Power distribution network elasticity improvement method based on distributed dynamic recovery

Abstract

The invention relates to the technical field of power distribution networks of power systems, in particular to a power distribution network elasticity improvement method based on distributed dynamic recovery, which comprises the following steps: step 1, acquiring operation data of a power distribution network; 2, establishing a topology reconstruction and rolling optimization dual-stage recovery model, taking maximized load recovery as a target, considering network constraints and resource operation constraints, and converting a probability model into a deterministic mixed integer linear programming problem through opportunity constraint conversion; 3, decomposing the model into distributed sub-problems by adopting an alternating direction multiplier method, introducing a boundary variable compensation mechanism to enhance the robustness under communication interruption, and realizing a multi-period dynamic recovery decision based on rolling optimization; step 4; an integer variable processing method combining projection and relaxation iteration is designed, and an induced acceleration objective function is constructed, so that the solving efficiency of a mixed integer sub-problem is improved, and the convergence and calculation speed of a distributed algorithm are ensured.

Description

Technical Field

[0001] This invention relates to the field of power system distribution network technology, specifically a distribution network resilience enhancement method based on distributed dynamic recovery, which is used to improve the self-healing capability and power supply reliability of the distribution network under extreme events or fault conditions. Background Technology

[0002] The high proportion of renewable energy integration provides crucial support for enhancing the resilience and flexibility of distribution systems under extreme fault conditions. However, the significant randomness and uncertainty of renewable energy output pose serious challenges to the formulation and implementation of distribution network load restoration strategies. For the load restoration problem of flexible distribution networks incorporating smart soft switches, existing research has proposed stochastic and robust restoration methods to address the impact of renewable energy uncertainties. However, stochastic restoration methods typically rely on extensive scenario modeling, resulting in high computational complexity and difficulty in meeting the real-time requirements of practical engineering applications. While robust restoration methods can guarantee the feasibility of restoration strategies under uncertain conditions, their strong conservatism can easily lead to reduced system resource utilization, thus limiting the flexibility and adaptability of restoration strategies.

[0003] Furthermore, most existing load restoration methods that consider the uncertainties of renewable energy adopt a centralized optimization model. In multi-regional interconnected distribution networks, this requires the aggregation of global information, making it difficult to balance privacy protection and communication reliability, and resulting in poor system scalability. Meanwhile, the flexible distribution network restoration problem typically involves a large number of discrete decision variables related to network reconfiguration, load switching, and equipment operating status, leading to strong non-convex characteristics in the optimization model. Existing distributed restoration models still have significant shortcomings in terms of convergence and computational efficiency, requiring further improvement. Summary of the Invention

[0004] To address the aforementioned issues, this invention proposes a distributed robust recovery method for flexible distribution networks that incorporates iterative and acceleration technologies. This method introduces intelligent soft switches with integrated energy storage and combines them with local distributed energy resources to achieve coordinated recovery control of multiple power sources and flexible equipment. During the recovery strategy formulation process, a distributed robust optimization method is introduced to promote the participation of intermittent renewable energy in load recovery without significantly increasing the computational burden, thereby improving the overall system resilience and resource utilization efficiency.

[0005] To address the coordinated restoration requirements of multi-regional distribution networks, this invention constructs a distributed robust restoration framework based on the alternating direction multiplier method. This framework enables the established robust restoration model to be solved in a distributed manner, achieving coordinated restoration and information decoupling among multiple regions. To ensure the good computability of the restoration model containing a large number of integer decision variables, this invention designs a solution process combining relaxation iteration and induced acceleration. This achieves the fusion and nesting of the distributed robust restoration model and the alternating direction multiplier method, significantly improving the algorithm's convergence performance and solution efficiency, and enhancing the method's engineering applicability in complex flexible distribution network restoration scenarios.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A method for enhancing the resilience of a distribution network based on distributed dynamic recovery includes the following steps: Step 1: Obtain power distribution network operation data; Step 2: Construct a two-stage dynamic recovery model, establish a two-stage recovery model of topology reconstruction and rolling optimization, with the goal of maximizing load recovery, consider network constraints and resource operation constraints, and transform the probabilistic model into a deterministic mixed-integer linear programming problem through chance constraint transformation; Step 3: Design a distributed recovery and boundary coordination strategy. The model is decomposed into distributed subproblems using the alternating direction multiplier method. A boundary variable compensation mechanism is introduced to enhance robustness under communication interruption. Multi-time dynamic recovery decision is realized based on rolling optimization. Step 4: Propose an efficient distributed solution algorithm, design an integer variable handling method that combines projection and relaxation iteration, and construct an induced acceleration objective function to improve the solution efficiency of mixed integer subproblems and ensure the convergence and computation speed of the distributed algorithm.

