Power outage restoration in distribution systems assisted by state estimation

Abstract

Embodiments described herein provide for power distribution control systems and methods. The power distribution control systems and methods estimate one or more signal measurements of one or more nodes of a power distribution network using state estimation, identify a location of an outage in the power distribution network, identify a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation, and output control signaling to control the one or more switches based on the identified status.

Description

TXTU-0005PC POWER OUTAGE RESTORATION IN DISTRIBUTION SYSTEMS ASSISTED BY STATE ESTIMATION CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims benefit of and priority to U.S. Provisional Application No. 63 / 659,164 filed June 12, 2024, which is hereby expressly incorporated by reference herein in its entirety as if fully set forth below and for all applicable purposes. BACKGROUND Field

[0002] Embodiments described herein generally relate to techniques for enhancing automatic outage restoration solutions. More specifically, embodiments described herein relate to state estimation (SE) techniques to estimate state variables of a power distribution network based on network topology, pseudo measurements, and actual field measurements. Description of the Related Art

[0003] Electric power delivery to end users is one of the primary responsibilities of distribution systems (DSs). Built with a meshed topology, DSs are traditionally operated in a radial configuration, as their intrinsic asymmetry and unbalancing make the meshed operation challenging. Customers are connected through single-, two- and three-phase arrangements throughout the network and phases, making the system unbalanced. To electrically connect these customers, the DSs have single-, two- and three-phase branches, creating unsymmetrical and, consequently, unbalanced properties. Based on these characteristics, the DS may be implemented with a substantial infrastructure and control equipment investments to operate reliably in a meshed configuration while ensuring stability. With the predominant radial operation, compounded by several pieces of equipment and presenting a higher resistance to reactance (R / X) ratios compared to meshed transmission systems, DSs have substantially higher losses and specific challenges in their modeling and operation. The recent trend of integrating Distributed Energy Resources (DER) over the DS at a customer level has also brought up challenges related to power quality and reliability.TXTU-0005PC SUMMARY

[0004] In one embodiment, a method for power distribution control is provided. The method includes estimating one or more signal measurements of one or more nodes of a power distribution network using state estimation, identifying a location of an outage in the power distribution network, identifying a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation, and outputting control signaling to control the one or more switches based on the identified status.

[0005] In another embodiment, a non-transitory computer-readable medium is provided. The non-transitory computer-readable medium stores instructions that, when executed by a processor, cause a computer system to perform the steps of estimating one or more signal measurements of one or more nodes of a power distribution network using state estimation, identifying a location of an outage in the power distribution network, identifying a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation, and outputting control signaling to control the one or more switches based on the identified status.

[0006] In yet another embodiment, an apparatus for power restoration is provided. The apparatus generally includes at least one memory comprising computer- executable instructions and one or more processors configured to execute the computer-executable instructions and cause the apparatus to: estimate one or more signal measurements of one or more nodes of a power distribution network using state estimation; identify a location of an outage in the power distribution network; identify a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation and the location of the outage; and output control signaling to control the one or more switches based on a identified status.TXTU-0005PC BRIEF DESCRIPTION OF THE DRAWINGS

[0007] So that the manner in which the above recited features of the present disclosure can be understood in detail, a more particular description of the disclosure, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this disclosure and are therefore not to be considered limiting of its scope, for the disclosure may admit to other equally effective embodiments.

[0008] Figure 1 illustrates an example distribution system.

[0009] Figure 2 is graph illustrating a resiliency curve of a system during an outage event.

[0010] Figure 3A is a diagram illustrating an example outage scenario with miscoordination.

[0011] Figure 3B is a graph illustrating the benefit of distribution systems state estimation (DSSE).

[0012] Figure 4 is a diagram illustrating how a distribution system (DS) can be separated into three layers.

[0013] Figure 5 is a flow diagram illustrating example operations for outage restoration which may be performed by a state machine, in accordance with certain aspects of the present disclosure.

[0014] Figure 6 illustrates a synthetic spanning tree network.

[0015] Figure 7 illustrates a synthetic spanning tree network exemplifying graph theory changes when switches are operated.

[0016] Figure 8 illustrates a network branch model and associated electrical parameters.

[0017] Figure 9 illustrates a network schematic after a reference node addition.TXTU-0005PC

[0018] Figure 10 is a diagram illustrating radial over-current protection with two relays.

[0019] Figure 11 is a diagram 1100 illustrating radial over-current protection to coordinate operations.

[0020] Figure 12 is a flow diagram illustrating example operations for power restoration, in accordance with certain aspects of the present disclosure.

[0021] Figure 13 depicts aspects of an example computing device.

[0022] To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the figures. It is contemplated that elements disclosed in one embodiment may be beneficially utilized on other embodiments without specific recitation. DETAILED DESCRIPTION

[0023] Certain aspects of the present disclosure are directed towards techniques for power outage restoration in a power distribution system using state estimation. In some aspects, a fault (e.g., short circuit) in the distribution system may be identified between switches of the distribution system. The fault may be located and one or more switches may be controlled to isolate the fault in the distribution system. For example, the fault may be located between two switches of the distribution system, where the two switches may be opened to isolate the fault between the switches. Once the fault is isolated, restoration techniques may be performed to identify switches that could be closed and / or open and restore power to certain customers.

[0024] State estimation may be used to determine the state of switches for power restoration in an attempt to maximize the number of customers that can be restored with power while ensuring a reliable and stable system after restoration. State estimation is a process that uses available data and a mathematical model of a system to estimate a state of the system, even if not all states are directly measurable. For power restoration, state estimation may be performed based on information indicating an amount of power being consumed from different nodes of the distribution system. The actual amount of power consumed from each node may not be known at the time of the outage. Thus, state estimation may receive real measurements (e.g., actualTXTU-0005PC sensor measurements such as power measurement immediately before the power outage occurred) as well as pseudo measurement data (e.g., prediction of power consumption from each node using historical data) to identify the state of the system (e.g., electrical characteristics at different nodes of the system). The state of the system may be used to control switches in an attempt to maximize the number of customers being restored power while meeting power quality metrics for the distribution system. The real measurements may include voltages on different sides of each switch as well a current across each switch that can be used to identify power delivery at the switch. In some cases, the real measurements may include one or more flags, such as a flag indicating whether the current across the switch has passed a certain threshold indicating a fault. The real measurements may also include meter information, such as voltage and current from customer meters. Pseudo measurements may include predicted measurements which may be based on historical measurement data used to forecast future node states.

[0025] State estimation may weight different types of measurements such as actual power measurements and pseudo measurements. For example, a greater weight may be given to actual power measurements as real power measurements are more reliable than pseudo measurements. The state estimation relates the different types of measurements and iteratively adjusts the estimated state of the system to minimize (or at least reduce) the difference between the model's predictions and the measurements, aiming for a state that "fits" the data as accurately as possible.

[0026] In some aspects, state estimation may be performed prior to the occurrence of the fault in order to reduce the amount of time it takes for restoration. That is, the state estimation may take a few minutes to complete. In an attempt to restore power to customers in a short amount of time (e.g., under 3 minutes), state estimation may be performed prior to the occurrence of an outage. Once the outage occurs, the state of the system as estimated prior to the outage may be used to perform system restoration, reducing the amount of time it takes to restore power to customers. Introduction to Distribution Systems

[0027] Distribution Systems (DSs) are mounted with different passive and active elements, such as switches, step-voltage regulators, capacitor banks, and meters.TXTU-0005PC Regarding switches, DSs can have a large variety of types, such as circuit breakers, reclosers, load breakers, motor-operated switches, and sectionalizers, which differ from each other in operational aspects. Even though circuit breakers and reclosers are able to extinguish short-circuit current levels, load breakers are designed only to interrupt load current levels. On the other hand, motor-operated switches and sectionalizers do not have the capability to extinguish the arc current and, by this, should not be operated under load. The integration of switching devices is primarily intended to reduce costs related to power delivery. Whenever customers receive poor power quality or face unplanned outages, the utility is subjected to monetary penalties by its regulatory agent.

[0028] As a result of the need to meet customer demands for reliable power delivery, utilities are encouraged to improve traditional distribution management systems’ monitoring and control capabilities, expanding it to Advanced Distribution Management Systems (ADMS). The ADMS concept embraces several online and offline monitoring and control features to ensure power delivery with reliability, continuity, and quality. This solution is usually centralized at the Distribution Operations Center (DOC) but may also have distributed elements. By having data about the network topology, such as electrical and physical information of connections, line segments, equipment, and customers, ADMS can obtain real-time field measurements to perform further analysis on the network to automatically respond to events or support operators’ decisions with more detailed information.

[0029] Figure 1 illustrates an example distribution system 100 showing the interaction between DS and DOC. Distribution Network Reconfiguration (DNR) is one of the primary ADMS features. Given a network topology, its operational configuration can be changed through the maneuvering of switches. The reconfiguration brings flexibility to the complex distribution networks by enabling transitions between operating configurations. Demand levels can drastically change over the day, week, months, and seasons, and especially on feeders that have customers subjected to Demand Response (DR) programs. Based on that, DNR is used as a planning feature to improve the power flow in a given time window according to the system’s demand levels. During DNR, it is important to understand the switches’ control capabilities, which can be divided into tele-controlled switches (TCS) and manually controlledTXTU-0005PC switches (MCS). Nowadays, many circuit breakers, reclosers, load breakers, and motor-operated switches are TCSs, while sectionalizers are MCSs and involve linemen’s intervention to be maneuvered. With newer and more accessible ADMS technologies, the grid modernization movement has made it more important to improve energy delivery by incorporating Intelligent Electronic Devices (IED) in general. IEDs are devices allocated over the network that can provide electrical and status measurements through communication venues and protocols. Some IEDs, such as TCSs, capacitor banks, and step-voltage regulators, may also perform actions based on remote controls. Even though having one TCS per network segment or node is not technically and financially feasible, increasing the number of TCSs enables more efficient DNR.

[0030] The DNR concept is also used for outage events. Instead of maintaining the systems de-energized according to the post-event scenario until the damage is repaired, switch maneuvers can isolate faults and partially restore de-energized customers. With this approach, only the faulted region would be kept de-energized until the structural damage is repaired by linemen. This method has been used for years in DSs, where DOC would dispatch crews to maneuver MCSs while operators would be remotely maneuvering TCSs. However, when TCSs operations happen in an automated manner based on a pre-defined logic, the restoration action is commonly called Self-Healing (SH) or, most recently, Fault Location Isolation and Service Restoration (FLISR) or Fault Detection, Isolation and Restoration (FDIR). Self-healing is able to utilize available field measurements to locate the fault and then compute a sequence of switching commands that can isolate the damaged area and restore as many customers as possible. According to the level of autonomy, the SH actions can be automatic or guided by operators’ supervision. The main goal of SH is to provide a fast response to an outage event that can reduce the number of impacted customers and facilitate the linemen’s efforts.

