System and method for anomaly detection in an electrical network

The present subject matter enables early or real-time detection of anomalies in electric networks. In various applications, the system detects anomalies, such as electricity theft, electricity surge, etc. It solves the difficult-to-detect problems in an electrical network, where anomalies like electricity theft or electrical surge may not be found until it has raised numerous concerns or complaints, or has created a significant impact on infrastructure functionality, service quality, or cost. In addition, the present subject matter decreases the requirement for large number of sensors and provides more cost effective and scalable solutions. The present subject matter provides a method for determining where a detected anomaly is occurring within an electrical network. Variations of the present subject matter include anomaly identification systems for addressing anomalies in large networks. Various applications of the present subject matter provide guidance or effective placement of sensors in the electrical network.

TECHNICAL FIELD

The present disclosure relates to methods and systems of modeling, analyzing, monitoring, and detecting electrical networks.

BACKGROUND

The monitoring of electrical networks is important to provide safe and effective operation of such networks. Problems can arise in networks that are not monitored or are poorly monitored. An example is the annual loss to electricity theft happening in India which is about $16 B, ⅓ of total generated power, and about $6 B in the US, where electricity is the third most stolen item.

Small networks can be monitored and controlled using a small number of sensors, and placement of the sensors in a small system is typically straightforward. However, large networks can be a challenge to monitor because, depending on the number of nodes of the system, it may be impractical to monitor every node. Even if every node could be monitored, the amount of time it would take to take measurements at each node, store them, and analyze them would create a vast amount of data and processing would be prohibitive in very large networks. In large networks where sensors are limited, the placement of such sensors is difficult to determine. The difficulties are compounded by the fact that networks can involve bidirectional information, nodes are added and subtracted over time, nodes can experience failures and other anomalies, and communications over the network and its sensors are jeopardized when equipment servicing a node fails. Network operators need to know about anomalies and other events occurring in large networks which impact network performance.

Therefore, there is a need in the art for monitoring large networks. Such monitoring should provide an indication of an anomaly, and preferably provide a way to isolate the portions of the network experiencing the anomaly. There is a further need in the art for various anomaly handling systems for large networks. There is also a further need for guidance about effective placement of sensors in the network.

SUMMARY

The present subject matter enables early or real-time detection of anomalies in electric networks. In various embodiments, the system detects anomalies, such as electricity theft, electricity surge. It solves the difficult-to-detect problems in an electrical network, where anomalies like electricity theft or electrical surge may not be found until it has raised numerous concerns or complaints, or has created a significant impact on infrastructure functionality, service quality, or cost. In addition, the present subject matter decreases the requirement for large number of sensors and provides more cost effective and scalable solutions while removing the hassle of working with huge and unnecessary data.

In various embodiments, the present subject matter provides a method for determining where a detected anomaly is occurring within the electric network. In various embodiments, the present subject matter includes anomaly handling systems for addressing anomalies in large networks. In various embodiments, the present subject matter provides guidance or effective placement of sensors in the electrical network.

In various embodiments, the present subject matter provides sensor data and network structural information and modeling with artificial intelligence to detect and localize the source of anomaly and predict the impact on the rest of the network.

Various aspects and advantages of exemplary embodiments of the disclosure have been summarized. The above Summary is not intended to describe each illustrated embodiment or every implementation of the present certain exemplary embodiments of the present disclosure. The Drawings and the Detailed Description that follow more particularly exemplify certain preferred embodiments using the principles disclosed herein.

In the drawings, like reference numerals indicate like elements. While the above-identified drawing, which may not be drawn to scale, sets forth various embodiments of the present disclosure, other embodiments are also contemplated, as noted in the Detailed Description. In all cases, this disclosure describes the presently disclosed disclosure by way of representation of exemplary embodiments and not by express limitations. It should be understood that numerous other modifications and embodiments can be devised by those skilled in the art, which fall within the scope and spirit of this disclosure.

DETAILED DESCRIPTION

The present disclosure provides methods and systems to enable early or real-time detection of problems (e.g., electricity theft, power surge, equipment failures, etc.) in an electrical network, and enable preemptive actions including predicting when and where an issue/problem in an electrical network may be likely to occur. Electrical networks described herein may include a power distribution network, such as for a state or municipality, industrial plant, office complex, or other large electrical power network, etc. Methods and systems described herein can solve the problem of difficult-to-detect problems in an electrical network, where anomalies like electricity theft or electrical surge may not be found until it has raised numerous concerns or complaints, or has created a significant impact on infrastructure functionality, service quality, or cost.

The present subject matter, in various embodiments, detects and/or localizes an anomaly (e.g., electricity theft) in an electrical network with limited sensor measurements before and after an anomaly has occurred. For example, in various embodiments the present subject matter can estimate an amount of stolen power and its location given that it occurs within the sensor coverage area. Additionally, various embodiments of the present system can estimate the voltage/current level of other non-observing nodes of the network to identify the impacted locations or the locations at high risk. In various embodiments, the present subject matter can identify the coverage of sensors for anomaly localization as well as sensor placement, meaning the minimal number of sensors and their best or nearly best location to fulfill an intended network goal.

In various embodiments, the present subject matter provides sensor data and network structural information and modeling with artificial intelligence to detect and localize the source of anomaly and predict the impact on the rest of the network.

FIG. 1illustrates a flow diagram of a method100of monitoring anomaly of interest in an electrical network. At110, one or more sensors are provided for the electrical network. The sensors can be any types of sensors capable of collecting data related to one or more parameters of the electrical network such as, for example, a current sensor, a voltage sensor, a frequency sensor, a line detection sensor, an equipment failure or fault sensor, etc. The sensors can be located at various nodes and in various locations of the electrical network. For example, sensors can be provided at the location of every home in a power distribution network, at every building in a power distribution network at every transformer in a power distribution network, at every power distribution center in a power distribution network, etc. The present disclosure provides methods on how to determine sensor placement in the electrical network and methods on how to determine sensor coverage in the electrical network based on modeling the electrical network, which will be described further below. The method100then proceeds to120.

At120, the sensors are instructed by a processor to collect the data. The processor can be located in a remote computer (e.g., server or the Cloud) out of the electrical network. In some embodiments, sensors may be provided for current or voltage use at nodes that are distributed in the electrical network, and data collected by the sensor may indicate status or trending of current or voltage of the respective nodes. In some embodiments, each sensor can be instructed to collect data at an initial time, which can be stored in a database as initial values. The method100then proceeds to130.

