Regularization techniques for improving efficiency of time domain protection

Abstract

One component of time domain protection in a power system is to estimate the location of a fault. In an embodiment, a multi-objective problem is proposed that includes a non-smoothness penalty function that drives a primary objective function for fault location estimation towards a solution that complies with smoothness between an input and an output of a machine learning model. Under the background of time domain protection and for other regression tasks, the technology improves the accuracy, blind area and speed of related technologies. In additional or alternative embodiments specific to time domain protection, the multi-objective problem may include a phasor deviation penalty function that drives a primary objective function toward a solution that minimizes deviation in phasor values. The trained machine learning model may be executed in the line protection system to determine whether to trip a circuit breaker of the power line.

Description

Background of the Invention Technical Field

[0002] Embodiments described herein relate generally to machine learning, and more particularly to an improved technique for training a machine learning model using heterogeneous data with high dimensionality to estimate fault locations, for example, for time domain protection. Background Art

[0003] Protection applications in power systems can be designed to isolate faulted sections of the power system in order to limit physical damage to the infrastructure of the power system. For example, time domain protection can protect overhead power lines by detecting faults within a protected area (e.g., up to a set range) and tripping a circuit breaker as quickly as possible once a fault is detected within the protected area. Detection of faults is typically performed using only local voltage and current measurements.

[0004] An integral part of time domain protection is estimating the location of the fault. The estimation can be performed as a sequence-to-sequence transformation of each sample. In particular, each measurement sample within the time series, including incoming voltage and current measurements, is processed as input to estimate the fault location for each sample. A decision on whether to trip the associated circuit breaker can then be made based on the estimated fault location.

[0005] This decision must be performed with a high degree of safety. This means that all faults should be isolated with high reliability, which requires insensitivity to faults or other events outside the protected area. An example of such an event is a fault at a busbar or at any other power line close to the protected power line. The final decision on whether to trip the relevant circuit breaker can be based on multiple factors such as fault detection, fault direction, arrival calculations, and safety counters.

[0006] The main challenge of estimating the fault location is that the estimation of the fault location must be highly accurate and fast. The method of the related art is to estimate the equivalent system equations for each sample by: solving a set of differential equations; estimating the instantaneous impedance of the power line, which contains information about the fault location; and then comparing the estimated instantaneous impedance with the relevant margin. The present disclosure is directed to overcoming one or more problems discovered by the inventors. Summary of the invention

[0007] Thus, disclosed are systems, methods, and non-transitory computer-readable media for training a machine learning model using heterogeneous data having high dimensionality, for example, to estimate fault locations for time domain protection.

[0008] In an embodiment, a method for performing manifold regularization while training a machine learning model includes using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set in the multiple feature sets includes feature values ​​for each of a plurality of features and is labeled with a target value for each of one or more targets, wherein each feature set in the multiple feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line; generating an optimization problem including an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error generated by the training data set in the optimization problem.

[0009] In a further embodiment, a method for performing manifold regularization while training a machine learning model includes using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set of the multiple feature sets includes feature values ​​for each of a plurality of features and is labeled with a target value for each of one or more targets; generating an optimization problem including an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0010] The non-smoothness penalty function may include a Laplace norm of the output value.

[0011] The non-smoothness penalty function can approximately calculate the gradient of the machine learning model on the data manifold of the machine learning model.

[0012] Training the machine learning model may include: clustering the training data set into a plurality of clusters, wherein the non-smoothness penalty function associates a weight with each of the plurality of clusters; and adaptively learning the weights of the plurality of clusters. Adaptively learning the weights of the plurality of clusters may include, for each of a plurality of external iterations: determining the weights of the plurality of clusters; and for each of a plurality of internal iterations, adjusting the machine learning model using the determined weights of the plurality of clusters in the non-smoothness penalty function to minimize the estimation error generated by the training data set in the optimization problem. The weights of the plurality of clusters may be determined in each of the plurality of external iterations using a multi-class classification algorithm. The machine learning model may be adjusted in each of the plurality of internal iterations using a regression algorithm.

[0013] The method may further include, before acquiring the training data set, acquiring data having a first dimension using the at least one hardware processor, and embedding the data in a lower dimensional space having a second dimensionality lower than the first dimensionality, wherein the training data set is acquired from the data in the lower dimensional space.

[0014] Each of the plurality of feature sets may represent a set of measurement signals of a power line, wherein each target value represents a fault location on the power line. The method may further include, by a protection device: receiving measurement signals from a plurality of sensors connected to the power line; applying a trained machine learning model to the measurement signals to estimate a fault location on the power line; comparing the estimated fault location with a threshold; and when the estimated fault location meets the threshold, tripping at least one circuit breaker to electrically isolate the fault location on the power line.

[0015] Each of the plurality of feature sets may represent a set of measured signals of a power line, wherein each target value represents a decision to trip or not trip a circuit breaker on the power line.

[0016] Each of the plurality of feature sets may represent a state of an electric power system, wherein each target value represents a value of a continuous variable of the electric power system, and wherein the method further comprises estimating the value of the continuous variable using a trained machine learning model.

[0017] The method may further include using the trained machine learning model for one of topology estimation, parameter estimation, power flow estimation, or load forecasting.

[0018] Each feature set in at least one subset of the plurality of feature sets in the training data set may be labeled with a reference true phasor value, wherein the optimization problem further comprises a phasor deviation penalty function configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in the at least one subset of the plurality of feature sets in the training data set increases. The at least one subset of the plurality of feature sets in the training data set may consist of only a portion of the plurality of feature sets in the training data set, the portion comprising a time series covering at least a predefined time length for each of the plurality of features. The encoder network may be an artificial neural network.

[0019] In an embodiment, a method for performing manifold regularization while training a machine learning model includes using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set in the multiple feature sets includes a feature value for each feature in a plurality of features and is labeled with a target value for each of one or more targets, wherein each feature set in the multiple feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line, and wherein each feature set in at least one subset of the multiple feature sets is labeled with a reference true phasor value; generating an optimization problem including an objective function and a phasor deviation penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the phasor deviation penalty function is configured to increase the estimated error in the optimization problem as the difference between the estimated phasor value output by an encoder network and the reference true phasor value for each feature set in the at least one subset of the multiple feature sets in the training data set increases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error generated by the training data set in the optimization problem.

[0020] In a further embodiment, a method for performing manifold regularization while training a machine learning model includes using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set of the multiple feature sets includes feature values ​​for each of the multiple features and is labeled with target values ​​for each of one or more targets, and wherein each feature set in at least one subset of the multiple feature sets is labeled with a reference true phasor value; generating an optimization problem including an objective function and a phasor deviation penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the phasor deviation penalty function is configured to increase the estimated error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value for each feature set in the at least one subset of the multiple feature sets in the training data set increases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0021] The at least one subset of the plurality of feature sets in the training data set may consist of only a portion of the plurality of feature sets in the training data set, the portion comprising, for each feature of the plurality of features, a time series covering at least a predefined time length.

[0022] The following aspects are preferred embodiments of the present disclosure.

[0023] 1. A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set of the multiple feature sets includes feature values ​​for each feature of a plurality of features and is labeled with a target value for each of one or more targets; generating an optimization problem including an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0024] 2. The method of aspect 1, wherein the non-smoothness penalty function comprises the Laplace norm of the output value.

[0025] 3. A method as described in aspect 1, wherein the non-smoothness penalty function approximately calculates the gradient of the machine learning model on the data manifold of the machine learning model.

[0026] 4. A method as described in aspect 1, wherein training the machine learning model includes: clustering the training data set into multiple clusters, wherein the non-smoothness penalty function associates a weight with each of the multiple clusters; and adaptively learning the weights of the multiple clusters.

[0027] 5. A method as described in aspect 4, wherein adaptively learning the weights of the multiple clusters includes, for each external iteration in multiple external iterations: determining the weights of the multiple clusters; and for each internal iteration in multiple internal iterations, using the determined weights of the multiple clusters in the non-smoothness penalty function to adjust the machine learning model to minimize the estimation error generated by the training data set in the optimization problem.

[0028] 6. The method of clause 5, wherein the weights of the plurality of clusters are determined in each of the plurality of outer iterations using a multi-class classification algorithm.

[0029] 7. A method as described in aspect 6, wherein the machine learning model is adjusted in each of the multiple internal iterations using a regression algorithm.

[0030] 8. The method as described in aspect 1 further includes, before obtaining the training data set, using the at least one hardware processor to obtain data having a first dimension, and embedding the data in a lower dimensional space having a second dimension lower than the first dimension, wherein the training data set is obtained from the data in the lower dimensional space.

[0031] 9. The method of aspect 1, wherein each of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line.

[0032] 10. The method as described in aspect 9 further includes, by a protection device: receiving measurement signals from multiple sensors connected to the power line; applying a trained machine learning model to the measurement signals to estimate a fault location on the power line; comparing the estimated fault location with a threshold; and when the estimated fault location meets the threshold, tripping at least one circuit breaker to electrically isolate the fault location on the power line.

[0033] 11. The method of aspect 1, wherein each of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a decision to trip or not trip a circuit breaker on the power line.

[0034] 12. A method as described in aspect 1, wherein each of the multiple feature sets represents a state of an electric power system, wherein each target value represents a value of a continuous variable of the electric power system, and wherein the method further includes using a trained machine learning model to estimate the value of the continuous variable.

