relation obtained above indicates the existence of a relative movement between sub-strokes. This relative movement is the key feature of a multi-touch gesture against a mono-touch gesture. Plenty of previous works (Kenrick Kin et al., 2012) made effort to extract the motion information from the movement. Inspired from the work by (Chi-Min [START_REF] Oh | MRF-based Particle Filters for Multi-touch Tracking and Gesture Likelihoods[END_REF], we measure three kinds of motion from the movement: translation, rotation and scaling. Figure 2 illustrates the computation of the three types of motion between sub-strokes. Note that it is unnecessary to compute these motion features between all two sub-strokes. The relative movement is meaningful only if two sub-strokes are written synchronously. In other words, these motion features are computed when a synchronization edge is existed between two nodes. Therefore, we assign these three values as the attributes of synchronization edge, , , . On the other hand, in order to capture the sequential relation between the sequential written strokes, we introduce a sequence edge that directs from the last sub-stroke of previse stroke to the first sub-stroke of the following stroke in our graph. This edge has an attribute as , which indicates the time delay between sequential written strokes.

The example in figure 3 considers a bimanual multi-touch gesture performed by a holding with left hand and writing 'z' character with right hand synchronously. The preprocessing step firstly resamples the gesture to 4 sub-strokes ( , … , ). Then, we convert each sub-strokes into node in the graph with its shape and duration information as its attribute, i.e.

, , , . Finally, two link edges and three synchronization edges (since the two strokes are performed synchronously, no sequence edge is built for this gesture) are built between nodes.

Graph matching algorithm

We adopt graph edit distance to compare the dissimilarity between two graphs. Since each graph has multiple sub-graphs, the first step is to find which sub-graph should be compare to which one from one graph to the other. This is a typical assignment problem that can be solved by Munkres' algorithm. The distance of each sub-graph pair is computed by DTW algorithm based on the sequence of nodes , , . Optimal matching is then found by Munkres' algorithm that total cost of the sub-graph assignment is minimized. Once we have the nodes' alignment path achieved by DTW, the edit operations on nodes and edges could be inferred from the nodes' DTW path. Consequently, the edit distance of two graph is defined as:

, , ,
where , denotes the edit cost for nodes pair, , is the edit cost for edges pair, denotes the assignment function which maps the nodes and edges of the first graph to the second graph .

Experiments

We describe below a series of experiments to evaluate our graph modeling and graph matching algorithm. The recognition system and database we use will be firstly introduced in this section.

Since we use the graph edit distance to measure the dissimilarity between graphs, a simplest recognition method is the k-Nearest Neighbors (kNN) which classifies an object by a majority vote of its neighbors. Meanwhile, we also implement a graph embedding algorithm introduced in (Kaspar [START_REF] Riesen | Graph Classification and Clustering Based on Vector Space Embedding[END_REF] to represent the graphs in vector space. The basic idea of this graph embedding algorithm is to represent a graph with a number of dissimilarities to a set of prototypical graphs. Prototypes are iteratively selected from training set under a prototype selection principles named Targetsphoere (also introduced in (Kaspar [START_REF] Riesen | Graph Classification and Clustering Based on Vector Space Embedding[END_REF]). A SVM classifier is then trained for the feature vector classification.

We designed a specific gesture database of 4346 samples collected from 21 participants. Based on these gestures, we defined a rotation dependent set (Set45) with 45 different classes and a rotation independent set (Set31) with 31 different classes (Figure 3). Both gesture sets have same content but different labels on some special cases. These gestures are distinguished from shape, temporal and motion information and can be generally separated to two categories. Category A contains bimanual or unimanual multi-touch gestures in which the trajectories are synchronously performed. Gestures in this category vary in shape, numbers of touch finger and motion. To validate the modeling of temporal information, category B contains 4 kinds of multi-stroke gestures which have same shape as some cases in category A but vary in written order. We conducted our experiments in a 5-fold cross validation and writer independent scheme. In each fold, samples from 6 writers are used for training and 15 writers for test.

Results

We compare the recognition accuracy between kNN and graph embedding regarding different numbers of prototypes. Results are given in figure 5. The experiments on set45 in Figure 5(a) show that with the increasing of the prototype number, the accuracy of graph embedding system keeps stable around 97% and performs significantly better than kNN system, which increases from 84.93% to 94.68%. Note that even with a limited number of prototypes, the graph embedding with SVM classifier could still obtain a good performance while kNN is not efficient enough. Same trend is also viewed in the result on set31. The graph embedding system performs as good as in set45 while accuracy of kNN system has a slightly decreasing from 94.68% to 93.59%. Results prove that graph modeling , matching and embedding framework is powerful to extract the discrimination information for multi-touch gesture pattern recognition.

Conclusion

In this paper, we propose a new method for modeling the shape, relative temporal and motion information in multi-touch gesture by a model of graph. We extract these information and quantify them as attribute on the nodes and in the graph. A way to make a comparison between graphs is also offered that can be used for graph classification. Results show that with the help of graph embedding and SVM classifier, our graph model has the ability to effectively distinguish different multi-touch gestures. Base on this model, in our future work we aim at developing a strategy to detect the pattern of multi-touch gesture at runtime as in (Kenrick Kin, 2012).

Figure 1 .

 1 Figure 1. Relative temporal relation between two strokes

Figure 2 .

 2 Figure 2. Three types of motion between sub-strokes, p and p represent the start points pair and end points pair, respectively. Since all the gestures are normalized inside a unit square bounding box, these three values have the same range in [0,1].

Figure 3 .

 3 Figure 3. Illustration of the graph representation. (a) Raw input multi-touch gesture. (b) Resampling the raw data to 4 substrokes using polygonal approximation. (c) Each sub-stroke is represented by a node with its features ( , , , ) as attribute. (d) Link edge and synchronization edge are built between nodes.

Figure 4 .

 4 Figure 4. Samples from the gesture database. Category A contains gestures in which the trajectories are synchronously performed. Note that the last two gestures Pinch (I) and Pinch (II) are allocated to different classes in rotation dependent set (Set45) but integrated in the same class in rotation independent set (Set31). Category B consists of two sequential mono-touch gestures which are distinguished from written order.

Figure 5 .

 5 Figure 5. Comparisons of the two recognition systems on both datasets. Accuracies are given as a function of the number of prototypes per class.


S Akkari
email: samy.akkari@supelec.fr

M Petit
email: marc.petit@supelec.fr

J Dai

X Guillaud
email: xavier.guillaud@ec-lille.fr

Interaction between the Voltage-Droop and the Frequency-Droop Control for Multi-Terminal HVDC Systems

Keywords: HVDC, grids, MTDC, voltage droop, frequency droop

MTDC systems need to provide a high level of reliability as well as efficient support to the interconnected AC grids. This can be achieved by equipping each converter with a dual controller combining both the voltage-droop and the frequency-droop control techniques. In this article, the theory behind the coupling between the two droops is investigated and an electromagnetic transient study of a 5-terminal HVDC grid is discussed to validate the theoretical results. The use of the dual controller allows the whole system to be more flexible. However, the frequency droop coefficient must be corrected to comply with the TSO's requirements.

Introduction

The ever growing installed capacity of offshore wind farms is pushing coastal nations to develop the Multi-Terminal HVDC (MTDC) technology, which is admittedly the most feasible solution to connecting the wind farms to onshore grids and interconnecting asynchronous AC systems [START_REF]The European Wind Energy Association[END_REF][START_REF]Renewable energy: Processing towards the 2020 target[END_REF]. However, the master-slave control technique used to operate the existing HVDC links is not suitable for meshed HVDC grids. The voltage-droop control technique, depicted in [START_REF] Akkari | Modélisation, simulation et commande des systèmes VSC-HVDC multi-terminaux[END_REF][START_REF] Haileselassie | Control of multi-terminal VSC-HVDC systems[END_REF][START_REF] Beerten | Modeling and control of multi-terminal VSC HVDC systems[END_REF], seems to be the best solution to safely and efficiently controlling MTDC systems.