At130, the collected data are transferred from the sensors and received by the processor. The data can be transmitted directly or indirectly via suitable technologies such as, for example, Wi-Fi, Bluetooth, Bluetooth Low Energy (BLE), cellular, Ethernet, etc. The collected data can be transferred in real time from the sensors, or be transferred at a later date to provide a retrospective indication of the electrical network. Data can also be collected at regular time intervals or an adapted time schedule based on a contextual situation. For example, data can be collected or transferred at a relatively higher frequency after a storm that may have damaged the electrical network, or when equipment at nodes are predicted to be near end-of-life. The method100then proceeds to140.

At140, the received data are analyzed by the processor to generate results. In some embodiments, the received data can be analyzed based on a model of the electrical network, such as that provided by network information170. In some embodiments, a model of the electrical network can include directionally connected nodes representing electrical infrastructure disposed in the electrical network. The nodes can be ordered as a partially order set, where order of nodes may change depending on directions of electrical flow between nodes. For example, electrical flow direction may change where there are fluctuations in “usage” at some of the nodes. The model can be representation(s) of the electrical network including, for example, a directed acrylic graph (DAG). The representations of the electrical network (e.g., a directed acrylic graph) can be stored or processed by the processor as a matrix data structure such as, for example, an adjacency matrix, a reachability matrix, etc.

In some embodiments, historical data or other data related to the electrical network can be combined with the received real-time data from the sensors and analyzed by the processor. Such other types of data may include, for example, previous issues/problems in the electrical network (e.g., voltage or current thresholds exceeded, faults or failures detected, etc.), weather (thunderstorms, tornadoes, flooding, etc., that might be the cause of a damage on the electrical network in certain areas), current flow or conductivity variations, and reported damage (e.g., fallen tree, electrical lines, brownouts and their correlation with power network damage). In some embodiments, baseline measurements can be conducted to determine whether there are any initial power (voltage or current or intermittent connections, etc.) issues during installation of the electrical network infrastructure. In some embodiments, historical data from the sensors related to the electrical network can be analyzed in terms of time, geography, etc., to derive an anomalous pattern. The data can be stored in, for example, a database associated with the processor or in a Cloud. The generated results including, for example, current or voltage or power draw trends, analysis reports, alerts, alarms, etc. The method100then proceeds to150.

At150, based on the analysis of the data, it is determined, via the processor or an operator/user, whether there is anomaly of interest in the electrical network. Possible anomalies of interest may include, for example, voltage or current drops or spikes, exceptionally high power draws, shorts or other events causing breakers to trip and transformers to fail or become compromised, or combinations thereof, etc. In some embodiments, the processor may further determine possible locations of the anomaly inside the electrical network. When there is an anomaly in the electrical network, the method100proceeds to160to generate an output in the form of, for example, an alarm, an alert, a report, etc. When there is no anomaly in the electrical network, the method100proceeds back to120. In some embodiments, the output may include prediction of future electrical network problems/issues such as, for example, power delivery failures based on similar environments by classifying the electrical network based on trends, size, usage, etc. In some embodiments, the output may include whether someone is stealing resources from the electrical network via a noticeable difference in network characteristics (e.g., a new “node” in the model being detected). In some embodiments, the output may be used to provide prioritization of power sourcing, shutdown, and/or diversion based on the infrastructure it is serving. For example, if a high power draw is detected for air conditioning etc., power delivery may be prioritized to a hospital over other infrastructure. In some embodiments, the output may be provided to homeowners by comparing their historical usage to their neighbors (e.g., to determine whether an anomalous current or voltage or power draw is localized to the house).

FIG. 2illustrates a detection system200for determining an anomaly of interest in an electrical network10by implementing, for example, the method100, according to one embodiment. The electrical network10can include a power distribution network such as, for example, a state or city power company, a power plant for a business, hospital, or school, a specialized power system such as for an office complex or office tower, a construction or manufacturing site, etc. One or more sensors12are provided for various locations inside the electrical network10.

The detection system200includes the sensors12, a computation component226, and one or more input/output devices216. The sensors12can be any types of sensors capable of collecting data related to one or more parameters of the electrical network such as, for example, a current sensor, a voltage sensor, a power sensor, a line fault or failure sensor, a frequency sensor, a consumption sensor, etc. Exemplary sensors may include passive, wireless sensors. The sensors12may include a radio-frequency identification (RFID), which can identify individual electrical infrastructure (e.g., an air conditioning system at a node) and its related information (e.g., size, model, usage, time of installation, status, etc.).

In some embodiments, the electrical network10may be an electrical network provided with its own power conditioning equipment, breakers, safety equipment, etc. The sensors12can be provided for the equipment of the network.

In the embodiment ofFIG. 2, the computation component226includes a processor212and a memory214. The computation component226is functionally connected to the sensors12, receives signals or data related to the electrical network10from the sensors12, and analyzes the received signals/data to generate results including, for example, analysis reports, alerts, alarms, etc. In some embodiments, the data received from the sensors12can be stored in the memory214. In some embodiments, a model can be created to represent the electrical network10. The model may include directionally connected nodes representing infrastructure of the electrical network. The model can include, for example, a directed graph or a partially ordered set, which is stored in the memory214as data in the form of an adjacency matrix. The processor212can analyze the model by interpreting and executing instructions from a software program associated with the processor212.

The memory214stores information. In some embodiments, the memory214can store instructions for performing the methods or processes described herein. In some embodiments, data related to the electrical network, or the model of the electrical network can be pre-stored in the memory214.

The memory214may include any volatile or non-volatile storage elements. Examples may include random access memory (RAM) such as synchronous dynamic random access memory (SDRAM), read-only memory (ROM), non-volatile random access memory (NVRAM), electrically erasable programmable read-only memory (EEPROM), and FLASH memory. Examples may also include hard-disk, magnetic tape, a magnetic or optical data storage media, a compact disk (CD), a digital versatile disk (DVD), a Blu-ray disk, and a holographic data storage media. Data may also be stored in a Cloud computing environment. It is understood that combinations of the foregoing memory214may be employed as well.