[0007] Further optimization of this technical solution includes obtaining the branch resistance and reactance parameters of the distribution network, the active / reactive power data of the load, the probability distribution data of wind power and photovoltaic power output, the access location and allowable upper and lower limits of distributed power sources, and the access location and port capacity of smart soft switches with energy storage.

[0008] Further optimization of this technical solution involves step 2, which specifically includes constructing a two-stage recovery framework encompassing topology reconfiguration and rolling optimization, and establishing a corresponding mathematical model. This model aims to maximize load recovery and minimize the weighted sum of voltage deviations. Its constraints cover power balance equations considering line switching, branch voltage and capacity safety constraints, distributed power source operation constraints, energy storage system charging and discharging constraints, intelligent soft-switching operation constraints, and network radial operation constraints based on spanning trees. To address the uncertainty of renewable energy output, a Gaussian mixture model is used to describe its randomness and correlation. Furthermore, a chance-constrained programming method is employed to transform the probabilistic power balance constraints into a deterministic linear equivalent form, thereby transforming the original non-convex probabilistic problem into a deterministic mixed-integer linear programming problem.

[0009] Further optimization of this technical solution involves designing a distributed dynamic recovery strategy based on ADMM decomposition in step 3. This strategy uses the alternating direction multiplier method to decompose the centralized recovery model into multiple regional sub-problems. For the coupling boundary variables between sub-regions, consensus variables and Lagrange penalty terms are introduced to decouple global constraints, constructing a fully distributed optimization model. To improve the robustness of the strategy during communication interruptions, a boundary variable compensation mechanism is designed, dynamically selecting optimized or preset compensation values ​​for information exchange based on the communication link status. Simultaneously, the tie-line switching logic is constrained to ensure that the interconnected system meets the requirements for radial operation. At the dynamic recovery level, the first stage optimizes and determines the full-cycle network topology and energy storage scheduling plan. The second stage employs rolling time-domain optimization, dynamically adjusting the output of distributed power sources and intelligent soft switches based on the latest renewable energy probability distribution to continuously optimize the load recovery effect.

[0010] Further optimization of this technical solution involves step 4, which proposes an improved distributed solution algorithm combining projection, relaxation iteration, and induced acceleration. To efficiently handle a large number of integer variables in the model and ensure algorithm convergence, a "projection-relaxation iteration" mechanism is designed: First, the boundary integer variables are relaxed into continuous variables, and the projection operator is used in the ADMM iteration to make them approximate 0 or 1. After the boundary variables converge and are fixed, the internal integer variables are solved using relaxation-iteration interaction, alternately optimizing the continuous relaxation model and the mixed integer linear programming sub-model after the boundary is fixed, until the convergence criterion is met. To accelerate the mixed integer linear programming solution process of each region's subproblem, the induced factor is calculated based on the current relaxed solution, and an objective function containing auxiliary induced terms is constructed to drive the integer variables to converge quickly to feasible integer solutions, thereby significantly improving the overall distributed computing efficiency.