[0035] 13. The method of aspect 1 further comprises using the trained machine learning model for one of topology estimation, parameter estimation, power flow estimation or load forecasting.

[0036] 14. A method as described in aspect 1, wherein each feature set in at least one subset of the multiple feature sets in the training data set is labeled with a reference true phasor value, wherein the optimization problem further includes a phasor deviation penalty function, which is configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in at least one subset of the multiple feature sets in the training data set increases.

[0037] 15. A method as described in aspect 14, wherein at least one subset of the multiple feature sets in the training data set is composed of only a portion of the multiple feature sets in the training data set, and the portion includes a time series covering at least a predefined time length for each feature of the multiple features.

[0038] 16. The method of clause 14, wherein the encoder network is an artificial neural network.

[0039] 17. A system comprising: at least one hardware processor; and software, the software being configured to: when executed by the at least one hardware processor, obtain a training data set comprising multiple feature sets, wherein each feature set of the multiple feature sets comprises feature values ​​for each feature of a plurality of features and is labeled with a target value for each of one or more targets; generate an optimization problem comprising an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and train the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0040] 18. A system as described in aspect 17, wherein each feature set in at least one subset of the multiple feature sets in the training data set is labeled with a reference true phasor value, wherein the optimization problem further includes a phasor deviation penalty function, which is configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in at least one subset of the multiple feature sets in the training data set increases.

[0041] 19. A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set in the multiple feature sets includes feature values ​​for each feature in a plurality of features and is labeled with target values ​​for each target in one or more targets, and wherein each feature set in at least one subset of the multiple feature sets is labeled with a reference true phasor value; generating an optimization problem including an objective function and a phasor deviation penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the phasor deviation penalty function is configured to increase the estimated error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value for each feature set in the at least one subset of the multiple feature sets in the training data set increases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0042] 20. A method as described in aspect 19, wherein the at least one subset of the multiple feature sets in the training data set is composed of only a portion of the multiple feature sets in the training data set, and the portion includes a time series covering at least a predefined time length for each feature of the multiple features.

[0043] It should be understood that any feature of the above method can be implemented alone or in any combination with any subset of the other features. Therefore, even if the attached claims indicate that there are specific dependencies between the features, the disclosed embodiments are not limited to these specific dependencies. On the contrary, any feature described herein can be combined with any other feature described herein, or implemented in any combination of any features without any one or more other features described herein. In addition, any method described above and elsewhere herein can be embodied in an executable software module of a processor-based system (such as a server) and / or in executable instructions stored in a non-transitory computer-readable medium, either alone or in any combination. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The details of the invention, both as to structure and operation, may be gleaned in part by examination of the accompanying drawings, in which like reference numerals refer to like parts, and in which:

[0045] Figure 1 illustrates an example infrastructure in which one or more processes described herein may be implemented according to an embodiment;

[0046] Figure 2 illustrates an example processing system that may be used to perform one or more of the processes described herein according to an embodiment;

[0047] Figure 3 illustrates the operation of a line protection system according to an embodiment;

[0048] Figure 4 illustrates a process for training a machine learning model according to an embodiment;

[0049] Figure 5 illustrates an example of a training sub-process utilizing a two-layer optimization according to an embodiment;

[0050] Fig. 6A and Figure 6B illustrates an encoder-decoder architecture according to an alternative embodiment;

[0051] Fig. 7A and Figure 7B The blind spots of the machine learning model trained with Laplace regularization and the blind spots of the machine learning model not trained with Laplace regularization according to the embodiment are respectively illustrated; and

[0052] Figure 8 A process for making a tripping decision according to an embodiment is illustrated. DETAILED DESCRIPTION

[0053] In an embodiment, a system, method, and non-transitory computer-readable medium for training a machine learning model using heterogeneous data with high dimensionality to, for example, estimate a fault location for time domain protection are disclosed. After reading this specification, it will be clear to those skilled in the art how to implement the present invention in various alternative embodiments and alternative applications. However, although various embodiments of the present invention will be described herein, it should be understood that these embodiments are presented only by way of example and illustration and not by way of limitation. Therefore, this detailed description of various embodiments should not be interpreted as limiting the scope or breadth of the present invention as set forth in the appended claims.

[0054] 1. System Overview

[0055] 1.1. Infrastructure

[0056] Figure 1 An example infrastructure in which one or more disclosed processes may be implemented according to an embodiment is illustrated. The infrastructure may include a platform 110 (e.g., one or more servers) that hosts and / or performs one or more of the various functions, processes, methods, and / or software modules described herein. The platform 110 may include a dedicated server, or alternatively may be implemented in a computing cloud, where the resources of one or more servers are dynamically and elastically allocated to multiple tenants based on demand. In either case, the servers may be collocated and / or geographically distributed. The platform 110 may also include or be communicatively connected to a server application 112 and / or one or more databases 114. In addition, the platform 110 may be communicatively connected to one or more user systems 130 via one or more networks 120. The platform 110 may also be communicatively connected to one or more external systems 140 (e.g., other platforms, websites, etc.) via one or more networks 120.

[0057] The network(s) 120 may include the Internet, and the platform 110 may communicate with the user system(s) 130 over the Internet using standard transfer protocols, such as Hypertext Transfer Protocol (HTTP), HTTP Secure (HTTPS), File Transfer Protocol (FTP), FTP Secure (FTPS), Secure Shell FTP (SFTP), etc., as well as proprietary protocols. Although the platform 110 is illustrated as being connected to various systems via a single set of network(s) 120, it should be understood that the platform 110 may be connected to various systems via a different set of one or more networks. For example, the platform 110 may be connected to a subset of the user systems 130 and / or external systems 140 via the Internet, but may also be connected to one or more other user systems 130 and / or external systems 140 via an intranet. Furthermore, although only a few user systems 130 and external systems 140, one server application 112, and a set of (multiple) databases 114 are illustrated, it should be understood that the infrastructure may include any number of user systems, external systems, server applications, and databases.

[0058] The user system(s) 130 may include any one or more types of computing devices capable of wired and / or wireless communication, including but not limited to desktop computers, laptop computers, tablet computers, smartphones or other mobile phones, servers, game consoles, televisions, set-top boxes, electronic self-service terminals, and / or point-of-sale terminals, etc. However, it is generally contemplated that the user system 130 will be a personal computer or workstation of an administrator responsible for training a machine learning model and / or deploying a trained machine learning model (e.g., to one or more line protection systems included in the external system 140). Each user system 130 may include or be communicatively connected to a client application 132 and / or one or more local databases 134.

[0059] The platform 110 may include a web server that hosts one or more websites and / or web services. In an embodiment where a website is provided, the website may include a graphical user interface, including, for example, one or more screens (e.g., web pages) generated in hypertext markup language (HTML) or other languages. The platform 110 transmits or provides one or more screens of a graphical user interface in response to a request from (multiple) user systems 130. In some embodiments, these screens may be provided in the form of a wizard, in which case two or more screens may be provided in a sequential manner, and one or more sequential screens may depend on the interaction of the user or user system 130 with one or more previous screens. Requests to the platform 110 and responses from the platform 110 (including screens of a graphical user interface) may be transmitted through (multiple) networks 120 using standard communication protocols (e.g., HTTP, HTTPS, etc.), which may include the Internet. These screens (e.g., web pages) may include a combination of content and elements such as text, images, videos, animations, references (e.g., hyperlinks), frames, inputs (e.g., text boxes, text areas, check boxes, radio buttons, drop-down menus, buttons, tables, etc.), scripts (e.g., JavaScript), etc., including elements composed of or derived from data stored in one or more databases (e.g., database(s) 114) accessible locally and / or remotely to the platform 110. It should be understood that the platform 110 may also respond to other requests from the user system(s) 130.

[0060] The platform 110 may include, be communicatively coupled to, or otherwise access one or more databases 114. For example, the platform 110 may include one or more database servers that manage the one or more databases 114. Server applications 112 executing on the platform 110 and / or client applications 132 executing on user systems 130 may submit data (e.g., user data, table data, etc.) to be stored in (multiple) databases 114, and / or request access to data stored in (multiple) databases 114. Any suitable database may be utilized, including, but not limited to, MySQL. TM 、Oracle TM 、IBM TM , Microsoft SQL TM 、Access TM , PostgreSQL TM 、MongoDB TMetc., including cloud-based databases and proprietary databases. For example, data can be sent to the platform 110 using the well-known POST request supported by HTTP, via FTP, etc. The data and other requests can be processed, for example, by server-side network technologies executed by the platform 110, such as server applet or other software module (e.g., included in the server application 112).

[0061] In an embodiment providing a network service, the platform 110 may receive requests from (multiple) external systems 140 and provide responses in extensible markup language (XML), JavaScript object notation (JSON), and / or any other suitable or desired format. In such an embodiment, the platform 110 may provide an application programming interface (API) that defines the manner in which (multiple) user systems 130 and / or (multiple) external systems 140 may interact with the network service. Thus, (multiple) user systems 130 and / or (multiple) external systems 140 (which may themselves be servers) may define their own user interfaces and rely on the network service to implement or otherwise provide the backend processes, methods, functions, and / or storage described herein, etc. For example, in such an embodiment, a client application 132 executing on one or more user systems 130 may interact with a server application 112 executing on the platform 110 to perform one or more or a portion of one or more of the various functions, processes, methods, and / or software modules described herein. In an embodiment, the client application 132 may utilize a local database 134 to store data locally on the user system 130 .

