E Zio

L R Golea

Analyzing the topological, electrical and reliability characteristics of a power transmission system for identifying its critical elements

Keywords: critical infrastructures, vulnerability assessment, electrical transmission system, network analysis, reliability, connectivity degree

The subject of this paper is the analysis of an electrical transmission system with the objective of identifying its most critical elements with respect to failures and attacks. The methodological approach undertaken is based on graph-theoretical (topological) network analysis. Four different perspectives of analysis are considered within the formalism of weighed networks, adding to the purely topological analysis of the system the reliability and electrical characteristics of its components. In each phase of the analysis: i) a graph-theoretical representation is offered to highlight the structure of the most important system connections according to the particular characteristics examined (topological, reliability, electrical or electrical-reliability); ii) the classical degree index of a network node is extended to account for the different characteristics considered. The application of these concepts of analysis to an electrical transmission system of literature confirms the importance of different perspectives of analysis on such a critical infrastructure.

Introduction

Engineered critical infrastructures are the systems of interest in this work. The motivation is that they provide the continuous flow of essential goods (e.g. energy, water, data) and services (e.g. banking, health care, transportation) which the welfare and security of our nations rely on. These critical infrastructures are subject to a set of multiple hazards and threats which must be identified and analyzed for optimal protection. Among the engineered critical infrastructures, the focus of this work is on the electrical transmission system and its analysis to identify the importance of the individual elements. The motivation is that the infrastructure for electrical transmission of most of the world"s countries is aging and failing, with funding often not sufficient to repair or replace it; in this situation, there is a growing demand for a rational, risk-based approach to its operation and maintenance. A number of recent studies have addressed the assessment of vulnerability in electric power systems, by graph-theoretical topological investigations as in [START_REF] Albert | Structural vulnerability of the North American power grid[END_REF]- [START_REF] Zio | Vulnerability analysis of a power transmission network[END_REF], physics-based models as in [START_REF] Yu | A practical approach for integrated power system vulnerability analysis with protection failures[END_REF]- [START_REF] Koonce | Bulk power risk analysis: Ranking infrastructure elements according to their risk significance[END_REF], agent-based modeling as in [START_REF] Schlä Pfer | Reliability Analysis of Electric Power Systems Using an Object-oriented Hybrid Modeling Approach[END_REF]. These studies all refer to the transmission system and are based on different conceptualizations of vulnerability [START_REF] Vaurio | Modelling the loss and recovery of electric power[END_REF]. Also, more sophisticated techniques such as polyhedral dynamics [START_REF] Degtiarev | System analysis: mathematical modelling and approach to structural complexity measure using polyhedral dynamics approach[END_REF] and artificial intelligence-based search methods [START_REF] Salazar | Optimization of Constrained Multiple-Objective Reliability Problems Using Evolutionary Algorithms[END_REF], [START_REF] Cadini | Optimal expansion of an existing electrical power transmission network by multi-objective genetic algorithms[END_REF] have been proposed to find critical elements and define vulnerability indices. The topological approach to vulnerability analysis is quite popular because in spite of its relative simplicity it offers the capability of identifying elements of structural vulnerability, i.e. network edges and nodes whose failure can induce a severe structural damage to the network through the physical disconnection of its parts. However, the methods based on such approach are limited from the point of view of the physical analysis of the electrical transmission systems, which the graphical networks represent; the limitations come from the fact that the analysis focuses only on the topological features of the network, thus neglecting its physical characteristics [START_REF] Bompard | Assesment of structural vulnerability for power grids by network performance based on complex networks[END_REF]- [START_REF] Zio | Identification of betweenness-central groups of components in a complex network infrastructure by genetic algorithms, accepted, ESREL Safety and Reliability for Managing Risk[END_REF]. In this respect, it is important to verify the extent of these limitations and possibly overcome them by complementation with more detailed physical analyses on critical parts of the network [START_REF] Eusgeld | The role of network theory and object-oriented modeling within a framework for the vulnerability analysis of critical infrastructures[END_REF], [START_REF] Bompard | Analysis of structural vulnerabilities in power transmission grids[END_REF]. In this paper, the formalism of weighed networks is exploited to provide different graph-theoretical representations and analyses of a power transmission system. The aim is to contribute to reducing the gap between the highly conceptualized (and abstract) analyses based purely on considerations of the system topology and the highly detailed (and computationally demanding) simulations of system behavior, in order to render the overall vulnerability assessment more feasible and robust. The "weights" appended to the network elements are intended to capture relevant electrical and reliability properties of the system, so as to overcome the classical simplifying but unrealistic assumption that electrical flow occurs along the shortest and failure-free paths of connections. The rest of the paper is organized as follows: Section 2 describes the theoretical basis for the proposed perspectives of analysis. In Section 3, the work is presented from a practical point of view with reference to the popular IEEE RTS 96 system [START_REF] Billinton | Reliability Assessment of Electric Power Systems Using Monte Carlo Methods[END_REF]. Section 4 contains a critical comparison of the four perspectives of analysis with an outlook on other perspectives. Conclusions are drawn in Section 5.

Different perspectives of analysis

In the present study, the system is modeled as a stochastic, weighted, undirected, connected network in which each electric bus is transposed into a node, linked by edges representing the overhead lines connecting consecutive buses. In this respect, this representation focuses on the actual topological structure of the power transmission network. Mathematically, the topological structure of the network can be represented as an undirected graph ( , ) G V E where V represents the set of vertexes (or nodes, or components) ( dim( ) NV 

is the number of nodes) and E represents the set of edges ( , )

ij ( dim( ) KE 
is the number of edges).

The connections are specified in an N×N adjacency matrix {a ij } whose entries are 1 if there is an edge joining node i to node j and 0 otherwise.

Graph-theoretical topological analysis

The transmission network is first analyzed from a purely topological point of view. In the topological representation, no specification of the physical length of the edges is given. Each link is considered having a length equal to one and thus the distance between two nodes i and j is represented solely by the number of edges travelled in the path from i to j. On the basis of {a ij }, it is possible to compute the matrix of the shortest path lengths {d ij } whose generic entry d ij is the number of edges making up the shortest path linking i and j in the network. The fact that G is assumed to be connected implies that d ij is positive and finite ij  and that there are ( , the degree being defined as the number of edges incident to the node [START_REF] Nieminen | On the Centrality in a Graph[END_REF]: , 1, 2,...,

i ij jN k a i N    (1)
Intuitively, the degree of a node measures its influence in the graph with respect to the size of its immediate environment.