[0031] Power systems faults are categorized into temporary and permanent faults. Temporary faults are short circuits that last a couple of seconds, usually happening in non-insulated overhead branches whenever the vegetation touches the conductor or when cables that lack proper spacers touch each other. Through reclosing coordination, protective devices are programmed to protect these temporary faults andTXTU-0005PC attempt to reclose, usually three or four times based on the open interval timeout for each shot. Approximately 80% of the DSs’ faults are temporary and are eliminated by themselves after the reclosing attempts. The remaining 20% are permanent faults caused mainly by damage or failure of power delivery structures, such as distribution towers and poles, which usually originate from traffic accidents, weather events, vegetation, and equipment failure. Due to the radial configuration, a fault event implies a power outage to customers connected downstream to the tripped relay or blown fuse. Ideally, the closest protection device to the fault would be operating before any other device operates. However, due to the high loading currents and small short- circuit levels, the coordination between recloser-recloser and recloser-fuse becomes challenging in DSs, and the chances for miscoordinations increase. Depending on the levels of Distributed Energy Resources (DER) capacity and generation in the system, parts of the DS can be subjected to reverse power flow. Considering that traditional DSs were designed for radial operation, a reverse power flow can significantly compromise control and protection schemes, increasing the chances of protection misbehaving.

[0032] With the intention to monitor and measure the quality of power delivery service, reliability indexes are factors used to represent the utilities’ performance. These metrics are computed based on different parameters and conditions defined by the IEEE 1366-2012 Guide for Electric Power Distribution Reliability Indices. Even though there can be planned and unplanned interruptions, the unplanned ones are the most challenging events. Planned interruptions are events when the utility is to perform maintenance or another type of service that involves part of the system to be de- energized for safety purposes. In these scenarios, customers are notified in advance, and a detailed plan for reconfiguration and de-energization is developed. An unplanned interruption is an unpredictable event where the utility may act fast enough to reduce its impact, independent of the condition. The standard defines two types of unplanned interruptions: temporary and sustained. Temporary interruptions last less than 5 minutes, while sustained last more than that. This time threshold is important for restoration analysis, as only sustained outages are counted for the reliability indexes, meaning that if the utility is able to restore as many customers as possible under a permanent fault event through operators or SH actions, it will significantly reduce the event’s impact on the reliability indexes.TXTU-0005PC

[0033] There is a need in the art for a fast and assertive restoration decision without relying on operators’ influence that can deal with uncertain operation conditions and unreliable protection schemes. The embodiments described herein relate to a Distribution Outage Restoration System (DORS) capable of addressing observability and protection miscoordination challenges, while providing an assertive restoration solution to re-energize customers and reduce the outage region.

[0034] Unlike transmission systems, distribution networks present an unbalanced and unsymmetrical topology that lacks observability and controllability. Being traditionally designed and operated in a radial configuration, any outage event implies the de-energization of downstream customers. Distribution Network Reconfiguration (DNR) is a practice that modifies the network configuration by opening and closing switches to enhance one or more operational aspects, such as power outage restoration. To make fast and assertive decisions that reduce costs and prevent cascading outages, restoration techniques have been automated to avoid operator intervention and take action within the first minutes of the event. The recent methodologies for automatic restoration solutions assume the availability, knowledge, and accuracy of node-by-node measurements, which is not a reality in actual distribution systems. Even though the current trend in grid modernization has increased the integration of teleoperated Intelligent Electronic Devices (IEDs), such as relays, reclosers, meters, load sensors, and switches, to improve energy delivery quality, continuity, and reliability, obtaining reliable electrical measurements on every node is still technically and financially infeasible. Based on that, certain embodiments of the present disclosure provide techniques for automatic outage restoration by embedding State Estimation (SE) techniques, which can be used to estimate the best fit of the system’s state variables based on the network topology and a reduced number of actual field measurements. The methodology described herein may be capable of making the network observable through SE to then perform fault location and isolation, detect and correct miscoordinations, improve restoration, and restore switching sequences. Even though compounded by time-consuming features, the techniques described herein provide time efficiency while also accounting for DER presence in the network. The methodology is developed and validated by Software- In-the-Loop (SIL) and Hardware-In-the-Loop (HIL) simulations using benchmarkTXTU-0005PC distribution networks with different sizes and topologies for validation, comparison, and evidence of effectiveness.

[0035] The challenges related to the current restoration state-of-art can be categorized into the interaction of distributed controls and lack of system awareness. On one side, distributed controls act locally with a limited amount of information and make decisions over the system operation. These isolated operations are usually fast and do not rely on communication links but may impact as well as be impacted by other control schemes. Even though protection is an ancillary service that provides reliable operation and protects the system from faults, distribution protection schemes are traditionally based on local and independent decisions. With that, wrong coordination, settings, and lack of system-wide information can imply a misoperation, increasing the interruption area and amount of switching maneuvers to reduce the outage impact.

[0036] Figure 2 is graph 200 illustrating a resiliency curve of a system during an outage event and compares the resiliency with the impact of miscoordination. As shown, the resilience curve can be defined by different factors, such as energized elements and loads or unserved power demand based on actions to recover the system. Considering the resilience curve based on the unserved demand, ^^^^^^, and switching operations ^^^^^^, graph 200 compares scenarios with an assertive protection operation and miscoordination. During an outage event, the system has a pre-event state defined by the network configuration and demand levels. As soon as an event happens at ^^^, there is a short time window where protective devices operate according to their pre-defined protection pickups betweenand ^^ଶ. The protection scheme performance may define the amount of de-energized loads. The assertive operation may only de-energize the actual number of customers to extinguish the short-circuit, ^^^^^ହ^. Whenever a miscoordination happens, a protective device that is not the closest one to the fault operates and de-energizes customers that are not part of the fault zone ^^^^^ଶ^. It is important to identify miscoordinations and correct them before starting the restoration process. Otherwise, customers that could be re- energized may be kept de-energized until the damage is repaired. For this, there is a temporization from the first protective device lockout to allow the protection scheme operation completion before computing a solution, as represented between ^^ଶand ^^ଷ.TXTU-0005PC As an illustration, between ^^ଷand ^^ସthe system performs the isolation of the fault, and then, between ^^ସand ^^ହ, the miscoordination is corrected, restoring some of the impacted customers and matching the outage level with the actual. Due to the inappropriate protection scheme performance in a miscoordination scenario, the system may perform additional switching operations, ^^^^^ହ^, to successfully restore the customers. The restoration actions start at ^^ହ, where automatic solutions and operators’ actions restore as many remaining customers as possible. ^^^^^^^ is reached when restoration actions are exhausted. The amount of unserved power in ^^^^^^^ may then only be restored by the intervention of linemen to repair the structural damage(s) and additional maneuvers to reconfigure the system back to its normal state.

[0037] Having in mind that the repair is constrained by the crews’ location at the service dispatch moment, skills, equipment, available information to locate the damage, and the severity of the damage, it is imperative that the restoration process is as efficient and fast as possible because the repair may take hours to be completed. Ideally, most of the customers would be re-energized within 5 minutes from the fault event to reduce the impact on reliability indexes. Aiming for a fast response to outages, academia and industry have been developing solutions that use a reduced amount of data to quickly compute actions to isolate and restore customers. However, power networks are complex systems and operate under several constraints, which are usually disregarded during these solutions.

[0038] The biggest challenge may be that distribution networks are traditionally known for lacking visibility. With extensive line segments, several pieces of equipment, and customer connections, distribution systems may not have enough measurement points to make the network observable. Even though not enough yet, the increasing integration of IEDs has been improving the distribution systems’ visibility, but it is still economically and technically infeasible to make the system fully observable. With sufficient measurement points, electrical power systems can use State Estimation (SE) tools to obtain estimated values of the network’s electrical quantities. The SE results provide a reliable state of the network loads, and even though these values may not be exact, they are close enough to the network operation point to provide an assertive restoration solution in highly meshed network topologies that lack sufficient number of field measurement points.TXTU-0005PC

[0039] With the intention to address gaps in DS restoration solutions, this disclosure is an integrated solution of miscoordination and SE techniques along with improvements for restoration of outages.

[0040] Figure 3A is a diagram 300 illustrating an example outage scenario with miscoordination, showing the importance of SE information to properly make decisions. Switches S1, S2, S4, and S5 are reclosers, and switch S3 is a motor- operated switch. In this example, the five switches define four cells (S1-S2, S2-S3, S3-S4, and S4-S5), which are sets of branches and nodes bounded by TCSs.

[0041] Initially at step (a), the network is in its pre-fault state and configuration, where customers are being supplied power. With a permanent fault in a lateral branch between S3 and S4, the short-circuit fault is supplied by the substation labeled “SS1”, and the current passes through S1, S2, and S3. In the illustrative miscoordination scenario, the recloser at S1 trips, even though S2 is the protective device closer to the fault, as presented in step (b). From the instantaneous and temporized over-current protection coordination, S2 should have tripped for the proposed fault. However, due to the miscoordination, S2 doesn’t trip or S1 trips faster than S2, and customers located within cells S1-S2, S2-S3, S3-S4, and S4-S5 are de-energized. With that, customers within cell S1-S2 would still be energized if the protection scheme had operated as expected.

[0042] Considering an advanced restoration solution capable of accounting for miscoordinations, the miscoordination and isolation plan analyses are performed in step (c). By understanding that S1, S2, and S3 sensed the fault currents and that S3 is the one most downstream of the feeder, the isolation plan identifies that S3 and S4 can isolate the fault and that S1 miscoordinated with S2. With the fault isolated at step (c), the miscoordination can be corrected by closing S1 and re-energizing customers within cells S1-S2 and S2-S3 at step (d). It is important to highlight that until this step, no restoration action has been taken, only isolation and miscoordination detection and correction. After these steps, it is possible to proceed with the restoration to re- energize customers downstream of the fault cell S3-S4. At step (e), it is possible to see that the adjacent feeder (substation labeled “SS2”) was able to pick up the customers within the cell between S4 and S5 by closing switch S5.TXTU-0005PC

[0043] Even though this scenario aims to illustrate the importance of considering miscoordination in an outage event and highlight the current research gap, several aspects should be considered in a restoration process. For instance, before transferring cell between S4-S5 to the adjacent feeder, it is important to ensure that the load transfer will not originate unacceptable levels of loading and voltage to the system’s backbone and laterals. To deal with this challenge, the restoration solution should use network topology and demand data through power flow equations to reliably provide a solution that respects these constraints. Pseudo data is not reliable enough to ensure a correct decision, as it may have several assumptions and considerably large errors related to these values. Certain aspects provide a Distribution Systems State Estimation (DSSE) technique for reconfiguration analysis.

[0044] Given a set of measurements with aggregated errors and system variables to be known, the SE can compute a state of variables based on a linear equation that represents the measurements in terms of the state variables.,

[0045] Figure 3B is a graph 350 illustrating the benefit of DSSE considering a 7- node distribution system with a substation at node 0. Three different results for nodal voltage magnitude and power demand are compared. The actual (AC) demand (labeled “AC-demand”) and the voltage profile based on the actual demand (e.g., labeled “AC voltage”) are shown. This information represents the actual state of the network. The pseudo / forecasted (PS) power demand (labeled “PS-demand”), and the voltage profile based on the PS demand is shown, where pseudo nodal demands are used in a power flow computation to obtain a pseudo voltage profile. Ideal measurement devices at every node may be implemented to obtain the network’s actual state. Based on that, pseudo measurements from asynchronous measurements and forecasting may be used and trusted for development, even though their accuracy is low. However, the DSSE is capable of considering field measurements and pseudo measurements according to their measurement errors to obtain an estimated state of the network. The graph 350 shows the SE computation results including the state- estimated demand (labeled “SE-Demand”), and the associated voltage profile labeled “SE-Voltage”.