[0062] The client applications 132 may be "thin," in which case the processing is primarily performed on the server side by the server applications 112 on the platform 110. A basic example of a thin client application 132 is a browser application that simply requests, receives, and renders web pages at the user system(s) 130, while the server applications 112 on the platform 110 are responsible for generating the web pages and managing database functions. Alternatively, the client applications may be "thick," in which case the processing is primarily performed on the client side by the user system(s) 130. It should be understood that the client applications 132 may perform the amount of processing relative to the server applications 112 on the platform 110 at any point along this spectrum between "thin" and "thick," depending on the design goals of a particular implementation. In any case, the software described herein, which may reside entirely on the platform 110 (e.g., in which case the server application 112 performs all processing) or on (multiple) user systems 130 (e.g., in which case the client application 132 performs all processing) or distributed between the platform 110 and (multiple) user systems 130 (e.g., in which case both the server application 112 and the client application 132 perform processing), may include one or more executable software modules that include instructions for implementing one or more processes, methods, or functions described herein.

[0063] 1.2. Example Processing Equipment

[0064] Figure 2 is a block diagram illustrating an example wired or wireless system 200 that may be used in conjunction with the various embodiments described herein. For example, the system 200 may be used as or in conjunction with one or more functions, processes, or methods described herein (e.g., for storing and / or executing software), and may represent components of the platform 110, the user system(s) 130, the external system(s) 140, the line protection system, and / or other processing devices described herein. The system 200 may be a server or any conventional personal computer, or any other processor-enabled device capable of wired or wireless data communications. It will be apparent to those skilled in the art that other computer systems and / or architectures may also be used.

[0065] The system 200 preferably includes one or more processors 210. The processor(s) 210 may include a central processing unit (CPU). Additional processors may be provided, such as a graphics processing unit (GPU), an auxiliary processor to manage input / output, an auxiliary processor to perform floating point math operations, a special microprocessor with an architecture suitable for fast execution of signal processing algorithms (e.g., a digital signal processor), a slave processor subordinate to a main processing system (e.g., a back-end processor), an additional microprocessor or controller for a dual-processor or multi-processor system, and / or a coprocessor. Such auxiliary processors may be discrete processors or may be integrated with the processor 210. Examples of processors that may be used with the system 200 include, but are not limited to, any processor provided by Intel Corporation of Santa Clara, California (e.g., a Pentium 400 processor ... TM , Core i7 TM , Xeon TM any processor provided by Advanced Micro Devices, Inc. (AMD) of Santa Clara, California, any processor provided by Apple Inc. of Cupertino (e.g., A-series, M-series, etc.), any processor provided by Samsung Electronics Co., Ltd. of Seoul, South Korea (e.g., Exynos TM ), any processor provided by NXP Semiconductors of Eindhoven, the Netherlands, etc.

[0066] The processor 210 is preferably connected to a communication bus 205. The communication bus 205 may include a data channel for facilitating information transfer between storage devices and other peripheral components of the system 200. In addition, the communication bus 205 may provide a set of signals for communicating with the processor 210, including a data bus, an address bus, and / or a control bus (not shown). The communication bus 205 may include any standard or non-standard bus architecture, such as a bus architecture that conforms to the Industry Standard Architecture (ISA), the Extended Industry Standard Architecture (EISA), the Micro Channel Architecture (MCA), the Peripheral Component Interconnect (PCI) local bus, bus architectures that conform to standards promulgated by the Institute of Electrical and Electronics Engineers (IEEE) including the IEEE 488 General Purpose Interface Bus (GPIB), IEEE 696 / S-100, and the like.

[0067] The system 200 preferably includes a main memory 215, and may also include a secondary memory 220. The main memory 215 provides instructions and data storage for programs executed on the processor 210 (such as any software discussed herein). It should be understood that the programs stored in the memory and executed by the processor 210 can be written and / or compiled according to any suitable language, including but not limited to C / C++, Java, JavaScript, Perl, Visual Basic, NET, etc. The main memory 215 is typically a semiconductor-based memory, such as a dynamic random access memory (DRAM) and / or a static random access memory (SRAM). Other semiconductor-based memory types include, for example, synchronous dynamic random access memory (SDRAM), Rambus dynamic random access memory (RDRAM), ferroelectric random access memory (FRAM), etc., including read-only memory (ROM).

[0068] The secondary memory 220 is a non-transitory computer-readable medium having computer executable code (e.g., any software disclosed herein) and / or other data stored thereon. The computer software or data stored on the secondary memory 220 is read into the main memory 215 for execution by the processor 210. The secondary memory 220 may include, for example, semiconductor-based memories such as programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable read-only memory (EEPROM), and flash memory (block-oriented memory similar to EEPROM).

[0069] The secondary storage 220 may optionally include internal media 225 and / or removable media 230. The removable media 230 is read and / or written in any well-known manner. The removable storage medium 230 may be, for example, a tape drive, a compact disk (CD) drive, a digital versatile disk (DVD) drive, other optical drives, a flash memory drive, etc.

[0070] In alternative embodiments, the secondary memory 220 may include other similar means for allowing computer programs or other data or instructions to be loaded into the system 200. Such means may include, for example, a communication interface 240 that allows software and data to be transferred to the system 200 from an external storage medium 245. Examples of external storage medium 245 include an external hard drive, an external optical drive, an external magneto-optical drive, and the like.

[0071] As mentioned above, the system 200 may include a communication interface 240. The communication interface 240 allows software and data to be transferred between the system 200 and an external device (e.g., a printer), a network, or other information source. For example, computer software or executable code may be transferred from a network server (e.g., platform 110) to the system 200 via the communication interface 240. Examples of the communication interface 240 include a built-in network adapter, a network interface card (NIC), a Personal Computer Memory Card International Association (PCMCIA) network card, a cardbus network adapter, a wireless network adapter, a universal serial bus (USB) network adapter, a modem, a wireless data card, a communication port, an infrared interface, an IEEE 1394 FireWire, and any other device capable of interfacing the system 200 with a network (e.g., network(s) 120) or another computing device. The communication interface 240 preferably implements industry-published protocol standards, such as Ethernet IEEE 802 standards, Fibre Channel, Digital Subscriber Line (DSL), Asynchronous Digital Subscriber Line (ADSL), Frame Relay, Asynchronous Transfer Mode (ATM), Integrated Services Digital Network (ISDN), Personal Communications Service (PCS), Transmission Control Protocol / Internet Protocol (TCP / IP), Serial Line Internet Protocol / Point-to-Point Protocol (SLIP / PPP), etc., but customized or non-standard interface protocols may also be implemented.

[0072] Software and data transmitted via communication interface 240 are typically in the form of electrical communication signals 255. These signals 255 may be provided to communication interface 240 via communication channel 250. In embodiments, communication channel 250 may be a wired or wireless network (e.g., network(s) 120), or any other communication link of any kind. Communication channel 250 carries signals 255 and may be implemented using a variety of wired or wireless communication means, including wire or cable, optical fiber, conventional telephone line, cellular telephone link, wireless data communication link, radio frequency ("RF") link, or infrared link, to name a few.

[0073] Computer executable code (e.g., a computer program, such as the disclosed software) is stored in the main memory 215 and / or the secondary memory 220. The computer executable code may also be received via the communication interface 240 and stored in the main memory 215 and / or the secondary memory 220. Such a computer program, when executed, enables the system 200 to perform various functions of the disclosed embodiments as described elsewhere herein.

[0074] In this specification, the term "computer-readable medium" is used to refer to any non-transitory computer-readable storage medium used to provide computer executable code and / or other data to or within the system 200. Examples of such media include main memory 215, auxiliary memory 220 (including internal memory 225 and / or removable media 230), external storage media 245, and any peripheral devices (including network information servers or other network devices) communicatively coupled to communication interface 240. These non-transitory computer-readable media are means for providing software and / or other data to system 200.

[0075] In embodiments implemented using software, the software may be stored on a computer-readable medium and loaded into the system 200 via the removable media 230, the I / O interface 235, or the communication interface 240. In such embodiments, the software is loaded into the system 200 in the form of an electrical communication signal 255. The software, when executed by the processor 210, preferably causes the processor 210 to perform one or more of the processes and functions described elsewhere herein.

[0076] In an embodiment, the I / O interface 235 provides an interface between one or more components of the system 200 and one or more input and / or output devices. Example input devices include, but are not limited to, sensors, keyboards, touch screens or other touch-sensitive devices, cameras, biometric sensing devices, computer mice, trackballs, pen-based pointing devices, and the like. Examples of output devices include, but are not limited to, other processing devices, cathode ray tubes (CRTs), plasma displays, light emitting diode (LED) displays, liquid crystal displays (LCDs), printers, vacuum fluorescent displays (VFDs), surface conduction electron emitter displays (SEDs), field emission displays (FEDs), and the like. In some cases, input and output devices may be combined, such as in the case of a touch panel display (e.g., in a smartphone, tablet computer, or other mobile device).

[0077] System 200 may also include an optional wireless communication component that facilitates wireless communication over a voice network and / or a data network (e.g., in the case of user system 130). The wireless communication component includes antenna system 270, radio system 265, and baseband system 260. In system 200, radio frequency (RF) signals are transmitted and received over the air by antenna system 270 under the management of radio system 265.