Reliability analysis

While some studies witnessed a reasonable association between the topology of the power grids and the robustness and stability of the power transmission systems [START_REF] Albert | Structural vulnerability of the North American power grid[END_REF], [START_REF] Latora | Efficient Behavior of Small-World Networks[END_REF], [START_REF] Volkanovski | Application of the fault tree analysis for assessment of power system reliability[END_REF], the relationship between network structure and system reliability is also of relevance. In this respect, the formalism of weighted networks [START_REF] Latora | Efficient Behavior of Small-World Networks[END_REF] can be undertaken to account for the reliability properties of the transmission network system. More precisely, a reliability matrix {p ij } can be introduced to describe the network reliability properties at a local level [START_REF] Zio | From Complexity Science to Reliability Efficiency: A New Way of Looking at Complex Network Systems and Critical Infrastructures[END_REF]; the generic element p ij represents the probability of successful transmission along the edge that connects node i and j. Since the graph is not fully connected, the reliability matrix tends to be sparse (p ij =0 for all pairs of nodes i and j that do not share a direct physical connection). In order to obtain a non-sparse matrix containing the complete information on the reliability of connection between any two pairs of nodes, the reliability ij p  of connection between i and j through any connecting path ij  is computed by a method based on a combination of cellular automata (CA) and Monte Carlo (MC) sampling [START_REF] Zio | A combined Monte Carlo and cellular automata approach to the unreliability analysis of binary network systems[END_REF]. In this method, the reliability ij p  , i.e. the probability of a successful connection from i to j, is computed by MCsampling a large number M of random realizations (MC trials) of the states of the connecting arcs and by CA-computing, for each realization, if a path from i to j exists; the ratio of the number of successful ij  paths over the total number of realizations computed gives the connection reliability from node i to node j. As indicator of the importance of the nodes from the reliability point of view, a reliability degree can be introduced as: , 1, 2,...,

ij r i jN k p i N     (2)
From the reliability analysis, a matrix   r ij s is computed on the basis of the S K most reliable paths. The matrix element r ij s is equal to 1 if the connection from node i to j is one of the S K most reliable connections, and 0 otherwise.   r ij s can be thought of as the reliability equivalent of the topological adjacency matrix {a ij }.

Electrical analysis

As mentioned in the Introduction, in the case of electrical transmission networks of interest here, the existing literature on vulnerability analysis largely takes a topological approach to identify the critical components in the network [START_REF] Albert | Structural vulnerability of the North American power grid[END_REF], [START_REF] Crucitti | Locating critical lines in high-voltage electrical power grids[END_REF], [START_REF] Zio | Vulnerability analysis of a power transmission network[END_REF]. Even though such analyses are capable of identifying elements of structural vulnerability, they are limited from the point of view of the physical analysis of the electrical transmission system, which the networks represent. These limitations are all related to the fact that the analysis performed focuses only on the topological features of the network, thus neglecting its physical characteristics; this is not realistic for electrical transmission networks in which: i) the "electrical" length of a path differs from the topological length, depending on the difficulty (resistance) of transmission; ii) the electrical power is not necessarily routed through the shortest paths: rather, the transmission of power is determined by physical rules, e.g. Kirchoff"s laws, nodal voltages etc. To practically overcome these limitations, an electrical connectivity metric was introduced within the weighed network formalism to capture the properties of node centrality, relative to metrics based on node-edge connectivity [START_REF] Hines | A Centrality Measure for Electrical Networks[END_REF]. The electrical characteristics of the individual network elements and their interconnection relationships can be expressed in terms of the bus admittance matrix, bus Y . Inverting the sparse bus admittance matrix ( 0 bus ij Y  for all pairs of nodes i and j that do not share a direct physical connection) that incorporates Kirchoff"s laws, a non-sparse matrix, known as impedance matrix, can be obtained.

The matrix of electrical distances is then given by the magnitude   ij m of the entries of the matrix bus Z . Admittance is a complete expression of the extent to which a circuit allows a current to flow; as the absolute value of the complex admittance becomes larger, the absolute value of the complex impedance becomes smaller and therefore smaller s is equal to 1 if the connection from node i to j is one of the S K values, and 0 otherwise. The computation of the connection distances between nodes i and j takes into account all possible paths between nodes, as in the case of the reliability analysis, and not only the shortest paths, as in the topological analysis. Based on the electrical information contained in the bus Z matrix, an indicator of the importance of the nodes can be introduced as:

1 , 1, 2,...,

e i ij jN k i N m    (3)
The electrical centrality measure of interest to electrical engineers introduced in [START_REF] Hines | A Centrality Measure for Electrical Networks[END_REF] and repeated in the paper quantifies that the importance of a substation as being inversely proportional to the Thé venin equivalent circuit seen in the substation. In other words, the electrical equivalent distance (Z Thé venin) between the generation nodes and the substation determines its importance. Thus, the importance of a substation is represented by its maximum level of short circuit power. Choosing this criterion to identify the importance of a node or substation in a power system implies depending on the number of the operating generators (node admittance). Therefore, the adopted definition of node importance accounts also for the actual generation dispatch.

Electrical-reliability analysis

As the physical reliability and electrical properties of the transmission network act together in shaping the behavior of the system, an analysis more complete than the previous ones would be one that combines the reliability and electrical characteristics together. In this view, a electricalreliability distance can be associated to each pair of nodes ij, i.e. the product between the reliability ij p  of connection between i and j and the corresponding electrical distance ij m .

The S K shortest connections define a matrix   er ij s , where the generic element er ij s is equal to 1 if the connection from node i to j is one of the S K values, and 0 otherwise. As an indicator of the importance of nodes from the electrical and reliability point of view, the electrical-reliability degree can be defined as:

  1 , 1, 2,..., ij er i ij jN k i N mp     (4)
As an alternative to the electrical-reliability degree, the expected electrical distance could be used to evaluate the importance of different network components. This measure is computed as:

  , 1, 2,..., ij ij i jN e m q i N     (5)
where ij q  is the probability of different combinations of failures when multiple ij  paths between the pair of nodes i and j exist, and ij m  is the electrical distance corresponding to the ij  path.

As previously stated, the measure based on electrical distances is relevant also for dispatching problems aimed at reduction of power losses. In the context of the electricalreliability analysis where short-term contingencies and supply continuity are the crucial factors, a complementary measure could be obtained by combining the reliability of lines together with the corresponding maximal transmission capacity. The further development of these indicators is beyond the purpose of this paper.

Results and discussion

Let us consider the power transmission network system IEEE RTS 96 of Figure 1 [START_REF] Billinton | Reliability Assessment of Electric Power Systems Using Monte Carlo Methods[END_REF]. The network consists of 24 bus locations connected by 38 lines and transformers. Some edges (4 out of 38) are constituted by double physical lines; for the purpose of the analysis, however, they are treated as a single edge of communication so that effectively 34 K  . The corresponding graph is shown in Figure 2.

Graph-theoretical topological analysis

Obviously, in the case of 34 The first column of Table 1 reports the ranking of the nodes based on the values of their degree. As defined in Eq. ( 1), the most important nodes from a degree point of view have the largest number of connections to other nodes in the network; thus, nodes 9 and 10, characterized by the largest number of incident edges (five) are placed in the first position of the topological ranking; on the contrary, node 7 with only one connection, falls in the last position of the ranking. 2, first and second column; it is 0 otherwise. The representation of the connection pattern of the 34 most reliable paths is shown in Figure 3, with the node size proportional to the number of incident edges. It can be seen that the most reliably-connected nodes do not necessarily share a direct physical connection. The reliability-driven representation of the network connectivity shows a different pattern than the topological one; this is due to the fact that the computation of the reliability of the connection from node i to j takes into account all the possible paths linking nodes i and j, not just the shortest topological one. This conceptual representation of the network shows that there are several hubs identified as important from the reliability point of view: they are traversed by the majority of the most reliable paths. The most reliably connected area is found to be the central part of the network, concentrated around nodes 9, 10, 11 and 12 (which have also the highest degree, as reported in Table 1), due to direct physical connections, i.e. edges 9-11, 10-11, 9-12, and indirect connections, i.e. 9-10 and 11-12. On the other hand, several nodes appear to be disconnected from the network, forming clusters of size one or two. Based on these findings, the network could be made more robust by increasing the reliability of these clusters of critical hubs. Table 1, third column, provides the ranking of the reliability degree for each node. Nodes 9, 10, 11 and 12 have the most reliable connections. The ranking is similar to the one obtained with the topological degree (Table 1, first column), with nodes 9, 10, 11 and 12 being the most important (reliable), and nodes 7 and 24 the least. This points to a strong correlation between the network structure and the system reliability due to the large values of connection reliability ij p  .