As the use of renewable energy sources increases, their participation in the onshore grids frequency regulation is under consideration, especially given the fact that polluting conventional generation units participating in frequency regulation are gradually shut down and replaced by green energy sources such as offshore wind farms in order to reduce greenhouse gas emissions. There arises the necessity to develop a new control technique that would allow the AC/DC converters of the HVDC grid to participate in the primary frequency regulation of the AC systems. This can be achieved through the power electronics of the converters by using a frequency-droop controller based on the same principle as the frequency regulation controllers used in conventional generation units in AC grids [START_REF] Haileselassie | Primary frequency control of remote grids connected by multi-terminal HVDC[END_REF][START_REF] Rault | Investigation on interactions between AC and DC grids[END_REF][START_REF] Chaudhuri | System frequency support through multi-terminal DC (MTDC) grids[END_REF]. The injected/withdrawn power of the converter into/from the AC grid should correspond to a precise value with regards to the frequency deviation of the grid, and must comply with the AC grid Transmission System Operator (TSO)'s requirements. This paper considers a control strategy that combines both the voltage-droop and the frequency-droop control techniques for a Voltage Source Converter (VSC) HVDC. This dual control technique helps stabilizing the DC grid thanks to the DC voltage droop and allows the MTDC system to participate in the frequency regulation of the different AC grids interconnected by the meshed HVDC grid thanks to the frequency droop.

Only a few papers used the two droop control techniques at the same time on a single VSC-HVDC converter. While [START_REF] Chaudhuri | System frequency support through multi-terminal DC (MTDC) grids[END_REF] focuses on the eigenvalues of the system and [START_REF] Vrana | Active power control with undead-band voltage & frequency droop for HVDC converters in large meshed DC grids[END_REF] describes the use of a dead-band control for both droops, those two papers do not discuss interactions between the two droops. Without going into further details, [START_REF] Haileselassie | Primary frequency control of remote grids connected by multi-terminal HVDC[END_REF] reports that the frequency droop does not give the desired value for a given frequency deviation. [START_REF] Rault | Investigation on interactions between AC and DC grids[END_REF] is the only paper that really tried to understand the interactions between the AC grids and the DC grid through a modal analysis, whose conclusion was that the use of both droops resulted in a modification of the DC voltage dynamics and small changes in frequency.

This article points out the coupling between the two droops of the dual controller. The interactions between the frequency droop and the voltage droop are mathematically quantified in order to properly choose the slopes of the droops allowing the system to behave as expected. Then a simulation using the software EMTP-RV is run to validate the theoretical study.

The paper is outlined as follows. First, the principle of the voltage-droop and the frequency-droop control techniques as well as the functioning of the dual controller are discussed in Section 2. In Section 3, a theoretical analysis of the coupling between the two droops is conducted. Finally in Section 4, a simulation validating the theoretical study illustrates the behaviour of a 5-terminal MTDC system, which interconnects two offshore wind farms and three asynchronous AC grids, after an AC fault. Conclusions are drawn in Section 5.

Combining the voltage-droop and the frequency-droop control techniques

This section briefly explains the functioning principle of both the voltage-droop and the frequency-droop control. Then, it considers a control strategy combining the advantages of the two control techniques.

Droop-based control techniques directly modify the power reference of the converter. Figure 1 shows the global command strategy of a VSC-HVDC converter. In this section, the i th converter of a n-converter MTDC system is considered.

The voltage-droop control technique

The basic principle of the voltage-droop control is as follows. If a DC voltage variation is detected by a converter equipped with a voltage droop, this controller will generate a linear deviation of the power reference of the converter that helps stabilizing the MTDC system by balancing the power on the DC grid, thus preventing the DC voltage from dropping to critical levels.

The literature furnishes several methods of voltage-droop control. The one described in [START_REF] Akkari | Modélisation, simulation et commande des systèmes VSC-HVDC multi-terminaux[END_REF][START_REF] Beerten | Modeling and control of multi-terminal VSC HVDC systems[END_REF][START_REF] Rault | Dynamic Modeling and Control of Multi-Terminal HVDC Grids[END_REF] seems to be the best feasible solution thanks to its simplicity and efficiency.

A converter equipped with a voltage-droop control technique modifies its power reference according to the DC voltage level by moving its operating point along the characteristic line whose slope is 1 kv , where k v is the voltage-droop parameter, as depicted in [START_REF]The European Wind Energy Association[END_REF].

∆P * vi = 1 k vi ∆u si (1) where:
-∆P * i is the total power reference deviation of the i th VSC-HVDC converter. A positive P i corresponds to a power injection from the DC grid to the AC grid.

k vi is the voltage-droop parameter of the i th converter, k vi < 0 . -∆u si is the DC voltage variation at the i th converter terminal with regard to its voltage reference:

∆u si = u si ref -u si .
Each controller is also equipped with limiters that prevent the converter from moving too far away from its original operating point.

The block diagram of the voltage-droop control technique is depicted in Figure 2.

+ + P ref P * + - u s u sref Voltage Droop

The frequency-droop control technique

The primary frequency control is currently used by conventional generation units in AC grids. If the frequency drops below (respectively rises above) its nominal value, the conventional units inject more (resp. less) power into the AC grid in order to stabilize the frequency of the system.

The frequency droops proposed in [START_REF] Haileselassie | Primary frequency control of remote grids connected by multi-terminal HVDC[END_REF][START_REF] Rault | Investigation on interactions between AC and DC grids[END_REF][START_REF] Chaudhuri | System frequency support through multi-terminal DC (MTDC) grids[END_REF] are similar although there are some differences within the global control of the VSC-HVDC (see Figure 1 for the global control of the converter used in this paper).

The solution adopted in the present paper is the same as [START_REF] Rault | Investigation on interactions between AC and DC grids[END_REF]. Its operating principle is very similar to the one of the voltage droop of Section 2.1. A converter equipped with a frequencydroop control participates in the frequency regulation of the AC grid it is connected to and modifies its power reference by following the characteristic line whose slope is 1 k f , where k f is to the frequency-droop parameter:

∆P * fi = 1 k fi ∆f i (2)
where:

k fi is the frequency-droop parameter of the i th converter, k fi > 0. -∆f i is the frequency variation of the AC grid connected to the i th converter with regard to its frequency reference:

∆f i = f i ref -f i .
Each controller is also equipped with limiters that prevent the converter from moving too far away from its original operating point.

The block diagram of the voltage-droop control technique is depicted in Figure 3.

+ + P ref P * + - f f ref Frequency Droop ∆f ∆P * f ∆P ∆f
Fig. 3: Block diagram of the frequency-droop technique

The dual control technique

The voltage-droop control technique provides a strong and reliable DC grid, while the frequency-droop control technique provides assistance to the interconnected AC grids. An ideal MTDC system should have both : a strong and reliable DC grid as well as a good support to the interconnected AC grids.

The combination of those two control techniques leads to a dual controller depicted in Figure 4. This dual controller holds both the advantages of the frequency-droop control technique and the voltage-droop control technique. However, the combination of the two droops also leads to undesirable side-effects which will be further discussed in Section 3.

With this dual controller the converter's operating point does not move along a single line any longer but obeys [START_REF] Akkari | Modélisation, simulation et commande des systèmes VSC-HVDC multi-terminaux[END_REF].