[0078] In an embodiment, the antenna system 270 may include one or more antennas and one or more multiplexers (not shown) that perform switching functions to provide transmit and receive signal paths for the antenna system 270. In the receive path, the received RF signal may be coupled from the multiplexer to a low noise amplifier (not shown) that amplifies the received RF signal and sends the amplified signal to the radio system 265.

[0079] In an alternative embodiment, the radio system 265 may include one or more radios configured to communicate on various frequencies. In an embodiment, the radio system 265 may combine a demodulator (not shown) and a modulator (not shown) in one integrated circuit (IC). The demodulator and modulator may also be separate components. In the incoming path, the demodulator removes the RF carrier signal, leaving a baseband receive audio signal that is sent from the radio system 265 to the baseband system 260.

[0080] If the received signal contains audio information, the baseband system 260 decodes the signal and converts it into an analog signal. The signal is then amplified and sent to a speaker. The baseband system 260 also receives analog audio signals from a microphone. These analog audio signals are converted into digital signals and encoded by the baseband system 260. The baseband system 260 also encodes the digital signals for transmission and generates a baseband transmit audio signal, which is routed to the modulator portion of the radio system 265. The modulator mixes the baseband transmit audio signal with an RF carrier signal to generate an RF transmit signal, which is routed to the antenna system 270 and can pass through a power amplifier (not shown). The power amplifier amplifies the RF transmit signal and routes it to the antenna system 270, where the signal is switched to the antenna port for transmission.

[0081] The baseband system 260 is also communicatively coupled to the processor(s) 210. The processor(s) 210 may access the data storage areas 215 and 220. The processor(s) 210 are preferably configured to execute instructions (i.e., computer programs, such as the disclosed software) which may be stored in the main memory 215 or the secondary memory 220. The computer programs may also be received from the baseband processor 260 and stored in the main memory 210 or the secondary memory 220, or executed upon receipt. Such computer programs, when executed, may enable the system 200 to perform the various functions of the disclosed embodiments.

[0082] 1.3. Line protection system

[0083] Figure 3The operation of the line protection system 330 according to an embodiment is illustrated. The power line 310 can be set between two substations 320A and 320B. One or more voltage measurement units 312 can measure the voltage on the power line 310 and output the voltage measurement results to the line protection system 330. In addition, one or more current measurement units 314 can measure the current on the power line 310 and output the current measurement results to the line protection system 330.

[0084] The line protection system 330 may include a controller 332 that may implement the system 200 or some subset thereof. For example, the controller 332 may include one or more processors 210 and store a machine learning (ML) model 334 in a memory (e.g., the main memory 215 and / or the auxiliary memory 220). The controller 332 may receive voltage measurements from the voltage measurement unit(s) 312, current measurements from the current measurement unit(s) 314, and / or other measurements from one or more other sensors or devices associated with the power line 310. The controller 332 may also derive one or more additional measurements from the received measurements.

[0085] The controller 332 may apply the machine learning model 334 to the measurement results as part of a time domain protection scheme. For example, the measurement results may be input to the machine learning model 334 to estimate the location of the fault 340 or output a decision on whether to trip (i.e., disconnect) the circuit breaker 316 on the power line 310. If the location of the fault 340 is within the protection zone of the line protection system 330 defined by the range setting 336, the controller 332 may determine to trip the circuit breaker 316 (i.e., switch the circuit breaker 316 to an open state). Otherwise, the controller 332 may determine not to trip the circuit breaker 316 (i.e., maintain the circuit breaker 316 in a closed state). When the controller 332 determines to trip the circuit breaker 316, the controller 332 may send a control signal to the circuit breaker 316 (e.g., directly or via a relay to control the circuit breaker 316) to disconnect the circuit breaker 316, and thereby isolate the fault 340 from the substation 320A. It should be understood that there may be a similar line protection system 330 on the other end of the power line 110, which can perform the same function to isolate the fault 340 from the substation 320B if the location of the fault 340 is within its corresponding protection zone.

[0086] 2. Process Overview

[0087] An embodiment of a process for training a machine learning model 334 using heterogeneous data with high dimensionality, for example, to estimate fault locations for time domain protection, will now be described in detail. It should be understood that the described process may be embodied in one or more software modules executed by one or more hardware processors (e.g., processor 210), for example, as a software application (e.g., server application 112, client application 132, and / or a distributed application including both server application 112 and client application 132), which may be executed entirely by the processor(s) of the platform 110, entirely by the processor(s) of the user system(s) 130, or may be distributed across the platform 110 and the user system(s) 130, such that some parts or modules of the software application are executed by the platform 110 and other parts or modules of the software application are executed by the user system(s) 130. The described process may be implemented as instructions represented in source code, object code, and / or machine code. These instructions may be executed directly by the hardware processor(s) 210, or alternatively, may be executed by a virtual machine running between the object code and the hardware processor(s) 210. Additionally, the disclosed software may be built upon or interfaced with one or more existing systems.

[0088] Alternatively, the described process can be implemented as a hardware component (e.g., a general purpose processor, an integrated circuit (IC), an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA) or other programmable logic device, a discrete gate or transistor logic, etc.), a combination of hardware components, or a combination of hardware components and software components. In order to clearly illustrate the interchangeability of hardware and software, various illustrative components, blocks, modules, circuits and steps are generally described herein according to their functions. Whether such functions are implemented as hardware or software depends on the specific application and the design constraints imposed on the entire system. The technician can use different ways to implement the described functions for each specific application, but such implementation decisions should not be interpreted as causing departure from the scope of the present invention. In addition, the functional grouping within the components, blocks, modules, circuits or steps is for ease of description. Without departing from the present invention, a specific function or step can be moved from one component, block, module, circuit or step to another.

[0089] In addition, although the processes described herein are illustrated by certain arrangements and sequencing of subprocesses, each process can be implemented by fewer, more, or different subprocesses and different arrangements and / or sequencing of subprocesses. In addition, it should be understood that any subprocess that is not dependent on the completion of another subprocess can be performed before, after, or in parallel with the other independent subprocess, even if the subprocesses are described or illustrated in a particular order.

[0090] 2.1. Heterogeneous data with high dimensionality

[0091] Regression problems in certain contexts (such as arrival computation for time domain protection) often use data with high dimensionality due to the large number of features that must be considered. For example, a feature may include many channels that include buffers of previous values. As a result, the data manifold (i.e., the space in which the data resides) may be quite complex, which makes it difficult to perform supervised learning (such as regression). Therefore, in an embodiment, information about the smoothness of the data manifold and the interconnectivity of the features in the data is added to the learning of the input-output mapping of the machine learning model 334.

[0092] In addition, such high-dimensional data is typically extremely heterogeneous. This is caused by varying simulation parameters, varying collection processes, different data sources, etc. The heterogeneity of the data may hinder the estimation of the fault location due to the limitations of the model in capturing the complexity of the information hidden in the data. In the context of time domain protection, this situation may occur when data is collected from several data sources with widely varying system parameters and signals (e.g., different towers or other sources in a power transmission system). Therefore, in an embodiment, the heterogeneous data is divided into relatively more homogeneous subgroups or "clusters" to facilitate the training of the machine learning model 334 (e.g., to reduce underfitting).

[0093] 2.2. Multi-objective optimization

[0094] In order to improve the machine learning model 334 (expressed as function f θ ) should minimize its estimation error. In addition, when the target of the machine learning model 334 is a continuous variable (for example, when the machine learning model 334 estimates the fault location), the function f θ It should be smooth on the input features. For example, in the context of time domain protection, the function f θ The estimated fault location (ie, target) should change smoothly with the change of the measurement sample (ie, input feature). Therefore, in an embodiment, the machine learning model 334 is trained by minimizing the following multi-objective optimization problem:

[0095]

[0096] Where X represents the input feature set in the training dataset, Y represents the benchmark true target set in the training dataset, and g θ It should be understood that the training data set may include feature X, which includes a plurality of feature sets {x1, x2, ...x n}. In other words, for each x i ={Feature 1, Feature 2, ...Featurem In addition, the training dataset includes a benchmark real target Y, which includes multiple target values ​​{y1, y2, ...y n}. Each feature set x in X i Use the corresponding reference ground truth target y in Y i It is worth noting that problem P is a multi-objective optimization problem because there are multiple objective functions ||Yf θ (X)|| 2 and g θ (X) is minimized.

[0097] Additionally or alternatively, the multi-objective optimization problem can be regularized to approximate the known equations that can be used to calculate f θ (X) The amount of prediction tasks performed. In particular, known voltage phasors and current phasors are available for time domain protection. However, in order to reliably calculate the phasors from a time series of voltage and current measurements, longer time series are required than are typically feasible at the speed at which a trip decision must be made. In an embodiment, to overcome this problem, the machine learning model 334 is trained by minimizing the following multi-objective optimization problem:

[0098]

[0099] Among them, h θ Denotes the phasor deviation penalty function. Similarly, problem P is a multi-objective optimization problem because there are multiple objective functions ||Yf θ (X)|| 2 and h θ (X) is minimized.