[0046] It is possible to observe that the computation based on pseudo measurements provides operating levels considerably far from the actual state of theTXTU-0005PC system. On the other hand, the SE solution could fit the state variables based on pseudo and actual measurements to obtain an operating point close enough to the real system’s levels. While having a fully measured DS is not feasible, by using a DSSE mathematical formulation, it is possible to efficiently estimate the state of the system based on a reduced set of measurements. Distribution Systems Model

[0047] Active distribution networks can be organized into layers, where each layer is mounted with specific types of equipment / devices and connections.

[0048] Figure 4 is a diagram 400 illustrating how DS can be separated into three layers, the physical, data, and model layers, used as a reference for the DORS development. The physical layer is compounded by physically active and passive elements of the DS, such as cabling, switches, infrastructure, loads, etc. The physical layer also contains the IEDs, which create the bridge between the Physical and Data Layers. IEDs are capable of measuring electrical quantities at their connection location and even perform actions such as opening and closing switches. As soon as this data is measured, it can be sent to the DOC through a bi-directional communication network, which can be defined by different venues. With the measurement data available at a centralized platform, information such as the network topology, containing the IEDs’ locations and network connectivity parameters, can define a system model in the model layer. This layer merges the field measurements into the network model to perform the different ADMS features.

[0049] As IEDs can provide different types of measurements according to the manufacturing and model, this disclosure considers TCSs to provide both source and load side three-phase voltage measurements and three-phase current flow measurements. For customers’ meters, voltage and complex power measurements are provided according to the number of phases supplying the customer. With the increasing availability of time-synchronization solutions, the measurements are assumed to be synchrophasors. Whenever a measurement device has a high-quality time synchronization, the device uses the accurate time to compute the phasor for the measured sine wave. The main benefit of synchrophasors is that phasors angles can be compared, as they were computed based on the same time reference withTXTU-0005PC negligible time difference errors. Considering the current trend in using load sensors to support DS measurement and fault location, this disclosure accounts for load sensors’ metering capabilities. Load sensors are placed in series over the distribution network and measure the phase current, which can be converted to complex power and then transmitted. In some cases, it may not be possible to obtain enough measurements from the DSs to make the DSSE solvable. Thus, pseudo measurements may be used to complement the remaining measurements and achieve a SE solution. Pseudo measurements are obtained from load and generation forecasting based on the customers’ installed demand and present a large error. Algorithm Structure

[0050] To develop a restoration solution capable of addressing the presented challenges with a fast enough response to restore customers in a short amount of time (e.g., within 3 seconds) from the outage event, the DORS algorithm structure is defined by a set of functions that run asynchronously. This asynchronous operation is designed based on a state machine approach, where one main flag variable defines which state the DORS is currently operating, besides having common functions before and after the actual state is computed.

[0051] Figure 5 is a flow diagram illustrating example operations 500 for outage restoration which may be performed by a state machine (e.g., DORS), in accordance with certain aspects of the present disclosure. The operations 500 may be performed for each computation cycle and may end in different conditions. Assuming that the network topology data is already available, such as nodes, branches, customers’ and devices’ locations, connections, and parameters, the DORS starts by collecting field measurements from the TCSs and customers’ meters at block 502.

[0052] Initially, the state machine is idle (e.g., is in state 0), where the system is monitoring for faults at block 504. A fault is characterized by having at least one recloser with the lockout and fault indicator flags on. If a fault is identified, DORS saves the time when the fault was identified at block 506, changes the state machine to state 1, and concludes the cycle. Otherwise, at block 508, the DORS checks for network configuration changes if there is no fault. In case a switch status changes, the network configuration is updated at block 510. Based on the latest network configuration,TXTU-0005PC DORS will be computing DSSE to obtain the state of the network at block 512, which is stored in a database 514, and then DORS will conclude the computation cycle.

[0053] Distribution systems protection may take a couple of seconds to finalize its coordination scheme. Thus, in state machine 1, DORS starts a temporization mode, where the system cycles for 10 seconds with the intention to allow the network to reach a new steady state and final post-fault configuration. This temporization is performed so that the miscoordination and restoration analysis are computed based on the actual post-fault network state. As soon as the temporization is completed at block 516, DORS will update the network configuration at block 518 and create two network models, the actual (post-fault) and previous (pre-fault) configurations, and then proceed to state 2.

[0054] At state 2, the DORS will start the miscoordination analysis to detect miscoordination between protective devices (e.g., switches) at block 520 and, if so, identify which of the protective devices should be closed back or open. With this information, at block 522, the fault location is processed next to determine which cell the fault is located in properly. By finding the fault cell, at block 524, the isolation process identifies the switches bounding the fault cell so they can be opened to isolate the faulted region. The isolation and miscoordination information is then used to generate an Isolation Control Sequence (ICS) at block 526. As soon as the ICS is generated, DORS changes the state machine to state 3 and proceeds to perform a switch control sequence (SCS) at block 528, where the ICS commands 530 will be dispatched.

[0055] As the different SCS steps may take seconds or even minutes to be finalized, instead of waiting for the ICS to be completed, the DORS may start state 3 by using the ICS information to create a network model considering the miscoordination correction and ICS changes. With this approach, at block 532, the restoration is computed based on the expected network configuration after isolation while the ICS is actually happening in parallel. For the restoration solution, the DSSE database (e.g., database 514 where state estimation results are stored) may be accessed to use the latest nodal power demand levels obtained from the SE analysis to compute a restoration configuration such that the isolation switches are maintained open, and other operational conditions are respected. The network configuration,TXTU-0005PC defined by open and closed switches, is then organized in control steps to generate a Restoration Control Sequence (RCS) at block 534, and the state is changed to 4. In some cases, before ending the computation cycle, the RCS and ICS may be merged as part of the SCS. In other words, once the ICS is complete, switches may be controlled in accordance with the RCS.

[0056] At state 4, DORS may check the statuses of the switches, according to the step-by-step commands sent by the SCS, until the restoration network configuration is achieved. Once the SCS is completed at block 536, the system returns to state 0.

[0057] DSSE may be computationally demanding. With that, it may not be feasible to perform DSSE when the fault happens, as the delay originated by the DSSE computation may impact how long it takes to perform the restoration. Also, configuration processing is another time-consuming task, as the DSs tend to be large, and graph theory techniques should be used to identify the network configuration and node-to-node relation.

[0058] DORS may be designed for computational time processing efficiency. By using state machines, the system can compute the functions according to the network state and restoration step. The algorithm functions are defined beforehand based on a set of inputs and outputs. Common functions may be computed before and after the state machine, while specific functions are only computed under a pre-defined machine state. A condition expresses the state change, and the system can stay within the same state during several cycles or only one. For every algorithm cycle, field measurements and status are read before the state machine, and the SCS is computed after the state machine. This allows the algorithm to run with the latest field information and send commands at every cycle instead of being limited to one state machine.

[0059] Another important feature of DORS is the interaction between ICS and RCS. By internally using the ICS information and avoiding waiting for the system to reach its isolation configuration, DORS can start computing the restoration beforehand based on the expected network configuration after the isolation. This saves time and improves the algorithm’s time efficiency. The DORS may obtain a restoration solution before the ICS is completed, but the RCS commands may be only dispatched whenTXTU-0005PC the TCSs’ statuses achieve the ICS control commands. If one of the TCSs does not respond to its command or report its status, DORS may not proceed to the next set of commands as it may create network loops or even re-energize the fault.

[0060] Another important feature of DORS is to keep computing the DSSE whenever the system is in normal operation, and when an event happens, the latest DSSE data will be available for use. DORS may be designed to expect one or multiple system faults and perform the entire miscoordination analysis, fault location, fault isolation, and restoration, considering all of the faults. The integrated temporization allows this consideration as lockouts caused by different faults may take different times to reach and be identified by DORS.

[0061] The DORS methodology is developed for medium-voltage distribution feeders. Even though low-voltage customers may provide important information, the DORS assumes equivalent models for low-voltage customers. The techniques described herein may be structured to consider unsymmetrical and unbalanced loads, along with robust power flow, DSSE, and optimization approaches that can handle large distribution network models. BACKWARD / FORWARD SWEEP POWER FLOW

[0062] Over time, the network configuration can drastically change, especially when restoration solutions are deployed in the system. This change in switches’ statuses can create reverse power flow and transfer segments to another feeder. Based on that, it is important to consider the current network configuration, which nodes are electrically connected to each substation, and the relationship between nodes before performing the power flow or any other analysis based on the network configuration. Even though the nodes and branches that compound a feeder may change with the reconfiguration and restoration, in radial systems, it may be assumed that there is only one substation connected per feeder, and the nodes are arranged in a radial configuration.

[0063] Figure 6 illustrates a synthetic spanning tree network 600 at a normal state. Considering ^^ as the substation node, it is possible to observe that each node has only one parent, but nodes can have none, one or multiple child nodes. As an example, the substation node, ^^, has ^^ and ^^ as child nodes, at the same time that ^^ doesn’tTXTU-0005PC have a parent node as it is a substation node but is the parent of nodes ^^ and ^^. Similarly, node ^^ is the parent node of ℎ and ^^, while ℎ and ^^ are the child nodes of ^^. Lastly, node ^^ is an edge node, with ^^ as its parent node. In distribution systems, the parent-child relation also implies which node is sending power and which node is receiving power. In a radial network configuration, each node should receive power from one other node, its parent, but it can send power to none, one, or multiple nodes, its child nodes. The graph theory concepts categorize the network nodes by levels, representing the node’s depth in relation to the substation node. The level represents the number of other nodes in the path between the node and the substation node. Another important piece of information is the relationship between parent and child nodes in each branch. Every time there is a change in the network switch status, the nodes’ depth and parent-child relation may be computed again.

[0064] Figure 7 illustrates a synthetic spanning tree network 700 exemplifying graph changes when switches are operated. It is possible to observe that the switching operation maintained the spanning tree configuration by opening one and closing the other device. In this new configuration, node ^^, which was in level 3 before, is now in level 4. In the normal state, node ^^ had node ^^ as a parent, but with the reconfiguration, ^^ is the new parent of node ^^. Line Model

[0065] Distribution systems’ line branches are commonly modeled as short or medium-length transmission lines. The admittance is usually pure capacitive, i.e., only compounded by susceptance. Even though the consideration of mutual capacitance doesn’t significantly impact the distribution systems’ power flow results, the shunt capacitance presents a significant contribution as well as series and mutual impedances. Based on that, mutual capacitance is not considered in the medium- length transmission line model. If the total shunt admittance of the line, assumed as purely capacitive, is divided into two equal parts and placed at the sending and receiving end nodes, the circuit is called nominal-^^.