[0011] The technical solution is further optimized, and step 2 is as follows: Step 2.1: Resume framework construction and build a "two-stage" fully distributed dynamic DSR framework: Step 2.1.1 Topology Reconstruction Phase: Implement network reconstruction strategies by controlling traditional TSS to optimize the overall topology and restore connectivity to the power outage area; Step 2.1.2 Rolling Optimization Phase: The rolling optimization method is used to iteratively determine the multi-source scheduling decision based on the latest operating state of the system, with the aim of maximizing load recovery; Step 2.2: Establishing the Recovery Model Step 2.2.1, Objective Function: The primary objective of the recovery model is to ensure power supply to the affected area. Voltage deviation is also a key indicator for evaluating system operating status. Therefore, a linearly weighted objective function is adopted, defined as: (1) in A set of time periods representing the power outage or restoration process. ; It is a set of nodes in a power distribution network. ; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active power load of node i during time period t; This represents the voltage amplitude at node i at time t; Indicates the upper and lower limits of the voltage amplitude; and Let be the weight coefficient, and satisfy... The first term in equation (1) aims to maximize load recovery. The second term represents the voltage, the degree to which it deviates from the desired range. ; Step 2.2.2, Constraints: The main constraints considered include power flow, DER operation, and network radiality, and their mathematical formulas are as follows: (2) (3) (4) (5) (6) (7) (8) (9) (10) in Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; This represents the distribution of active and reactive power flow between nodes during a period; This represents the active and reactive power output of the DG within period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active and reactive loads of node i during time period t; This indicates the active and reactive power output of port SOP within period t; These represent the active power outputs of ESS (Energy Storage System), WT (Wind Power), and PV (Photovoltaic Power) at time t, respectively. This indicates the active and reactive power output of a distributed energy source (DER). This represents the voltage amplitude at node i at time t; These are the resistance and reactance of the branch circuit; This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates the upper limit of active and reactive power capacity of the branch; A large positive number, used for logical constraint linearization; Indicates the upper and lower limits of the voltage amplitude; Constraints (2)-(5) represent the power balance equations. Ignoring power loss, considering line switching, the branch voltage constraint is captured in equations (6)-(7), while the safety constraint is specified by equations (8)-(10). The operating constraints of distributed generator sets (DG), ESS, and flexible SOPs follow the following equation: DG should comply with power and capacity limits: (11) (12) in This represents the active and reactive power output of the DG within period t; This indicates the maximum active power output and apparent power capacity of the DG; ESS includes charge / discharge limits: (13) (14) (15) (16) (17) in The ESS is in discharge mode; while The ESS is in charging mode; This represents the active power released and stored in the ESS within period t; Indicates the maximum permissible charge and discharge power of the ESS; This indicates the state of charge of the ESS at a certain moment; This indicates the discharge and charging efficiency of the ESS. These represent the lower limit of the low-charge state and the upper limit of the high-charge state of the ESS, respectively. Indicates ESS capacity; SOP considers port power limitations: (18) (19) (20) in This indicates the active and reactive power output of port SOP within period t; Represents the set of nodes that connect soft openings; This indicates the maximum reactive power output at SOP. Indicates SOP capacity; To enhance system resilience, topology reconstruction is employed. Therefore, a spanning tree model is used to formulate the conditions for network radiality, as shown in equations (21)-(23). (twenty one) (twenty two) (twenty three) in This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; Represents the set of root nodes; introduces binary variables. and To indicate different directions of trends; Describe the power flow state in a certain direction between nodes i and j within time period t (1 indicates that there is power flow in that direction, and 0 indicates that there is no power flow). Step 2.2.3, Model Transformation: In order to address the uncertainty of regeneration, a chance-constrained transformation method is introduced, which enhances the solvability of the model through linearization; (twenty four) in Represents the set of nodes that connect soft openings; Represents a set of DG, ESS, and WT / PV; These represent the active power outputs of distributed energy (DG), energy storage system (ESS), wind power (WT), and photovoltaic (PV) at time t, respectively. This represents the active power load of node i during time period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the upper limit of the active power output of WT / PV at node r at time t, that is, the maximum active power that the device can output during this period. This is the probability threshold.