Reliability analysis

Electrical analysis

Table 2, column five provides the most important paths with respect to the electrical distance. The edge 7-8 was identified as the most important, i.e. having the shortest electrical distance and therefore a larger propensity for power to flow between its connecting nodes. The tendency of power to flow through this edge is a consequence of the network"s structure; as it can be seen from Figure 1, node 7 is a generator connected to the rest of the network only by node 8. As in the reliability case, both direct physical connections, i.e. edges 7-8 and 1-2, and indirect connections, i.e. 18-22 and 17-21 are ranked as the most important connections. Figure 4 shows the connection pattern realized by the 34 shortest electrical paths. The resulting graph has the same size as the original network in Figure 2 (24 nodes and 34 edges), and the node size is proportional to the number of incident edges. The electrical representation of the network suggests a very different structure with respect to the topological and the reliability perspectives. The most vulnerable area from an electrical perspective is the north-west side of the network, where the power flow concentrates due to Kirchoff"s Laws. Indeed, this area contains several important generators (i.e. nodes 15, 16, 18, 21 and 22), and therefore there is a strong exchange of power among the nodes. These electrical hubs appear to be potentially more vulnerable with respect to faults or malicious attacks than the other network nodes. Table 1, fifth column provides the ranking of the nodes with respect to the electrical degree and shows a different prioritization of the network nodes with respect to the topological and reliability degrees. Nodes 18, 17 and 21 are identified as the most important from an electrical point of view, having the highest degree of electrical connectivity. They are ranked as being the most vulnerable because power is more likely to flow through these electrical hubs. In spite of that, they have small reliability degrees (Table 1, forth column), which expose the network to potential severe outages. On the contrary, node 7 is among the less critical nodes according to the electrical degree, and consistently, it has the smallest reliability degree.

Electrical-reliability analysis

The electrical-reliability representation of the connection pattern of the 34 most reliable electrical paths is shown in Figure 5, with the node sizes adjusted according to the number of incident edges. The most important edge from the electrical and reliability point of view is 7-8, which also represents a physical connection of the electrical transmission network analyzed (Figure 1). This edge, ranked as first in Table 2, sixth column is characterized by the smaller electrical distance ( 78 0.1030 m 

) and a relatively small reliability of 0.7409. Figure 5 suggests that the majority of the most vulnerable paths are those connecting the south-east and the north sides of the network through nodes 7 and 22; these paths are characterized by small values of the electrical-reliability distance. Some of the 34 S K  connections include also paths with short electrical distance and average reliability, located in the north area of the network system. Table 1, seventh column provides the ranking of the network nodes with respect to the reliability and electrical properties of the network, which bears resemblance to the reliability ranking (Table 1, third column). This fact suggests that the reliability (and structural) characteristics have a larger influence upon the vulnerability of the network than the electrical characteristics. In this view, node 7 having the smallest reliability degree (Table 1, forth column) and a relatively large electrical degree (Table 1, sixth column) is ranked among the most critical nodes (Table 1, eighth column). Nonetheless, the reliability-electrical analysis highlights criticalities complementary to the electrical analysis. In particular, node 16, characterized by a relatively large reliability degree (Table 1, forth column) and large electrical degree (Table 1, sixth column) is ranked among the critical nodes when combing the two characteristics (Table 1, eighth column). From this perspective, the core made of nodes 9, 10, 11 and 12, concentrated in the central part of the network, is found to be the most robust, with node 10 having the smallest electrical-reliability degree. On the contrary, the north side of the network appears to be the most vulnerable, with nodes 17, 18 and 22 placed in the first positions of the electrical-reliability ranking.

Choosing among the perspectives

The four perspectives on network analysis can be framed into the broader scenario of risk analysis against random failures and malevolent attacks [START_REF] Bier | Optimal resource allocation for defense of targets based on differing measures of attractiveness[END_REF]- [START_REF] Solé | Robustness of the European power grids under intentional attacks[END_REF]. In this view, a comparative evaluation is proposed to determine which perspective to choose for a given circumstance. Table 3 provides a starting point for choosing among several perspectives on network analysis based on mutual comparison criteria [START_REF] Eusgeld | Comparative evaluation of modeling and simulation techniques for interdependent critical infrastructures[END_REF], [START_REF] Kroger | Vulnerable Systems[END_REF]. The major drawback of the purely topological analysis is that it relies solely on the network structure to assess the connectivity performance and nodes importance, and does not consider any physical properties nor system dynamics (first three columns of Table 3). By considering the connections reliability among the nodes based on the failure probabilities of the edges, the reliability-driven representation highlights the most reliable indirect connections, and the most reliably connected areas of the network. Nevertheless, the topological and reliability analyses fail to explicitly incorporate the physical laws governing the electrical flow. An electrical-or electrical-reliability-driven representation of the network indeed highlights the electrical hubs that appear to be more vulnerable, due to the smaller electrical-or electrical-reliability distances of the paths that traverse them. The incapacitation of these vulnerable paths causes the electrical flow to be routed along alternative paths, possibly triggering cascades of overloads and extensive blackouts. The quantification of system vulnerability indicators and the identification of its critical elements are the two main outputs of a vulnerability assessment. While providing complementary information, vulnerability indicators are parameters encompassing the static and/or dynamic characteristics of the whole system, whereas the identification of critical elements comes from their ranking with respect to their individual connectivity efficiency and/or their contributions to the propagation of failures, with their effects, through the system [START_REF] Zio | Vulnerability assessment of critical infrastructures[END_REF]. To properly quantify the importance and criticality of the network components, various weighted indicators can be defined as complements to the topological indicators. From a topological viewpoint, various measures of the importance of a network element (edge or node) can be found in the literature [START_REF] Watts | Collective dynamics of "small-world" networks[END_REF]. This centrality measures can be further extended by considering suitable connections weights to identify the critical components with respect to the reliability and the electric features [START_REF] Rocco | A Flow Importance Measure with Application to an Italian Transmission Power System[END_REF] of the networks components. By further considering the "reliability distances" or the "electrical-reliability distances" among network nodes in terms of the probabilities of failure of the interconnecting links, or in terms of a combination of the reliability and the electrical properties of the links, vulnerability indicators can be defined for use in the analysis of the robustness of network systems. The "data needs" criterion refers to the quantity and quality of input data needed to properly perform the analysis. By associating the high scale to the electrical and electrical-reliability analysis we consider that this approaches strongly depend on a high quantity and quality of input data to provide reasonable outputs. On the contrary, graph-theoretical and reliability analyses rely on a minimum amount of information such as the adjacency matrix and the failure rate of the lines. The level of maturity of each perspective can be measured by the amount of available literature review [START_REF] Eusgeld | Comparative evaluation of modeling and simulation techniques for interdependent critical infrastructures[END_REF]. Many application examples are found in scientific literature for the graph-theoretical and the electrical analysis so we assume a high level of maturity. On the contrary, a poor level of maturity is assigned to the electrical-reliability perspective due to the scarce experience in this field. We assume for the reliability analysis a middle level of maturity. Complexity criteria takes into account the difficulty of the algorithms and programs of the respective methodological approach. Two levels of complexity can be distinguished: a low level of complexity assigned to the graph-theoretical perspective due to the simplicity of the algorithms necessary to perform such an analysis and a medium level of complexity for the remaining perspectives.