[0066] Figure 8 illustrates a network branch model 800 and associated electrical parameters. The model 800 may be used to explain power flow backward and forward actions based on a nominal-^^ model.TXTU-0005PC Load Model

[0067] Each node can be compounded by complex power consumption,^ ^^^^^, and^^^,థ,ఎgeneration, ^^^,థ,ఎ ^ ^^^^^,థ,ఎ. In a polynomial (ZIP) load model, the load is modeled as a composition of three components: constant impedance, constant current, and constant power. The constant impedance term varies its demand proportionally to the square of the voltage magnitude, and the constant current varies the power demand directly proportional to the voltage magnitude, while a constant power term doesn’t vary its demand based on the voltage levels.

[0068] Following a ZIP load model, the node’s active and reactive power demand, ^^^ௗ,థ,ఎand ^^^ௗ,థ,ఎ, respectively, is then obtained by Eq. (4.1)

[0069] The ZIP factors for constant impedance, current, and power load models are defined by ^^^, ^^^and ^^^, respectively, which respect the condition of Eq. (4.2). For this analysis, the factors are considered constant over time and the same across phases. It is also important to highlight that generation is considered to be constant power, and by this, it doesn’t have ZIP factors.

[0070] As the shunt admittance of branch ij, ^^^^ ൌ ^^^^^^^^,థ, andis the susceptance, which is assumed as purely capacitive for nominal-^^ model and the shunt power is also strictly reactive. Being Γ^the set of branches connected to node i, the total shunt and reactive power,, for a given node ^^, phase ^^ and iteration in a nominal-^^ transmission line can be obtained by Eq. (4.3).∀^^ ∈ ^^ே, ^^ ∈ ^^Ф

[0071] Where the superscript indicates the conjugate operator, ^^^,థ,ఎis thevoltage phasor, ^^ே is the set of nodes and ^^Ф is the set of phases ^^^, ^^, ^^^. The nodalTXTU-0005PC operational active and reactive power, ^^^^^,థ,ఎand ^^^^^,థ,ఎ, respectively, will then depend on the node’s demand and the total shunt capacitive reactive contribution from the connected lines, as presented by Eq. (4.4).

[0072] With the graph theory parameters computed for a given network configuration and the line and load models defined, it is possible to proceed and compute the backward / forward sweep power flow. The application of this method is restricted to radial networks, and for this, it is widely used for distribution systems applications. The iterative process is defined by two steps, one backward from the edge nodes to the substation node and one forward from the substation to the edge nodes. Both actions are repeated until the convergence condition is achieved. As an initial condition of the iterative method, the nodal voltage phasors may be guessed. Traditionally, this initialization is made with a flat approach, where nodal voltages are assumed as 1 per-unit (p.u.) of magnitude and 120° between phase angles. However, flat initialization can also be based on the substation voltage measurements, known through TCS measurements, which can improve the convergence fastness. The substation flat initialization is used in the developed DORS. For the sake of simplicity, variables, parameters, and formulations below are based on p.u. values. Backward Action

[0073] The first step of the backward action is to compute the operational active and reactive power for each node based on Eq. (4.1)-(4.4). The computation then proceeds to obtain the total equivalent power for each node, which is defined by the summation of the node’s operational power, the equivalent power of each child node, and the power losses of each line connecting the node to its child nodes. Based on that, the computation may start by analyzing the edge nodes and then move through the network levels until the substation node is reached. For edge nodes, the equivalent active and reactive power, ^^^^andis given by Eq. (4.5)

[0074] As soon as the equivalent power for each edge node is obtained, it is possible to compute the upstream node’s equivalent power. Being ^^ the parent nodeTXTU-0005PC of node ^^, the phasor current flow of branch ^^^^, ^^^^,థ,ఎ, is defined by Eq. (4.6) and the branch active and reactive power losses, and ^^^^^,థ,ఎ, respectively, are defined by Eq. (4.7).∀^^^^ ∈ ^^^ , ^^ ∈ ^^Ф

[0075] Where, ^^^is the set of branches, ^^^^and ^^^^are the resistance and reactance matrices for branch ^^^^, the operatorindicates matrices multiplication, and the subscript “^^” indicates the vector line referent to the phase ^^. Considering the ^^^as the set branches connecting node ^^ to its child nodes, the equivalent power of nodes that are not edge nodes is obtained by Eq. (4.8).Forward Action

[0076] Once nodal equivalent active and reactive powers are calculated up to the substation node, it is possible to perform the forward action. As the substation nodal voltage is known, it is possible to use the equivalent power of the child node to compute the branch voltage drop and then obtain the child node’s voltage level. This computation starts with the substation node and ends in each edge node. Being ^^^^ the branch under computation, the branch’s current flow is defined by Eq. (4.9).∀^^^^ ∈ ^^^ , ^^ ∈ ^^ФTXTU-0005PC

[0077] Being, ^^^^,ఎ ൌ ^^^^^,^,௧ ^^^^,^,ఎ ^^^^,^,ఎ^், where the superscript “^^” indicates thetranspose operator, the child node ^^ voltage phasor is obtained through the parent’s voltage phasor subtracted by the voltage drop on branch ^^^^, as presented by Eq. (4.10).

[0078] This computation happens until edge nodes are reached. With the new voltages for each node, the convergence test is performed, and in case the convergence is not satisfied, the backward and forward steps are computed again based on the latest voltage values. Convergence Conditions

[0079] In iterative methods, convergence is obtained when the maximum difference between the current and the previous iteration results is smaller than the predefined condition, ^^. In the power flow problem, the convergence variable is the voltage phasors on each node, which can be formulated by Eq. (4.11).

[0080] A maximum number of iterations may be defined. In case the computation reaches the maximum number of iterations before the convergence, the computation is stopped without providing a reliable result. In a converged power flow, the main results will be each node’s voltage magnitude and angle.TXTU-0005PC DISTRIBUTION SYSTEMS STATE ESTIMATION Concept and Considerations

[0082] Given a system topology with a set of measurements and their errors as well as a set of variables to be known, the DSSE can compute a desired state of these variables based on an equation that represents the measurements in terms of the state variables. The SE problem is defined by Eq. (5.1). ^^ ൌ ^^^^^^ ^ ^^ (5.1)

[0083] Being ^^ the state variables, and ^^ the measurements that have different levels of errors, ^^. The measurement function, ^^^^^^, is defined by a set of linear equations relating the measurements to the state variables. In the sequence, the DSSE formulation used for this disclosure. Formulation of a Problem to be Solved

[0084] A reference node may be used to solve the problem in power systems state estimation. Even the substation node may be used as the reference node, the reference node is expected to be a balanced source, with 1 p.u. of voltage magnitude and 120° between phase angles, being phase a at 0°. In a Branch-Current (BC) DSSE, branches’ phase current flow are state variables alongside the reference node voltage phasor. Being ℝ^∙^ and ^^^∙^ the real and imaginary operators of a complex number, for a given node ^^ or branch ^^^^ and phase ^^, ^^^^^,థ,௧ ൌthe state vector, ^^, is defined by Eq. (5.2).∀^^^^ ∈ ^^^Where, is the set of branches, and ^^ is the reference node.

[0085] The measurement vector, ^^, is compounded by nodal voltage, branch current flow, and nodal current injection phasors obtained as field, virtual, or pseudo measurements. For a BC formulation, power injection and power flow measurements are converted to current injection and current flow equivalent measurements,TXTU-0005PC respectively. Based on that, ^^ is then represented by a function of ^^, ^^^^^^, as equivalent measurements. Being ^^ே^the set of nodes with voltage measurements, ^^ேௌset of nodes with power demand measurements and ^^^ௌset of branches with power flow measurements, ^^^^^ఎ^ based on the state variables for iteration ^^, ^^ఎ, is given by Eq. (5.3).Where, ^^^,థis the current phasor injection at node ^^,

[0086] Even though adding voltage measurements to the DSSE doesn’t significantly increase the accuracy and performance of the BC-DSSE, more field measurements are preferable. For this disclosure, field measurements are obtained from TCS, meters, and load sensors but could also be obtained from step-voltage regulators, capacitor banks, and other IEDs. For TCS, voltage from both the source and load sides is commonly available along with the power flow, while load sensors only provide power flow measurements. On the other hand, a customer meter can provide the nodal power demand and voltage measurements. The nature of the field voltage measurements depends on the IED capability and accessibility for time synchronization. In this disclosure, it is assumed that every TCS and meter has access to high-accuracy time synchronization. Based on that, these devices are able to provide synchrophasor voltage measurements. In case time synchronization is not available, the DSSE would only be able to consider the voltage magnitude measurement, as the angle would not be reliable for the computation. Virtual measurements are used to constrain the DSSE according to the network topology and configuration, i.e., virtual measurements are used to define zero-injection nodes, which are nodes without load connection. As the third type of measurement, pseudo measurements originate from historical and forecasting load flow and demand profiles, which are known for their lack of accuracy. Virtual measurements are modeled with high weight, small variance, or even an equality constraint. In contrast, pseudo measurements present a low weight and high variance due to the lack of accuracy in forecasting models. Field measurements’ variance depends on several factors relatedTXTU-0005PC to the Current Transformer (CT) and Potential Transformer (PT), equipment, and communication quantization and accuracy but usually presents a variance higher than the virtual measurements and considerably lower than the pseudo measurements. The error related to each measurement is represented by the ^^^, which is independently distributed according to a normal distribution with zero mean and variance given byThe variance matrix, ^^, is a diagonal matrix defined by the variances of each measurement, as shown in Eq. (5.4). ^^ ൌ ^^^^^^^^^^^ଶ௭భ , ⋯ ^^௭ଶ^ಾ^ (5.4)

[0087] The Weight matrix, ^^, is defined as the inverse of the variance matrix, ^^ ൌ^^ି^. Where the superscript “-1” represents the inverse operator. One of the most traditional formulations to solve the SE problem is through the Weighted Least Square (WLS) method. The WLS solution is obtained by minimizing Eq. (5.5).

[0088] The Jacobian matrix, ^^, represents the network model and is built based on the partial derivative of the measurement function ^^^^^^ with respect to each state variable. The process of building the Jacobian is discussed in the next section. The gain matrix, ^^, is then obtained by squaring the Jacobian matrix, as presented by Eq. (5.6). ^^ ൌ ^^் ∗ ^^ ∗ ^^ (5.6)

[0089] By using the Gauss-Newtown method, the state vector, ^^, that minimizes Eq. (5.5) is obtained by solving Eq. (5.7) and (5.8) iteratively.

[0090] Equations (5.7) and (5.8) together are commonly known as the normal equation. As the Jacobian matrix is constant, the normal equation can be reduced to Eq. (5.9).