∆P * i = ∆P * vi + ∆P * fi = 1 k vi ∆u si + 1 k fi ∆f i (3)

Interactions between the two droops

This section points out the existing coupling between the two droops used by the dual controller presented in Subsection 2.3 and theoretically quantifies their interactions. However, there is a coupling between the two droops. For example, if the DC voltage decreases (resp. increases), the voltage droop modifies the amount of power injected/withdrawn by the converter so that there is either less (resp. more) power withdrawn from the DC grid or more (resp. less) power injected into the DC grid. This, in turn, decreases (resp. increases) the frequency of the AC grid connected to the converter and the frequency droop opposes the action of the voltage droop, resulting in interactions between the AC grids and the DC grid.

This coupling is problematic since it degrades the performances of both the voltage droop and the frequency droop of the converter equipped with the dual controller. However, when the AC grid needs frequency support, priority should be given to the frequency droop since the TSO of the concerned AC grid expects a rigorous value of the frequency droop for the VSC-HVDC converter. This means the impact of the voltage droop on the frequency droop must be quantified and the frequency droop parameter must be corrected in order to comply with the TSO's requirements. The impact of the frequency droop on the voltage droop also needs to be quantified in order to be taken into account when managing the DC grid.

Theoretical study on the impact of the voltage droop on the frequency droop

The following theoretical study gives the direct relation between the frequency deviation of the grid and the power deviation of the converter.

In order to simplify the equations, the following simplifying assumption is made:

• DC voltage variations are identical at all nodes of the HVDC grid. This is equivalent to neglecting the variation of the sum of power losses in the DC grid.

Hence, if we consider an MTDC system with n converters: ∀j ∈ {1, . . . , n} , ∆u sj = ∆u s (4)

After an AC event such as the loss of a generation unit or a load modification, in steady state, the effective power deviation of each one of the n converters obeys the power reference deviation of (3), which gives (5):

∀j ∈ {1, . . . , n} , ∆P j = 1 k vj ∆u s + 1 k fj ∆f j (5)
Note: Equation ( 5) gives a general expression of the power deviation of each VSC-HVDC converter. If the considered converter j is not equipped with a voltage-droop controller then k vj = -∞ kV/MW. If the considered converter j is not equipped with a frequency-droop controller then k fj = ∞ Hz/MW.

The sum of the n equations ( 5) gives :

n j=1 ∆P j = n j=1 1 k vj ∆u s + n j=1 1 k fj ∆f j (6)
Since the variation of the sum of power losses is neglected and since the power on the DC grid is balanced by the converters equipped with a voltage droop, n j=1 ∆P j = 0. Thus (6) becomes:

n j=1 1 k vj ∆u s = - n j=1 1 k fj ∆f j (7)
and since ∆u s is considered identical for all VSC-HVDC converters, (7) leads to a new expression of ∆u s :

∆u s = -1 n j=1 1 kv j n j=1 1 k fj ∆f j (8)
Let us now consider the i th converter of the MTDC system. Replacing ∆u s in ( 5) by ( 8) yields:

∆P i = -1 k vi n j=1 1 kv j   n j=1 1 k fj ∆f j   + 1 k fi ∆f i (9)
Equation [START_REF] Vrana | Active power control with undead-band voltage & frequency droop for HVDC converters in large meshed DC grids[END_REF] shows that the power deviation of the i th converter consists of:

The power deviation created by the sum of the frequency droops of every converter on the DC grid weighted by

-1 kv i n j=1 1 kv j
. This coefficient depends on the action of the voltage droop of the i th converter and also depends on the contribution (voltage-droop parameters k vj ) of each converter participating in the DC voltage regulation.

The power deviation created by the frequency droop of the i th converter.

Since the maximum power deviation of an AC/DC converter is limited and since the total power transmitted through a converter is usually small in regard to the AC grid total active power, then if there is no AC fault happening at the same time on the other AC grids, ∆f j for j = i are negligible and ( 9) becomes:

∆P i = 1 k fi   1 - 1 k vi n j=1 1 kv j   ∆f i ( 10
)
This means that the actual relation between the AC grid frequency and the converter connected to this grid is now described by [START_REF] Rault | Dynamic Modeling and Control of Multi-Terminal HVDC Grids[END_REF] and not by ( 2) anymore. In order to honour the announced frequency droop of the converter to the TSO in ( 2) in spite of the interaction between the voltage droop and the frequency droop, the frequency droop parameter k f of ( 10) must be corrected.

Case study

This section illustrates the behaviour of a 5-terminal MTDC system similar to the one used in [START_REF] Sarlette | Cooperative frequency control with a multi-terminal high-voltage DC network[END_REF] that interconnects two offshore wind farms and three asynchronous AC grids after a disturbance. Simulations are run with the electromagnetic transients program EMTP-RV. The DC grid includes six DC lines, as depicted in Figure 5. The very simplistic AC grid 1 used in this simulation is also depicted in Figure 5. All grid parameters are listed in Appendix. In this section, only the VSC-HVDC converter 1 is equipped with the dual controller, meaning only the frequency of the AC grid 1 is supported by the MTDC system.

Converters 2 and 3 are only equipped with a voltage-droop controller, meaning they participate in the DC voltage regulation (along with converter 1) but they do not support the frequency regulation of their AC grids.

Converters 4 and 5 are offshore converters and are connected to wind farms. Since the time lapse of this study is small (about 30 seconds), the power output of the wind farms can be supposed constant. Converters 4 and 5 always inject into the DC grid the maximum output power of the wind farms. For this reason they are equipped with neither a frequency-droop controller nor a voltage-droop controller.

The modelling and the control of the VSC-HVDC converters is depicted in Figure 1.

The initial power reference of each VSC-HVDC converter as well as their droop coefficients are shown in Table 1.

Converter 1 2 3 4 5 P ref (MW) 150 200 -50 -100 -200 k v (kV/MW) -0.2 -0.2 -0.2 -∞ -∞ k f (Hz/MW) 0.005 ∞ ∞ ∞ ∞
Table 1: Power reference values and droop coefficients of the VSC-HVDC converters

Case of an AC generation unit loss

In this simulation, the AC grid 1 loses a generating station of 100 MW at t = 3s (generator 2 of Figure 5).

Interaction between the frequency droop and the voltage droop

Figure 6 shows the converter 1's power reference modification due to the voltage droop and the frequency droop. Since more power is withdrawn from the DC grid and injected into the AC grid 1 due to the activation of the frequency-droop controller, the DC voltage drops as well, which activates the voltage droop of the dual controller. The power deviation created by the voltage droop is caused by this interaction between the AC and the DC grid and hampers the frequency droop, preventing it from functioning at its full efficiency. • The voltage-droop controller is activated but the frequencydroop controller is not activated (dashed curve). • The voltage-droop controller is not activated but the frequency-droop controller is activated (dashdotted curve). • The dual controller is fully activated (solid curve).

Figure 7 shows that the DC voltage remains the same when the frequency-droop controller is deactivated (dashed curve): the DC grid does not participate in the frequency regulation of AC grid 1. When the dual controller is fully activated (solid curve), the DC voltage does not drop as low as when the voltage-droop is deactivated (dashdotted curve) because the voltage-droop controller tries to limit the voltage drop. However, in this case the activation of the voltage-droop controller is not desired because it hampers the functioning of the frequency droop, and is solely caused by the interactions between the DC and the AC grid.

Figure 8 shows that the dual controller participates in the frequency regulation (solid curve) and stabilizes the frequency at a level above the scenario where the frequency droop is deactivated (dashed curve). However, the frequency droop is not as efficient as it is in the scenario where the voltage droop is deactivated (dashdotted curve).