[0100] In an embodiment, the multi-objective optimization problem can be solved by θ and the phasor deviation penalty function h θ The combination of the two is as follows:

[0101]

[0102] Figure 4 A process 400 for training a machine learning model 334 according to an embodiment is illustrated. The process 400 can be implemented within a server application 112 on a platform 110, within a client application 132 on a user system 130, and / or distributed between the server application 112 and the client application 132.

[0103] In subprocess 410, a training data set is obtained. The training data set includes a plurality of feature sets X, each of which includes a feature value for each of a plurality of features and is labeled with a target value for each of one or more targets Y. In the context of time domain protection, each feature set in X may represent a set of measurement signals of the power line 310 (e.g., a voltage measurement result from the voltage measurement unit 312, a current measurement result from the current measurement unit 314, etc.). Each target in Y may represent a reference true fault location or a reference true trip decision on the power line 310.

[0104] In an embodiment, data having a first dimensionality is obtained and embedded in a lower dimensional space having a second dimensionality lower than the first dimensionality. A training data set is then obtained from the data in this lower dimensional space. In other words, the dimensionality of the data is reduced for the training data set. In this case, it should be understood that during operation of the machine learning model 334, the dimensionality of the input data can be similarly reduced.

[0105] In subprocess 420, the objective function (e.g., ||Yf θ (X)|| 2 ) generates an optimization problem, where f θ represents the machine learning model 334. In addition, the optimization problem is to use the non-smooth penalty function g θ Or the phasor deviation penalty function h θ One or both of , as discussed above.

[0106] In subprocess 430, the machine learning model 334 is trained by minimizing the error in the optimization problem generated in subprocess 420. It should be understood that the error is represented by an objective function that calculates the difference between the ground truth and the output of the machine learning model 334. In addition, a non-smoothness penalty function g is included in the optimization problem. θ In the case of θ The smoothness of decreases and increases. Including the phasor deviation penalty function h in the optimization problem θ In the case of , this error increases as the deviation from the reference true phasor value increases. Therefore, minimizing the error in the optimization problem involves minimizing f θ non-smoothness and / or minimize the deviation of phasor values.

[0107] Once the machine learning model 334 has been trained, the machine learning model 334 can be deployed. For example, the machine learning model 334 can be stored in a memory of a controller 332 of one or more line protection systems 330. The controller 332 can apply the machine learning model 334 to real-time inputs (e.g., measurement signals obtained from measurement units 312 and 314) to estimate the location of a fault on the power line 310, make a decision about whether to trip the circuit breaker 316, or otherwise inform a time domain protection scheme implemented by the controller 332.

[0108] 2.3. Non-smoothness penalty

[0109] Non-smoothness penalty function g θ Manifold regularization is provided by assuming that the data manifold M in which the data is embedded is inherently smooth. The non-smoothness penalty function g θ is configured to measure the function f θ The degree of non-smoothness relative to the data, where higher values ​​indicate greater non-smoothness and lower values ​​indicate greater smoothness. Therefore, the non-smoothness penalty function g θ Penalty function f θ It should be understood that the smoothness of a function is a property measured based on the number of continuous derivatives that the function has over a certain domain. Conceptually, smoothness means that small changes in the input of the machine learning model 334 do not produce large changes in the output of the machine learning model 334.

[0110] In an embodiment, the non-smoothness penalty function g θ Including function f θ However, the non-smoothness penalty function g θ The output can be as f θ Any function that increases as the non-smoothness in increases. For example, the non-smoothness penalty function g θ can alternatively be implemented as a Lipschitz continuity constraint.

[0111] In the non-smooth penalty function g θ is the function f θ An example of the Laplace norm is:

[0112]

[0113] in, represents the Laplacian of the data matrix represented in X. In other words, is the Laplacian matrix associated with the neighborhood graph computed on X. Given a simple graph G with n nodes, the Laplacian matrix L associated with the graph G is nxnis defined as L:=DA, where D is the degree matrix and A is the adjacency matrix of G. The expression is the Laplace norm of the prediction target (e.g., fault location):

[0114]

[0115] Non-smoothness penalty function g θ This formulation of can be viewed as a special case of the Lagrangian relaxation of the smoothness constrained regression problem, where the Lagrangian multiplier λ=1.

[0116] In this embodiment, the manifold regularization becomes Laplace regularization. The Laplace operator of the input features X captures the spectral composition of the data manifold M in which these input features X are embedded. When the input data is dense, the smoothing function changes slowly as the input data changes. Therefore, for the function f θ Penalizing the non-smoothness of θ The Laplace norm of . Minimize the function f θ The Laplace norm of (representing the prediction of the machine learning model) ensures that the predicted target (e.g., estimated fault location) obeys the correspondence (i.e., smoothness) with the input data (e.g., features obtained from measurement samples) and its spectral distribution.

[0117] When the distances between data points in a training dataset are interpreted as a graph, the Laplacian matrix of this graph can help estimate the marginal distribution of the data manifold M in which the data points are embedded and which is otherwise unknown. As the number of data points increases, the Laplacian matrix L converges to the Laplacian-Beltrami operator ΔM, which is the gradient Therefore, The gradient of the objective function f on the data manifold M is approximately calculated as given by:

[0118]

[0119] Among them, dP x (x) represents a small slice of the probability density of x.

[0120] 2.4. Clustering for improved robustness

[0121] In Laplace regularization, the entire data matrix is ​​used to compute the Laplace norm of the machine learning model’s predictions. While this forces the machine learning regression model to learn a smooth function f θ , but learning is based on the spectral distribution of the entire input data as a whole. For complex training datasets with significant heterogeneity, machine learning models will generally not be able to focus solely on different parts of the data manifold with different levels of smoothness. This may affect the learning function fθ Robustness, as machine learning models may be sensitive to the heterogeneity of input data.

[0122] In an embodiment, to account for the potential heterogeneity in the data manifold, the input data in the training dataset is first clustered instead of using the entire training dataset as a whole during training. In particular, the training dataset is clustered so that each cluster has an increased intra-cluster homogeneity, and then the weighted sum of the Laplace norms of all clusters is used as the non-smoothness penalty function g θ The weights on the Laplace norm of the clusters train the machine learning model 334 to place unequal focus on different isomorphic subgroups of the training dataset. Mathematically, the multi-objective optimization problem to be solved becomes:

[0123]

[0124] or

[0125]

[0126] Where K is the predefined number of clusters, X k is the input feature set belonging to the kth cluster, f θ (X k ) T Yes θ (X k ), and λ k represents the weight of the kth cluster, where ∑ k λ k = 1. Each weight λ k Can be predefined or learnable. In the case of static weights, the weights λ can be predefined using grid search during training, as in hyperparameter tuning.

[0127] Any suitable clustering technique may be used, including but not limited to K-means clustering, spectral clustering, information-based clustering, etc. In K-means clustering, the optimal number of clusters is found using the elbow method on cluster inertia. The elbow method involves plotting the change as a function of the number of clusters and taking the elbow of the curve as the number of clusters to use. The inertia of a cluster is the average over all clusters of the sum of the squared distances of the points in the cluster from the centroid of the cluster.

[0128] Spectral clustering uses the spectra (eigenvalues) of a special matrix (i.e., graph Laplacian matrix) generated from k-nearest neighbors or an affinity graph built from the data. While spectral clustering performs better at clustering data than K-means clustering (where all members of each cluster are not close to each other in the Euclidean sense), spectral clustering can also be much slower than K-means clustering in achieving convergence. This makes spectral clustering prohibitive for large data sets.

[0129] Given the data, information-based clustering maximizes the entropy of the target and minimizes the target. Formally, the training loss to be minimized is given by:

[0130]

[0131] Where B is the number of mini-batches, X i is the i-th mini-batch of the input dataset X, H represents entropy, and p clust is the clustering probability, and γ∈(0,1). In a specific embodiment, information-based clustering produces the best results. Therefore, in an embodiment, clustering is performed using information-based clustering.

[0132] 2.5. Two-layer training with adaptive weights

[0133] In an embodiment, the weight λ of each cluster is learned dynamically k , in order to adapt the overall training process of the machine learning model 334 using meta-learning. Meta-learning, or learning to learn, is the process of systematically observing how different machine learning methods perform on a wide range of learning tasks and then learning from this experience or metadata to learn new tasks faster than otherwise possible. In the context of multi-task learning, the meta-learning algorithm ensures that the performance of each task improves with experience and the number of tasks. In an embodiment, meta-learning is applied to the disclosed multi-objective optimization problem to improve the robustness of the machine learning model by dynamically learning the cluster weights λ during a two-layer training process. However, in alternative embodiments, predefined cluster weights λ may be used.

[0134] For learning a smooth objective function f θ For problem P (e.g., for estimating fault locations), the weight λ for each cluster k is k can be considered as hyperparameters of the machine learning model. Meta-learning can be applied to these hyperparameters to adaptively learn each λ k , where λ k =f ω (X k ).

[0135] To this end, a two-layer optimization loop is performed. In the inner loop, the base learner is updated as a regressor with parameters θ. In the outer loop, the meta-learner is updated as a K-class classifier with parameters ω. In this case, the multi-objective optimization problem with a non-smoothness penalty function becomes:

[0136]

[0137] in,

[0138]

[0139] Here, argmin refers to the independent variable of the minimum value. In an alternative embodiment, a joint optimization may be used instead of the bi-level optimization.