[0091] Based on that, the BC-DSSE problem starts by defining the equivalent measurements in terms of the state variables. In the sequence, the measurementTXTU-0005PC functions, ^^^^^^, are defined for the Jacobian matrix construction purposes. The Weight matrix, ^^, construction is presented to successfully obtain the Gain matrix, ^^. Equivalent Measurements Reference Node

[0092] Even though the substation node may be used as the reference node, it may be assumed that the node is balanced, with voltage angles displaced 120º and nominal magnitudes. As DSs may be unbalanced, the Thévenin equivalent is used to model the power system’s circuit upstream of the substation node. By definition, the Thévenin equivalent is compounded by a balanced three-phase source and a series impedance that is obtained by the substation three-phase and single-phase short-circuit levels. To illustrate this approach,

[0093] Figure 9 illustrates a network schematic 900 after a reference node addition. It is important to highlight that by using the Thévenin approach, an extra node will be added to the network for the DSSE purpose, the internal balanced node. The number of additional branches will depend on the number of substation nodes in the system, where the Thévenin impedance between each substation and the reference node will depend on each substation’s short-circuit level. With the addition of six state variables based on the reference node’s real and imaginary voltage parts, it is possible to add six new virtual measurements to each substation node for zero-injection constraint.

[0094] Setting the Thévenin internal bus phase a at 0° and being ^^^^,థthe equivalent complex voltage phasor in per-unit, the three-phase reference node voltages are defined by Eq. (5.10).

[0095] These equations may be used as virtual measurements in the formulation, with small variance or equally constrained. By having the three-phase and single-TXTU-0005PC phase short-circuit levels for each substation, the Thévenin equivalent positive and negative sequence impedance is obtained by Eq. (5.11).

[0096] Where, ^^^^^ and ^^^ଶ^ are the positive and negative sequence impedance between the reference node ^^ and the substation node ^^, being ^^ே,ோthe set ofsubstation nodes. ^ത^^ is the nominal phase-to-phase voltage in kV and ^^^ௌ,ଷ^^^is the three-phase short-circuit level in MVA at node ^^. Equation (5.12) presents the zero- sequence impedance, ^^^^^ , computation based on the single-phase short-circuit level in MVA at node ^^, ^^^ௌ,^^^^.

[0097] With the sequence impedance computed based on the short-circuit levels, it is possible to obtain the phase impedance matrix, ^^^^, through symmetrical components transformation, as presented in Eq. (5.13), where ^^ is the phasor rotation operator, with a value of 1∡120°.Complex Voltage Measurement

[0098] In the case of synchrophasor voltage measurement, the real and imaginary parts of this complex measurement are represented directly in the equivalentmeasurement vector, ^^ ^^^. Beingand ^^^,థ,௧the real and imaginary parts of synchrophasor measurement obtained at time ^^, the equivalent complexmeasurement,and ^^^^,థ,ఎ, is obtained by Eq. (5.14)

[0099] In some cases, it may be assumed that voltage measurements are synchrophasors, and by this, there is no need to compute equivalent voltageTXTU-0005PC measurements. Still, to compute the Jacobian terms for complex voltage measurements, it is important to formulate the measurement function for each of these measurements. Being Π^^the set of branches connecting node i to node j, the complex voltage measurement function is based on Kirchhoff’s voltage law and given by Eq. (5.15).

[0100] Performing the matrices multiplication and decoupling real and imaginary parts, it is possible to obtain Eq. (5.16) and (5.17) from Eq. (5.15).

[0101] By taking the derivative of the real and imaginary parts of the equivalent voltages with respect to each state variable, we obtain Eq. (5.18)-(5.25).TXTU-0005PCComplex Current Flow Measurement

[0102] For a given phase ^^ and time ^^,are the measured active and reactive power flow at branch ^^^^, respectively. The equivalent branch current flow phasor for iteration ^^,can be obtained in terms of the measured power flow and estimated node voltage, as presented by Eq. (5.26).

[0103] voltage phasor of node ^^ obtained from the previous iteration. Given that the power measurements are converted into equivalent current measurements, the variances of the measurements may be adjusted through the error propagation theory. Being ^^ଶ^^್ ^,ഝand ^^ଶ್ ೕொ^ೕ^,ഝthe variances related to the power flow measurements, the matrix of equivalent variances is obtained by Eq. (5.27) subjected to the factors ^^^^,థ,ఎand , which are formulated by Eq. (5.28) and (5.29)

[0104] Where, ^^ଶூ^^^ ೕ^,ഝand ^^ଶூ^^^ ೕ^,ഝcompound the diagonal terms as the intrinsic variance of the real and imaginary parts, respectively.is the covariance between^^^^^^^,థ,ఎ, which is placed outside the diagonal according to the line and column of the realTXTU-0005PC and imaginary parts. It is important to highlight that at the first iteration, the estimated voltage is set with a flat initial value, and then the equivalent branch current flow is updated for each iteration based on the most recent estimated voltage phasor. As branch currents are part of the system state variables, their measurement function is defined by Eq. (5.30) and (5.31).

[0105] By taking the derivative of the real and imaginary parts of the current flow measurement functions with respect to each state variable, we obtain Eq. (5.32)- (5.35).Complex Current Injection Measurement

[0106] Considering the nominal-^^ line model, presented in Figure 8, the lines’ shunt capacitance may be accounted for. For a given phase ^^ and time ^^, ^^ௗandare the measured active and reactive power demand at node ^^, respectively. As this is a measurement and the load has a ZIP composition, it is important to compute the new demand based on the ZIP, besides accounting for the total adjacent lines’ shunt capacitance. Being ^^^^,థ,ఎି^the voltage phasor obtained from the previous iteration, thenodal equivalent active and reactive power, ^^ௗ, respectively, is by Eq. (5.36).TXTU-0005PC

[0107] The equivalent current injection phasor for iteration ^^,can be obtainedin terms of the measured power demand and estimated node voltage, as presented by Eq. (5.37).

[0108] It is important to highlight that power demand is defined as current drawn from the node, where generated power is negative and consumed power is positive. However, for the DSSE formulation, the nodal currents are defined as injection, where generated power is positive and consumed is negative. For this reason, a negative sign is added in Eq. (5.37). Similarly to the power flow measurements, as the power demand measurements are converted into equivalent current measurements, the variances of the measurements may be adjusted through the error propagation theory. Being ^^ଶ^ the variances related to the power demand measurements, theொ^^,ഝmatrix of equivalent variances is obtained by Eq. (5.38) subjected to the factors

[0109] Where, ^^ଶூ^^^ ^,ഝand ^^ଶூ^^^ ^,ഝcompound the diagonal terms as the intrinsic variance of the real and imaginary parts, respectively.is the covariance between ^^^^^, which is placed outside the diagonal according to the line and column of theTXTU-0005PC and imaginary parts. Being Eq. (5.41) the current injection measurement function, it can be decoupled into real and imaginary parts, as presented by Eq. (5.42) and (5.43), respectively.

[0110] The derivative of the real and imaginary parts of the current injection functions with respect to each state variable is defined by Eq. (5.44)-(5.47).Jacobian

[0111] With equivalent measurement functions formulated based on state variables and then derivative with respect to each state variable and phase, the final Jacobian matrix, ^^. As the formulation is done for a three-phase system, the Jacobian is mounted by 9 submatrices, where each submatrix represents the relation between the measurements phase and the state variable phase. Equation (5.48) presents a submatrix, where phase ^^ is defined by the measurement phase, and phase ^^TXTU-0005PC represents the state variable phase. With all nine submatrices structured, the Jacobian is defined by Eq. (5.49).

[0112] The proper placement of the partial derivative elements depends on the arrangement of the state vector, ^^, and the equivalent measurements vector, ^^^^^^, presented in Eq. (5.2) and (5.3), respectively. Computation Algorithm

[0113] As the Jacobian and Gain matrices are constant based on the network configuration, they may be only computed whenever there is a change in the switches’ statuses. With this approach, the DORS saves computational time while obtaining DSSE results. On the other hand, equivalent measurements and the weight matrix are updated every iteration as they depend on the estimated nodal voltages obtained from the previous iteration. Bearing in mind that the presented formulation is based on BCs, the state variables obtained by Eq. (5.5) are the voltage of the reference node and the BCs for network branches. With that, a one-iteration forward sweep action may be performed to obtain the nodal voltage based on the estimated BCs. The approach is based on the formulation for the backward / forward sweep power flow method, Eq. (4.10). The DSSE convergence is achieved when the maximum absolution delta forTXTU-0005PC the state variables is less or equal to the convergence condition, ^^, as is defined by Eq. (5.50). ^^^^^^^|^^ఎା^ െ ^^ఎ|^ ^ ^^ (5.50)

[0114] The WLS BC-DSSE problem is computed based on the set of actions to initialize parameters and a set of actions to be computed iteratively. Algorithm 1 presents the WLS-based BC-DSSE with the main goal of obtaining the optimally estimated nodal power demand, where ^^^^௫is the maximum number of iterations allowed for computation. 1 WLS-based BC-DSSE 1: voltage phasors 2: Build Jacobian matrix based on Eq. (5.48) and (5.49) 3: Set iteration counter to ^^ ൌ 14: while ^^ ^ ^^^^௫5: Compute Weight matrix based on Eq. (5.4), Eq. (5.27), and (5.38) 4: Compute Gain matrix based on Eq. (5.6) 7: Compute equivalent measurements based on Eq. (5.14), (5.26), and (5.36) 8: Compute state variables based on Eq. (5.9) 9: Compute nodal voltages based on a forward sweep action 10: if Eq. (5.50) then 11: break 12: else 13: ^^ ൌ ^^ ^ 1State Estimation Results

[0115] As soon as the BC-DSSE algorithm converges to its conditions, theestimated active and reactive power demand on each node,are computed based on the estimated current flows,As discussed, the final state variable vector, ^^ఎ, will result in the optimally estimated real and imaginary parts ofthe network’s branch currents,With this result, the estimated current injectionat each node is initially obtained by Eq. (5.51).

[0116] With the estimated current injection and nodal voltage phasors, it is possible to compute the power demand through Eq. (5.52). As commented before, demand is defined as negative for generation and positive for consumption, but the DSSE nodal currents are considered positive for generation and negative for consumption.TXTU-0005PC Because of this, a negative sign may be added in Eq. (5.52). Also, as the nodal current injection is accounting for the lines’ capacitive shunt, it is important to remove this contribution to accurately obtain the nodal power demand.(5.52) ∀^^ ∈ ^^ே , ^^ ∈ ^^Ф

[0117] The most recent estimated complex power demand for each node is then used for the outage restoration solution. The main benefit of having an estimated power demand is to increase the accuracy of the restoration decisions by having nodal demand levels close enough to the actual system’s operation at time ^^. FAULT LOCATION AND ISOLATION Power Systems Protection

[0118] A protection scheme aims to ensure that the electrical system is maintained stable and reliable by isolating components that are under fault while leaving as much of the network as possible in operation. Power outages are primarily originated from equipment or infrastructure damage, which creates short-circuits and high current levels. Even though many devices may operate during such events, only digital relays and IEDs are able to report measurements and status through communication protocols. Based on that, this disclosure only focuses on digital relay devices as protective devices, but there are other elements responsible for protecting power systems, such as fuses and surge arresters. Protection Functions

[0119] Different protection functions have been established to effectively protect power systems from various electrical disturbances, where each protection function or relay has a specific responsibility and setting. The ANSI / IEEE Standard C37.22022 defines different function numbers, acronyms, and contact designations for power system devices. With a protection scheme set, the action of each relay function provides enough information to understand what happened in the system and if each function acted as expected.TXTU-0005PC

[0120] Most protection functions are defined as instantaneous, temporized, or remote. An instantaneous operation will trip as soon as the measured value reaches the pickup value or after a constant delay. However, there is still a time delay related to the command and circuit-breaker operation. A temporized element will have a tripping time inversely proportional to the measured level, besides accounting for the time delay related to the command and circuit-breaker operation, i.e., the higher the value, the faster the operation will be. Remote operation is mostly defined as an instantaneous operation that is triggered from a remote device other than by local measurements. Even though protection systems for transmission are considerably different from distribution systems, below are some of the most common protection functions used on DSs that may also be applied to transmission systems:

[0121] Under-Voltage Relay (ANSI 27): The ANSI 27 protects the system from under-voltage levels that can damage equipment or create critical operational conditions. This function can be set as instantaneous, temporized, or even remote. Instantaneous and temporized tripping will act according to the local voltage measurements.