[0012] Equation (24) is a chance constraint, which means that the probability that all available generator outputs can meet the restored load demand must be greater than the confidence level. The original DSR model, i.e., equations (1)-(24), is a non-convex probability problem, requiring model transformation to ensure its solvability. The inherent randomness and correlation of wavelet transform and photovoltaic power output pose challenges to accurately representing them using standard probability distribution models. It is assumed that the actual power output of renewable energy is determined by the predicted value, accompanied by superimposed prediction errors. Then, the probability density function PDF of wavelet transform and photovoltaic transformation is modeled using GMM, as follows: (25) In the formula is the joint PDF of WT and PV; M is the number of Gaussian components; The weight of the m-th Gaussian component ×with its mean vector Covariance Matrix ; So, given Cumulative Distribution Function (CDF): (26) Cumulative distribution function The core meaning is the probability that a random variable takes a value less than or equal to y, which is mathematically expressed as a probability density function: M represents the number of components that make up the normal distribution; This represents the weight of the m-th normal component; Let represent the normal cumulative distribution function of the i-th normal component at y.

[0013] Using equation (26), the equivalent transformation of the chance constraint is obtained: (27) in It is a set of nodes in a power distribution network. ; This represents the active power load of node i during time period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. Let represent the active power output of distributed energy source (DG) and energy storage system (ESS) at time t, respectively. Represents the set of DG and ESS; express The inverse function is calculated using the bisection method; This is the probability threshold.

[0014] Following this process, the recovery model is transformed into a deterministic mixed-integer linear programming (MILP) problem, which can be solved using commercial solvers.

[0015] This technical solution is further optimized, and step 3 is as follows: Step 3.1: Centralized DSR Model Decomposition Step 3.1.1: Satisfy Subnet Radial Constraints: In this step, the physical connections between adjacent regions are isolated, and each subnet performs DSR independently to ensure radial configuration; Step 3.1.2: Distributed Optimization of the Entire System: Using the subnet topology obtained in Step 1, a fully distributed model is developed to determine the inter-subnet reconstruction strategy and DER scheduling results. Furthermore, a boundary variable compensation mechanism is introduced to ensure the reliability of the recovery strategy in the event of a communication failure. For area a, this is represented as: (28) in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. Introducing consensus continuity and binary variables and To achieve model decomposition. and As dual variables; It is the set of adjacent regions of region a; This is the local objective function for region a, which typically includes the operating costs of that region (such as DER output costs and network loss costs), constraint penalty terms, etc., and is used to measure the economic efficiency and rationality of the operation of region a itself. Represented as It is the augmented Lagrange penalty coefficient. It is the penalty term coefficient of consistency constraints in distributed algorithms (such as ADMM), used to enhance the consistency of boundary variables between subnets; In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a.

[0016] A communication link failure prevented the control center from exchanging optimization boundary variables, causing the algorithm to fail to converge. Constraints were then added. (29) in For boundary decision variables; This indicates whether the communication link between adjacent areas a and b is valid. Then, the boundary variables for inter-regional exchange are determined through distributed optimization; if Then give the compensation value. The compensation strategy allows for the use of predetermined optimal boundary values ​​between adjacent regions without the need for information exchange. In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a.

[0017] Meanwhile, to determine the switching decision for the communication line, two sets are defined: the set of feasible communication areas F and the set of unfeasible communication areas Y. The communication line between these two areas is subject to the following constraints. (30) in and Indicates the region a and B The status of the first line between them It is a region a and B The set of all branches between them. Indicates excluding branches The set of remaining branches, as shown in equation (30), indicates that region a and region B Only one connecting line is required between them; Indicates the branch between regions a and B The on / off state variable (binary variable) takes a value of 1 to indicate that the branch is closed and connected, and a value of 0 to indicate that the branch is open. This indicates the relationship between a and B, excluding branches. The on / off state variables (binary variables) of other branches v, and their value rules are the same as those of other branches v. Consistent.

[0018] Step 3.2: Dynamic Service Recovery Strategy

[0019] To address short-term fluctuations in renewable energy, a dynamic recovery strategy was adopted, which consists of two phases. In the first stage, Equation (28) is optimized to determine the system topology reconfiguration strategy and ESS scheduling results for the entire recovery cycle. This stage involves topology optimization to maximize the connectivity between the power outage area and the local DER support area, thereby enhancing system resilience. In the second stage, the spanning tree constraints (21)-(23) are related to the topology and ESS constraints. That is, equations (13)-(17) are removed, resulting in the following distributed service recovery model: (31) It is constrained by equations (2)-(12), (18)-(20), (27) and (29).