[0140] Figure 5 An example of a training subprocess 430 utilizing a two-layer optimization according to an embodiment is illustrated. Initially, in subprocess 431, the training data set is clustered as discussed above. Then, the two-layer training begins. In particular, subprocesses 432 and 435 represent an outer loop or meta-learner, while subprocesses 433 and 434 represent an inner loop or base learner. Meta-learning enhances the robustness of multi-objective learning so that the performance of the trained model does not vary widely with small changes in the input data set.

[0141] In subprocess 432, the weight λ of each cluster is determined k In an initial iteration of subprocess 432, the cluster weights λ may be initialized to an initial value. In each subsequent iteration of subprocess 432, the cluster weights λ may be updated. In this case, any suitable meta-learning algorithm may be used to determine new cluster weights λ to identify a potential set of cluster weights λ that may further minimize the optimization problem. Examples of suitable techniques include: model-agnostic meta-learning that encodes prior knowledge into trainable weight initializations, as described in Finn et al., "Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks", International Conference on Machine Learning (ICML) 2017, arXiv:1703.03400, which is incorporated herein by reference as if fully set forth; and meta-learning that uses random initialization to improve the weight update rule to produce better generalization to new tasks during training, as described in Baik et al., "Meta-Learning with Adaptive Hyperparameters", NeurIPS 2020, arXiv:2011.00209, which is incorporated herein by reference as if fully set forth.

[0142] In subprocess 433, the parameters of the machine learning model 334 are determined and the error of the optimization problem is calculated. The parameters of the machine learning model 334 may include the machine learning model 334 (i.e., f θ) and / or other variables in subprocess 432. It should be understood that these weights are different from the weights λ assigned to the clusters in subprocess 432. In the initial iteration of subprocess 433, the parameters of the machine learning model 334 can be initialized to initial values. In each subsequent iteration of subprocess 433, the parameters can be updated. Similarly, any suitable technique (such as Pareto optimization) can be used to determine new parameters to identify potential parameter sets that can further minimize the optimization problem. Examples of suitable techniques are disclosed in Sener et al., "Multi-Task Learning as Multi-Objective Optimization", Neural Information Processing Systems (NeurIPS) 2018, arXiv:1810.04650, which is incorporated herein by reference as if fully set forth.

[0143] In subprocess 434, it is determined whether the first stop condition of the inner loop is met. The first stop condition may include that the number of iterations (i.e., training epochs) of the inner loop reaches or exceeds a threshold, the change in the error of the optimization problem relative to the previous iteration reaches or drops below a threshold, the error of the optimization problem reaches or drops below a threshold, the duration of the inner loop reaches or exceeds a threshold, etc. If the first stop condition is met (i.e., "yes" in subprocess 434), the training subprocess 430 proceeds to subprocess 435. In this case, the current iteration of the inner loop has ended. If the first stop condition has not been met (i.e., "no" in subprocess 434), the training subprocess 430 returns to subprocess 433 to update the parameters of the machine learning model 334.

[0144] In subprocess 435, it is determined whether the second stop condition of the outer loop is met. The second stop condition may include the number of iterations of the outer loop reaching or exceeding a threshold, the change in the error of the optimization problem relative to the previous iteration reaching or falling below a threshold, the error of the optimization problem reaching or falling below a threshold, the duration of the outer loop reaching or exceeding a threshold, etc. If the second stop condition is met (i.e., "yes" in subprocess 435), the training subprocess 430 ends. In this case, the machine learning model 334 has been trained and can be verified, tested, retrained, and / or deployed. If the second stop condition has not been met (i.e., "no" in subprocess 435), the training subprocess 430 returns to subprocess 432 to update the cluster weights λ and perform another iteration of the inner loop.

[0145] 2.6. Phase deviation penalty

[0146] Phase deviation penalty function h θManifold regularization specific to time domain protection is provided. As described above, in order to reliably calculate phasors from a time series of voltage and current measurements, longer time series are required than are typically feasible at the speed at which a trip decision must be made. Therefore, a training data set (e.g., acquired in subprocess 410 of process 400) can be enhanced with longer time series that can be reliably used to calculate phasors, the training data set including short time series for training the machine learning model 334 to operate accurately within a short time frame. In particular, based on real measurements of noise levels, the lengths of these longer time series are selected to be long enough (e.g., covering at least a predefined time length) for good estimates of the voltage phasors and current phasors to be calculated. These longer time series are labeled with reference true phasor values, for example, with each reference true phasor value representing a voltage phasor and a current phasor.

[0147] Phase deviation penalty function h θ Can be based on an encoder-decoder architecture. Fig. 6A An encoder-decoder architecture 600A according to a first embodiment is illustrated. According to the architecture 600A, an input time series 610 is input to an encoder network 620, which generates a potential representation 630 of phasor values ​​(e.g., voltage phasors and current phasors) that approximate the input time series 610. A decoder network 640 can receive the potential representation 630 as input and output a prediction 650. Both the encoder network 620 and the decoder network 640 can include an artificial neural network (ANN). It is worth noting that:

[0148] f θ (x) = d θ (e θ (x)

[0149] is equivalent to:

[0150]

[0151] Among them, e θ is the encoder network 620, and d θ is the decoder network 640 .

[0152] Figure 6BThe encoder-decoder architecture 600B according to the second embodiment is illustrated. According to the architecture 600B, the input time series 610 is input to two encoder networks 620 and 620'. The encoder network 620, the potential representation 630, the decoder network 640 and the prediction 650 can be the same as in the architecture 600A. The encoder network 620' can generate a potential representation 630', which captures more information contained in the input time series 610 than the potential representation 630, such as complex impedance, admittance, apparent power and sequence components. The decoder network 640' can receive both the potential representation 630 and the potential representation 630' as inputs, and outputs a prediction 650'. Both predictions 650 and 650' can be composed of the same target, but prediction 650' will generally be more accurate because prediction 650' is based on more information than prediction 650. Therefore, the final output of the architecture 600B is prediction 650'.

[0153] In an embodiment, the phasor deviation penalty function h θ The encoder network 620 is used to obtain a latent representation 630 representing an estimated phasor value that can be compared with a reference true phasor value. In particular, the phasor deviation penalty function h θ It can be defined as follows:

[0154]

[0155] where Φ is a subset of the training dataset consisting of time series that are long enough (e.g., covering at least a predefined time length) to compute the phasor, and x i is the feature set of data instance i in the training subset Φ, φ i is the ground truth phasor value of data instance i in the training subset Φ, and e θ (x i ) is the latent representation 630 produced by the encoder network 620.

[0156] Phase deviation penalty function h θ is a cost function that penalizes the estimated phasor value e θ (x i ) and the reference true phasor value φ i Mathematically, the multi-objective optimization problem to be solved becomes:

[0157]

[0158] Or, if with the non-smooth penalty function g θ (X) Combination:

[0159]

[0160] It can be expressed as

[0161]

[0162] Phase deviation penalty function h θ Using encoder network 620 (ie, e θ ) to obtain approximate phasor values ​​and calculate the difference between those approximate phasor values ​​and the reference true phasor values. In order to minimize the estimation error of the optimization problem, in addition to the machine learning model 334 (i.e., f θ ), the encoder network 620 will also be trained to produce a more accurate approximation of the phasor value given an input time series. Once trained, the encoder network 620 will be able to approximate the phasor value regardless of the length of the input time series.

[0163] The above non-smooth penalty function g θ As discussed in the previous section, the Laplace regularization is implemented using the phasor deviation penalty function h θ The domain-specific regularization of can be used for the regression of fault location. However, in the context of time domain protection, this domain-specific regularization can alternatively be used for different learning tasks. For example, this domain-specific regularization can be used to train the machine learning model 334 to output a binary decision of whether to trip the circuit breaker 316.

[0164] Phase deviation penalty function h θ is described as calculating the deviation between the estimated phasor value and the reference true phasor value. However, in an additional embodiment, the phasor deviation penalty function h θ The deviation between the estimated values ​​of other obtained quantities (such as complex impedance, admittance, apparent power and sequence components, etc.) and the reference true values ​​can also be calculated.

[0165] 3. performance

[0166] Experiments were conducted using the machine learning model 334 with Laplace regularization for the fault location task in time domain protection. In particular, the regression model for estimating the fault location was trained with and without Laplace regularization using process 400. In addition, for training with Laplace regularization, training was performed with adaptive weights and predefined weights, the adaptive weights being set by the meta-learner in Figure 5 The following table depicts an overall comparison of the mean squared error (MSE) of the fault location estimation on the validation dataset using 5-fold cross validation for each of these trained models:

[0167]

[0168] As demonstrated by the above results, the model trained with Laplace regularization and adaptive weights achieved the lowest average MSE on the validation data set. In addition, the two models trained with Laplace regularization produced a lower average MSE than the model trained without Laplace regularization. The error variance (as defined by standard deviation) of the two models trained with Laplace regularization was also lower relative to the model trained without Laplace regularization. This reflects the robustness of the model trained with Laplace regularization for the changes in initial conditions and data sets. It is noteworthy that the performance of the model trained with Laplace regularization and adaptive weights is more consistent between folds compared to the model trained with Laplace regularization and predefined weights.