[0122] Directional Power Relay (ANSI 32): The ANSI 32 mostly protects the system from reverse power flow levels that may impact other subsystems or protection reliability. This function can be set for both active and reactive power flows and is usually instantaneous based on a maximum power level pickup.

[0123] Instantaneous Over-Current Relay (ANSI 50): The ANSI 50 protects the system from over-current levels caused by short-circuits or overloading that can damage equipment or create critical operational conditions. This is an instantaneous element, which will trip as soon as the measured current exceeds the pickup or after a constant time delay.

[0124] Temporized Over-Current Relay (ANSI 51): The ANSI 51 protects the system from over-current levels caused by short-circuits or overloading that can damage equipment or create critical operational conditions. This temporized element allows coordination between other protection devices. As soon as the measured current exceeds the pickup, the temporization begins and lasts according to the current level.TXTU-0005PC

[0125] Over-Voltage Relay (ANSI 59): The ANSI 59 protects the system from overvoltage levels that can damage equipment or create critical operational conditions. This function can be set as instantaneous, temporized, or even remote. Instantaneous and temporized tripping will act according to the local voltage measurements.

[0126] Automatic Reclosing Relay (ANSI 79): The ANSI 79 defines time intervals to perform automatic reclosing operations. It usually has up to 4 reclosing attempts, where each one has its own pre-defined open interval time. Defining different ANSI 50 / 51 curves for each reclosing is a common practice. By using fast and delay curves, coordination between reclose-reclose or reclose-fuse can be obtained. Protection Operation

[0127] While ANSI 27 / 59 mostly operates locally, based on the protective device nodal voltage measurements, ANSI 50 / 51 operates based on current flow measurements. As the short-circuit level is higher closer to the substation and lower at the end of the feeder according to the equivalent impedance to the substation, over- current protection is set to protect zones. Distribution protection usually follows a zone scheme, where each protective device covers a predetermined zone and serves as a backup to other zones.

[0128] Figure 10 is a diagram 1000 illustrating radial over-current protection with two relays and their zones. In this scenario, Relay 1 (^^^) is connected upstream Relay 2 (^^ଶ), where ^^^primarily protects the zone between nodes 1 and 2 and then provides backup protection to ^^ଶ’s primary zone. The short-circuit level increases from right to left, i.e., from the end of the feeder to the substation. This means that a fault between ^^^and ^^ଶwill have a higher short-circuit level than a fault between ^^ଷand ^^ସ. At the same time, a fault upstream ^^ଶwill only be measured and seen by ^^^and not by ^^ଶ, while a fault downstream ^^ଶwill be seen by ^^^and ^^ଶ, as the substation will be feeding this short-circuit. Based on that, ^^^can serve as backup for ^^ଶin case it misses or fails in operating.

[0129] Figure 11 is a diagram 1100 illustrating radial over-current protection to coordinate ^^^and ^^ଶoperations and protect the feeder with primary and backup protection zones. The ANSI 50 is usually designed to cover 70-90% of the primaryTXTU-0005PC zone, so measurement errors can be accounted for and misoperation is avoided. Then, ANSI 51 is used to protect the remaining 10-30% of the section and also works as a backup to the downstream zone. As presented in Figure 11, ANSI 51 pickups, ^^ௌR,ହ^, are set low and even outside the feeder reach, which allows high-impedance fault sensing. However, ^^ଶhas a faster operation than ^^^for faults between ^^ଷand ^^ସ, where ^^ଶhas instantaneous tripping for faults within 70-90% of its zone and then ANSI 51 for the remaining parcel. Regarding the zone between ^^^and ^^ଶ, ^^ଶis looking into this zone, but there are protection schemes where ^^ଶcould also provide reverse protection in case of DERs connection in ^^ଷand ^^ସ. Disregarding DERs for now, ^^^would trip instantaneously for a fault within 70-90% of its zone and then temporized for the remaining part. Even though not shown in Figure 11, it is most likely that ^^^has a backup relay located upstream of its location, which will provide a first backup to ^^^and a secondary backup to ^^ଶ.

[0130] It is important to highlight that protective relays are designed to properly identify and interrupt short-circuit currents, and in case of a misoperation, there are other backup protective devices that will ensure the fault current is extinguished, even though with a temporization delay. Field protection schemes are set to protect in different directions, which is important when DNR reconfiguration and DERs are considered. Even though DSs are operated radially, DNR can change its radial configuration over time, which can drastically change protection zones and the relation between protective devices. The network configuration change may create an operation scenario where the voltage levels are outside the ANSI 27 / 59 settings, creating trippings. New network configurations present different loading and short- circuit levels. In case the loading is increased to unexpected levels to the protective relay, the ANSI 50 / 51 protection functions may trip for load conditions. The reconfiguration may provide new short-circuit levels to the zones. If the new minimum short-circuit level for a given zone is smaller than its relay pickup, the relay may not operate under a fault event. ANSI 32 is another element used to avoid reverse power flow in the system, which may trip under a new reconfiguration that results in providing reverse power at the device’s point of connection. This disclosure accounts for ANSI 27 / 59, 50 / 51, and 79, as these are the most common functions in a DS protection scheme, although other relay types may be used.TXTU-0005PC Fault Detection

[0131] As soon as a fault happens in the system, relays may trip according to their protection settings and pickups, which will raise a tripping flag. If an Automatic- Reclosing protection function (ANIS 79) is set, relays may also attempt to reclose to reestablish service in case of temporary faults. In this scenario, the relay will have a successful reclosing, and no further action is needed as there is no outage. However, relays may cycle as many times as they have been set to for permanent faults until reaching a lockout state where the relay stays open. An outage is first identified by DORS through the lockout of one or multiple protective devices, which may take a couple of seconds for the information to reach the Supervisory Control and Data Acquisition (SCADA). It is common practice to have another time delay from receiving the first lockout to start any analysis. This allows enough time to collect information on the post-fault network configuration when the reclosers’ cycles and the protection scheme are completed. Besides the lockout flag, a fault indicator flag is used by DORS. As soon as a fault happens, protective relays that are in the path of this fault will measure high-current values, meaning that the short-circuit is being fed through its path. The fault indicator flag identifies which switches sensed these short-circuit current levels during the event. Fault Location, Miscoordination Analysis, and Fault Isolation

[0132] Due to the usual high load current levels and the trend of increased numbers of reclosing devices to improve reliability, protection coordination is becoming more challenging in distribution networks. Even with advanced protection philosophies, miscoordination between protective devices is commonly expected to happen. The miscoordination is due to a lockout of a protective device outside the fault zone, which implies that the device expected to trip for the fault may cycle, but an upstream device locks out first. Techniques have been developed to identify miscoordination from a protection point-of-view. However, it is important to integrate miscoordination detection into fault isolation, as many customers may not be re-energized when a protection scheme misbehave occurs.

[0133] TCSs may or may not be protective devices, which implies the possibility of a non-protective TCS being the closest device to the fault location. This aspect bringsTXTU-0005PC the concept of cells instead of zones. Cells are a set of interconnected nodes and branches bounded by a set of TCS, while zones are bounded by protective relays. Assuming that new switches won’t be installed in the network during the analysis period, sets of nodes and branches forming each cell and zone can be computed beforehand with graph theory techniques.

[0134] It is possible to develop a centralized control model to locate the exact TCS that should be responsible for isolating the fault independent of which relay has locked out. Even though not all TCS may have fault indicators,and ^^^^^^,௧, this information can be obtained based on the available electrical measurements from these devices. ^^^^^^,௧indicates if the TCS has sensed the short-circuit current. Being ^^^andthe maximum current level, respectively, through Eq. (6.1), it is possible to compute ^^^^^^,௧.

[0135] Where, ^^^^^,థ,௧is the TCS measured current phasors. It is important to highlight that although reclosers can have a single-pole or three-phase operation, this disclosure considers only a three-phase operation, where all three phases are open or closed together. The reason to assume this is that most utilities use three-phase operation, as single-pole operations can bring severe issues with network unbalancing.

[0136] Being, Π^^the set of branches connecting the substation node s to node j based on the pre-fault network configuration, and ^^ௌthe set of TCSs’ branches, Algorithm 2 presents the computation sequence to locate the closest TCS to the fault per feeder. The main outputs are two sets of switches, one to be open and another to be closed, ^^ௌூ^,ைௌ, and ^^ௌூ^,^ௌ, respectively, as part of the ICS. Algorithm 2 Fault Location and Miscoordination Analysis Input ^^^^^^,௧, ^^^^^^,௧Output ^^ௌூ^,ைௌ, ^^ௌூ^,^ௌProcedure 1: ^^ூ^ௌௌ,ை ൌ ^ ^2:^^ ூ^ௌௌ,^ ൌ ^ ^3: ^^^^^^_^^^^^^^^ℎ ൌ 0TXTU-0005PC 4: ^^^^^^_^^^^^^^^^^ℎ ൌ 05: for ^^^^ in ^^ௌdo 4: if ^^^^^^,௧then 7: if |Π^^ | ^ ^^^^^^_^^^^^^^^ℎ t8:9:10:11: ^^^^^^,௧12: if ^^^^ ൌ ^^^^^^_^^^^^^^^^^append ^^^^ to ^^ௌூℎ^then13:,ைௌ14: else 15: append ^^^^ to ^^ௌூ^,^ௌ

[0137] From lines 5 to 9, the algorithm identifies the furthest downstream TCS, in reference to the pre-fault network configuration, that reported a fault indication. The remaining algorithm locates and appends the furthest downstream switch to the ^^ௌூ^,ைௌ, and the remaining switches in that feeder that felt the fault should be appended to closed. With that, the miscoordination reclosers will be part of the ^^ௌூ^,^ௌ, which will be closed back in the appropriate control sequence. The SCS will then be responsible for checking the switch status and only sending the command in case the command is to change its current status.