[0020] in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; Equation (31) is executed by the rolling horizon framework, which is dynamically reformulated as time progresses from t to t+1 and incorporates the latest probability distribution results of renewable energy. The DSR strategy is dynamically adjusted during the optimization process, thereby improving its operational reliability. The presence of a wide range of binary decision variables in equations (28) and (31) often leads to serious computational challenges, including inefficiency and non-convergence.

[0021] This technical solution is further optimized, and step 4 is as follows: Step 4.1: Optimize the framework design The presence of binary variables related to the operational control of the interior and boundary regions complicates the direct application of ADMM to solving the DSR model. To address this, a novel optimization framework is proposed that can iterate the projection and relaxation of boundary variables. The method is combined with the traditional ADMM to solve the convergence problem in the distributed DSR model. On this basis, induced acceleration techniques are used to improve the computational efficiency of subproblems within the distributed optimization framework. Step 4.2: Converging Projection and Iterative Process The binary variables in equation (28) are of two types, namely To handle these binary variables, different subproblems are formulated and their optimal values ​​are determined sequentially. Subproblem 1: Binary Variables The existence of boundary constraints hinders the smooth convergence of the ADMM iteration process. To address this, an auxiliary continuous variable is introduced. Relax the Boolean constraints, and then modify equation (28) using the following update strategy: (32a) (32b) (32c) (32d) (32e) In the formula, B represents the coupled network cluster; As a regularization factor; Indicates in The projection on the screen is used to round entries to 0 or 1. The penalty factor for ADMM; It refers to the update value of the intermediate variable at the boundary within the coupled network set B. and dual variables The mean of the sums.

[0022] In equation (32), the binary variables are relaxed to continuous variables, resulting in a convex formula that ensures the effective convergence of ADMM. During the iteration process, a projection-based strategy is used to relax the variables. Push to Boolean values, as described in equation (32c); then, apply the consensus variable. The optimized solution is projected onto the closest continuous solution. Given the binary value of the algorithm, under the constraint of the augmented Lagrangian function, the convergence induction condition of the algorithm is... This ensured The optimal solution is a binary solution; In the ADMM step, gradually increase Beneficial for driving For stable convergence to Boolean values, the update strategy is as follows: (33) Where c is a constant; It is a conditional operator; and Let the original residual and the dual residual be represented by , respectively, in iteration k. They converge to: (34) (35) in and For predetermined tolerance; Subproblem 2: The solution to subproblem 1 fixes the boundary binary variables. Meanwhile, internal load switching variables The requirement is for a Boolean value, since it is an integer variable. The existence of this model indicates that it belongs to a non-convex class, which leads to a convergence problem in ADMM. To address this, a solution strategy based on relaxation iteration is introduced. During the relaxation phase of integer variables, Relaxation is continuous This forms a convex model, which is then solved using ADMM with convergence. (36) It conforms to equations (2)-(20), (27) and (29); and The update strategy is shown in equation (32). When solving equation (36), the boundary variables can be fixed and a completely independent model can be developed for each region. (37) It conforms to equations (2)-(20) and (27); in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; obtained in equation (37) After finding the optimal value, this value is kept fixed and used to reformulate equation (36); equations (36) and (37) are optimized alternately until convergence is achieved. The optimal service recovery strategy is determined by solving subproblems 1 and 2 in sequence. The convergence criterion is: (38) in To summarize the objectives for all regions in iteration z. For predetermined tolerance; Then, since there are no boundary binary variables, equation (31) in stage 2 can be solved by the relaxation iteration method; Step 4.3: Acceleration Strategy Acceleration Strategy: The computational efficiency of the algorithm largely depends on the solutions of the independent models (37). Each model is expressed as a MILP problem. To address this issue, an induced acceleration strategy is adopted to improve its computational performance. By adding an induced objective function, equation (37) is restated as follows: (39) It conforms to equations (2)-(20) and (27); in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. In the formula for The induced function is expressed as: (40) In the formula, As a weighting factor, In equation (37) The relaxed solution, in which the binary variables are relaxed to continuous values ​​during the optimization process; if ,So , exist Time is more than in The time is smaller; for minimization problems, It will tend towards 0; similarly, if ,but Will be induced to 1, this is This trend indicates that this method accelerates the solution process.