The processor212may include, for example, one or more general-purpose microprocessors, specially designed processors, application specific integrated circuits (ASIC), field programmable gate arrays (FPGA), a collection of discrete logic, and/or any type of processing device capable of executing the techniques described herein. In some embodiments, the processor212(or any other processors described herein) may be described as a computing device. In some embodiments, the memory214may be configured to store program instructions (e.g., software instructions) that are executed by the processor212to carry out the processes or methods described herein. In other embodiments, the processes or methods described herein may be executed by specifically programmed circuitry of the processor212. In some embodiments, the processor212may thus be configured to execute the techniques for analyzing data related to an electrical network described herein. The processor212(or any other processors described herein) may include one or more processors.

Input/output device216may include one or more devices configured to input or output information from or to a user or other device. In some embodiments, the input/output device216may present a user interface218where a user may control the assessment of an electrical network. For example, user interface218may include a display screen for presenting visual information to a user. In some embodiments, the display screen includes a touch sensitive display. In some embodiments, a user interface218may include one or more different types of devices for presenting information to a user. The user interface218may include, for example, any number of visual (e.g., display devices, lights, etc.), audible (e.g., one or more speakers), and/or tactile (e.g., keyboards, touch screens, or mice) feedback devices. In some embodiments, the input/output devices216may represent one or more of a display screen (e.g., a liquid crystal display or light emitting diode display) and/or a printer (e.g., a printing device or component for outputting instructions to a printing device). In some embodiments, the input/output device116may be configured to accept or receive program instructions (e.g., software instructions) that are executed by the processor112to carry out the embodiments described herein.

The detection system200may also include other components and the functions of any of the illustrated components including the processor212, the memory214, and the input/output devices216may be distributed across multiple components and separate devices such as, for example, computers. The system200may be configured as a workstation, desktop computing device, notebook computer, tablet computer, mobile computing device, or any other suitable computing device or collection of computing devices. The system200may operate on a local network or be hosted in a Cloud computing environment. The illustrated components ofFIG. 2are shown merely to explain various aspects of the present disclosure and the addition or removal of components would be apparent to one of skill in the art.

The detection system200allows a user to determine anomalies in an electrical network in real time. In some embodiments, the collected data from an electrical network can be automatically analyzed in real time, via the processor, based on a model of the electrical network to generate results for output. The detection system200further allows a user to predict possible issues/problems that an electrical network may have in the future.

FIG. 3Aillustrates a diagram of an electrical network10as an exemplary electrical network, according to one embodiment. Power is supplied from power supply2. Various electrical device infrastructures3are connected by power lines4. Sensors are provided at selected locations A through G in the electrical network10. The sensors can be functionally connected to a processor of a detection system such as the system200ofFIG. 2, and configured to collect data at the various locations in the network10, which can then be transmitted to the processor of the detection system to analyze. Based on the sensor data received from the nodes at locations A-G, the processor of the detection system can analyze the data and determine that the nodes at locations A-C, F and G have “good” power condition, while the nodes at locations D and E have “bad” power condition in the illustrated example. The processor can further determine, based on a model of the network10, that the possible cause of anomaly (e.g., short or open circuit) may be located at node31in the network10between the locations B and D.

FIG. 3Billustrates a diagram of an electrical network10′ as another exemplary electrical network. Power is supplied from a power supply2′ to the power line network10′. Various electrical device infrastructures3′ are connected by power lines4′. The various electrical device infrastructures3′ can be represented by nodes that are directionally connected where arrows correspond to current or power flow directions in the electrical network. As an example, electrical device infrastructure located at nodes A, B and C are directionally connected as shown in the left inset, where node C is a dependency-connection between node A and node B, and nodes A and B are therefore not dependency (d)-separated Node C is defined as a d-separator in the path from node A to node B.

In the present disclosure, models are created to represent various electrical networks. A model of electrical network can include directionally connected nodes representing the electrical device infrastructure disposed in the electrical network. Properties of current or power flow such as, e.g., flow directions, flow rate, within the electrical network can be measured by flow sensors or derived using Kirchoff's Current Law (KCL), Kirchoff's Voltage Law (KVL), power conservation principles, or based on factors such as, for example, trended power, voltage or current data, equipment utilization, network expansion or contraction, special network demands (e.g., extremely hot weather and commensurate power draws for air conditioning, etc.), system failures (open or short circuits, such as due to damage or breaker operation or shunting of power or current, etc.), specialized metering or measurement at a node or location, etc. In some embodiments, a model can be representation(s) of the electrical network including, for example, a directed acrylic graph (DAG). The representations of the electrical network (e.g., a directed acrylic graph) can be stored or processed by a processor as a matrix data structure matrix such as, for example, an adjacency matrix, a reachability matrix, etc.

FIG. 4Aillustrates a model20of an electrical network, according to one embodiment. In the model20, various electrical device infrastructure are represented by nodes1through17. For example, node1may represent a power supply. Nodes1-17are directionally connected to form a directed graph, which is a partially ordered set. The arrows inFIG. 4Acorrespond to power or current flow directions in the electrical network. It is to be understood that an electrical network can be represented by various directed graphs. A directed graph of an electrical network can further be represented by a matrix data structure such as, for example, an adjacency matrix, reachability matrix, etc. The matrix data structure can be stored and/or analyzed by a processor.

The present disclosure provides methods to analyze, via a processor such as processor212ofFIG. 2, a model of electrical network. In some embodiments, the model or representations of the model can be analyzed to evaluate whether a node or a set of nodes satisfies one or more localizability criteria. In some embodiments, the localizability criteria may include, for example, for a given node or node set, evaluating whether there are at least two sensors disposed downstream of the given node or node set which have the respective paths not sharing any d-separator with respect to the given node or node set. When a given node satisfies the localizability criteria, no sensor is provided to the given node. When the given node does not satisfy the localizability criteria, one or more sensors are provided to the given node. Instructions associated with the processor can be interpreted and executed to analyze the model or representations of the model. The above process of analyzing may include dynamic programming including a bottom-up approach starting from the lowest level in the representations of the model, as further explained by one embodiment as shown inFIG. 4B.

FIG. 4Bis a flow diagram of a method300for determining sensor placement in an electrical network, according to one embodiment. At310, an electrical network is represented by a model. The model can be, for example, a directed graph such as the directed graph20shown inFIG. 4A. The directed graph can be a partial order set where nodes are ordered in different levels. The method300then proceeds to320. At320, one or more nodes or node sets at the lowest level of the model are evaluated to determine at330whether the node or node set satisfies localizability criteria, e.g., whether there are at least two sensors besides the given node or node set which have the respective paths not sharing any d-separator with respect to the given node or node set. When the node or node set satisfies the localization criteria, the method300proceeds to340. When the node or node set does not satisfy the localizability criteria, the method300proceeds to350. At340, sensor(s) is added to the node or node set, and the model (e.g., the graph) is updated with the added sensor(s). At350, next mode or mode set at the same level or an upper level is evaluated in the same manner. The process continues until the node(s) at the upper most level of the electrical network is evaluated.