[0169] Unlike typical machine learning evaluations, accuracy alone does not reflect the need to make appropriate tripping decisions as quickly as possible. The problem is causal and temporal in nature, which creates asymmetry in the decision at each sample, because the model can only use past trajectories and a single crossing of the threshold (e.g., at the fault location) to determine when to trip the circuit breaker 316. Therefore, the overall tripping decision should be used to evaluate accuracy. In an embodiment, since the output of the model is the location of the fault, this output is compared with the threshold (e.g., range setting 336) that defines the protection zone. If the actual fault location is close to the range setting 336, the decision result may not be perfect. Therefore, each model is evaluated by calculating how far the fault must be from the range setting 336 to make the overall tripping decision perfect. The range around the range setting 336 where the decision made by the model is not perfect can be referred to as the "blind zone" of the range setting 336. In the blind zone, there is at least one input that produces a miss operation (i.e., a false negative) or a false operation (i.e., a false positive). The blind zone captures the total width of the range, which covers the wrong estimate representing a false positive or false negative. It will be appreciated that a more accurate model will produce a narrower blind zone than a less accurate model. A blind zone of width zero would indicate a model that makes perfect decisions for all situations.

[0170] Fig. 7A FIG. 3 illustrates the blind spot of a machine learning model 334 trained using Laplace regularization and adaptive weights according to an embodiment. Figure 7BThe blind spot of a model trained without Laplace regularization is illustrated. The model was tested for different range settings and fault locations. Each band of the blind spot indicates a percentile of the error estimate. The illustrated bands represent percentiles greater than 5%, between 1% and 5%, between 0.1% and 1.0%, and greater than 0% but less than 0.1%. The dashed lines represent user-specified boundaries on the error estimate, and the thick black line represents instances where the fault location is exactly equal to the range setting 336. The situation above the thick black line represents an instance where the fault location is greater than the range setting 336 (i.e., outside the protection zone) and therefore the line protection system 330 should not trip the circuit breaker 316. Conversely, the situation below the thick black line represents an instance where the fault location is less than the range setting 336 (i.e., within the protection zone) and therefore the line protection system 330 should trip the circuit breaker 316. As can be seen from Fig. 7A and Figure 7B It is clear that the machine learning model 334 trained with Laplace regularization and adaptive weights outperforms the model trained without Laplace regularization in terms of having a narrower blind spot. Although not shown, the machine learning model 334 trained with Laplace regularization and adaptive weights also outperforms the machine learning model 334 trained with Laplace regularization and predefined weights in terms of having a narrower blind spot.

[0171] Improvements were also achieved in terms of speed. Specifically, the average inference speed for the model using Laplace regularization and adaptive weights was 3.34 milliseconds, which is faster than the average speed of 3.45 milliseconds for the model using Laplace regularization and predefined weights and the average speed of 3.58 milliseconds for the model without Laplace regularization. These times include the delay caused by applying a safety counter to the output of the tested model.

[0172] In summary, the model trained with Laplace regularization and adaptive weights outperforms the model trained with Laplace regularization and predefined weights, and both models trained with Laplace regularization outperform the model trained without Laplace regularization, in terms of both accuracy and speed.

[0173] 4. Example Embodiments

[0174] The disclosed embodiment formulates the training of the machine learning model 334 as a multi-objective optimization problem. In addition to the main objective function of calculating the error of the machine learning model 334 (e.g., as a regression task), the multi-objective optimization problem also includes a non-smoothness penalty function g θ, the non-smoothness penalty function drives the primary objective function to a solution that respects smoothness between the input and output of the machine learning model 334. In particular, portions of the data that contribute to the non-smoothness of the data manifold (which results in high gradients) are penalized during training. To facilitate this, the gradients of neighboring graphs formed by the data can be added to the MSE loss that measures the reconstruction error. As discussed herein, training the machine learning model 334 in this manner improves the performance of the machine learning model 334 in at least four ways: (i) lower mean squared error; (ii) narrower blind spots; (iii) enhanced robustness; and (iv) lower average speed.

[0175] This uses a non-smooth penalty function g θ Laplace regularization (with or without adaptive weights for clustering) can be used in any regression task where the data is heterogeneous and the prediction regression function is smooth on the data (i.e., the regression function does not change abruptly with data points). In addition to applications in fault location in time domain protection, other potential application areas include, but are not limited to, power state estimation, topology estimation, parameter estimation (e.g., grid inertia), power flow estimation, load forecasting, etc. In this case, Laplace regularization will be particularly advantageous if the data is highly heterogeneous (e.g., the input signals are collected from different towers or other sources).

[0176] As a non-smooth penalty function g θ Additionally or alternatively, the multi-objective optimization problem may include a phasor deviation penalty function h θ , the phasor deviation penalty function utilizes the encoder network 620 to drive the primary objective function to a solution that minimizes the deviation of the phasor values ​​and optionally other derived quantities. This domain-specific regularization improves the regressor by learning to derive the phasors from time series (e.g., time series of voltage and current). This information is valuable for specific applications of time domain protection, but is difficult to compute from short time series corresponding to the fast decision times expected for time domain protection.

[0177] Figure 8A process 800 for making a trip decision according to an embodiment is illustrated, wherein a machine learning model 334 is used to estimate the location of a fault. In subprocess 810, a controller 332 of a line protection system 330 acting as a protection device may receive measurement signals from a plurality of sensors connected to a power line 310. For example, these measurement signals may include or consist of voltage values ​​measured by a voltage measurement unit 312 and current values ​​measured by a current measurement unit 314. In subprocess 820, the controller 332 may apply the trained machine learning model 334 to the measurement signals to estimate the location of a fault 340 on the power line 310. Then, in subprocess 830, the controller 332 may compare the estimated fault location with a predetermined threshold representing a range setting 336. For example, the controller 332 may determine whether the estimated fault location is less than or equal to the range setting 336. When it is determined that the estimated fault location meets the threshold (i.e., "yes" in subprocess 830), indicating that the fault 340 is within the protection zone defined by the range setting 336, the controller 332 can trip the circuit breaker 316 in subprocess 840 to electrically isolate the fault location on the power line 310. Otherwise, when it is determined that the estimated fault location does not meet the threshold (i.e., "no" in subprocess 830), the controller 332 determines not to trip the circuit breaker 316 in subprocess 850.

[0178] In an alternative embodiment where the output of the machine learning model 334 is a trip decision rather than an estimated fault location, the controller 332 can apply the trained machine learning model 334 to the measurement signal to determine whether to trip the circuit breaker 316. It should be understood that in this case, the measurement signal can be the same as described above. During the training of the machine learning model 334, the feature set will represent this set of measurement signals, and each target value will represent a decision to trip or not trip the circuit breaker. Therefore, in this embodiment, the machine learning model 334 is a binary classifier.

[0179] Embodiment 1: A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to perform the following operations: obtaining a training data set comprising multiple feature sets, wherein each feature set of the multiple feature sets comprises feature values ​​for each of the multiple features and is labeled with a target value for each of one or more targets; generating an optimization problem comprising an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0180] Embodiment 2: The method as described in Embodiment 1, wherein the non-smoothness penalty function includes the Laplace norm of the output value.

[0181] Embodiment 3: A method as described in any of the preceding embodiments, wherein the non-smoothness penalty function approximately calculates the gradient of the machine learning model on the data manifold of the machine learning model.

[0182] Embodiment 4: A method as described in any of the preceding embodiments, wherein training the machine learning model comprises: clustering the training data set into a plurality of clusters, wherein the non-smooth penalty function associates a weight with each of the plurality of clusters; and adaptively learning the weights of the plurality of clusters.

[0183] Embodiment 5: A method as described in Embodiment 4, wherein adaptively learning the weights of the multiple clusters includes, for each external iteration in multiple external iterations: determining the weights of the multiple clusters; and for each internal iteration in multiple internal iterations, using the determined weights of the multiple clusters in the non-smoothness penalty function to adjust the machine learning model to minimize the estimation error generated by the training data set in the optimization problem.

[0184] Embodiment 6: The method of Embodiment 5, wherein the weights of the plurality of clusters are determined in each of the plurality of outer iterations using a multi-class classification algorithm.

[0185] Embodiment 7: The method of embodiment 6, wherein the machine learning model is adjusted in each of the multiple internal iterations using a regression algorithm.

[0186] Embodiment 8: The method as described in any of the preceding embodiments further includes, before obtaining the training data set, using the at least one hardware processor to obtain data having a first dimension, and embedding the data into a lower dimensional space having a second dimension lower than the first dimension, wherein the training data set is obtained from the data in the lower dimensional space.

[0187] Embodiment 9: The method as described in any of the preceding embodiments, wherein each feature set of the plurality of feature sets represents a set of measurement signals of a power line, and wherein each target value represents a fault location on the power line.

[0188] Embodiment 10: The method as described in Embodiment 9 further includes, by a protection device: receiving measurement signals from a plurality of sensors connected to the power line; applying a trained machine learning model to the measurement signals to estimate a fault location on the power line; comparing the estimated fault location with a threshold; and when the estimated fault location meets the threshold, tripping at least one circuit breaker to electrically isolate the fault location on the power line.

[0189] Embodiment 11: The method as described in any preceding embodiment, wherein each feature set of the plurality of feature sets represents a set of measurement signals of a power line, and wherein each target value represents a decision to trip or not trip a circuit breaker on the power line.

[0190] Embodiment 12: A method as described in any of the preceding embodiments, wherein each of the multiple feature sets represents a state of an electric power system, wherein each target value represents a value of a continuous variable of the electric power system, and wherein the method further includes using a trained machine learning model to estimate the value of the continuous variable.