[0023] Algorithm program: Provide a flowchart to illustrate the order of operations in the algorithm.

[0024] Step ① Determine the radial topology of all independent subnets. Step ② Determine the topology, DSR strategy, and dynamic service recovery scheme.

[0025] ①. Initialization phase: Initialize the relevant parameters of the Distributed Service Recovery (DSR) algorithm, including the upper limit of the number of iterations, the convergence error threshold, and the length of the rolling time domain; ②. Radial Topology Determination Step ①: Each sub-region network independently performs DSR calculations locally, and determines its own radial topology by solving the corresponding distributed optimization model; The original problem is broken down into two subproblems: Subproblem I: Determine the switching state of the boundary branch using the projection method (32); Subproblem II: By relaxing integer variables, construct the integer relaxation stage model (36) and the independent model solution stage (37) respectively, and iterate and update between the two stages; ③. Stage I Result Output: In Phase I, the network topology and static DSR strategy are determined, providing a foundation for subsequent dynamic recovery optimization. ④. Dynamic DSR Optimization Step ②: Based on the given stage I topology, the rolling time-domain framework is used to execute equation (31), and the relaxed-iterative method is combined to solve the distributed DSR optimization problem, thereby obtaining the dynamic DSR strategy of the system. ⑤. Acceleration Strategies and Iterative Updates: An acceleration strategy is introduced during the solution process to coordinate and update boundary variables and solutions to subproblems, thereby improving the efficiency and convergence speed of distributed solution. ⑥. Stage II Results Output: In Phase II, a dynamic DSR strategy that takes into account the characteristics of time evolution is obtained; ⑦. Termination Criterion: If the preset convergence condition is met, the calculation is terminated and the final optimization result is output; otherwise, return to step 2 and continue to perform iterative calculation.

[0026] Unlike existing technologies, the above technical solution has the following beneficial effects: 1. Achieve fully distributed recovery scheduling across regions, reducing communication volume and protecting data privacy.

[0027] 2. The adoption of opportunity constraint + GMM significantly improves the reliability of recovery strategies under regenerative volatility.

[0028] 3. The three-stage ADMM significantly improves the convergence performance of mixed-integer DSR models.

[0029] 4. Boundary compensation ensures that a stable recovery strategy can still be generated when communication is interrupted.

[0030] 5. Topology reconfiguration and rolling optimization should be coordinated to improve load recovery rate and system resilience. Attached Figure Description

[0031] Figure 1 This is a flowchart of a method for improving the resilience of distribution network topology based on distributed dynamic recovery. Figure 2 Here is the flowchart of the proposed solution algorithm; Figure 3 This is the topology diagram of the improved IEEE 33 bus test system; Figure 4 These are the optimized results of the system voltage distribution diagrams under five different conditions; Figure 5 It is the effect of regularization parameters on the algorithm; Figure 6 This refers to the convergence performance of the iterative ADMM method; Figure 7 This refers to the computation time of CPLEX and the acceleration method. Detailed Implementation

[0032] To explain in detail the technical content, structural features, objectives, and effects of the technical solution, the following description is provided in conjunction with specific embodiments and accompanying drawings.

[0033] To address the dynamic service recovery problem of multi-regional distribution networks under extreme fault conditions, this invention proposes a distributed dynamic elastic recovery method based on iteration and acceleration techniques. First, distribution network topology parameters, load data, distributed resource operating limits, and the probability distribution characteristics of wind / solar power are collected. Then, the uncertainty of renewable energy is modeled based on a Gaussian mixture model.

[0034] Secondly, to address the service restoration problem that is highly coupled with network topology reconfiguration and multi-source coordinated scheduling, a distribution network service restoration model is constructed with the objective functions of maximizing load restoration and minimizing voltage deviation. By introducing opportunity constraints and rolling horizon optimization methods, intermittent renewable energy sources are allowed to participate in the restoration process under uncertain conditions, thereby improving system restoration efficiency and operational reliability.