In accordance with the agreement signed with the TSO, the VSC-HVDC converter must participate in the frequency regulation with a frequency droop parameter of 5 mHz/MW. Because of the interaction between the two droops, the actual relation between the power deviation of the converter and the frequency of the AC grid is ∆f1 ∆P1 = 7.7 mHz/MW. Hence the frequency droop does not meet the requirements agreed with the TSO and needs to be corrected.

Correction of the frequency droop

Equation [START_REF] Rault | Dynamic Modeling and Control of Multi-Terminal HVDC Grids[END_REF] indicates that the relation between the AC grid's frequency and the total power deviation of converter 1 is:

∆P 1 = 1 k f1   1 - 1 k v1 5 j=1 1 kv j   ∆f 1 (11)
The original engagement with the TSO was for the frequency droop to obey (12):

∆P 1 = 1 k f1 ∆f 1 (12)
In order to fulfil the TSO's requirements, the original frequency droop parameter k f1 must be corrected.

By replacing k f1 in (11) by the corrected value k f1 in (13), the effective frequency droop finally corresponds to the agreed value with the TSO.

k f1 = k f1   1 - 1 k v1 5 j=1 1 kv j   (13)
According to (13), k f1 needs to be rescaled by 1-

1 kv 1 5 j=1 1 kv j
which corresponds to 67% of the original frequency droop parameter.

Figure 9 shows that, initially, the converter equipped with a dual controller does not meet the requirements imposed by the TSO: the dash curve does not fit the dashdotted curve. However, with the corrected value of the frequency droop, the dual controller allows the converter to fulfil the TSO's requirements (the solid curve fits the dashdotted curve) while still participating in the voltage regulation of the DC grid through an active voltage droop.

Conclusions

After reviewing the basic functioning of the voltage droop and the frequency droop for VSC-HVDC converters, this paper discussed the use of a dual controller using both droops at the same time. The coupling between the two droops was pointed out and a theoretical study showed that this coupling makes it necessary for the frequency droop parameter to be corrected in order to comply with the TSO's requirements.

Simulations using the software EMTP-RV showcases the impact of the voltage droop on the frequency droop after an AC fault and validates the corrected value of the frequency droop parameter obtained in the theoretical study, allowing the converter to comply with the TSO's exigences despite the interactions between the two droops.

The dual controller is a viable alternative to single-droop controllers since it offers the advantages of the two droops at the same time, thus a more reliable system. However, the interactions between the AC grids and the DC grid must not be overlooked since they degrades the efficiency of both droops.

Fig. 2 :

 2 Fig. 2: Block diagram of the voltage-droop technique

Fig. 4 :

 4 Fig. 4: Block diagram of the dual controller

Fig. 5 :

 5 Fig. 5: Single-line diagram of the studied 5-terminal MTDC system and the AC grid 1

Figure 7 and

 7 Figure 7 and Figure 8 respectively show the voltage of the DC grid and the frequency of the AC grid 1 in three different scenarios with regards to the dual controller of the converter 1:

1 Fig. 6 :Fig. 7 :k f 1 =Fig. 8 :

 16718 Fig. 6: Power reference modification due to the voltage droop and the frequency droop

Corrected k f 1 Fig. 9 :

 19 Fig. 9: Frequency of the AC grid 1

Generator data:


S Akkari
email: samy.akkari@centralesupelec.fr

J Dai

M Petit

X Guillaud

Coupling between the Frequency Droop and the Voltage Droop of an AC/DC Converter in an MTDC System

Keywords: frequency control, HVDC, MTDC, voltage control

MTDC systems are required to be reliable, and in certain cases, should participate in the AC grids frequency regulation. This can be achieved by providing AC/DC converters with a dual controller combining both the voltage-droop and the frequency-droop control techniques. In this paper, the coupling between the two droops is theoretically investigated and quantified for both a DC side fault and an AC side fault. A 5-terminal HVDC grid is simulated to validate the theoretical results.

I. INTRODUCTION

It is anticipated that the North Sea will hold an installed wind energy capacity of more than 100 GW by 2030 [START_REF]The European Wind Energy Association[END_REF], [START_REF]Renewable energy: Processing towards the 2020 target[END_REF]. The Multi-Terminal HVDC (MTDC) technology seems to be the most feasible solution to connecting the wind farms to onshore grids as well as interconnecting asynchronous AC systems [START_REF] Kirby | HVDC transmission for large offshore wind farms[END_REF]. Over the last decade, voltage-droop-based control schemes have become the preferred option to safely and reliably coordinate the power exchange between the terminals [START_REF] Haileselassie | Control of multi-terminal VSC-HVDC systems[END_REF], [START_REF] Beerten | Modeling and control of multi-terminal VSC HVDC systems[END_REF].

With the shut-down of polluting conventional generation units participating in the frequency regulation, MTDC systems will most likely be solicited to support the AC grids' frequency. This can be achieved through the power electronics of the converters by using a frequency-droop controller [START_REF] Rault | Investigation on interactions between AC and DC grids[END_REF], [START_REF] Chaudhuri | System frequency support through multi-terminal DC (MTDC) grids[END_REF], based on the same principle as the speed governor used in conventional generation units in AC grids.

Ideally, any converter connected to an onshore AC grid should be equipped with a dual controller implementing both droops. However, [START_REF] Akkari | Interaction between the voltage-droop and the frequency-droop control for multi-terminal HVDC systems[END_REF] shows that using both droops at the same time leads to interactions between the AC grids and the MTDC system that degrade the efficiency of each droop. These interactions need to be theoretically quantified and the frequency droop parameter must be corrected so that the frequency droop still fulfils the agreement with the TSO despite the coupling between the two droops. The impact of these interactions on the DC voltage variations also needs to be quantified in order to allow the correct design of the DC voltage-droop parameters as well as the DC grid.

II. THE DUAL CONTROL TECHNIQUE

A. Control of the VSC-HVDC converter

The modelling and the control of the VSC-HVDC converter has already been covered in the literature (see [START_REF] Beerten | Modeling and control of multi-terminal VSC HVDC systems[END_REF] and [START_REF] Rault | Dynamic Modeling and Control of Multi-Terminal HVDC Grids[END_REF]). One common model is depicted in Figure 1, where two droops modify the power reference of the active power loop of the converter.

B. The dual control technique

The functioning of the voltage droop and the frequency droop is fully detailed in [START_REF] Akkari | Interaction between the voltage-droop and the frequency-droop control for multi-terminal HVDC systems[END_REF]. The combination of the frequency-droop and the voltage-droop control techniques leads to a dual controller depicted in Figure 2. This dual controller enjoys the advantages of the two droops. However, their combination also leads to undesirable side-effects which will be further discussed in the following section.

+ + P * g P * ∆P * + - f f * Frequency Droop ∆P * f = 1 kf ∆f ∆f ∆P * f ∆P ∆f + - + + u s u * s Voltage Droop ∆P * v = 1 kv ∆u s ∆u s ∆P * v ∆P ∆us

Fig. 2. Working principle of the dual controller

With this dual controller, the i th converter's operating point obeys

∆P * i = ∆P * vi + ∆P * fi = 1 k vi ∆u si + 1 k fi ∆f i (1)
where:

-∆P * i is the total power reference deviation of the i th VSC-HVDC converter. A positive P i corresponds to a power injection from the DC grid to the AC grid.

k vi is the voltage-droop parameter of the i th converter, k vi < 0. -∆u si is the DC voltage control error at the i th converter terminal with regard to its voltage reference:

∆u si = u si ref -u si .
k fi is the frequency-droop parameter of the i th converter, k fi > 0. -∆f i is the frequency control error of the AC grid connected to the i th converter with regard to its frequency reference:

∆f i = f i ref -f i .
C. The dual nature of the controller

Using both droops at the same time creates a coupling between them. In fact, if the DC voltage decreases, the voltage droop modifies the power operating point of the converter to retain more energy inside the DC grid. As a consequence, the balancing is modified in the AC areas. The frequency droop is then activated and opposes the action of the voltage droop by re-modifying the power operating point in the opposite direction in order to inject more power into the AC grids.