Application of the method300ofFIG. 4Bto the electrical network represented by the model20ofFIG. 4Aproduces results that are shown inFIGS. 4C-D. The evaluation starts from the lowest level, e.g., node16or17in the directed graph20. It is found that nodes16and17do not satisfy the localizability criteria. There are no sensors downstream of node16or17. Sensors are added at nodes16and17, respectively, and the directed graph20is updated with the added sensors at nodes16and17. Then, node14or15at the next level can be evaluated. It is found that nodes14and15satisfy the localizability criteria. There are two sensors downstream at nodes16and17, and the respective paths from nodes14and15to the two sensors at nodes16and17do not sharing any d-separator. No sensors will be provided to nodes14and15. This bottom-up approach continues and each node can be evaluated. It is found that nodes11-13,8,9, and1-7satisfy the criteria and no sensors will be provided to these nodes. Node10does not satisfy the localizability criteria. The respective paths from node10to the two sensors at nodes16and17, e.g.,10-13-14-16and10-13-15-17, can share a d-separator (e.g., node13). A sensor is provided to node10and the directed graph20is updated.

With sensors being placed at nodes16,17and10, it is sufficient to cover the whole electrical network. That is, by analyzing the data from the sensors located at nodes10,16and17, a processor of a detection system can explicitly determine the status at each node in the electrical network.FIG. 4Dillustrates a reformatted version of the directed graph ofFIG. 4Cprovided with sensors at selected locations. In some embodiments, when there is an anomaly in the electrical network, the processor can explicitly determine the location of anomaly by analyzing the sensor data. For example, when sensors data indicate that nodes11,14and16have a fault or detected anomaly and the remaining nodes have no detected fault or anomaly, the processor can determine that the cause of fault or anomaly is located at node11.

When an electrical network requires n sensors to completely cover the electrical network, a complexity index of the network can be expressed as the ratio of the required number n of sensors and the number of nodes in the electrical network. For example, the complexity index of the electrical network20ofFIG. 4Ais3/17.

It is to be understood that in some embodiments, one or more nodes of a model can be grouped into respective node sets before evaluation. For example, nodes14and15ofFIG. 4Acan be grouped into a single node set which can be evaluated. Each node set can include one or more adjacent nodes, and each node set can be evaluated, in the same manner as shown inFIGS. 4B-C, to determine whether the node sets or a combination of nodes and node sets satisfy the localizability criteria.

FIG. 5Aillustrates a model30of an electrical network, according to another embodiment. The electrical network is represented by a directed graph. Node1may represent a power supply. The model30can be analyzed in a manner similar as shown inFIGS. 4A-Dfor the model20, as discussed above. The model30includes74nodes among which thirty-two nodes do not satisfy the localizability criteria and sensors are provided for the thirty-two nodes (e.g., circled nodes inFIG. 5A) to completely cover the whole electrical network30. The complexity index of the electrical network30ofFIG. 5Ais32/74.FIG. 5Billustrates a reformatted version of the directed graph ofFIG. 5Aprovided with sensors at selected locations.

In some embodiments, the nodes of the model30can be first grouped into respective node sets before evaluation.FIG. 5Cshows that some adjacent nodes are grouped into the same node set32(e.g., within the same circle). In this manner, the number of required sensors to completely cover the whole electrical network can decrease from 32 to 15, as compared toFIG. 5B. The tradeoff is that when one specific node set is determined to be the cause of an anomaly, it may not tell the explicit location (e.g., which node) of anomaly within the specific node set.

In some embodiments, an electrical network can be modified to reduce the number of sensors required for full coverage of the electrical network. As shown inFIG. 5D, one or more connections34(e.g., power lines) can be added to directionally connect electrical device infrastructure represented by nodes. When the connections34are added inFIG. 5D, the number of required sensors to completely cover the whole electrical network can decrease from 32 to 11, as compared toFIG. 5B.

The present disclosure further provides methods of determining sensor coverage in an electrical network. The methods can include building a model of the electrical network. The model can include directionally connected nodes representing electrical device infrastructure disposed in the electrical network. One or more sensors can be positioned at one or more selected locations in the electrical network. The model or representations of the model can be stored in, for example, a processor. The model or its representation can be analyzed, via the processor, to evaluate whether each node satisfies one or more localizability criteria. The localizability criteria may include, for example, for a given node, evaluating whether there are at least two sensors other than the given node which have the respective paths not sharing any d-separator with respect to the given node.

Based on results of the evaluation, the nodes of the model can be assigned to one of a localization area, a detection area, and an out-of-reach area. A localization area refers to an area in an electrical network where the location of anomaly (e.g., at a specific node) can be explicitly determined. A detection area refers to an area in an electrical network where data/signal related to an anomaly may be detected, but the exact location of the anomaly is unknown. An out-of-reach area refers to an area in the electrical network where no information related to the anomaly can be obtained. When a given node satisfies the localizability criteria, the given node is assigned to the localization area, and when the given node does not satisfy the localizability criteria, the given node is assigned to the detection area or the out-of-reach area.

When the given node does not satisfy the localizability criteria, the given node can be further evaluated to determine whether a sensor is located at or away from the given node. If there are no sensors, the area corresponding to the given node is assigned to the out-of-reach area, otherwise it is assigned to the detection area.

In some embodiments, evaluation of paths between the given node and other nodes can be performed by using suitable algorithms regarding reachability of such as an advanced Markov chain method. An exemplary Markov chain method was described in Golnari et al., “Pivotality of Nodes in Reachability Problems Using Avoidance and Transit Hitting Time Metrics,” 7th Annual Workshop on Simplifying Complex Networks for Practitioners SIMPLEX 2015, May. 2015, which is incorporated herein by reference in its entirety. It is to be understood the evaluation of paths can be performed by any other suitable algorithms.

In some embodiments, the sensitivity of sensors (e.g., sensitivity on measuring anomalous electrical parameters) and an absorption (hitting) probability matrix Q can be analyzed to determine a minimum detectable change in power, current, or voltage level for at least some of the nodes. The absorption (hitting) probability matrix Q of an electrical network will be described further below.