[0035] Next, to address the problem that the centralized model is difficult to solve directly due to the coupling of boundary variables between regions in a multi-regional power distribution network, a distributed modeling method based on the alternating direction multiplier method is adopted to decouple the boundary coupling variables such as power and switch status between adjacent regions. By introducing Lagrange penalty terms and consensus constraints into the objective function of each regional sub-model, the original centralized dynamic recovery model is transformed into a distributed dynamic service recovery model, realizing multi-regional coordinated recovery and information decoupling.

[0036] Finally, to address the computational complexity and convergence difficulties caused by integer decision variables such as network reconstruction and load switching, which are prevalent in service recovery models, a distributed solution framework combining variable relaxation, projection update, and interactive iteration is proposed to ensure the convergence of the distributed hybrid integer recovery model. Furthermore, an induced acceleration technique is introduced to guide integer variables to rapidly approximate the optimal discrete solution, significantly improving the solution efficiency and engineering applicability of the recovery model, and enhancing the robustness and feasibility of the recovery strategy in communication-constrained environments.

[0037] A preferred embodiment of the present invention provides a method for enhancing the resilience of a distribution network based on distributed dynamic recovery, the specific implementation steps of which are as follows: Step 1: Obtain power distribution network operation data It can acquire parameters such as the resistance and reactance of distribution network branches, active / reactive load data, probability distribution data of wind and solar power output, access locations and allowable upper and lower limits of distributed power sources, as well as access locations and port capacities of smart soft switches with energy storage.

[0038] Collect basic data for power distribution network restoration model

[0039] The system collects basic topology and real-time operational data of the distribution network, specifically including: resistance and reactance parameters of each branch; active and reactive load data of each node; predicted values ​​and probability distribution characteristics of wind and solar power output (used to construct Gaussian mixture models); access locations of distributed generation (DG) and their upper and lower limits of active / reactive power output; access locations, capacity, charging and discharging power limits and efficiency of energy storage systems (ESS); access port locations, transmission capacity and reactive power support capabilities of smart soft switches (SOPs); and safe operating limits of system voltage.

[0040] Step 2: Construct a two-stage dynamic recovery model

[0041] A two-stage recovery model of topology reconstruction and rolling optimization is established with the goal of maximizing load recovery. It takes into account network constraints and resource operation constraints, and transforms the probabilistic model into a deterministic mixed-integer linear programming problem through chance constraint transformation.

[0042] Construct a two-stage dynamic recovery model and perform deterministic transformation

[0043] A two-stage recovery framework incorporating topology reconfiguration and rolling optimization is constructed, and a corresponding mathematical model is established. The model aims to maximize the load recovery and minimize the weighted sum of voltage deviations. Its constraints cover power balance equations considering line switching, branch voltage and capacity safety constraints, distributed power source operation constraints, energy storage system charging and discharging constraints, smart soft-switching operation constraints, and network radial operation constraints based on spanning trees. To address the uncertainty of renewable energy output, a Gaussian mixture model is used to describe its randomness and correlation. Chance-constrained programming is employed to transform the probabilistic power balance constraints into a deterministic linear equivalent form, thereby transforming the original non-convex probabilistic problem into a deterministic mixed-integer linear programming problem.

[0044] Step 2.1: Restore the framework build

[0045] Constructing a "two-stage" fully distributed dynamic DSR framework: Step 2.1.1 Topology Reconstruction Phase: Implement network reconstruction strategies by controlling traditional TSS to optimize the overall topology and restore connectivity to the power outage area.

[0046] Step 2.1.2 Rolling Optimization Phase: The rolling optimization method is adopted to iteratively determine the multi-source scheduling decision based on the latest operating status of the system, with the aim of maximizing load recovery.

[0047] Step 2.2: Establishing the Recovery Model

[0048] Step 2.2.1, Objective Function: The primary objective of the recovery model is to ensure power supply to the affected area, and voltage deviation is also a key indicator for evaluating the system's operating status. Therefore, this paper adopts a linearly weighted objective function, defined as: (1) in A set of time periods representing the power outage or restoration process. ; It is a set of nodes in a power distribution network. ; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active power load of node i during time period t; This represents the voltage amplitude at node i at time t; Indicates the upper and lower limits of the voltage amplitude.

