DETERMINING A NETWORK TOPOLOGY OF A HIERARCHICAL POWER SUPPLY NETWORK

The invention relates to a method and apparatus for determining a network topology of a hierarchical network, in particular of a hierarchical power supply network. A network which is a power supply network or power transmission network can comprise a plurality of network nodes connected to each other hierarchically. Distribution system operators DSO in power grids or power supply networks are often confronted with the situation that their knowledge of the network topology of the power supply network is incomplete, defective or even absent. This is due to the fact that a distribution grid infrastructure has often grown over decades and there can be many modifications to the original setup due to extensions or repair/replacement activities which have not been sufficiently documented. Due to the large dimensions of the power supply network and the plurality of the network nodes, a manual verification of the underlying network topology has become out of reach or can only be undertaken under large expenses. Furthermore, distribution systems or power supply systems generally lack sufficient monitoring infrastructure which can indicate inconsistencies in the estimated network topology of the respective power supply network. Consequently, the identification of errors in an estimated network topology of a hierarchical network is so far unsystemat- ic and happens incidentally.

Accordingly, it is an object of the present invention to provide a method and an apparatus for determining a network topology of a hierarchical network sufficiently.

This object is achieved according to a first aspect of the present invention by a method for determining a network to- pology of a hierarchical network comprising the features of claim 1.

The invention provides according to the first aspect of the present invention a method for determining a network topology of a hierarchical network,

the method comprising the steps of

providing an outage state matrix representing an outage state of leaf nodes at the lowest hierarchical level of said hier- archical network,

decomposing the state matrix into a first probability matrix indicating for each inner node of said hierarchical network the probability that the respective inner node forms the origin of an outage at the lowest hierarchical level of said hierarchical network and into a second probability matrix indicating for each leaf node at the lowest hierarchical level of said hierarchical network the probability that an inner node forms a hierarchical superordinate node of the respective leaf node at the lowest hierarchical level of said hier- archical network and

evaluating the second probability matrix to determine the network topology of said hierarchical network.

In a possible embodiment of the method according to the first aspect of the present invention a hierarchical network is a hierarchical power supply network.

In a still further possible embodiment of the method according to the first aspect of the present invention the hierar- chical power supply network comprises at its lowest hierarchy level leaf nodes formed by smart meters generating event data streams indicating a current outage state of the leaf nodes of said power supply network. In a possible embodiment of the method according to the first aspect of the present invention the generated event data streams are supplied to a processing unit which derives the outage state matrix from the received event data streams . In a still further possible embodiment of the method according to the first aspect of the present invention each event data stream generated by a smart meter forming a leaf node of this hierarchical power supply network consists of event data triples .

In a possible embodiment of the method according to the first aspect of the present invention each event data triple of an event data stream comprises

a timestamp indicating a time when an event is detected, a smart meter identifier identifying the smart meter generating the respective event data stream and

an event identifier identifying an event type of the respec- tive event.

In a possible embodiment of the method according to the first aspect of the present invention an outage state matrix derived from event data streams is a two-dimensional matrix, wherein the first dimension of said outage state matrix represents time and

wherein the second dimension of said outage state matrix represents smart meters forming leaf nodes at the lowest hierarchical level of said power supply network.

In a still further possible embodiment of the method according to the first aspect of the present invention each column of the two-dimensional outage state matrix represents states of a respective smart meter at particular points in time and wherein each row of said outage state matrix represents the states of all smart meters at a particular point in time.

In a still further possible embodiment of the method according to the first aspect of the present invention the outage state matrix is a binary state matrix,

wherein a first logical value indicates that the respective smart meter is in the outage state at a particular point in time and wherein a second logical value indicates that the respective smart meter is not in the outage state at a particular point in time.

In a further possible embodiment of the method according to the first aspect of the present invention event data streams generated by smart meters forming leaf nodes at the lowest hierarchical level of said hierarchical power supply network are transmitted continuously or periodically via a data network to a remote controller comprising a processing unit which derives the outage state matrix from the received event data streams and stores the derived outage state matrix in a memory unit for further processing.

In a still further possible embodiment of the method accord- ing to the first aspect of the present invention the determined network topology of the hierarchical network is output via a data or user interface.

The invention further comprises according to a second aspect an apparatus for determining a network topology of a hierarchical network comprising the features of claim 12.