The "requested time" criterion describes the effort employed in developing the algorithms for the analysis, and the simulation speed. A low level can be assigned to the graph-theoretical and reliability analyses because of the relative small amount of time needed to build-in the algorithms and a fast simulation speed, while we assume a medium level for the other two analyses. A number of alternative approaches for the vulnerability assessment of electrical networks have been reported in Table 3, fifth to eight row [START_REF] Kroger | Vulnerable Systems[END_REF]. The choice of the suitable approach depends on the type of system, the objective of the analysis, and the available information. For example, statistical analysis [START_REF] Casals | Analysis of major failures in Europe's power grid[END_REF] is suitable when rich data sets about the system operation and performance are available. It is characterized by a high level of maturity and low level of complexity. However, the structure of the network under analysis may be hidden by the fact that the data are often presented in an aggregate form [START_REF] Dekker | Simulating Network Robustness for Critical Infrastructure Networks, Conferences in Research and Practice in Information Technology[END_REF], [38]. Moreover, the effective use of the sets of data is difficult because they come from a variety of past operating conditions that may not fully reflect the situations of interest at present and in the future. Probabilistic risk assessment is another mature methodology that can be applied for analyzing network systems [START_REF] Patterson | Identification of critical locations across multiple infrastructures for terrorist actions[END_REF], [START_REF] Volkanovski | Application of the fault tree analysis for assessment of power system reliability[END_REF]. It integrates deterministic and stochastic tools to carry out a systematic and structured evaluation of the risk associated with every life cycle aspect of a complex engineered technological system, which may lead to undesired consequences triggered by an accident initiating event. The disadvantages of this methodology arise from its complexity that leads to significant efforts in logic modeling and quantification, and from the limited capability of providing an exhaustive analysis.

Agent-based modeling (ABM) is also a suitable approach, particularly useful for situations with sparse or non-existent macro scale information. ABM is able to use the rich sources of micro-level data to develop interaction forecasts. The main disadvantages of these simulation models lie in the complexity of the computer programs, which tends to obscure the underlying assumptions, and in the inevitable input subjectivity. The high level complexity leads also to a high level of requested time. Object-oriented modeling offer also an attractive modeling paradigm for describing the dynamic network behavior [START_REF] D"inverno | Understanding Agent Systems[END_REF]. One of the major advantages of an object-oriented approach is the possibility to include physical laws into the simulation. The level of modeling detail offered by the object-oriented approach allows analyzing a multitude of time-dependent availability aspects. The main problems are related to the slow simulation speed and the large number of parameters to be input in the analysis [START_REF] Eusgeld | The role of network theory and object-oriented modeling within a framework for the vulnerability analysis of critical infrastructures[END_REF]. However, by focusing specific safety aspects, the model can be simplified and the computational burden reduced. Based on this comparative evaluation, we propose a framework of analysis that incorporates the four perspectives presented in this work to be used as a preliminary screening analysis of the network vulnerabilities. This is reasonable since the first step in a general risk assessment is represented by the system characterization [START_REF] Moore | Application of the API/NPRA SVA methodology to transportation security issues[END_REF], [START_REF] Garrick | Confronting the risks of terrorism: making the right decisions[END_REF]. By performing such an analysis a better comprehension of the system is achieved, and the different criticalities within the network components are highlighted. Such framework could be used as a starting point for a more detailed analysis, focused on the previously identified critical components.

Conclusions

In this paper, the analysis of an electrical transmission network system has been undertaken with the objective of identifying the critical components, within a vulnerability assessment frame of work. The electrical transmission network IEEE RTS 96 has been taken as reference to present the development of the analysis from four different perspectives: topological, reliability, electrical and electrical-reliability. For each perspective, the structure of the most important system connections has been identified, and an extended degree index has been computed for the nodes based on this structure. The three weighted indicators quantify the importance of the network components with reference to the role played by the observed characteristic (topological, reliability, electrical or electrical-reliability). A comparative evaluation of the four perspectives also with respect to other approaches for the vulnerability assessment was performed and a mechanism for choosing among them was provided. In particular, we found that the reliability analysis identifies the clusters of critical hubs whose reliabilities have to be enhanced to ensure network robustness against random failures. Moreover, the electrical analysis ranks as vulnerable the buses that are traversed by the greater part of the power flow. These electrical hubs are more vulnerable to faults or malicious attacks because their malfunctions lead to a large redistribution of power to other electrical routes. By comparing the reliability and the electrical analyses, we were able to find inconsistencies in the network design. Namely, nodes with large electrical degree have small reliability degrees, which expose the network to potential severe outages. Therefore, we suggest network modifications that increase the reliability degree of the nodes which have a large electrical degree.

Tables and figures

K

  electrical distances, which can be used to define a matrix  

  S KK the smallest values in the matrix of the shortest path lengths are equal to 1, and correspond to the direct physical connections, i.e.     ij ij sa  (Figure2).

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. The IEEE RTS 96

Figure 5 .

 5 Figure 5. The IEEE RTS 96 redrawn to highlight the structure of the most reliable electrical connections

Table 2 ,

 2 third column lists the 34 defined on the basis of the smallest values of shortest path lengths in the topological analysis. The matrix element r ij s is 1 if the connection from node i to j is one of the 34

	S K 	most reliable connections, which can be used to define a
	matrix   r ij s analogous to  

ij s S K  listed in Table

Table 1 .

 1 Degree indicators of the 24 network nodes

	Topological Degree Reliability Degree Electrical degree Electrical-reliability
								degree
	node	degree	node	degree	node	degree	node	degree
	9	5	9	19.0939	18	0.2469	7	0.4215
	10	5	10	19.0928	17	0.2467	22	0.4149
	11	4	11	19.0926	21	0.2467	18	0.3781
	12	4	12	19.0912	15	0.2465	17	0.3622
	16	4	13	18.1300	16	0.2462	21	0.3592
	1	3	3	18.0415	22	0.2452	19	0.3515
	2	3	24	17.9313	19	0.2449	20	0.3440
	3	3	1	17.6274	24	0.2447	14	0.3317
	8	3	16	17.5905	20	0.2433	15	0.3304
	13	3	23	17.4177	23	0.2423	16	0.3238
	15	3	2	17.3933	14	0.2419	23	0.3198
	17	3	15	17.2740	13	0.2388	4	0.3176
	21	3	5	17.0725	11	0.2386	8	0.3149
	23	3	8	16.9606	12	0.2370	24	0.3133
	4	2	14	16.7865	3	0.2368	5	0.3030
	5	2	4	16.7335	9	0.2340	13	0.3029
	6	2	6	16.6817	8	0.2316	3	0.3021
	14	2	20	16.3159	7	0.2315	2	0.2985
	18	2	19	16.0972	4	0.2303	1	0.2951
	19	2	21	16.0233	1	0.2250	6	0.2914
	20	2	17	15.8838	2	0.2245	11	0.2872
	22	2	18	15.2564	5	0.2242	12	0.2858
	24	2	22	13.8271	10	0.2219	9	0.2820
	7	1	7	12.7517	6	0.2107	10	0.2678

Table 2 .