[0191] Embodiment 13: The method as described in any of the preceding embodiments further includes using the trained machine learning model for one of topology estimation, parameter estimation, power flow estimation or load prediction.

[0192] Embodiment 14: A method as described in any of the preceding embodiments, wherein each feature set in at least one subset of the multiple feature sets in the training data set is labeled with a reference true phasor value, wherein the optimization problem further includes a phasor deviation penalty function, and the phasor deviation penalty function is configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in at least one subset of the multiple feature sets in the training data set increases.

[0193] Embodiment 15: A method as described in Embodiment 14, wherein at least one subset of the multiple feature sets in the training data set is composed of only a portion of the multiple feature sets in the training data set, and the portion includes a time series covering at least a predefined time length for each feature of the multiple features.

[0194] Embodiment 16: A method as described in Embodiment 14 or 15, wherein the encoder network is an artificial neural network.

[0195] Embodiment 17: A system comprising: at least one hardware processor; and software, wherein the software is configured to perform the method as described in any one of Embodiments 1 to 16 when executed by the at least one hardware processor.

[0196] Embodiment 18: A non-transitory computer-readable medium having instructions stored thereon, wherein the instructions, when executed by a processor, cause the processor to perform the method as described in any one of Embodiments 1 to 16.

[0197] Embodiment 19: A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to perform the following operations: obtaining a training data set including multiple feature sets, wherein each feature set in the multiple feature sets includes feature values ​​for each feature in the multiple features and is labeled with a target value for each target in one or more targets, and wherein each feature set in at least one subset of the multiple feature sets is labeled with a reference true phasor value; generating an optimization problem including an objective function and a phasor deviation penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the multiple feature sets, and wherein the phasor deviation penalty function is configured to increase the estimated error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in the at least one subset of the multiple feature sets in the training data set increases; and training the machine learning model by adjusting the machine learning model to minimize the estimated error in the optimization problem generated by the training data set.

[0198] Embodiment 20: A method as described in Embodiment 19, wherein at least one subset of the multiple feature sets in the training data set is composed of only a portion of the multiple feature sets in the training data set, and the portion includes a time series covering at least a predefined time length for each feature of the multiple features.

[0199] Embodiment 21: A system comprising: at least one hardware processor; and software, wherein the software is configured to perform the method as described in Embodiment 19 or 20 when executed by the at least one hardware processor.

[0200] Embodiment 22: A non-transitory computer-readable medium having instructions stored thereon, wherein the instructions, when executed by a processor, cause the processor to perform the method as described in Embodiment 19 or 20.

[0201] The above description of the disclosed embodiments is provided to enable any person skilled in the art to implement or use the present invention. It will become apparent to those skilled in the art that various modifications to these embodiments will become apparent, and the general principles described herein may be applied to other embodiments without departing from the spirit or scope of the present invention. Therefore, it should be understood that the description and accompanying drawings presented herein represent the presently preferred embodiments of the present invention, and therefore represent the subject matter that the present invention is widely considered. It should be further understood that the scope of the present invention fully encompasses other embodiments that may be apparent to those skilled in the art, and therefore the scope of the present invention is not limited.

[0202] Combinations such as "at least one of A, B, or C", "one or more of A, B, or C", "at least one of A, B and C", "one or more of A, B, and C", and "A, B, C, or any combination thereof" described herein include any combination of A, B, and / or C, and may include multiple A, multiple B, or multiple C. Specifically, combinations such as "at least one of A, B, or C", "one or more of A, B, or C", "at least one of A, B, and C", "one or more of A, B, and C", and "A, B, C, or any combination thereof" may be only A, only B, only C, A and B, A and C, B and C, or A, B, and C, and any such combination may contain one or more members of its constituent elements A, B, and / or C. For example, the combination of A and B may include one A and multiple Bs, multiple A and one B, or multiple A and multiple Bs.

Claims

1. A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to perform the following operations: Obtain a training data set including multiple feature sets, where: Each feature set of the plurality of feature sets comprises a feature value for each feature of the plurality of features and is labeled with a target value for each of one or more targets, wherein each feature set of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line; generating an optimization problem including an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and corresponding output values ​​output by the machine learning model for the plurality of feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases; and The machine learning model is trained by adjusting the machine learning model to minimize the estimation error produced by the training data set in the optimization problem.

2. The method of claim 1, wherein: The non-smoothness penalty function includes a Laplace norm of the output value.

3. The method of claim 1, wherein: The non-smoothness penalty function approximately calculates the gradient of the machine learning model on the data manifold of the machine learning model.

4. The method of claim 1, wherein: Training the machine learning model includes: clustering the training data set into a plurality of clusters, wherein the non-smoothness penalty function associates a weight with each cluster in the plurality of clusters; and The weights of the plurality of clusters are adaptively learned.

5. The method of claim 4, wherein: Adaptively learning the weights of the plurality of clusters comprises, for each outer iteration of a plurality of outer iterations: determining weights of the plurality of clusters; as well as For each of a plurality of inner iterations, the machine learning model is adjusted using the determined weights of the plurality of clusters in the non-smoothness penalty function to minimize the estimation error resulting from the training data set in the optimization problem.

6. The method of claim 5, wherein: The weights of the plurality of clusters are determined in each outer iteration of the plurality of outer iterations using a multi-class classification algorithm.

7. The method of claim 6, wherein: The machine learning model is tuned in each of the plurality of inner iterations using a regression algorithm.

8. The method of claim 1, further comprising, before acquiring the training data set, using the at least one hardware processor to acquire data having a first dimension, and embedding the data into a lower dimensional space having a second dimension lower than the first dimension, wherein The training data set is obtained from the data in the lower dimensional space.

9. The method of claim 1, further comprising: receiving measurement signals from a plurality of sensors connected to the power line; applying a trained machine learning model to the measurement signal to estimate a fault location on the power line; comparing the estimated fault location to a threshold value; as well as When the estimated fault location satisfies the threshold, at least one circuit breaker is tripped to electrically isolate the fault location on the power line.

10. The method of claim 1, wherein: Each feature set of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a decision to trip or not trip a circuit breaker on the power line.

11. The method of claim 1, wherein: Each of the plurality of feature sets represents a state of an electric power system, wherein each target value represents a value of a continuous variable of the electric power system, and wherein the method further comprises estimating the value of the continuous variable using a trained machine learning model.

12. The method of claim 1, further comprising using the trained machine learning model for one of topology estimation, parameter estimation, power flow estimation, or load forecasting.

13. The method of claim 1, wherein: Each feature set in at least one subset of the multiple feature sets in the training data set is labeled with a reference true phasor value, wherein the optimization problem further includes a phasor deviation penalty function, which is configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in the at least one subset of the multiple feature sets in the training data set increases.

14. The method of claim 13, wherein: The at least one subset of the plurality of feature sets in the training data set consists of only a portion of the plurality of feature sets in the training data set, the portion comprising, for each feature of the plurality of features, a time series covering at least a predefined time length.

15. The method of claim 13, wherein: The encoder network is an artificial neural network.

16. A system comprising: at least one hardware processor; as well as software, the software being configured, when executed by the at least one hardware processor, obtaining a training data set comprising a plurality of feature sets, wherein each feature set of the plurality of feature sets comprises a feature value for each of the plurality of features and is labeled with a target value for each of one or more targets, wherein each feature set of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line, generating an optimization problem including an objective function and a non-smoothness penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the plurality of feature sets, and wherein the non-smoothness penalty function is configured to increase the estimated error in the optimization problem as the smoothness of the machine learning model decreases, and The machine learning model is trained by adjusting the machine learning model to minimize the estimation error produced by the training data set in the optimization problem.

17. The system of claim 16, wherein: Each feature set in at least one subset of the multiple feature sets in the training data set is labeled with a reference true phasor value, wherein the optimization problem further includes a phasor deviation penalty function, which is configured to increase the estimation error in the optimization problem as the difference between the estimated phasor value output by the encoder network and the reference true phasor value of each feature set in the at least one subset of the multiple feature sets in the training data set increases.

18. A method for performing manifold regularization while training a machine learning model, the method comprising using at least one hardware processor to: Obtain a training data set including multiple feature sets, where: Each feature set of the plurality of feature sets comprises a feature value for each feature of the plurality of features and is labeled with a target value for each of one or more targets, wherein each feature set of the plurality of feature sets represents a set of measured signals of a power line, and wherein each target value represents a fault location on the power line, and wherein each feature set of at least a subset of the plurality of feature sets is labeled with a reference true phasor value; generating an optimization problem including an objective function and a phasor deviation penalty function, wherein the objective function calculates an estimated error between the target value and a corresponding output value output by the machine learning model for the plurality of feature sets, and wherein the phasor deviation penalty function is configured to increase the estimated error in the optimization problem as a difference between an estimated phasor value output by an encoder network and a reference true phasor value for each feature set in the at least one subset of the plurality of feature sets in the training data set increases; and The machine learning model is trained by adjusting the machine learning model to minimize the estimation error produced by the training data set in the optimization problem.

19. The method of claim 18, wherein: The at least one subset of the plurality of feature sets in the training data set consists of only a portion of the plurality of feature sets in the training data set, the portion comprising, for each feature of the plurality of features, a time series covering at least a predefined time length.

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