This coupling is problematic since it degrades the performances of both droops. However, when the AC grid needs frequency support, priority should be given to the frequency droop since the TSO of the concerned AC grid expects that the VSC-HVDC converter should respect a contractual value of the frequency droop. Thus, the impact of the voltage droop on the frequency droop must be corrected in order to comply with the TSO's requirements. On the other hand, the impact of the frequency droop on the voltage droop in the event of a DC fault also needs to be quantified in order to choose the voltage-droop parameter correctly.

III. THEORETICAL STUDY ON THE MUTUAL IMPACT OF TWO DROOPS

A. Impact of the voltage droop on the functioning of the frequency droop

In this section, the impact of the voltage droop on the frequency droop in the event of an AC fault is studied.

This theoretical study tries to find the relation between the frequency deviation of the AC grid and the effective power deviation of the converter connected to the AC grid under the dual controller. It is assumed each AC grid is connected to the DC grid through a single VSC converter.

In order to simplify the equations, the following simplifying assumption is made: DC voltage variations are identical at all nodes of the HVDC grid. This is equivalent to neglecting the variation of the power losses in the DC grid.

Hence, for an MTDC system with n converters:

∀j ∈ {1, . . . , n} , ∆u sj = ∆u s (2)
After an AC event such as the loss of a generation unit or a load change, the effective power deviation of each one of the n converters obeys its reference described by [START_REF]The European Wind Energy Association[END_REF], which gives ∀j ∈ {1, . . . , n} ,

∆P j = 1 k vj ∆u s + 1 k fj ∆f j (3)
If the considered converter j is not equipped with a voltagedroop and/or a frequency-droop controller then k vj = -∞ and/or k fj = ∞.

The sum of the n equations (3) gives:

n j=1 ∆P j = n j=1 1 k vj ∆u s + n j=1 1 k fj ∆f j (4)
Since the variation of the power losses is neglected and that the power on the DC grid is balanced by the converters equipped with a voltage droop, we have n j=1 ∆P j = 0. Thus (4) becomes:

n j=1 1 k vj ∆u s = - n j=1 1 k fj ∆f j (5)
In addition, since ∆u s is considered identical for all VSC-HVDC converters, (5) leads to a new expression of ∆u s :

∆u s = -1 n j=1 1 kv j n j=1 1 k fj ∆f j (6)
Let us now consider the i th converter of the MTDC system. Replacing ∆u s in ( 3) by ( 6) yields:

∆P i = -1 k vi n j=1 1 kv j   n j=1 1 k fj ∆f j   + 1 k fi ∆f i (7)
Equation [START_REF] Chaudhuri | System frequency support through multi-terminal DC (MTDC) grids[END_REF] shows that the power deviation of the i th converter consists of:

The power deviation created by the sum of the frequency droops of every converter on the DC grid weighted by

-1 kv i n j=1 1 kv j
. This coefficient depends on the contribution of each converter participating in the DC voltage regulation of the DC grid (voltage-droop parameters k vj ). The power deviation created by the frequency droop of the i th converter. Since the maximum power deviation of an AC/DC converter is limited and that the total power transmitted through a converter is usually small compared to the AC grid total active power, then if there is no AC fault occurring at the same time on the other AC grids, ∆f j for j = i are negligible and ( 7) becomes:

∆P i = 1 k fi   1 - 1 k vi n j=1 1 kv j   ∆f i (8)
Hence, because of the interaction between the frequency droop and the voltage droop, the actual relation between the AC grid frequency and the converter connected to this grid is now described by [START_REF] Akkari | Interaction between the voltage-droop and the frequency-droop control for multi-terminal HVDC systems[END_REF]. This means the effective frequency-droop value, k fi (ratio of the frequency and the power extracted or withdrawn from the AC grid), is not equal to the initial frequencydroop value k fi since k fi = ∆fi ∆Pi =

k f i   1- 1 kv i n j=1 1 kv j   .
In order for the effective frequency droop to satisfy the contractual frequency-droop value k fi contractual required by the TSO, the initial frequency-droop parameter k fi must be corrected such that the corrected value takes into account the voltage-droop parameter of each converter of the DC grid:

k fi = k fi contractual   1 - 1 k vi n j=1 1 kv j   (9)
In this case, the effective frequency-droop value (k fi ) is actually equal to the contractual frequency-droop value (k fi contractual ) agreed with the TSO.

B. Impact of the frequency droop on the functioning of the voltage droop

In this section, the impact of the frequency droop on the voltage droop in the event of a DC fault is studied.

Even though priority should be given to the frequency droop for AC grid support, the voltage droop is important for regulating the voltage of the DC grid, especially when a DC fault occurs. In fact, a malfunctioning of the voltage droop could cause the whole MTDC system to collapse in the event of a DC fault.

Since a DC fault can lead to a converter loss, this study considers that n -1 converters are still connected to the DC grid after a DC fault while the k th converter is lost. The power deviation of each converter after the DC fault is as follows:

   ∀j ∈ {1, . . . , n} \ {k} , ∆P j = 1 k vj ∆u s + 1 k fj ∆f j ∆P k = ∆P fault k = -P k (10)
The sum of the n -1 equations (10) gives:

n j=1 j =k ∆P j = n j=1 j =k 1 k vj ∆u s + n j=1 j =k 1 k fj ∆f j (11)
Since the variation of the power losses is neglected and that the power on the DC grid is balanced by the converters equipped with a voltage droop,

n j=1 j =k ∆P j = P k .
Thus (11) becomes:

n j=1 j =k 1 k vj ∆u s = P k - n j=1 j =k 1 k fj ∆f j (12)
In addition, since the DC voltage deviations are supposed identical for all VSC-HVDC converters, (12) leads to a new expression of ∆u s :

∆u s = 1 n j=1 j =k 1 kv j     P k - n j=1 j =k 1 k fj ∆f j     (13)
Equation ( 13) shows that the DC voltage variation of the HVDC grid consists of the difference of the power initially injected or withdrawn by the lost converter and the power deviation created by the sum of the frequency droops of every converter on the DC grid. This difference is weighted by the coefficient

1 n j=1 j =k 1 kv j
which depends on the contribution of each converter still connected to the DC grid after the disturbance and participating in the DC voltage regulation (voltage-droop parameters k vj ).

IV. CASE STUDY

This section illustrates the behaviour of a 5-terminal MTDC system interconnecting two offshore wind farms and three asynchronous AC grids.

The AC grid 1 is either strong (such as a continental AC grid) or weak (such as an islanded AC grid) depending on the simulations, whereas AC grids 2 and 3 are always strong AC grids. Converters 1, 2 and 3 are equipped with dual controllers (frequency and voltage droops) whereas converters 4 and 5 do not have the dual controller since they are connected to offshore wind-farms and inject all the power harvested by the wind-farms into the DC grid. Since the time lapse of this study is about 20 seconds only, the power output of the wind farms are supposed constant.