 2 Ranking of network connections with respect to the reliability, electrical and reliability-electrical distances

	From	To	Reliability Rank Electrical	Rank Electrical-	Rank From	To	Reliability Rank Electrical	Rank Electrical-	Rank
	node	node			distance		reliability distance		node	node			distance		reliability
															distance
	1	2	0.8991	6	0.1505	15	-	-	10	24	0.8822	16	-	-	-	-
	1	9	0.8869	13	-	-	-	-	11	12	0.9996	2	-	-	-	-
	1	10	0.8870	12	-	-	-	-	11	13	0.9389	5	-	-	-	-
	1	11	0.8869	13	-	-	-	-	11	23	0.8730	22	-	-	-	-
	1	12	0.8868	14	-	-	-	-	11	24	0.8822	16	-	-	-	-
	2	9	0.8786	19	-	-	-	-	12	13	0.9391	4	-	-	-	-
	2	10	0.8787	18	-	-	-	-	12	24	0.8821	17	-	-	-	-
	2	11	0.8786	19	-	-	-	-	15	16	0.8900	10	0.1506	16	-	-
	2	12	0.8785	20	-	-	-	-	15	17	-	-	0.1484	11	-	-
	3	9	0.8919	7	-	-	-	-	15	18	-	-	0.1470	9	-	-
	3	10	0.8917	9	-	-	-	-	15	19	-	-	0.1584	28	-	-
	3	11	0.8918	8	-	-	-	-	15	21	-	-	0.1455	8	-	-
	3	12	0.8918	8	-	-	-	-	15	22	-	-	0.1477	10	0.1061	25
	3	22	-	-	-	-	0.1085	32	15	24	-	-	0.1535	20	-	-
	3	24	0.9862	3	0.1580	25	-	-	16	17	-	-	0.1493	12	-	-
	4	7	-	-	-	-	0.1027	19	16	18	-	-	0.1501	14	-	-
	4	22	-	-	-	-	0.1024	17	16	19	-	-	0.1551	21	-	-
	5	7	-	-	-	-	0.1076	30	16	20	-	-	0.1620	34	-	-
	5	22	-	-	-	-	0.1090	34	16	21	-	-	0.1506	17	-	-
	7	8	-	-	0.1030	1	0.0763	1	16	22	-	-	0.1512	18	0.1070	28
	7	14	-	-	-	-	0.1004	13	16	24	-	-	0.1606	33	-	-
	7	15	-	-	-	-	0.0999	12	17	18	-	-	0.1375	4	-	-
	7	16	-	-	-	-	0.1020	16	17	19	-	-	0.1575	24	0.1085	33
	7	17	-	-	-	-	0.0891	4	17	20	-	-	-	-	0.1062	26
	7	18	-	-	-	-	0.0850	3	17	21	-	-	0.1413	7	-	-
	7	19	-	-	-	-	0.0944	8	17	22	-	-	0.1392	6	-	-
	7	20	-	-	-	-	0.0980	11	17	24	-	-	0.1593	31	-	-
	7	21	-	-	-	-	0.0899	5	18	19	-	-	0.1582	26	0.1024	18
	7	22	-	-	-	-	0.0772	2	18	20	-	-	-	-	0.1006	15
	7	23	-	-	-	-	0.1083	31	18	21	0.8898	11	0.1366	2	-	-
	8	9	0.8731	21	-	-	-	-	18	22	-	-	0.1380	5	0.1037	21
	8	10	0.8731	21	-	-	-	-	18	23	-	-	-	-	0.1047	23
	8	11	0.8730	22	-	-	-	-	18	24	-	-	0.1583	27	0.1065	27
	8	22	-	-	-	-	0.1042	22	19	20	-	-	0.1522	19	-	-
	13	22	-	-	-	-	0.1031	20	19	21	-	-	0.1586	29	0.1075	29
	14	22	-	-	-	-	0.1004	14	19	22	-	-	0.1594	32	0.0930	7
	9	10	0.9997	1	-	-	-	-	19	23	-	-	0.1570	22	-	-
	9	11	0.9997	1	-	-	-	-	20	21	-	-	-	-	0.1060	24
	9	12	0.9997	1	-	-	-	-	20	22	-	-	-	-	0.0915	6
	9	13	0.9389	5	-	-	-	-	20	23	-	-	0.1493	13	-	-
	9	24	0.8826	15	-	-	-	-	21	22	-	-	0.1367	3	-	-
	10	11	0.9997	1	-	-	-	-	21	24	-	-	0.1572	23	-	-
	10	12	0.9997	1	-	-	-	-	22	23	-	-	-	-	0.0951	9
	10	13	0.9389	5	-	-	-	-	22	24	-	-	0.1592	30	0.0965	10

Table 3 .

 3 Evaluation of the four perspectives based on literature review and developed criteria

		Accounting	Accounting	Accounting	Identification	Identification of	Data needs	Maturity Complexity Requested
		for system	for	for	of importance	vulnerability				time
		structure	physical	dynamics	indicators	indicators
			properties
	Graph-
	theoretical	YES	NO	NO	YES	NO	LOW	HIGH	LOW	LOW
	analysis
	Reliability analysis	YES	YES	NO	YES	NO	LOW	MIDDLE MEDIUM	LOW
	Electrical analysis	YES	YES	YES	YES	YES	HIGH	HIGH	MEDIUM	MEDIUM
	Reliability-
	electrical	YES	YES	YES	YES	YES	HIGH	POOR	MEDIUM	MEDIUM
	analysis
	Statistical analysis	NO	NO	NO	YES	YES	HIGH	HIGH	LOW	LOW
	Risk analysis	YES	YES	YES	YES	YES	HIGH	HIGH	HIGH	HIGH
	Agent-based modeling	YES	YES	YES	YES	YES	LOW	HIGH	HIGH	HIGH
	Object-
	oriented modeling and	YES	YES	YES	YES	YES	HIGH	HIGH	HIGH	HIGH
	simulation

Acknowledgements

This work has been partially funded by the Foundation pour une Culture de Securite Industrielle of Toulouse, France, under the research contract AO2006-01.


Nicolas Perrin

PROBABILISTIC INTERPRETATION FOR THE NONLINEAR POISSON-BOLTZMANN EQUATION IN MOLECULAR DYNAMICS

The Poisson-Boltzmann (PB) equation describes the electrostatic potential of a biomolecular system composed by a molecule in a solvent. The electrostatic potential is involved in biomolecular models which are used in molecular simulation. In consequence, nding an ecient method to simulate the numerical solution of PB equation is very useful. As a rst step, we establish in this paper a probabilistic interpretation of the nonlinear PB equation with Backward Stochastic Dierential Equations (BSDEs). This interpretation requires an adaptation of existing results on BSDEs.

Résumé. En dynamique moléculaire, l'équation de Poisson-Boltzmann (PB) permet de décrire le potentiel électrostatique d'un système moléculaire composé d'une molécule dans un solvant. Ce potentiel électrostatique intervient dans les modèles de simulation numérique permettant de comprendre la structure, la dynamique et le fonctionnement des protéines. La résolution numérique de l'équation de PB est donc une étape importante de ces simulations. Aussi, nous proposons dans un premier temps, une interprétation probabiliste de l'équation de PB non-linéaire à l'aide des Equations Diérentielles Stochastiques Rétrogrades (EDSR). Cette interprétation nécessite une adaptation des résultats d'existence et d'unicité des solutions d'EDSR.

Introduction

In this paper, we establish a probabilistic interpretation for the solution of the nonlinear Poisson-Boltzmann (PB) equation -∇. ( (x)∇u(x)) + κ 2 (x) sinh {u(x)} = g(x), x ∈ R 3 , where and κ are piecewise constant functions from R 3 to (0, +∞), and g is a singular measure. Our probabilistic interpretation consists in the sum of a singular known function and the solution of a particular Backward Stochastic Dierential Equation (BSDE). We prove an existence and uniqueness result for this BSDE by extending to our situation the methodology developed in [BDH + 03] or [START_REF] Pardoux | weak convergence and homogenization of semilinear PDEs[END_REF].