FIG. 6Ais a flow diagram of a method400for determining sensor coverage in an electrical network, according to one embodiment. At410, an electrical network is represented by a model. The model can be, for example, a directed graph such as the directed graph20shown inFIG. 4A, or for bidirectional flow it can be graphs such as those shown inFIG. 14andFIG. 16. The method400then proceeds to420. At420, one or more nodes of the model are evaluated with respect to the sensor nodes to determine at430whether the node or node set satisfies localizability criteria, e.g., whether there are at least two sensors other than the given node or node set which have the respective paths not sharing any d-separator with respect to the given node or node set.

When the node or node set satisfies the localization criteria, the method400proceeds to440. When the node or node set does not satisfy the localizability criteria, the method400proceeds to450. At440, the node or node set is assigned to a localization area. The method400then proceeds to480.

At450, the node or node set is further evaluated to determine there are detectable sensor(s) at the node or node set or at other node(s). When there is a sensor located at or downstream of the node or node set, the method400proceeds to460. At460, the node or node set is assigned to a detection area. When there are no sensors located at or downstream of the node or node set, the method400proceeds to470. At470, the node or node set is assigned to an out-of-reach area. The method400then proceeds to480.

At480, next mode or mode set at the same level or an upper level in the model is evaluated in the same manner. The process continues until the node(s) at the upper most level of the electrical network is evaluated.

By applying the method400ofFIG. 6Ato an electrical network represented by the model20ofFIG. 4A, the results are shown inFIG. 6B. As shown inFIG. 6B, sensors are provided to nodes10and16, respectively. The evaluation starts from the lowest level, e.g., node16or17in the directed graph20. It is found that node16do not satisfy the localizability criteria. There are no sensors downstream of node16. Node16is further evaluated to determine whether there is a sensor at or downstream of node16. There is a sensor located at node16. Node16is assigned to a detection area. Nodes17is evaluated in the same manner. It is found that node17do not satisfy the localizability criteria. There are no sensors downstream of node17. Node17is further evaluated to determine whether there is a sensor at or downstream of node17. There is no sensor located at or downstream of node17. Node17is assigned to an out-of-reach area. Then, node14or15at the next level can be evaluated. It is found that nodes14and15do not satisfy the localizability criteria. There is only one sensor downstream at node16. Nodes14and15are assigned to a detection area, respectively. This bottom-up approach continues and each node can be evaluated. It is found that nodes1,3and6each satisfy the localizability criteria and assigned to a localization area. For example, for node3, one path from node3to the sensor at node16through3-5-9-12-14-16and one path from node3to another sensor at node10through3-6-10do not share any d-separator, and node3satisfies the localizability criteria. Other nodes are assigned to a detection area. With sensors being placed at nodes10and16, the electrical network20is partially covered, and divided into the localization area, the detection area, and the out-of-reach area. By analyzing the data from the sensors located at node10and16, a processor of a detection system can (i) explicitly determine the location of anomaly if the anomaly source is within the localization area, and (ii) detect the occurrence of an anomaly if the anomaly source is within the detection area.

FIG. 6Cillustrates a diagram of a model40where the nodes are assigned to a localization area61, a detection area62, and an out-of-reach area63, respectively, according to one embodiment. The model40ofFIG. 6Aare the same as the model30ofFIG. 5Bexcept that the number of sensors and their locations are different. Sensors are provided for the electrical network inFIG. 5Bsuch that the whole electrical network is completely covered, while inFIG. 6C, the electrical network is partially covered and the nodes are assigned to one of a localization area, a detection area, and an out-of-reach area.

FIG. 6Dillustrates a diagram of a model50where the nodes are assigned to a localization area61, a detection area62, and an out-of-reach area63, respectively, according to another embodiment. The model50ofFIG. 6Bis the same as the model40ofFIG. 6Cexcept that the number of sensors and their locations are different. InFIG. 6D, the electrical network is partially covered and the nodes are assigned to one of a localization area, a detection area, and an out-of-reach area. As compared toFIG. 6C, when the number and/or location of sensors change, the assignment of the nodes to different areas is changed accordingly.

The present disclosure further provides methods of detecting and localizing an anomaly of interest in an electrical network. The methods can determine the level of the anomaly and its location (e.g., at which node of the electrical network) when the anomaly occurs within a localization area of the electrical network. The methods can further detect the occurrence/existence of anomalies at other nodes (e.g., nodes in a detection area of the electrical network) and identify the locations at high risk within the electrical network.

The methods may include, for example, providing one or more sensors disposed at one or more selected locations in the electrical network. The sensors are configured to collect data from the electrical network at the respective locations. Data can be collected, via the one or more sensors, from the electrical network at the one or more locations on which the one or more sensors are disposed. The collected data can include sensor measurements before and after an anomaly has occurred. The data can then be received from the sensors, and analyzed, via the processor, based on a model of the electrical network. The model can include directionally connected nodes. In some embodiments, one or more imaginary nodes can be added between two adjacent nodes to generate a harmonic function of absorbing states based on the model.

In the case of electrical networks, Kirchhoff Current Law (KCL) and Kirchhoff Voltage Law are fundamental techniques for solving the current or voltage values at a node. KCL provides that the net sum of current for each node is zero (current is a signed (positive or negative) quantity reflecting direction towards or away from a node): ΣI=0. KVL provides that the net sum of voltages over a loop is zero: ΣV=0. One way to model it is to model an illegitimate load as an unknown resistance added to a node. Its value and location are unknown.

To demonstrate the application of electricity theft, voltage may be used as a metric to model a harmonic function in a network modeled as a Markov chain. In such network, states represent the nodes and transitions occur through network links. The power source and the ground are modeled as absorbing states. A transition probability matrix of states is computed from adjacency matrix (consisting of link resistors and node connection):
P=D−1Aj

The fundamental matrix computation and hitting probability derivation is as follows:

Matrix P is partitioned to four parts:

Where I represents interior and B represents boundary. Note that PBIis a matrix of all zeros and PBBis an identity matrix. Fundamental matrix and hitting probabilities are computed from P:
N=(I−PII)−1, Q=NPIB

A detection algorithm compares the model-based data (hitting probability Q) against the real sensor data (sensing voltage) and if the difference is larger than some threshold, an anomaly is identified.

The localization algorithm (back propagation and estimation method) provides that the absorption (hitting) probability by the new absorbing node s2can be found from the fundamental matrix belonging to the case when only node s1was absorbing:

There is no need to computing a new matrix inverse for any candidate of new absorbing node.