Simulations are run using EMTP-RV. The 5-terminal HVDC grid topology is depicted in Figure 3. The topology of the AC grid 1 with 5 generators used in these simulations as well as the AC and DC line parameters are detailed in the Appendix. The initial power reference of each VSC-HVDC converter as well as their droop coefficients1 are shown in Table I

TABLE I POWER REFERENCE VALUES AND DROOP COEFFICIENTS OF THE VSC-HVDC CONVERTERS

A. Case of an AC event

In this simulation, the AC grid 1 is a weak islanded AC grid (250 MW generation). The power withdrawn from the DC grid by the converter 1 is set at 200 MW (almost 45% of the total power of the AC grid). At t = 2 s, an additional load of 50 MW is connected, leading to a power imbalance on the AC grid.

Because of the imbalance between load and generation, the frequency of the AC grid 1 decreases. Figure 4 and5 show respectively the frequency of the AC grid 1 and the power transiting through the VSC converter when:

• Only the voltage droop of the dual controller of converter 1 is activated (no frequency droop). • The dual controller is fully activated and the frequencydroop parameter is the original value. • The dual controller is fully activated and the frequencydroop parameter is the corrected value. The dashed curve shows a larger frequency decrease since the frequency droop of the VSC converter is not activated and thus no power is being extracted from the DC grid and injected into the AC grid to stabilize the frequency. The blue curve corresponds to the case where the frequency droop is enabled but the frequency-droop parameter is the original parameter that has not been corrected. In this case, the disruptive interactions between the two droops prevents the frequency droop from functioning at its full capacity. In fact, the TSO of the AC grid 1 expects a frequency-droop parameter k f1 from the VSC converter such that k f1 = ∆f1 ∆P1 = 0.05 p.u., but according to Figures 4 and5, the effective frequency-droop parameter is k f1 = ∆f1 ∆P1 = 0.074 p.u. because of the disruptive interactions between the two droops.

The red curve corresponds to the case where the frequency droop is enabled and the corrected value of the frequencydroop parameter has been implemented so that the frequency droop compensates the disruptive interaction between the droops. The TSO of the AC grid 1 expects an effective frequency-droop parameter k f1 from the VSC converter such that k f1 = ∆f1 ∆P1 = 0.05 p.u. In order to comply with this condition, the corrected frequency-droop parameter k f1 implemented in the VSC converter is:

k f1 = k f1   1 - 1 k v1 3 j=1 1 kv j   = 1 30 (14)
and according to Figures 4 and5, the effective frequencydroop parameter is actually equal to k f1 = ∆f1 ∆P1 = 0.05 p.u., despite the disruptive interaction between the voltage and the frequency droops.

B. Case of a DC event

In this scenario, the VSC-HVDC converter 4 is lost at t = 2 s. Two simulations are run: in the first one, the AC grid 1 is a strong AC grid (20 GW generation), while in the second one the AC grid 1 is a weak AC grid (250 MW generation). The power withdrawn from the DC grid by the converter 1 is the same in both simulations and is set at 200 MW.

Figure 6 shows that after the loss of converter 4 originally working as a rectifier, the DC voltage decreases, causing the three converters equipped with a voltage-droop control to modify their power operating point in order to balance the MTDC system and stabilize the DC voltage. Figure 7 shows the frequency of the AC grid 1 and Figure 8 shows the voltage of the DC grid when:

• Only the voltage droop of the dual controller of converter 1 is activated (k f1 = ∞). • The dual controller is fully activated and the AC grid 1 is a weak AC grid. • The dual controller is fully activated and the AC grid 1 is a strong AC grid. The voltage droop of the converter 1 working as an inverter reduces the amount of power extracted from the DC grid. This creates a power imbalance on the AC grid 1 that results in a frequency decrease which, in turn, initiates the frequency droop of the dual controller. This is the coupling between the voltage droop and the frequency droop. In fact, the power set point variation due to the frequency droop is not equal to zero and partially cancels the power deviation created by the voltage droop. This causes the DC voltage to drop lower when the dual converter is fully activated than when the frequency droop of the dual controller is not activated: the frequency droop hampers the proper functioning of the voltage droop. In particular, if the AC grid 1 is a strong AC grid, the impact of the power imbalance on the frequency is minimal whereas it can be substantial if the AC grid 1 is a weak AC grid. This explains why the interaction between the two droops has little impact on the DC voltage if the AC grids are strong. However, if one of the AC grid participating in the DC voltage droop is a weak AC grid, then the disruptive interaction between the voltage droop and the frequency droop can become dangerous for the DC grid because it prevents the voltage droop from working correctly.

It is to be noted that the power set point variation due to the frequency droop reflects the dynamics of the frequency of the AC grid. This means initially the voltage droop is not impacted by the frequency droop until the frequency starts decreasing, as shown in the zoomed-in part of Figure 8.

To better illustrate this point, according to (13), the relation between the DC voltage, the frequency of the AC grids and the power deviation due to the loss of converter 4 can be expressed as:

∆u s = P 4 - 3 j=1 1 k f j ∆f j 3 j=1 1 kv j (15)
As shown in Figure 7, the frequency deviation created by the interaction between the droops for large AC grids is negligible, thus ∆f 2 and ∆f 3 can be neglected.

If the considered scenario corresponds to the case where the AC grid 1 is also a strong AC grid, then ∆f 1 is negligible as well (in fact ∆f 1 = 4.81E-5 p.u. according to Figure 7) and the DC voltage deviation calculated by (15) gives ∆u s = 0.087 p.u. (which is also confirmed by Figure 8). In this scenario, the DC voltage is not impacted by the disruptive interaction between the two droops, and the MTDC system behaves as if the frequency droop of converter 1 were deactivated.

However, if the considered scenario corresponds to the case where the AC grid 1 is a weak AC grid, then ∆f 1 = 2.16E-3 p.u. (according to Figure 7) and the DC voltage deviation calculated by (15) gives ∆u s = 0.094 p.u. (Figure 8 shows a final ∆u s = 0.096 p.u. This 2% error is generated by the limits of the simplifying assumption ( 2)). This shows that using a weak AC grid to participate in the DC voltage regulation of the MTDC system can be dangerous since it leads, in this case, to a 8% increase in DC voltage variation on the HVDC grid.

V. CONCLUSIONS

In an MTDC system, each VSC-HVDC converter connected to an AC grid can be equipped with a DC voltage-droop controller to participate in the DC voltage regulation of the HVDC grid, and with a frequency-droop controller to participate in the onshore grid frequency regulation. However, using both controllers at the same time (also called dual controller) creates a disruptive interaction between the two droops.

Because of this coupling between the voltage droop and the frequency droop, the initial frequency-droop parameter needs to be corrected in order to fulfil the TSO's requirements despite being hampered by the voltage droop. For this purpose, a corrected value of the frequency-droop parameter has been proposed and validated.

On the other hand, the frequency droop must not hamper the functioning of the DC voltage droop during a DC event. It has been shown in this paper that the frequency droop has a limited impact on the DC voltage of an MTDC system as long as the frequency of the AC grids shows very small deviations, which is generally the case for large AC grids. However, if a weak AC grid is connected to the MTDC system, the frequency droop of the converter connected to this weak AC grid can dangerously aggravate any possible DC voltage variation on the MTDC system. This increased voltage variation needs to be taken into account in the choice of the voltage-droop parameters and the design of the DC grid.

Fig. 3 .

 3 Fig. 3. Topology of the HVDC grid

Fig. 4 .Fig. 5 .

 45 Fig. 4. Frequency of the AC grid 1

Fig. 6 .

 6 Fig. 6. Power injection/withdrawal of the VSC-HVDC converters before and after the disconnection of converter 4 (P 1 and P 2 are superposed in this figure)

Fig. 8 .

 8 Fig. 8. DC voltage at the converter 1 terminal

  .