The PB equation is used to calculate electrostatic energies and forces in molecular simulation. The numerical methods which are already used are Finite Element, Boundary Element or Finite Dierence (cf. [START_REF] Baker | Implicit solvent electrostatics in biomolecular simulation[END_REF] for a discussion on these methods). The probabilistic interpretation we state in this work should provide theoretical foundations for the use of BSDEs simulation methods.

In Section 1, we introduce the Poisson-Boltzmann equation in molecular dynamics and underline three diculties we have to face to get a probabilistic interpretation of the nonlinear Poisson-Boltzmann equation.

In Section 2, we recall the probabilistic interpretation of divergence-form operators with discontinuous coefcients in term of SDE with weighted local time.

In Section 3, we very briey introduce BSDEs theory, before giving a general result for the existence and uniqueness of the solution of BSDEs with time-dependent monotonicity (Theorem 3.3), which we apply in order to get a probabilistic interpretation theorem (Theorem 3.5) of the solution of the corresponding Partial Dierential Equation (PDE).

In Section 4, we state our main result (Theorem 4.1). In order to prove it, we state an existence, uniqueness and regularity theorem (Theorem 4.2) for the solution v in H 1 (R d ) to the regularised Poisson-Boltzmann equation. Then we conclude in Section 4.3 by giving the steps of proof for our main result.

The Poisson-Boltzmann equation in Molecular Dynamics

The electrostatic potential of a biomolecular system in a medium of electric permittivity generated by a charge distribution ρ c is given by the Poisson equation (cf. [START_REF] Baker | Implicit solvent electrostatics in biomolecular simulation[END_REF])

-∇. ( (x)∇u(x)) = 4πρ c (x), x ∈ R 3 .
We consider a molecule immersed in an aqueous solvent. The charge distribution ρ c can be decomposed as a sum of the molecule contribution ρ m and the solvent contribution ρ s . The contribution of the molecule is modelled by a nite system of charges centred in the positions of the N atoms of the molecule

ρ m (x) = N i=1 q i δ xi (x), x ∈ R 3 ,
where q i ∈ R and x i ∈ R 3 are respectively the charge and the position of the ith atom and where δ x is the Dirac distribution at x ∈ R 3 . The solvent contribution is modelled by:

ρ s (x) = e c m j=1 nj z j exp {-e c z j u(x)/k B T e -V j (x)/k B T e }, x ∈ R 3 ,
where m is the number of dierent types of ions in the solvent, nj is the number of ions of the type j, z j is the valence of ions of the type j, e c is the elementary charge, k B is the Boltzmann constant, T e is the system temperature and V j models the steric interaction between the ions of the type j and the molecule.

For an electrolyte as NaCl where N = 2 with q 1 = 1 and q 2 = -1, the Poisson equation simplies to give the nonlinear Poisson-Boltzmann equation

-∇. ( (x)∇u(x)) + 8πe c ne -V (x)/k B Te sinh {e c u(x)/k B T e } = 4π N i=1 q i δ xi (x), x ∈ R 3 .
(1)

Let Ω int and Ω ext be two disjoint open subsets of R 3 , which model the molecule and the solvent respectively, such that Ωint ∪ Ωext = R 3 ; Ω int is supposed to be bounded with a smooth boundary Γ.

The dielectric permittivity is represented by a discontinuous function taking two values inside the molecule Ω int (vacuum permittivity) and outside Ω ext (solvant permittivity):

(x) = int > 0 si x ∈ Ω int , ext > 0 si x ∈ Ω ext .
In the following, we study the nonlinear Poisson-Boltzmann equation

-∇. ( (x)∇u(x)) + κ 2 (x) sinh {u(x)} = g(x), x ∈ R 3 .
(

) 2
where the modied Debye-Hückel parameter

κ 2 (x) = 0 si x ∈ Ω int , κ2 > 0 si x ∈ Ω ext .
(3)

approximates e -V (•)/k B Te 8πe 2 c I k B Te with I = 1 2 s ns z 2
s . We should emphasize that this approximation requires a hypothesis of low linear charge density as underlined in [START_REF] Baker | Implicit solvent electrostatics in biomolecular simulation[END_REF].

The probabilistic interpretation for the nonlinear PB equation we propose in this paper requires to handle three diculties:

(1) singularity of the source term (cf. Section 2.1);

(2) discontinuity of the coecient in the divergence-form operator (cf. Section 2.2);

(3) non-linearity of the equation (cf. Section 3).

Remark 1.1. If we suppose e c u(•) much smaller than k B T e , we obtain a linear version of the Poisson-Boltzmann

equation -∇. ( (x)∇u(x)) + κ 2 (x)u(x) = g(x), x ∈ R 3 , (4
) for which a probabilistic interpretation has been established in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF].

Handling of the singularity of the source term and the divergence-form operator

In this section, we handle the singularity of the source term and discontinuity of the coecient in the divergence-form operator.

The Poisson-Boltzmann equation with a regularised source terme

The singular source term is handled by writing the solution as a sum of a known singular function and the solution of a regularised equation.

The following regularisation is introduced in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]. Let χ be a C ∞ function with a compact support embedded in Ω int , such that χ(x) = 1 for x close to {x 1 , . . . , x N } and G = i G i where

G i (x) = q i int 1 |x -x i | , x ∈ R 3 , satisfying -∇.( int ∇G i ) = q i δ i .
We then study the function v = u -χG which is the solution of

-∇.( (x)∇v(x)) + κ 2 (x) sinh(v(x)) = g(x), x ∈ R 3 .
(5)

The regularised source term is then expressed as g(x) = int (G(x)∆χ(x) + ∇G(x).∇χ(x)) for x ∈ R 3 , since the support of χG is in Ω int while κ is null on Ω int . In particular, the function g is C ∞ .

The martingale problem associated with the divergence-form operator

In [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF], a probabilistic interpretation is introduced for divergence-form operators in R d , d 1 with discontinuous coecients of the form

L• = ∇( (x)∇•),
where is a piecewise constant function from R d to (0, ∞) with a smooth discontinuity manifold Γ.

We suppose that Ω int is a simply connected set with a boundary Γ of class C ∞ , and that

Ω ext = R d \(Ω int ∪Γ).
The smoothness of Γ allows us to construct a mapping π of class C 2 b (where C 2 b is the set of 2 time continuously

Ω int Ω ext Γ x π(x) n(x)
x i , q i

x 1 , q 1

x 3 , q 3

x 2 , q 2 ρ(x)

Figure 1 dierentiable functions with bounded derivatives of all order up to 2) from a neighborhood N of Γ to Γ such that

|x -π(x)| = d(x, Γ), ∀x ∈ N ,
where d(x, Γ) denotes the distance of x to Γ. Let n(x) be the unit vector normal to Γ at x ∈ Γ pointing in direction of Ω ext . Finally, let ρ be the signed distance to Γ positive in Ω ext and negative in

Ω int for all x ∈ N ρ(x) = (x -π(x)) • n(π(x)), x ∈ N ,
which can be extended to the whole Euclidean space R d in a function C 2 b still denoted ρ (cf. Fig. 1). The probabilistic interpretation of elliptic and parabolic PDEs driven by the operator L relies on the following martingale problem. Denition 2.1. Let (C, B) be the set C of continuous functions w from [0, ∞) to R d endowed with the Borel cylindrical σ-eld B and the canonical ltration (B t , t 0). A family of probability measures P x on (C, B) solves the martingale problem for the operator L if, for all x ∈ R d , one has