[0138] By identifying the switch responsible for the fault isolation and using the pre- fault network configuration, it is possible to locate the outage cell. The cells’ outage status variable, ^^^௫,௧, is then updated to 0 in case of an outage, otherwise to 1. The outage cell’s boundary switches are appended to ^^ௌூ^,ைௌ, so they can be opened to isolate the fault. With this proposed approach, the restoration will only maintain de- energized the actual outage cell and not the lockout protection zone(s). Switching Control Sequence

[0139] The SCS is a technique used to coordinate a sequential order of switching operations to achieve a new configuration from an initial state that respects constraints, such as looping the network and closing it into the fault. The present SCS technique is responsible for performing both ICS and RCS. Isolation Control Sequence

[0140] As soon as the faulted cell(s) is located, the SCS computes the steps to isolate the cell and then reestablish the miscoordinated devices. This process happensTXTU-0005PC based on ^^ௌூ^,ைௌand ^^ௌூ^,^ௌ. The algorithm initially opens the switches in the ^^ௌூ^,ைௌ, and monitors their state until the operations are completed. As soon as these TCSs are in an open state, the system sends closing commands to TCSs in ^^ௌூ^,^ௌ. The restoration optimization is computed while the ICS is happening, giving the system enough time to operate its TCSs and have the faulted cell(s) isolated before the outage restoration solution is obtained. Along with it, the restoration optimization is computed based on the fault isolation results. As soon as the switches responsible for the isolation and the miscoordinated devices are identified, the optimization computation uses this information to optimize the post-isolation network reconfiguration, even though the ICS is taking place in parallel with this computation. Restoration Control Sequence

[0141] The restoration process presented in the following section provides the state for each branch, ^^^^^,௧, which defines a desired restoration network configuration. Being the branches’ statuses after the isolation, a set of switches that should be open and closed, ^^ௌோ,^ைௌand ^^ௌோ,^^ௌ, to perform the restoration is computed by Algorithm 3. Algorithm 3 Restoration Control Sequence Input ^^^^^,௧, ^^^^^,௧ି^Output ^^ௌோ,^ைௌ, ^^ௌோ,^^ௌProcedure 1:^^ோ^ௌௌ,ை ൌ ^ ^2:^^ோ^ௌௌ,^ ൌ ^ ^3: for ij in ^^ௌdo 4:if ^^^^^,௧^ ^^^^^,௧ି^then 5: ij appended to ^^ௌோ,^ைௌ6:else if ^^^^^,௧^ ^^^^^,௧ି^then 7:ij appended to

[0142] The algorithm logic will compare the current and previous status of each branch that has a switch. In case the previous state is closed and the current is open, an open command is appended to ^^ௌோ,^ைௌ. Similarly, in case the previous state was open, and the current one is closed, then a close command is emitted by appending the switch

[0143] With the RCS computed, devices in ^^ௌோ,^ைௌwill first be opened, and then in ^^ௌோ,^^ௌwill be closed. This stage-by-stage SCS approach is capable of avoidingTXTU-0005PC operation scenarios that may cause system instability and / or cascade tripping, such as network looping, closing into the fault, and re-energizing customers that are not part of the restoration scenario. OUTAGE RESTORATION

[0144] The restoration problem aims to obtain the switches’ statuses to define a new network configuration that can reduce the total unserved kW and can be achieved with a reduced amount of switching maneuvers. The solution should provide radiality along with post-reconfiguration current flow and nodal voltage levels within the operational limits.

[0145] The approach is formulated as a deterministic Multi-Objective Optimization (MOO) Mixed-Integer Linear Programming (MILP) optimization, where the continuous parameters, such as power demand, are fixed and known through the DSSE results, while the current flow and nodal voltage will be determined by the solution along with the integer status of each switch. Having the nodal power demand as the main input parameter, the solution is constrained by equality and inequality relations that represent the network topology, power flow, and operation limitations according to each parameter and variable. The exact equations are linearized by techniques presented in the literature to provide a linear model that can be fast and easily solvable by commercial solvers without compromising the solution’s accuracy.

[0146] As discussed before, the restoration problem is computed based on the network configuration after the isolation, even though the isolation may not have yetbeen completed in the field. For that, ^^ െ 1 represents the time when the isolation andmiscoordination are completed, and ^^ represents the time for the restoration solution. As the DORS analysis is designed to be computed and successfully performed within5 minutes, the ^^^^ௗ,థ,௧is considered the same since the latest DSSE analysis, and bythis,

[0147] Being ^^^^^,௧the branch ^^^^ status and ^^^^,௧ the node ^^ status withactive power demand, Eq. (7.1) presents the optimization objective.TXTU-0005PC

[0148] To reduce both the amount of unserved power and the quantity of switching maneuvers, ^^^,^and ^^ଶweights are used to normalize the two objectives into costs. ^^^,^is the value of lost load (VOLL) of node i, which depends on several social- economical aspects of the region and varies according to the type of customers connected to node ^^. The VOLL was presented for different commercial and industrial (C&I) and residential customers during the winter of 2023. Below are the VOLL: ^ Medium and large C&I customers: $17.2 / kW ^ Small C&I customers: $132.3 / kW ^ Residential customers: $0.9 / kW

[0149] Medium and large C&I customers present a smaller VOLL than small C&I because they commonly have backup generation resources, while small C&I mostly rely on the utility supply. Residential customers have a small VOLL, but when considering the large number of customers connected per node, their outage event can significantly impact the total utility VOLL. ^^ଶis the cost per switching operation, which is not usually provided by the literature, but this disclosure calculates it based on the device’s initial cost, planning horizon, and endurance. A typical TCS has an average endurance of 10,000 operations, an initial cost of approximately $15,000, and an annual maintenance cost of 2% of the annualized investment. Considering 15 years as a planning horizon and an inflation rate of 3%, the cost for each operation is $1.86, as presented by (7.2). ^ହ15,000 ^^ ൌ∙ ^1 ௬ଶ10,000^ 0.02 ∙െ 0.03^ ^ ≅ 1.86 $ / maneuver (7.2)

[0150] It is important to highlight that even though the cost per switching operation is significantly smaller than the VOLL, during a restoration, the main goal is to reestablish power as fast as possible. Hence, this term is important to ensure a configuration closer to the pre-fault network configuration, which does not require too many maneuvers to be achieved.TXTU-0005PC Cell Constraints

[0151] Distribution networks are organized in cells, which are groups of nodes and branches bounded by TCS. This concept defines that the energization or de- energization of a cell implies the energization or de-energization of the nodes and branches within the cell. Considering that outage detection and isolation happen prior to the restoration computation, it is important to constrain the restoration model to de- energize branches and nodes within the fault cell. Beingthe binary variable to represent the energization status of cell ^^ at time ^^ and ^^^௫,௧is a binary parameter that represents the outage status of cell ^^ at the end of the ICS actions, time ^^, the outage constraint is defined by Eq. (7.3).

[0152] In case an outage is identified in cell ^^,will be 0, and then the cell status will be forced to become 0. In case cell ^^ doesn’t contain an outage, its status will be defined by the optimization algorithm as energized (^^^^,௧ ൌ 1), or de-energized(^^^^,௧ ൌ 0). Beingthe set of nodes within cell ^^, andthe binary variable for the energization status of node ^^, the nodes’ statuses based on the cell energization are defined by constraint in Eq. (7.4). Similarly, being ^^^^,ேௌthe set of non-switchable branches within cell ^^ and ^^^^^,௧the binary variable for the energization status of branch ^^^^, the branches’ energization statuses are defined by Eq. (7.5).

[0153] Beingthe set of switchable branches bounding cell ^^, the status of switches depends on the status of the cell’s outage and not on the cell’s energization status, as presented by Eq. (7.6).

[0154] With this relation, the TCSs bounding cell o will be forced to open (de- energized) whenever there is an outage within the cell. Otherwise, the TCSs’ statuses will be defined by the optimization solution as closed or open. Operational ConstraintsTXTU-0005PC

[0155] Whenever a node is energized, its voltage level may be maintained within acceptable limits. Similarly, branch currents may be maintained under overcurrent protection pickups and cabling thermal limits. As the nodal voltage magnitude is a variable, its squared value becomes a non-linearity in the model. Based on that, a new variable, can be used to linearize the model. Beingthe model is linear, and the results for voltage magnitude can be obtained by simply getting the squared root

[0156] Equation (7.7) defines nodal voltage for substation nodes based on the DSSE results. Being ^^^and ^^ெthe minimum and maximum operational voltage limits in p.u., respectively, Eq. (7.8) ensures that voltage levels of energized nodes are within these boundaries. According to ANSI, ^^^and ^^ெare defined as 0.95 and 1.05 p.u, respectively, but they can also be defined based on ANSI 27 / 59 settings.

[0157] by Eq. (7.9), where ^^^ெ^ is the maximum allowed current flow in branch ^^^^. Where, ^^^^,థis the branch’s phase connectivity indicator.

[0158] As the active and reactive power flows will be a result of the optimization, it is important to limit their values to the maximum apparent power flow,, which also depends on the conductor’s mechanical and thermal limitations, but it is not easily decoupled into active and reactive power limitations. However, the apparent power, , equation compounds a circle equation,^ ^^^^,థ,௧ ൌ,which is a non-linear constraint. A linearization approach starts by using a conic approach, defined by Eq. (7.10), which relates the voltage and current with active and reactive power flows.

[0159] A linearization model to limit the apparent power in terms of active and reactive power is present by approximating the circle by a hexagon. Initially, the high-TXTU-0005PC maximum apparent power,is computed based on the actual maximum apparent power flow, , of branch ^^^^, phase ^^, as presented in Eq. (7.11).

[0160] Where, ^^ is the number of vertices of the polygon, selected as 6 for a hexagon approximation. Note that,is still a constant based on the limitations of the branch conductors, and for that, it can be computed just once before any iteration. In this formulation, the branch’s active power,can still be limited

[0161] The branch’s reactive power, ^^^^^,థ,௧, is then limited by the polygon boundaries in terms of ^^ெ,ுas presented by Eq. (7.13)-(7.15)Power Flow Constraints

[0162] Based on the Linear Distflow formulation, Kirchhoff’s current law may be considered to ensure nodal power balance. The law defines that the total power leaving the node may equal the total power arriving at the node, accounting for branch power flows and nodal power demand. As the branch power flows are variables resulting from the optimization, the nodal power demand is a constant parameter defining the optimization solution. As the power balance constraint applies to the network nodes, energized or de-energized, it is extremely important to have accurate power demand value to the real state of the network. Otherwise, the problem will leadTXTU-0005PC to optimizing a scenario inconsistent with the actual network and customers’ consumption. Based on that, DORS uses the DSSE results for nodal power demand at this step to improve the outage restoration performance and ensure that the optimization is performed as accurately as possible to the actual system state. However, if DSSE data is unavailable, other demand levels can be used as soon as it is understandable that the solution may lead to a false solution, which doesn’t optimize the actual system.

[0163] By decoupling active and reactive power, Eq. (7.16) and (7.17) present the nodal power balance equations for active and reactive power, respectively, where power demand is defined as negative for generation and positive for consumption.Also, being ^^^^,௧ ൌ ^^^^^,^,௧ ^^^^,^,௧ ^^^^,^,௧^், the power losses between each adjacentbranch are accounted.