The superposition law (a modified version) provides that the value of node i, i.e. Ciafter, when node s1and s2are both boundary nodes with values Cs1and Cs2, is equal to its value when only node s1was the boundary node plus to its value when both s1and s2are boundary nodes but with values 0 and Cs2after−Cs2beforerespectively.

The following formulation is derived based on the superposition law and the lemma:
Csensorafter=Csensorbefore+(Csourceafter−Csourcebefore)p(sensor,source),

Where

Only one matrix inverse for the whole process is required

(N{ground_).
Csourceafteris found from the Least Squares Method.

Cascade prediction can be accomplished by (1) once the source of anomaly and its value is estimated, (2) it is added to the Markov model as a new absorbing state and the whole computations in the fundamental matrix computation and hitting probability derivation (above) are repeated to estimate the voltage level of the other non-observing nodes (to identify the locations in high impact or huge voltage drop).

For cases that have some extra effects in the network that no metric is harmonic, an imaginary node is added to network to model the extra effect and make the formulations harmonic (but which is not needed for an electricity theft example as demonstrated herein).

FIG. 7illustrates a method of modeling an electrical network by adding imaginary or decay nodes, according to one embodiment. The model of the simplified electrical network having an electrical directional aspect includes nodes1and2directionally connected to node3. Imaginary nodes4and5are added between the nodes1and3, and the nodes2and3, respectively. Without the imaginary nodes, the electrical directional aspect can be expressed by:
C3=(q13d13C1+q23d23C2)/(q13+q23)   (1a)

where Ci is the electrical directional aspect at node i, qijis an electrical flow rate from node i to j, dijis an electrical directional aspect decay parameter from node i to j.

The above equation (1a) is not harmonic, and there may be no-existing, not unique, or inefficient solutions.

By adding the imaginary nodes (e.g., nodes0,4and5), the electrical directional aspect levels (e.g., the level of power loss due to theft) at the real nodes (e.g., nodes1-3) of the electrical network can be calculated as one or more harmonic functions. The power loss decay, d, is set to zero at imaginary node0. Then, for example, C4may be represented as:
C4=(C1d13+C0(1−d13))/d13+(1−d13)   (1b)

Which can then reduce to the harmonic function:
C4=C1d13(1c)

This formulation may then be derived for the remaining nodes to gain a harmonic diffusion model. Given this harmonic diffusion model, and following Dirichlet's principle, the anomaly level of all real nodes may be found when the level of the anomaly (e.g., power loss level) at the sources (originating node and any nodes with a sensor) is known.

The power loss level at the imaginary or decay nodes can be set to be about zero. In some embodiments, when the location and amount of a power loss are determined, the power loss levels at other nodes (e.g., nodes in a detection area of the electrical network) of the electrical network can be calculated by:
Cinterior=QCboundary(2)

where Cinterioris the vector of power loss levels of all nodes except for the site of the power loss, Cboundaryis the vector of power loss level at the power loss site, and Q is an absorption (hitting) probability matrix which can be determined by a transition probability matrix. The transition probability matrix can be determined from an adjacency matrix constructed from weights, representative of power flow and power loss decay in the electrical network. The transition probability matrix is further partitioned to represent interior and boundary sources. The absorption probability matrix, Q, can be solved for a specific electrical network by one skilled in the art.

In some embodiments, a model of electrical network with imaginary nodes can be used to determine the location of one or more power loss sources based on data collected by sensors at some nodes, which can be further used to estimate power loss levels at other nodes and/or identify locations of failure in high risk.

In some embodiments, for a relatively simple electrical network, the location of a source of power loss can be determined by individually considering each node as a boundary node with an assumed anomaly level as potential source (of the loss), and comparing these modeled values to the sensor measurements. For relatively complex electrical networks, this method may be difficult to implement due to the complexity of the network.

In some embodiments, the location of a power loss can be determined by placing at least two sensors at downstream locations of the electrical network, analyzing data from the sensors before and after an anomalous event, and modeling the initial, non-anomalous state of a node. An absorption probability matrix and the level at the anomaly source can then be obtained using a fitting estimation method, such as, a Least-Squares Method, which is known in the art.

In some embodiments, data from the sensors can be analyzed to compute the level of power loss at nodes within a detection area of the electrical network. The methods of assigning nodes to a detection area has been discussed above. See, for example,FIG. 6B. The power loss levels at the nodes can be computed by determining an absorption probability matrix Q. The data can be further analyzed to localize one or more potential power loss sources based on the computed level of power loss at the nodes. Based on the information of the potential power loss sources, power loss levels at other nodes in the electrical network can be further estimated.

FIGS. 8A-Cillustrate how to determine the location of power loss sources with varying amounts of sensors in an electrical network. InFIG. 8A, one sensor is located at node82and the power loss level is measured at the node82. Using the model and power loss localization method discuss above, the nodes of the electrical network can be respectively assigned to a localization area, a detection area, and an out-of-reach area. The power loss levels for the nodes within the detection area can be calculated to determine potential power loss source(s). In this case, ten nodes (e.g., nodes with a square inFIG. 8A) are identified as potential anomalous sources, and are labeled with their respective power loss levels. InFIG. 8B, two sensors are located at nodes83aand83b,respectively. Similarly, the power loss levels for the nodes within the detection area can be obtained. In this case, four nodes have been identified as a potential sources of power loss, and are labeled with their respective power loss levels. InFIG. 8C, three sensors are located at nodes84a,84band84c,respectively. In this case, the location of source of power loss is determined to be at node88. InFIGS. 8A-B, a power loss associated with a subset of localized nodes can be determined based on the calculated power loss levels and sensor(s) locations, while inFIG. 8C, the described method can exactly localize the node with the source of the power loss.

FIG. 9Aillustrates a physical model of a multiple node electrical network to demonstrate the subject matter of the present disclosure. In the physical model power distribution is measured over the three sensor locations. Electricity theft is modeled as connecting an illegitimate load to one of the nodes in the network. The present anomaly detection approach can be used to localize the site of the illegitimate load (theft) using the three sensors using a digital twin model.FIG. 9Billustrates a digital twin model of the multiple node electrical network ofFIG. 9A. The resulting model of the system shows which nodes the sensors are located at and where the illegitimate load is located using the algorithm according to this example of one embodiment of the present subject matter.FIG. 10is a block diagram of the inputs and outputs of the digital twin model ofFIG. 9B, according to an embodiment. System information, such as node-to-node connectivity (e.g., an adjacency matrix), and source node identification (e.g., power, VCC) and voltage information are some possible inputs for voltage measurement applications. The sensor data is input with the system information to the digital twin which executes software to perform network analytics. The resulting network may produce, in various embodiments and combinations, one or more of: detection of an anomaly, localization of a detected anomaly, and/or anomaly cascade prediction. Other variations of detections are possible without departing from the scope of the present subject matter.