The invention provides according to a second aspect an apparatus for determining a network topology of a hierarchical network,

said apparatus comprising:

a memory unit adapted to store an outage state matrix representing an outage state of leaf nodes at the lowest hierarchical level of said hierarchical network and

a processing unit adapted to decompose the stored state matrix into a first probability matrix indicating for each inner node of said hierarchical network the probability that the respective inner node forms the origin of an outage at the lowest hierarchical level of said hierarchical network and into a second probability matrix indicating for each leaf node at the lowest hierarchical level of said hierarchical network the probability that an inner node forms a hierar- chical superordinate node of the respective leaf node at the lowest hierarchical level of said hierarchical network, wherein the decomposed second probability matrix is evaluated by the processing unit to determine the network topology of said hierarchical network.

The invention further provides according to a third aspect a network controller of a network comprising an apparatus according to the second aspect of the present invention, where in the processing unit of said network controller is adapted to derive the outage state matrix from event data streams re ceived from smart meters of said network.

The invention further provides according to a fourth aspect a hierarchical power supply network comprising the network controller according to the third aspect of the present invention,

wherein said hierarchical power supply network comprises at its lowest hierarchical level leaf nodes formed by smart me- ters adapted to generate event data streams indicating a current outage state of the leaf nodes of the hierarchical power supply network.

In a possible embodiment of the hierarchical power supply network according to the fourth aspect of the present invention the event data streams generated by smart meters forming leaf nodes at the lowest hierarchical level of said power supply network are transmitted via a data network to a data interface of the network controller of said hierarchical pow- er supply network.

In the following possible embodiments of the different aspects of the present invention are described in more detail with reference to the enclosed figures.

Fig. 1 shows a block diagram for illustrating a possible exemplary embodiment an apparatus for determining network topology of a hierarchical network according to an aspect of the present invention; shows a flowchart of a possible exemplary embodiment of a method for determining a network topology of a hierarchical network according to a further aspect of the present invention; shows a schematic data representation and a factorization scheme which can be employed by a method and apparatus for determining a network topology of a hierarchical network according to the present invention . As can be seen in fig. 1 an apparatus 1 for determining a network topology NT of a hierarchical network comprises in the illustrated embodiment a memory unit 2 and a processing unit 3. The memory unit 2 is adapted to store an outage state matrix X representing an outage state of leaf nodes at the lowest hierarchical level of a hierarchical network 4 as shown in fig. 1. The hierarchical network 4 can be a hierarchical power supply network which comprises a plurality of network nodes connected to each other hierarchically. Leaf nodes of such a hierarchical power supply network 4 are formed by smart meters 5 providing event data streams EDS to a processing unit 3 of the apparatus 1. The processing unit 3 is adapted to decompose the state matrix X stored in the memory unit 2 into a first probability matrix W and into a second probability matrix H. The first probability matrix W indicates for each inner node of the hierarchical network 4 the probability that the respective inner node forms the origin of an outage at the lowest hierarchical level of the hierarchical network. The second probability matrix H indicates for each leaf node at the lowest hierarchical level of the hierarchical network 4 the probability that an inner node forms a hierarchical superordinate node of the respective leaf node at the lowest hierarchical level of the hierarchical network 4. The processing unit 3 of the apparatus 1 is further adapted to evaluate the second probability matrix H to determine the network topology NT of the hierarchical network 4. The processing unit 3 is adapted to derive the outage state matrix X from event data streams EDS received from smart meters 5 of the hierarchical power supply network 4. The derived outage state matrix X can be stored by the processing unit 3 into the memory unit 2 for further processing.

The hierarchical power supply network 4 comprises at its low- est hierarchical level leaf nodes formed by smart meters 5 generating event data streams EDS indicating a current outage state of the respective leaf nodes of said power supply network 4. The generated event data streams EDS are supplied to the processing unit 3 which is adapted to derive the outage state matrix X from the received event data streams. In a possible embodiment the generated event data streams are supplied via a data network to the processing unit 3 of the apparatus 1. In a possible embodiment each event data stream EDS generated by a smart meter 5 of the power supply network 4 consists of event data triples. Each event data triple of the event data stream EDS can comprise in a possible embodiment a timestamp TS indicating a time when an event is detected, a smart meter identifier SM-ID identifying the respective smart meter 5 generating the respective event data stream EDS and an event identifier EV-ID identifying an event type of the respective event.

The state matrix X is derived by the processing unit 3 from the received event data streams EDS generated by multiple smart meters 5 within the power supply network 4. The derived outage state matrix X is in a possible embodiment a two- dimensional matrix, wherein the first dimension of said outage state matrix X represents time and wherein the second dimension of said outage state matrix X represents smart meters 5 forming leaf nodes of the lowest hierarchical level of the power supply network 4. Each column of the two-dimensional outage state matrix represents states of a specific smart meter 5 at particular points in time and each row of the outage state matrix X represents the states of all smart meters 5 at a particular point in time. In a possible embodiment the outage state matrix X is a binary state matrix, wherein a first logic value indicates that the respective smart meter 5 is in the outage state at a particular point in time and wherein a second logic value indicates that the respective smart meter 5 is not in the outage state of a particular point in time.