[0164] In this proposed MILP model, the load’s ZIP configuration is neglected. As presented in Eq. (4.1), the constant current and constant impedance ZIP load models depend on the voltage magnitude and its squared value, respectively. By using ଶ variable substitution for the voltage, ^^^,థ,௧= ห^^^,థ,௧ห , the square root of ^^^,థ,௧would be a non-linearity in the system. The literature has proposed optimization models to account for ZIP load models to improve voltage or power losses, which can be significantly impacted by the load model and shunt contribution. However, for restoration purposes, the goal is to re-energize as many customers as possible instead of accurately computing voltage and power losses.

[0165] Kirchhoff’s voltage law is another constraint that may be considered inpower systems optimization. Being, ^^^^,௧ ൌ ^^^^^் ^,^,௧^^^^^,^,௧^^^^^,^,௧൧ and ^^^^,௧ൌTXTU-0005PC ^^^,^,௧ ^^் ^^ ^^^,^,௧൧ , Eqs. (7.18) and (7.19) define the voltage drop between two nodes based on the branch characteristics and the active and reactive power flow.

[0166] To identify a satisfactory solution, the Big-M concept is utilized to relax the constraints (7.18) and (7.19). If branch ^^^^ is energized (^^^^^,௧ ൌ 1), and there is an actualconductor for phase ^^ൌ 1), then the M term will be disregarded, where thatbecomes zero. On the other hand, when the branch is de-energized (^^^^^,௧ ൌ 0) or thereis no phase conductor (^^^^,థ ൌ 0), ^^^^^,థ,௧andwill be zero, according to (7.12)- (7.15), and then Kirchhoff’s law is not applicable to the branch. In a Kirchhoff voltage law formulation for a three-phase unbalanced distribution systems optimizationproblem equivalent branch matrices of resistance, ^^^^^, and reactance, ^^^^^, efficientlydecouple voltage phase and magnitude. It is assumed that voltage magnitudes are similar between phases, and the phase unbalancing is not too severe, so the voltages are nearly balanced across the phases. This consideration enables the phase difference approximations presented by Eq. (7.20) and (7.21).

[0167] Where,is the current phase of branch ^^^^ for phase ^^, and ^^ is relative phase unbalance. Based on the approximations, Eq. (7.22) and (7.23) present the formulation to obtain the equivalent resistance and reactance matrices in a three- phase system with relatively small unbalancing, respectively.TXTU-0005PC

[0168] Where the superscript “^^” is the Hamiltonian operator, “^” denotes the Hadamard product (dot product) operator, and ^^ is the vector of relative phaseunbalance per phase, defined as ^^ ൌ ^1Radiality Constraints

[0169] Formulating radiality constraints in DSs can be challenging, as the network can have a highly meshed topology and connections to different substations. By defining two binary variables for each branch indicating forward,, and ^^^ି^,௧, power flow directions, it is possible to ensure a network that corresponds to a spanning tree regardless of the power flow direction. Equation (7.24) restricts reverse power flow on branches ^^^^ departing from substation nodes. Constraint (7.25) defines that each couple of nodes defining a branch connection should respect the parent and child relation if they are energized.

[0170] In case of node ^^ is the parent of node ^^ൌ 0) or node ^^the parent of node ^^ൌ 0 and ^^ି^^,௧ ൌ 1), then the node should be consideredenergized (^^^ ൌ 1), otherwise is de-energized (^^ ^ ି^^,௧ ൌ 0ൌ 0 and ^^^^,௧ ൌ 0)

[0171] Equation (7.26) imposes that every energized node has only one parent node, from which it receives power. At the same time, each node may have none, one, or multiple child nodes. Analogically, de-energized nodes do not have any parent or child nodes.

[0172] It is important to highlight that Eqs. (7.24)-(7.26) provide radiality in small and large networks that may have connectivity to one or several substations while respecting the de-energized nodes by the solution. However, nodes’ connectivity is imposed by the power flow equations, Eqs. (7.16)-(7.19).

[0173] Figure 12 is a flow diagram illustrating example operations 1200 for power restoration, in accordance with certain aspects of the present disclosure. TheTXTU-0005PC operations 1200 may be performed, for example, by a processing system, such as the processing system 1305 described with respect to Figure 13.

[0174] At block 1202, the processing system may estimate one or more signal measurements of one or more nodes of a power distribution network using state estimation. An estimated signal measurement may include an estimated electrical characteristic (e.g., voltage, current or power) at the one or more nodes. The one or more signal measurements may be estimated based on at least one of field electrical measurements, switch status information, forecasting data, historical data, or network topology information. The one or more signal measurements may be estimated based on different sets of node measurements for the power distribution network and weights associated with the different sets of node measurements. For example, the different sets of node measurements may include pseudo-node measurements for the power distribution network and field node measurements for the power distribution network. The pseudo node measurements may include at least one of historical node measurements or forecasting measurements. The one or more signal measurements may be estimated using an equation that relates, to each other, state variables associated with the different sets of node measurements based on the weights.

[0175] At block 1204, the processing system may identify a location of an outage in the power distribution network. The one or more signal measurements may be estimated prior to an occurrence of the outage. For example, the one or more signal measurements may be estimated using the state estimation stored in a database (e.g., database 514) before the occurrence of the outage. The processing system may retrieve the estimated one or more signal measurements from the database after the occurrence of the outage to identify the status of the one or more switches that form a network configuration to restore power to one or more cells of the power distribution network.

[0176] At block 1206, the processing system may identify a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation and the location of the outage.TXTU-0005PC

[0177] At block 1208, the processing system may output control signaling to control the one or more switches based on the identified status. For example, the one or more processors may output the control signaling to perform distribution network reconfiguration based on the location of the outage and the one or more signal measurements estimated using state estimation. Example Computing Device

[0178] Figure 13 depicts aspects of an example computing device 1300. The computing device 1300 includes a processing system 1305. The processing system 1305 may be configured to perform processing functions for outage restoration as described herein.

[0179] The processing system 1305 includes one or more processors 1310. The one or more processors 1310 are coupled to a computer-readable medium / memory 1325 via a bus 1340. In certain aspects, the computer-readable medium / memory 1325 is configured to store instructions (e.g., computer-executable code) that when executed by the one or more processors 1310, cause the one or more processors 1310 to perform the operations 1200 described with respect to Figure 12, or any aspect related to it. Note that reference to a processor performing a function may include one or more processors 1310 performing that function.

[0180] In the depicted example, computer-readable medium / memory 1325 stores code (e.g., executable instructions), such as code for estimating 1345, code for identifying 1350, and code for outputting 1355. Processing of the code for estimating 1345, code for identifying 1350, and code for outputting 1355 may cause the computing device 1300 to perform the operations 1200 described with respect to FIG. 12, or any aspect related to it.

[0181] The one or more processors 1310 include circuitry configured to implement (e.g., execute) the code stored in the computer-readable medium / memory 1325, including circuitry such as circuitry for estimating 1315, circuitry for identifying 1320, and circuitry for outputting 1321. Processing with circuitry for estimating 1315, circuitry for identifying 1320, circuitry for outputting 1321 may cause the computing device 1300 to perform the operations 1200 described with respect to FIG.12, or any aspect related to it.TXTU-0005PC

[0182] While the foregoing is directed to embodiments of the present disclosure, other and further embodiments of the disclosure may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.

Claims

TXTU-0005PC What is claimed is:

1. A method for power restoration, comprising: estimating one or more signal measurements of one or more nodes of a power distribution network using state estimation; identifying a location of an outage in the power distribution network; identifying a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation and the location of the outage; and outputting control signaling to control the one or more switches based on the identified status.

2. The method of claim 1, wherein the one or more signal measurements are estimated further based on at least one of field electrical measurements, switch status information, forecasting data, historical data, or network topology information.

3. The method of claim 1, wherein the control signaling is output to perform distribution network reconfiguration based on the location of the outage and the one or more signal measurements estimated using state estimation.

4. The method of claim 1, wherein the one or more signal measurements are estimated prior to occurrence of the outage.

5. The method of claim 1, wherein the one or more signal measurements are estimated based on different sets of node measurements for the power distribution network and weights associated with the different sets of node measurements.

6. The method of claim 5, wherein the different sets of node measurements include pseudo-node measurements for the power distribution network and field node measurements for the power distribution network.

7. The method of claim 6, wherein the pseudo-node measurements include at least one of historical node measurements or forecasting measurements.TXTU-0005PC 8. The method of claim 5, wherein the one or more signal measurements are estimated using an equation that relates, to each other, state variables associated with the different sets of node measurements based on the weights.

9. The method of claim 1, wherein the one or more signal measurements estimated using the state estimation are stored in a database before occurrence of the outage, the method further comprising retrieving the estimated one or more signal measurements from the database after occurrence of the outage to identify the status of the one or more switches that form a network configuration to restore power to one or more cells of the power distribution network.

10. A non-transitory computer-readable medium storing instructions that, when executed by a processor, cause a computer system to perform the steps of: estimate one or more signal measurements of one or more nodes of a power distribution network using state estimation; identify a location of an outage in the power distribution network; identify a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation and the location of the outage; and output control signaling to control the one or more switches based on an identified status.

11. The non-transitory computer-readable medium of claim 10, wherein the one or more signal measurements are estimated further based on the at least one of field electrical measurements, switch status information, forecasting data, historical data, or network topology information.

12. The non-transitory computer-readable medium of claim 10, wherein the control signaling is output to perform distribution network reconfiguration based on the location of the outage and the one or more signal measurements estimated using state estimation.TXTU-0005PC 13. The non-transitory computer-readable medium of claim 10, wherein the one or more signal measurements are estimated prior to occurrence of the outage.

14. The non-transitory computer-readable medium of claim 10, wherein the one or more signal measurements are estimated based on different sets of node measurements for the power distribution network and weights associated with the different sets of node measurements.

15. The non-transitory computer-readable medium of claim 14, wherein the different sets of node measurements include pseudo-node measurements for the power distribution network and field node measurements for the power distribution network.

16. The non-transitory computer-readable medium of claim 15, wherein the pseudo-node measurements include at least one of historical node measurements or forecasting measurements.

17. The non-transitory computer-readable medium of claim 14, wherein the one or more signal measurements are estimated using an equation that relates, to each other, state variables associated with the different sets of node measurements based on the weights.

18. The non-transitory computer-readable medium of claim 10, wherein: the one or more signal measurements estimated using the state estimation are stored in a database before occurrence of the outage; and the non-transitory computer-readable medium further store instructions that, when executed by the processor, cause the computer system to retrieve the estimated one or more signal measurements from the database after occurrence of the outage to identify the status of the one or more switches that form a network configuration to restore power to one or more cells of the power distribution network.

19. An apparatus for power restoration, comprising: at least one memory comprising computer-executable instructions; andTXTU-0005PC one or more processors configured to execute the computer-executable instructions and cause the apparatus to: estimate one or more signal measurements of one or more nodes of a power distribution network using state estimation; identify a location of an outage in the power distribution network; identify a status of one or more switches that form a configuration to restore power to one or more cells of the power distribution network based on the one or more signal measurements estimated using state estimation and the location of the outage; and output control signaling to control the one or more switches based on an identified status.

20. The apparatus of claim 19, wherein the one or more processors are configured to execute the computer-executable instructions and cause the apparatus to estimate the one or more signal measurements further based on at least one of field electrical measurements, switch status information, forecasting data, historical data, or network topology information.

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