	Converter	1	2	3	4	5
	P * g (MW)	200	200	-50	-162	-200
	kv (p.u.)	-0.4834	-0.4834	-0.4834	-∞	-∞
	k f (p.u.)	0.05	0.05	0.05	∞	∞

The voltage-droop parameter values were computed to achieve a time response of 100 ms, see[START_REF] Rault | Dynamic Modeling and Control of Multi-Terminal HVDC Grids[END_REF] for more information.

APPENDIX DC cable specifications:


NUMERICAL ANALYSIS OF THE ANCHORAGE OF MARITIME PINE TREES IN CONNECTION WITH ROOT STRUCTURE

The mechanical structure of 24 root systems was studied in 50 year old Maritime pine (Pinus pinaster Ait) trees after the 1999 storm, which resulted in the loss of >4 million hectares of forest in south-west France. Half of the study trees had been uprooted during the storm and the other study trees were left standing. Topology and geometry of the root systems were measured using a 3D digitising device. The spatial distribution of roots was then analysed. The 3D representation of each root system was used to calculate the mechanical behaviour of the structure in different compass directions, using a finite element model. Results show that resistance to windthrow can be provided by four different structural adaptations of root systems, and that root architecture may explain half of the sensitivity to uprooting. There was no significant difference in uprooting resistance between trees that were damaged by the storm, and those which remained upright. However, all root systems were found to be highly asymmetric with regard to morphology. This asymmetry influenced strongly the mechanical behaviour of anchorage, depending on the direction of loading.

Introduction

The big storm that hit Western Europe in December 1999 destroyed millions of trees in the French Aquitaine Maritime pine (Pinus pinaster, Ait.) forest, resulting in huge economic losses. As this kind of catastrophic event is predicted to become more frequent in the future, it is essential to better understand the mechanisms that are involved when trees are damaged. Uprooting was the main mode of failure observed in the study region, which is close to Bordeaux [START_REF] Cucchi | Wind-firmness in Pinus pinaster Ait. stands in Southwest France: influence of stand density, fertilisation and breeding in two experimental stands damaged during the 1999 storm[END_REF]. This complex phenomenon has already been examined in many countries that are exposed to strong winds. Investigations have often been supported by field observations, as well as experiments. Early studies in Scotland were concerned with Sitka spruce, which develops shallow root systems [START_REF] Coutts | Root architecture and tree stability[END_REF]. The role of anchorage components, i.e. resistance of roots in tension, resistance to bending, weight of root/soil plate and soil resistance were defined [START_REF] Coutts | Components of tree stability in Sitka spruce on peaty gley soil[END_REF]. It was shown that the root tensile strength is the most important factor for stability of Sitka spruce. However, there is a huge variability between trees, which can be explained by differences in root morphology. Subsequent research has addressed root anchorage in connection with an analysis of the root topology and geometry [START_REF] Ennos | The mechanics of root anchorage[END_REF][START_REF] Niklas | The biomechanics of Pachycereus pringlei root systems[END_REF], Mick-ovski et al 2002), focusing on particular elements such as taproot, laterals or forks. Nevertheless, no studies have been carried out which integrate the full description of the whole root system of mature trees.

Root architecture is defined by the kinetics of branching which, in addition to the morphogenetic program, is very controlled by environmental factors. In particular it is suspected that mechanical stresses can play a fundamental role on the development of root structures, causing significant modification of allocation rules in order to optimize tree stability [START_REF] Nicoll | Adaptive growth of tree root systems in response to wind action and site conditions[END_REF][START_REF] Stokes | Responses of young Sitka spruce clones to mechanical perturbation and nutrition: effects on biomass allocation, root development, and resistance to bending[END_REF]. On a related issue, an interest in analyzing the biomechanical behavior of roots is found in investigations of root growth in addition to tree stability. Techniques have been developed to describe and record tree topology and geometry under a Multi Tree Graphs (MTG) format, in order to analyze and model tree architecture. This method has been recently adapted and used to digitalize root systems [START_REF] Danjon | Characterisation of structural tree root architecture using 3D digitising and AMAPmod software[END_REF].

The aim of the current study was to describe and analyze the mechanical behavior of old Maritime pine tree root systems. Particular emphasis was placed on material that was available after the 1999 storm.

Material and methods

Studies were carried out on 24 root systems of 50 year old Maritime pine trees. These trees had been growing in the same stand at Bilos in the French Aquitaine region, where 10% of the trees were damaged during the 1999 storm. All of the trees fell in the eastward direction, and the prevailing wind in this region is typically from the west. The soil is a sandy podzol, with some hard pan, and a high water table in winter. Twelve root systems were from trees that were already uprooted, and the remaining 12 were extracted using a mechanical digger. The direction with regard to north was marked. The root systems were cleaned with an air spade and transported to a site close to the Pierroton INRA station where they were weighed. The morphology of each root system was described using a 3D Fastrak Polhemus digitalizer, adapting the technique which has been already used by Danjon et al 1999 to larger structures. Diameter of each measured segment was also recorded. Major and minor axes were measured when the cross section was found elliptical. Morphological data were recorded under Multi Tree Graph (MTG) format files [START_REF] Godin | A multiscale model of plant topological structures[END_REF]. These files were then used to visualize the 3D root systems on a computer (fig. 1) and to analyse statistically morphological parameters with the software AMAPmod [START_REF] Godin | Exploring plant topological structure with the AMAPmod software: an outline[END_REF]. In total, 150 000 root segments were measured. The spatial variability of the hard pan was high: in some places there was no hard pan, and in other places it was approximately located a depth of 80 to 130 cm. In places where the hard pan was absent, rooting depth was constrained by a high water table level. The trees had generally developed a dense windward root network, and a large chuck resting on the hard pan, which can be larger than the tap root. Large horizontal surface laterals were often highly tapered determining a "Zone of Rapid Taper" (ZRT) set to 80 cm radial distance from the main axis of the stump. About 150 summary variables per tree have been computed in order to perform a statistical analysis. They correspond to the branching characteristics, and to the distribution of volume or length of roots as a function of diameter, order, depth, azimuth, radial distance, inclination and type. Several types of roots were distinguished: (1) stump, (2) tap root, ( 3) & (4) horizontal and shallow roots [segments within the ZRT and outside the ZRT], ( 5) & ( 6) sinker roots [base within the ZRT and outside the ZRT], (7) deep roots, (8) other roots (oblique and at an intermediate depth). Fisher's parametric and Wilcoxon's non-parametric pair-wise tests were used to compare the pairs of uprooted and undamaged trees from the same diameter at breast height (dbh) class. The multivariate variability was assessed through a Principal Component Analysis (PCA) that has been carried out on the different root types considering geometric, topological and spatial variables.

A finite element model was developed under the Abaqus software (Hibbitt, Karlsson & Sorensen, Inc.) in order to calculate the mechanical behavior of root structures submitted to bending forces. Bernoulli 3D beam elements were used to discretize each structure. An interface was developed under the software MatLab that allows meshes to be generated from the MTG files, and Abaqus command files to be automatically generated. The model does not take into account the soil matrix, given the focus on behavior of the root structure itself. Embedded boundary conditions were applied at each branch extremity and a homogeneous elastic perfectly plastic material MPa 8000 E L = was considered for all beam elements. This constitutive law allows failure to occur in the elements reaching a fixed stress threshold MPa 28 L = σ [START_REF] Stokes | The biomechanics of tree root anchorage[END_REF]. Displacement increments were imposed in two perpendicular directions, i.e. East-West and South-North, to a node at a fixed distance above the tree base. Increments of reaction force were calculated at the displaced node. They were used to trace the force vs. displacement curves that represent the mechanical behavior of the whole root structure. Comparisons between the different root systems were used to determine a stiffness of the structure stiffness, which is defined by the slope of the tangent to the curve at its origin, and the strength which is the maximum force reached before the first failure (fig. 2).