P x {w ∈ C : w(0) = x} = 1,
and, for all φ satisfying

φ ∈ C 0 b (R d ) ∩ C 2 b (R d \Γ), ∇φ • (nπ) ∈ C 0 b (N ),
the process

M φ t (w) := φ(w(t)) -φ(w(0)) - t 0 Lφ(w(s))ds is a P x -(B t ) martingale.
It is shown in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] that the unique solution of this martingale problem is the law P x of all weak solution

(Ω, F x , (F x t ) t 0 , P x , {B t ; t 0} , {X x t ; t 0}) of the Stochastic Dierential Equation (SDE)

X x t = x + t 0 2 (X x s )dB s + ext-int 2 ext t 0 n(X x s )dL 0 s (D x ), D x t = ρ(X x t ), (6)
where B is a F x t -standard d-dimensional Brownian motion, and L 0 s (D x ) is the local time at 0 of the continuous semimartingale D x . It is actually shown in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] that all weak solutions to the SDE (6) are identical in law. The above process X x is used in the new probabilistic interpretation that we develop in next section.

A Probabilistic interpretation in terms of BSDEs

One of the probabilistic methods to handle nonlinear Partial Dierential Equations (PDEs) is based on Backward Stochastic Dierential Equations (BSDEs) (cf. [BDH + 03] or [START_REF] Pardoux | weak convergence and homogenization of semilinear PDEs[END_REF]). We give a result for BSDEs with a time-dependent monotonicity condition. Then we conclude by applying the above result to BSDEs driven by a SDE with weighted local time.

BSDEs with time-dependent monotonicity condition

In all what follows, we consider B be a standard d-dimensional Brownian motion dened on a probability space (Ω, F, P), where {F} is the natural augmented ltration of B. Let τ be a F t stopping time not necessarily inite, ξ a F τ -measurable random variable and f an application from

Ω × R + × R k × R k×d to R k . Denition 3.1. A solution of the BSDE (τ, ξ, f ) is a couple of progressively measurable processes (Y, Z) with values in R k × R d×k , which satises the following properties (1) t → 1 {t τ } f (t, Y t , Z t ) is in L 1 loc (0, ∞), and t → Z t is in L 2 loc (0, ∞);
(2

) Y t∧τ = Y T ∧τ + T ∧τ t∧τ f (s, Y s , Z s )ds - T ∧τ
t∧τ Z s dB s for all 0 t T ; (3) Y t = ξ and Z t = 0 on the event {t τ }. The application f is called the generator of the BSDE.

Let suppose that:

(h1) (a) f (•, y, z) is progressively measurable for all y and z;

(b) for all r > 0, for all n ∈ N , sup |y| r |f (t, y, 0) -f (t, 0, 0)| ∈ L 1 (]0, n[×Ω, dt ⊗ P); (c) Lipschitz continuity : |f (t, y, z) -f (t, y, z )| K(t) z -z for all t, y, z, z a.s., where K is a progressively measurable positive process which is upper bounded by a constant K, (d) monotonicity : for all t, y, y , z a.s.,

y -y , f (t, y, z) -f (t, y , z) µ(t)|y -y | 2
where µ is a progressively measurable process which is upper bounded by a constant μ and lower bounded by a constant µ, (e) integrability :

E τ 0 e R t 0 λ(s)ds |f (t, 0, 0)| 2 dt < ∞
where λ is a progressively measurable process such that, for all t in R + ,

2µ(t) + K 2 (t) < λ(t), ( h2
) the nal condition ξ is a F τ -measurable random variable such that E T 0 e R t 0 λ(s)ds |f (t, e -1/2 R t 0 λ(s)ds ξt , e -1/2 R t 0 λ(s)ds ηt )| 2 dt < ∞
where λ(t) := 2µ(t) + K 2 (t), ξ = e R T 0 λ(s)ds ξ, ξt = E ξ|F t and η are predictable processes such that

ξ = E ξ + ∞ 0 ηt dB t , E ∞ 0 |η t | 2 dt < ∞.
Denition 3.2. For a given progressively measurable process λ, the solution

(Y, Z) of BSDE (τ, ξ, f ) is said to be in L 2 λ(•) if (Y, Z) 2 λ(•) := E sup 0 t τ e R t 0 λ(s)ds |Y t | 2 + τ 0 e R t 0 λ(s)ds (|Y t | 2 + Z t 2 )dt < ∞.
We prove in [START_REF] Perrin | Stochastic methods in molecular dynamics[END_REF] the following existence and uniqueness result of the solution to BSDEs.

Theorem 3.3. If Assumptions (h1-h2) are true, there exists a unique solution (Y, Z) to the BSDE (τ, ξ, f ) such that, for all progressively measurable process λ such that λ(

•) -2µ(•) -K 2 (•) > λ > 0, (Y, Z) 2 λ(•) CE e R τ 0 λ(s)ds |ξ| 2 + τ 0 e R t 0 λ(s)ds |f (t, 0, 0)| 2 dt . (7)
The proof of this theorem follows closely the steps of the proof in [START_REF] Pardoux | weak convergence and homogenization of semilinear PDEs[END_REF] by adapting them to time dependent coecients.

BSDEs driven by SDEs with weighted local time

Let (Ω, F x , (F x t ) t 0 , P x , {B t ; t 0} , {X x t ; t 0}) be an arbitrary weak solution of the SDE (6). Consider the Forward-Backward SDE (∞, 0, f ), for all t, T , such that for all 0 t T ,

Y x t = Y x T + T t f (X x s , Y x s , Z x s )ds - T t Z x s dB s . ( 8
)
where

f is a generator from R d × R k × R k×d to R k such that (H1) f (•, y, z)
is measurable for all y and z; (H2) y → f (x, y, z) is continuous for all x, z;

(H3) Lipschitz continuity : |f (x, y, z) -f (x, y, z )| K(x) z -z for all x, y, z, z a.s., where K is measurable,positive and upper bounded by a constant K, (H4) monotonicity : for all x, y, y , z a.s.,

y -y , f (x, y, z) -f (x, y , z) µ(x)|y -y | 2
where µ is measurable, upper bounded by a constant μ and lower bounded by a constant µ,

(H5) integrability : E ∞ 0 e R t 0 λ(X x s )ds |f (X t , 0, 0)| 2 dt < ∞
where λ is measurable and such that 2µ(x)

+ K 2 (x) < λ(x) for all x in R d , (H6) sup |y| r |f (X x t , y, 0) -f (X x t , 0, 0)| ∈ L 1 (]0, n[×Ω, dt ⊗ P)
, for all n ∈ N * and r > 0. The following existence and uniqueness theorem for Eq. ( 8) is a corollary of Theorem 3.3 with τ = ∞ and ξ = 0.