In a possible embodiment the event data streams EDS generated by the smart meters 5 forming leaf nodes at the lowest hierarchical level of the hierarchical power supply network 4 are transmitted continuously to the processing unit 3 of the apparatus 1. In an alternative embodiment the event data streams EDS generated by the smart meters 5 forming leaf nodes at the lowest hierarchical level of the hierarchical power supply network 4 are transmitted periodically to the processing unit 3 of the apparatus 1. Transmission of the event data streams EDS can in a possible embodiment be performed via a separate data network. In a possible embodiment the apparatus 1 is integrated in a remote network controller of the power supply network 4. The processing unit 3 of the network controller is adapted to derive the outage state matrix X from the received event data streams EDS and stores the derived outage state matrix X in the memory unit 2 of the apparatus 1 for further processing. The processing unit 3 can load the stored outage state matrix X and perform a decomposition of the loaded state matrix into a first probability matrix W and into a second probability matrix H. Then the processing unit 3 evaluates the decomposed second probability matrix H to determine the network topology NT of the hierarchical power supply network 4. The calculated network topology NT of the hierarchical power supply network 4 can be output by the processing unit 3 via a user interface or via a data interface. Accordingly, the apparatus 1 can determine the network topology NT of the hierarchical network 4 automatically based on the received plurality of event data streams EDS. The present invention comprises according to a further aspect a network controller of a hierarchical power supply network 4 wherein the network controller is adapted to calculate the network topology NT of the power supply network 4 automati- cally based on event data streams EDS received from smart meters of the hierarchical power supply network 4 or leaf nodes of said hierarchical power supply network. The apparatus can perform a clustering applied to event data streams EDS of multiple smart meters 5 which identify hierarchical dependen- cies within the hierarchical network 4. Event data streams EDS are used to identify the hierarchical dependencies, i.e. dependencies between leaf nodes, parent or superordinate nodes up to a root node of the hierarchical power supply network 4. Identification of the hierarchical dependencies can be performed for each smart meter within a tree topology of the hierarchical power supply network 4, wherein each smart meter 5 forms a leaf node at the lowest hierarchical level of the power supply network 2. The total number of clusters can equal the number of network nodes in the tree topology of the power supply network 4 and the number of assigned clusters per smart meter 5 can equal the number of its parent nodes in the topology up to the root node of the hierarchical power supply network 4. The event data stream EDS of each smart meter 5 can consist in a possible embodiment of a triple (timestamp TS, smart meter ID and event ID) , wherein the event ID takes two values representing an outage event where the smart meter 5 detects a power loss and/or a restoration event where the smart meter 5 detects a restoration of power. In a possible embodiment the event data triples of all smart meters 5 can be sorted by the timestamp TS in an ascending order and converted into a sparse binary outage state matrix X∈ {0,l} ^mxn , wherein each column of the sparse binary outage state matrix X represents the state of the respective smart meter 5 at particular points in time. The first dimension of the state matrix X represents the time and the second dimension of the state matrix represents the smart meters. While iterating over the sorted event data triples, a new row can be appended at the bottom of the outage state matrix X whenever any of the smart meters 5 records an event and optionally repeated in regular time intervals in between provided at least one smart meter is in the outage state. The new row captures the state of all smart meters 5 at a particular instant according to a predetermined encoding scheme as described above. All zero rows of the outage state matrix X, i.e. where no smart meter 5 is in the outage state at a particular point in time, can be omit- ted. The resulting representation is typically very sparse, because only a small fraction of the smart meters 5 is in an outage state at a particular point in time. By storing only the non-zero values, i.e. high logic values indicating that the respective smart meter 5 is in the outage state, the data representation is feasible for encoding outage information of power supply networks even with a very large number of smart meters 5.