Results and discussion

A paired comparison (parametric or non parametric) yielded few significant differences of the mean between the windthrown and the undamaged trees (Table 1). The stump corresponds to approximately 20% of the coarse root volume both for the windthrown and the undamaged trees, and is not taken into account in the root volume computations because it has a specific role in the root system. Undamaged trees had more sinkers in the ZRT (25% of root volume vs. 17%), and a larger East chuck (7% of root volume vs. 3.5%). In contrast, windthrown trees had more oblique roots and roots between 40 and 60 cm depth (18% of root volume vs. 9%). Undamaged trees had also less root volume south (20% vs. 29%), and therefore a higher proportion of roots leeward and windward. Only the uprooted tree 520 yielded a very high negative loading on PC3 (14% of variability). This tree has a very low proportion of root volume in the ZRT and a high length of medium diameter roots. Three trees (496 et 761 windthrown, 100 undamaged) had exceptionally high loading on PC4 (10% of variability): they had a low dbh and a low windward network. It should be noticed that the undamaged tree 100 has a high loading for PC2 which may have counterbalanced its low windward network. Resistance to uprooting depends on the four first principal components which together, originate in the four following structural features: -a strong vertical rooting coming from various proportion of tap root, secondary sinkers and an east chuck; -a high proportion of root volume in the ZRT, especially the leeward ZRT; -a large windward network of roots; -a larger proportion of root volume windward and leeward.

One or a combination of these structural elements were poorly developed on 6 windthrown trees and strongly developed on 5 undamaged trees. These characteristics were moderately developed on the 13 remaining samples. Structural adaptation of root systems may explain about half of the resistance to uprooting, even if the location of the trees in the stand and sizes of crown and stem remain important parameters that must also be taken into consideration. In this study, pairwise comparisons were poorly adapted to examine differences between root architectures due to the large number of adaptation strategies to resist windthrow.

Results of the mechanical analysis show very complex responses and a significant variability in behavior (Fig. 6). This is attributed to the huge variability of root morphology and size. Strength of the root systems varies from 20kN to 150kN. Some response curves show one or more hardening zones, i.e. increasing parts of the curves, which could be due to initially curved elements that straighten out during deformation of the root system and then hold under tension. This assumption must be examined more closely for each individual case, given consideration of root morphology. Numerical calculations show significant differences between the mechanical behavior of a root system in the two directions of pulling, recognizing that the strength of uprooted trees is a highly variable resistance (Fig. 7). It is believed these anisotropic responses are related to the asymmetric growth that has been observed concerning both ramification ratio and root cross section areas. Nevertheless, strength and stiffness factors do not discriminate between uprooted and stable trees. Recognizing the prevailing winds are oriented in the West, North-West directions, the results appear to confirm the hypothesis of a biomechanical adaptation of trees in order to reinforce root anchorage [START_REF] Stokes | Responses of young Sitka spruce clones to mechanical perturbation and nutrition: effects on biomass allocation, root development, and resistance to bending[END_REF][START_REF] Coutts | The development of symmetry, rigidity and anchorage in the structural root system of conifers[END_REF]. Such anisotropic development has been already observed, for example with more ramified roots windward and woody root structures leeward. Furthermore, I and T cross section shapes are often encountered [START_REF] Nicoll | Adaptive growth of tree root systems in response to wind action and site conditions[END_REF], which optimize bending stiffness for a given available quantity of carbon.

Concluding remarks

A structural mechanical model has been developed in order to calculate the mechanical state of Maritime pine tree root systems. This tool can be used for any root system that has been extensively measured and recorded in a MTG file, including either real or simulated architectures [START_REF] Blaise | A model simulating interactions between plant shoot and root architecture in a non-homogeneous environment[END_REF]. Soil matrix was not taken into consideration, given the focus on the influence of the root structure only. Notwithstanding these simplifications, the results corroborate the idea that trees may adapt their root growth to achieved an improved their anchorage, as evident in the anisotropy of root structure that has been found.

It is possible to improve the model by taking into account MOE and MOR, which depend on root diameter. The model could also differentiate between the strength of roots in tension and in bending. These improvements would allow for a characterization of mechanical behavior of root systems in two opposite directions, i.e. windward and leeward. Companion analyses have also the soil component taken into consideration [START_REF] Dupuy | influence of morphological and mechanical properties on the anchorage of root systems[END_REF].

In conclusion, linking models of root architecture to mechanical state is a great challenge which merits further development, given some attempts that have not considered the roots as a 3D branching system [START_REF] Mattheck | Mechanical control of root growth: A computer simulation[END_REF]. Similar developments have been already published for the aerial part, which link growth simulations to the biomechanical behavior of trees (Fourcaud et al 2003a, Fourcaud et al 2003b).

Fig. 1 :

 1 Fig. 1: 3D representation of a digitized root system of a 50 year old Maritime pine tree.

Fig. 2 :

 2 Fig. 2: force vs. displacement curves calculated in two different directions

Fig. 4 :

 4 Fig. 4 : Principal component analysis of tree characteristics. Scores for PC1 and PC2 (a) loadings for the 18 original variables; (B) loadings for the 24 trees of the sample, P refers to the proportion, V to root volume, L to root length.

Fig. 5 :

 5 Fig. 5 : Principal component analysis of tree characteristics. Scores for PC3 and PC4 (a) loadings for the 18 original variables; (B) loadings for the 24 trees of the sample, P refers to the proportion, V to root volume, L to root length.

  Fig. 6: two examples of force vs. displacement curves showing anisotropic responses and hardening.

Fig. 7 :

 7 Fig. 7: root systems strength and stiffness ratio between West and North

Table 1 :

 1 Percentage of root volume in winthrown vs. undamaged trees. Paired tests: Fisher and Wilcoxon.

	Variable winthrown Undamaged coef.variation difference(%) fischer	wilcoxon
	vertical roots	19.1	27.3	0.432	-29.8	0.008**	0.009**
	ZRT sinkers	16.7	25.3	0.471	-33.9	0.006**	0.009**
	oblik and
	medium	17.9	9.52	0.767	88.5	0.047*	0.034*
	deep
	East chock	3.4	7.1	0.779	-51.7	0.021*	0.034*
	medium deep	26.4	20.1	0.323	31.6	0.032*	0.021*
	South	28.8	20.1	0.289	43.2	0.004**	0.005**
	North
	and	48.2	41.5	0.174	16.2	0.003**	0.002**
	South
	The multivariate variability was assessed through a PCA with 18 variables. The two first compo-
	nents correspond to nearly half of the variability. Four windthrown trees (32, 663, 583 and 761)
	have high negative loadings for the first component (PC1) which corresponds to a high propor-
	tion of oblique and moderately deep roots but a small tap root and a low root volume in the lee-
	ward ZRT. Five undamaged trees (370, 685, 374, 100 and 114) had exceptionally high loadings
	for the second component (PC2) which correspond to a low proportion of root volume perpen-
	dicular to the wind direction and a high volume of sinker roots, including the east chuck. All these
	9 trees had a high root volume / stem volume ratio.

Acknowledgements

This work has been founded by the Région Aquitaine and the French government via GIP-ECOFOR. Thanks are due to T. Carnus, M. Curtet, C. Espagnet, B. Issenhuth, F. Lagane, M. M'Houdoir, P. Pougné for their full contribution to the root system digitalizing. Gratitude also is due to J. Fannin for his comments and corrections.