Theorem 3.4. If Assumptions ( H1-H6) are true, the BSDE (8) has a unique solution (in the sense of Denition 3.1) such that

E sup 0 t e R t 0 λ(X x s )ds |Y x t | 2 + ∞ 0 e R t 0 λ(X x s )ds |Y x t | 2 + Z x t 2 dt CE ∞ 0 e R t 0 λ(X x s )ds |f (X x t , 0, 0)| 2 dt .
Sketch of proof. The key arguments of the proof are the following:

• The weak uniqueness of the solution (Ω, F x , (F x t ) t 0 , P x , {B t ; t 0} , {X x t ; t 0}) to the SDE (6) allows one to apply Theorem 4.29 in [START_REF] Jacod | of Grundlehren der Mathematischen Wissenschaften[END_REF] to (Ω, F x , (F x t ) t 0 , P x ) local martingales to get a process

2 (X x t ) in Z x t .
The BSDE (8) provides a probabilistic interpretation to elliptic semilinear PDEs driven by the dierential operator L:

-Lu(x) + f (x, u(x), (∇u √ 2 )(x)) = 0, x ∈ R d . (9)
Theorem 3.5. Let u belongs to C 2 (R d \Γ) a solution of Eq.( 9), such that for all

x in R d E ∞ 0 e R t 0 λ(X x s )ds (∇u √ 2 )(X x s ) 2 dt < ∞. (10)
Suppose there exists a function r in C 2 , such that ∀x ∈ N ,

r(x) = int ext -1 ∇ int u(π(x)) • n(π(x)),
and such that û belongs to

C 2 (R d \Γ) ∩ W 2,∞ loc (R d ), where û(x) := u(x) -r(x)[ρ(x)] + .
Then, for all x in R d , the couple u(X x t ), (∇u

√ 2 )(X x t )
is the solution of BSDE (8); in particular u(x) = Y x 0 . The proof consists in using a generalised Itô formula proved by Theorem 2.8 in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF].

Remark 3.6. We should emphasize that Y x 0 is deterministic as (Y x t ) t is F t -adapted.

Remark

The regularised PB equation (Eq. ( 5)), for d = 3, is associated to the BSDE (∞, 0, f ) of generator given by

f (t, y) = g(X x t ) -κ 2 (X x t ) sinh y, x ∈ R 3 , y ∈ R, t ∈ [0, ∞),
where X x t is the solution of the SDE given by Eq. ( 6), and :

Y x t = Y x T + T t g(X x s ) -κ 2 (X x s ) sinh Y x s ds - T t Z x s dB s , (11)
for all t, T as 0 t T .

Le us remind a result in [BDH + 03] :

Theorem 3.7 (Briand and al.). Under Assumptions (h1-h2) with K, µ and λ constants, there exists a unique solution to the BSDE (τ, ξ, f ). This result does not directly apply to the regularised PB equation (Eq. ( 5)), on the one hand because of the discontinuities of and on the other hand because assuming (h1d) and (h1e) with µ constant would impose µ 0 and λ > 0. It makes dicult (and wrong in general) the verication of the integrability hypothesis that E ∞ 0 e λt |g(X x t )| 2 dt is nite.

Remark 3.8. We should emphasize that a result for monotone BSDEs (with constant µ = 0) is obtained in [START_REF] Royer | BSDEs with a random terminal time driven by a monotone generator and their links with PDEs[END_REF], under the strong hypothesis that f (t, 0, 0) ≡ 0 a.s., which is not satised in our case.

(x) = χ(x)G(x) + Y x 0 , ∀x ∈ R 3 ,
where Y x 0 denotes the solution (Y x , Z x ) of the BSDE (11) corresponding to the regularised Poisson-Boltzmann equation (Eq. ( 5)).

To prove this result, we need to verify that the unique solution to the regularised Poisson-Boltzmann equation (Eq. ( 5)) satises the assumptions of Theorem 3.5.

Existence and uniqueness of the solution

In [START_REF] Perrin | Stochastic methods in molecular dynamics[END_REF] we show the following existence and uniqueness theorem of a weak solution in the subset M of

H 1 (R 3 ), M := v ∈ H 1 (R 3 )| sinh v ∈ L 2 (R 3 ) .
Theorem 4.2. The regularised Poisson-Boltzmann equation (Eq. ( 5)) has a unique weak solution v ∈ M . This solution belongs to C 0 b (R 3 ) ∩ C 2 (R 3 \Γ), and its restriction v |Γ belongs to C 3 (Γ). There also exists a function r in

C 2 (R 3 ) such that ∀x ∈ N r(x) = int ext -1 ∇ int v(π(x)) • n(π(x)), (*) and such that v(x) := v(x) -r(x)[ρ(x)] + (**) is in C 2 (R 3 \Γ) ∩ W 2,∞
loc . Furthermore, the gradient ∇v of v belongs to L ∞ .

Sketch of proof.

(1) The existence and uniqueness of the solution in H 1 (R 3 ) are obtained by an energy minimisation.

(2) The properties v in C 0 b (R 3 )∩C 2 (R 3 \Γ) and v in W 2,∞ loc are proved using similar arguments as in [BCMT10], Appendix A.

(3) The bounded condition of ∇v is shown in two steps. First, the previous step implies that ∇v ∈ L ∞ loc (R 3 ).

Second, we notice that for

k in {1, • • • , 3}, v satises: ext Ωext ∇∂ k v • ∇φ + κ Ωext ∂ k v cosh vφ = 0 ∀φ ∈ H 1 0 (Ω ext ).
By using the fact that κ cosh κ > 0, we can prove in a very similar way as for the proof of Theorem IX.27 by H. Brezis in [START_REF] Brezis | Analyse fonctionnelle[END_REF] that ∇u ∈ L ∞ (Ω ext ). The proof follows the following steps.

(1) The following lemma (proved in [START_REF] Perrin | Stochastic methods in molecular dynamics[END_REF]) allows us to verify the integrability Hypothesis (H5 ) for the generator f (x, y) = -κ 2 (x) sinh(y) + g(x) of the nonlinear PB equation. (2) In Section 4.2, we have stated an existence and uniqueness result (Theorem 4.2) for solutions v in H 1 to the regularised Poisson-Boltzmann equation. (3) Furthermore, the solution v satises the assumptions of Theorem 3.5 which makes the link between PDEs solutions and BSDEs solutions. In particular, Lemma 4.3 and the property ∇v ∈ L ∞ allow us to verify Assumption (10).

Conclusion

With this probabilistic interpretation, it should be possible to implement new numerical methods for the Poisson-Boltzmann equation. To this aim, an ecient way to simulate the forward process given by Eq. (6) has to be implemented. With more regularities hypothesis, simulation of a Forward-Backward SDE system can be done by quantication (cf. [START_REF] Delarue | A forward-backward stochastic algorithm for quasi-linear PDEs[END_REF]) or by a combination of Picard's iterations and an adaptive control variable (cf. [START_REF] Gobet | Solving BSDE with adaptive control variate[END_REF]). However, the adaptation to the PB equation has to be done.

Zx t ; t 0

 0 and express them with stochastic integrals • t .• The uniform ellipticity of allows one to change Zx

  t

4.

  Application to the nonlinear Poisson-Boltzmann equation 4.1. Main result It is then possible to give the following probabilistic interpretation to the nonlinear Poisson-Boltzmann equation. Theorem 4.1. The nonlinear Poisson-Boltzmann equation (Eq. (2)) admits the probabilistic interpretation u

4. 3 .

 3 Proof of Theorem 4.1

  Lemma 4.3. Let X x be a solution of the SDE (6) associated to the operator L• := ∇( ∇•). There exists a choice of p and q positive, such thatE x s )ds dt < ∞where λ(X x s ) = p1 X x s ∈Ωint -q1 X x s ∈Ωext . So the generator f satises Assumptions (H1 -H6 ).

TOSCA project-team, INRIA Sophia Antipolis, France. nicolas.perrin@inria.fr c EDP Sciences, SMAI