FIG. 11Ais a flow diagram of the system for anomaly detection of an electrical network using statistical modeling, according to an embodiment. Processing system600receives network structure information610and flow characteristic information620to generate a system model614and perform statistical modeling616and matrix computation617. An example of flow characteristic information for fundamental matrix computation includes Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law for solving the current or voltage values at a node:a. KCL: the net sum of current for each node is zero (current is a signed (positive or negative) quantity reflecting direction towards or away from a node): ΣI=0.b. KVL: the net sum of voltages over a loop is zero: ΣV=0.

In such cases, an illegitimate load is modeled as an unknown resistance added to a node. Its value and the location is unknown.

The resulting matrix computations are modified using model-based sensor data622and evaluated using sensor data630by evaluator function module624. The result provides an indication of detection of an anomaly636(e.g., electricity theft), according to an embodiment. The system600can detect an anomaly in the electrical network given measurements from a limited number of sensors before and after an anomaly has occurred. For example, in various embodiments the method estimates that power is being stolen using estimates of the voltage/current level of other non-observing nodes of the network to identify the impacted locations or the locations at high risk. Electrical networks are typically modeled as undirected (bidirectional) networks.

FIG. 11Bis a flow diagram of a system for anomaly detection of an electrical network using Markov chain statistical modeling, according to an embodiment. InFIG. 11Bstatistical modeling module616is a Markov chain modeling module616. For the application of electricity theft, we pick voltage as the metric to model our harmonic function. The network is modeled as a Markov chain, where states represent the nodes and transitions are happen through network links. The power source and the ground are modeled as the absorbing states. The transition probability matrix of states is computed from adjacency matrix (consisting of link resistors and node connection):
P=D−1Aj.

Processing system600uses Markov chain modeling based on harmonic functions616and fundamental matrix computation618to perform a hitting probabilities derivation612and use model-based sensor data622to evaluate the system information (network structure610and flow characteristics620) in view of the sensor data630by evaluator module624to provide anomaly detection636.

FIG. 11Cis a flow diagram of a system for anomaly detection and anomaly localization of an electrical network using Markov chain statistical modeling, according to an embodiment. Various embodiments provide localization of the anomaly638by adding a back propagation module626to the system ofFIG. 11B. This provides localization of the detected anomalies.

FIG. 11Dis a flow diagram of a system for anomaly detection, anomaly localization, and anomaly cascade prediction of an electrical network using Markov chain statistical modeling, according to an embodiment. Various embodiments additional perform Markov chain modification634, fundamental matrix modification632and hitting probabilities modification628to provide anomaly cascade prediction642. Anomaly cascade prediction642provides a value estimation of metrics or variables for the other nodes of the electrical system after an anomaly cascade. This gives users of the electrical network further insight as to how the system is tolerating the anomaly on a node-by-node basis, and the number and localization of the nodes impacted by the anomaly.

FIG. 12is a flow diagram demonstrating one example of a data analysis of sensor data and network information in an electrical network, according to an embodiment. In process flow diagram700, assume that sensors are placed on an electrical grid and that digitized information709of the electrical network grid is input with digitized sensor information collected from sensors on the network grid701to operation node702. The expected value at a sensor location (Vs) is provided by solving system equations710and can be compared with the real value obtained from the sensor location (VSi) using comparison module703. It is understood that in various applications the expected versus real value measurements may be voltage, power, current, electron flow, etc. If the comparison does not exceed a threshold703, the next node is checked as flow returns to module702. If the comparison exceeds a threshold703then an anomaly detection is output by anomaly detection module704. In various embodiments, anomaly detection module calculates a probabilistic, binary confidence interval. In embodiments involving anomaly localization, the output of the anomaly detection module704triggers step705which checks each node “i” and value (anomaly size) ΔU(i)as the candidate for an anomaly. The solving system equation module706uses outputs from step705and provides ΔVs(i). The candidate with the smallest error is picked (i*=argmin|ΔVs−ΔVs(i)|) at step707, and anomaly localization is output at module708. Any changes to the network information are applied to the network information module709which is applied to solving system equations710. The anomaly cascade prediction module711predicts the impact of the anomaly on various affected nodes of the system and outputs them in embodiments featuring an anomaly cascade prediction output.

In various embodiments, the system ofFIG. 13is performed, for example, by the analysis step140ofFIG. 1. It is understood that the order of analysis and steps and modules involved may be performed in hardware, software, firmware, and in combinations thereof. Variations in order and performed functions may vary without departing from the scope and spirit of the present subject matter. In various embodiments, the algorithms and software operating on the systems ofFIG. 10,FIGS. 11A-11D, andFIG. 13may be performed by the system ofFIG. 2, such as processor212and memory214of computation component226. In various embodiments, distributed processing and storage, such as cloud embodiments, may be employed with the system ofFIG. 2, or in combination. Outputs are provided by the I/O device216or user interface218, or on other connected apparatus, depending on the particular implementation and application.

FIG. 13is a block diagram showing a data analysis system800having sensor data802and network information801inputs and various outputs804(anomaly detection),805(anomaly localization), and806(cascade prediction) using artificial intelligence and network theory803, according to an embodiment. In various embodiments, network information801includes network topology (e.g., an adjacency matrix A, as shown inFIG. 15, which is symmetric for bidirectional flow), flow model providing boundary (e.g., absorbing) node information, and providing flow characteristics (e.g., power start; ground end).

FIG. 14is a node diagram showing various nodes in an example of a network structure of an electrical network, according to an embodiment. Power (Vcc, node1) is interconnected to nodes2-6, and ground node7as shown. Various measurements between the nodes can be performed and represented in a table, such as the matrix inFIG. 15.FIG. 15is an adjacency matrix (matrix A is symmetric for bidirectional flow) showing measured resistance values of the links of the structure ofFIG. 14, according to an embodiment. In this matrix, all of the links have a resistance of either 100 Kohm or 1000 Kohm (if referenced to ground). (The zeros reflect empty entries.) In a voltage measurement example, the boundary nodes are Vcc (node ID=1), value=5 Volts; and Ground (node ID=7), value=0 Volts. The electrical network can be characterized in terms of current and voltage using Kirchoff's Current Law and Kirchoff's Voltage Law. The sensors are installed at node ID's=2 and 4, and the sensor reads voltage.