The tree topology of the hierarchical network 4 can be iden- tified from the outage state matrix X by performing a matrix factorization. In the matrix factorization performed by the processing unit 3 of the apparatus 1 the outage state matrix X is decomposed into two different matrices The first

probability matrix indicates for each inner node of the hierarchical network 4 the probability that the respective inner node forms the origin of an outage at the lowest hierarchical level of said hierarchical network 4. The second probability matrix H indicates for each leaf node at the lowest hierarchical level of the hierarchical network 4 the probability that an inner node forms a hierarchical superor- dinate node of the respective leaf node at the lowest hierarchical level of said hierarchical network 4. and

with r being the number of non-leaf nodes or super-

ordinate nodes in the tree topology of the hierarchical net- work, wherein i.e. the outage state matrix. Each row

of contains only non-negative values and sums up to 1. The first probability matrix represents the probability that the kth non-leaf node is the root cause or origin of an outage at time index i. Constraining implies mutually exclusive outage root causes which is an admissible assumption for a power supply network. The value represents the proba

bility that the kth non-leaf node is a superordinate node of the j th smart meter forming a leaf node at the lowest hierarchical level. Thus, is the probability that an

outage has been observed at the jth smart meter 5 at time index i. To ensure that the value ranges of and as well as the row sum constraint enforced on W are satisfid, the two matrices are p ^arameterized as and

The values of the parameter matrices W and H can be determined in a possible embodiment through maximum log- likelihood estimation by minimizing the cross-entropy between the target value Xj _j and the estimated value _j i.e. the differentiable objective is minimized

via a gradient-based optimization method. Scalability of the method and robustness of the optimization algorithm against convergence to local optima can be achieved by performing lock-free multithreaded stochastic optimization and by introducing additional noise to the update direction similar to simulated annealing. The following table depicts a learning algorithm and can be used by a method according to an aspect of the present invention in a possible implementation. function FACTORIZE(X∈ {0,1} ^mxn , ] ])

Initialize and at random

while not converged do

Draw i∈ {1, ... , m} uniformly at random

Draw j∈ {1, n} uniformly at random

Draw β∈ {-1,1} at random with Ρ[β = 1] = p

Compute forward pass

where r indicates the number of non-leaf nodes, α is the learning rate, and , β determines the sign of the current update direction under the above-described noisy update scheme.

The above algorithm performs a matrix factorization for a tree topology identification based on binary event data. The updates can be executed in parallel without locking. After the optimization process has converged, the network topology NT can be inferred from the cluster memb ership matrix

forming the second probability matrix.

The set of edges of the network topology NT of the hierarchical network 4 can be determined using the following algorithm. In a possible implementation of the method and apparatus according to the present invention the following algo- rithm is employed to infer the network topology NT of the power supply network 4 from the second probability matrix H. The algorithm can yield the edges of a graph where the non- leaf nodes are labelled {l,...,r} and the leaf nodes are la- belled { r+1 , r+n} . function INFERT0P0L0GY

Binarize using thresholdy

Get frequency of each cluster

Set of edges in the graph denoted by tuples (parent, child)

Store indices of nonzero entries in

Store frequencies of the entries in P

Compute child node

for argsort(fp) in ascending order of f _p do

return E

Fig. 3 depicts a data representation as well as a factorization scheme that can be employed by the method and apparatus according to the present invention for a simple exemplary hierarchical network comprising eight network nodes connected to each other in three hierarchical levels. As can be seen in fig. 1, the exemplary simple network comprises a root node 1 connected to two intermediate superordinate nodes 2, 3 each having leaf nodes, wherein leaf nodes 4, 5, 6 are connected to superordinate nodes 2 and leaf nodes 7, 8 are connected to superordinate node 3. The intermediate nodes 2, 3 are both connected to the root node 1. In fig. 3 columns 1 to 3 of the left-hand side matrix represent observations made by the non- leaf nodes of the network which are not accessible in practice. Only using columns 4 to 8 which comprise observations made by the smart meters forming leaf nodes of the hierarchical network, the first probability matrix W and the second probability matrix H can be obtained through factoriza- tion using the algorithm depicted in the first table above.

After having decomposed the state matrix X, the second probability matrix is evaluated automatically to determine or calculate the network topology NT of the hierarchical network. This is performed in a possible implementation using the algorithm as depicted in table 2 above.

The calculated network topology NT can be output at a data interface of the processing unit 3 to a further evaluation unit for further processing of the derived network topology NT of the hierarchical network 4. Further, the calculated network topology NT can be output via a user interface to an operator of the power supply network 4. The method and apparatus according to the present invention can be used for determining the network topology NT of any kind of hierarchical network. The data generated by the multiple smart metering devices of the power supply network provides a massive amount of data that can be used by the method according to the present invention to calculate automatically the network topology NT of the respective network. The method identifies auto- matically the tree topology of a hierarchical power supply network 4 based on outage events recorded by the smart meter devices 5 forming leaf nodes of the hierarchical power supply network 4.