FIG. 16is a node diagram showing various node measurements for calculations of the network structure ofFIG. 14, according to an embodiment. To demonstrate the measurement algorithm, according to one embodiment, first, Absorbing Markov Chain modeling with transition probability matrix P, where P=D−1A and D=diag (sum(A, 2)) is performed.

Next, it is proved that the voltage at node I is equal to the absorption probability by Vcc node multiplied by the Vcc voltage, starting from node “i”. (See for example, Snell & Doyle, “Random Walk and Electrical Networks,” https://arxiv.org/abs/math/0001057, January 2000, and https://math.dartmouth.edu/˜doyle/docs/walks/walks.pdf, Jul. 5, 2006.). A fundamental matrix (F) is calculated from which the absorption probabilities are computed: F=(I−Pa)−1, where Pais the matrix resulted from removing absorbing nodes' rows and columns (e.g., the 1stand 7throws and columns in this example).

The present subject matter can be applied to a variety of different applications including, but not limited to electrical networks, power distribution systems, fiber optic networks, battery fuel cells.

Exemplary embodiments of the present disclosure may take on various modifications and alterations without departing from the spirit and scope of the present disclosure. Accordingly, it is to be understood that the embodiments of the present disclosure are not to be limited to the following described exemplary embodiments, but is to be controlled by the limitations set forth in the claims and any equivalents thereof.

Listing of Exemplary Embodiments

Exemplary embodiments are listed below. It is to be understood that any of embodiments in lists I and II can be combined.

Embodiment List I

Embodiment 1 is a system for detecting anomalies in a bidirectional electrical network with a plurality of sensors distributed throughout nodes of the electrical network, comprising:

a network processor, comprising:

a network topology storage having a network input for receipt of network node information, including a description of electrical node requirements of a plurality of nodes of the network, including node-to-node connectivity;

a sensor data storage including a sensor data input for receipt of sensor measurements from the plurality of sensors and information associating each measurement with a sensor node location within the network;

an electrical system model storage including relationships between the sensor measurements and electrical node requirements at each node of the electrical network;

a network matrix calculator configured to receive the network node information and electrical node requirements, and to generate a statistical expected measurement range at each sensor node location; and

a comparison module configured to compare the sensor measurements with the statistical expected measurement range for each sensor node location of the electrical network, and to generate an anomaly detection for each node where the sensor measurements fall outside the statistical expected measurement range.

Embodiment 2 is the system of embodiment 1, further comprising an anomaly localization module configured to identify a node or nodes associated in the network associated with an anomaly detection.

Embodiment 3 is the system of embodiment 2, further comprising an anomaly cascade prediction calculator configured to recalculate statistical expected measurement ranges for each node of the electrical network based on an anomaly detection.

Embodiment 4 is the system of any one of embodiments 1-3, wherein the network matrix calculator includes a Markov chain modeling module to predict network node measurements based on harmonic functions.

Embodiment 5 is the system of any one of embodiments 1-4, wherein the anomaly cascade prediction calculator includes a Markov chain modification module to recalculate network node measurements based on a detected anomaly.

Embodiment 6 is the system of any one of embodiments 1-5, wherein the electrical system model storage relationships are based on Kirchoff's Current Law (KCL).

Embodiment 7 is the system of any one of embodiments 1-5, wherein the electrical system model storage relationships are based on Kirchoff's Voltage Law (KVL).

Embodiment 8 is the system of embodiment 6, wherein the anomaly cascade prediction calculator is used to identify non-monitored nodes which will experience high current drops due to the detected anomaly.

Embodiment 9 is the system of embodiment 7, wherein the anomaly cascade prediction calculator is used to identify non-monitored nodes which will experience high voltage drops due to the detected anomaly.

Embodiment List II

Embodiment 10 is a method for detecting anomalies in an electrical network using a plurality of sensors distributed throughout nodes of the electrical network, the method comprising:

receiving network node information, including receiving a description of electrical node requirements of a plurality of nodes of the network;

receiving sensor measurements from the plurality of sensors and information associating each measurement with a sensor node location within the network;

receiving an electrical system model including relationships between the sensor measurements and electrical node requirements at each node of the electrical network;

using the network node information and electrical node requirements to calculate a statistical expected measurement range at each sensor node location;

comparing the sensor measurements with the statistical expected measurement range for each sensor node location of the electrical network, and generating an anomaly detection for each node where the sensor measurements exceeded the statistical expected measurement range.

Embodiment 11 is the method of embodiment 10, further comprising identifying a node or nodes associated in the network associated with an anomaly detection.

Embodiment 12 is the method of embodiment 11, further comprising recalculating statistical expected measurement ranges for each node of the electrical network based on an anomaly detection.

Embodiment 13 is the method of any one of embodiments 10-12, further comprising using Markov chain modeling to predict network node measurements based on harmonic functions.

Embodiment 14 is the method of any one of embodiments 10-13, further comprising using Markov chain modeling to recalculate network node measurements based on a detected anomaly.

Embodiment 15 is the method of any one of embodiments 10-14, wherein the electrical node requirements are based on Kirchoff s Current Law (KCL).

Embodiment 16 is the method of any one of embodiments 10-14, wherein the electrical node requirements are based on Kirchoff s Voltage Law (KVL).

Embodiment 17 is the system of embodiment 15, further comprising using Markov chain modeling to identify non-monitored nodes which will experience high current drops due to a detected anomaly.

Embodiment 18 is the system of embodiment 16, further comprising using Markov chain modeling to identify non-monitored nodes which will experience high voltage drops due to a detected anomaly.

While the specification has described in detail certain exemplary embodiments, it will be appreciated that those skilled in the art, upon attaining an understanding of the foregoing, may readily conceive of alterations to, variations of, and equivalents to these embodiments. Accordingly, it should be understood that this disclosure is not to be unduly limited to the illustrative embodiments set forth hereinabove. In particular, as used herein, the recitation of numerical ranges by endpoints is intended to include all numbers subsumed within that range (e.g., 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.80, 4, and 5). In addition, all numbers used herein are assumed to be modified by the term “about.”

Furthermore, various exemplary embodiments have been described. These and other embodiments are within the scope of the following claims.