[0049] and Let be the weight coefficient, and satisfy... The first term in equation (1) aims to maximize load recovery. The second term represents the voltage, the degree to which it deviates from the desired range. .

[0050] Step 2.2.2, Constraints: The main constraints considered include power flow, DER operation, and network radiality, and their mathematical formulas are as follows: (2) (3) (4) (5) (6) (7) (8) (9) (10) in Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; This represents the distribution of active and reactive power flow between nodes during a period; This represents the active and reactive power output of the DG within period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active and reactive loads of node i during time period t; This indicates the active and reactive power output of port SOP within period t; These represent the active power outputs of ESS (Energy Storage System), WT (Wind Power), and PV (Photovoltaic Power) at time t, respectively. This indicates the active and reactive power output of a distributed energy source (DER). This represents the voltage amplitude at node i at time t; These are the resistance and reactance of the branch circuit; This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates the upper limit of active and reactive power capacity of the branch; A large positive number, used for logical constraint linearization; Indicates the upper and lower limits of the voltage amplitude.

[0051] Constraints (2)-(5) represent the power balance equations, where power losses are neglected. Considering line switching, branch voltage constraints are captured in equations (6)-(7). Safety constraints are specified by equations (8)-(10).

[0052] The operating constraints of distributed generator sets (DG), ESS, and flexible SOPs follow the following equation: DG should comply with power and capacity limits: (11) (12) in This represents the active and reactive power output of the DG within period t; This indicates the maximum active power output and apparent power capacity of the DG.

[0053] ESS includes charge / discharge limits: (13) (14) (15) (16) (17) in The ESS is in discharge mode; while The ESS is in charging mode; This represents the active power released and stored in the ESS within period t; Indicates the maximum permissible charge and discharge power of the ESS; This indicates the state of charge of the ESS at a certain moment; This indicates the discharge and charging efficiency of the ESS. These represent the lower limit of the low-charge state and the upper limit of the high-charge state of the ESS, respectively. Indicates the ESS capacity.

[0054] SOP considers port power limitations: (18) (19) (20) in This indicates the active and reactive power output of port SOP within period t; Represents the set of nodes that connect soft openings; This indicates the maximum reactive power output at SOP. This indicates the SOP capacity.

[0055] This embodiment considers topology reconfiguration to enhance system resilience. Therefore, a spanning tree model is used to formulate the conditions for network radiality, as shown in equations (21)-(23).

[0056] (twenty one) (twenty two) (twenty three) in This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; Represents the set of root nodes; introduces binary variables. and To indicate different directions of trends. Describe the power flow state in a certain direction between nodes i and j within time period t (1 indicates that there is power flow in that direction, and 0 indicates that there is no power flow). Step 2.2.3, Model Transformation: In order to address the uncertainty of regeneration, an opportunity constraint transformation method is introduced, which enhances the solvability of the model through linearization.

[0057] (twenty four)

[0058] in Represents the set of nodes that connect soft openings; Represents a set of DG, ESS, and WT / PV; These represent the active power output of DG (distributed energy), ESS (energy storage system), WT (wind power), and PV (photovoltaic) at time t, respectively. This represents the active power load of node i during time period t; This indicates that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the upper limit of the active power output of WT / PV at node r at time t, that is, the maximum active power that the device can output during this period. This is the probability threshold.

[0059] Equation (24) is a chance constraint, which means that the probability that the output of all available generators can meet the restored load demand must be greater than the confidence level. Therefore, using a probabilistic method to evaluate the power balance constraint is beneficial for the participation of uncertain renewable energy sources in service restoration.

[0060] The original DSR model, i.e., equations (1)-(24), is a non-convex probability problem, requiring model transformation to ensure its solvability. The inherent randomness and correlation of wavelet transform and photovoltaic power output pose challenges to accurately representing them using standard probability distribution models. It is assumed that the actual power output of renewable energy is determined by predicted values, accompanied by superimposed prediction errors. Then, the probability density function (PDF) of wavelet transform and photovoltaic transformation can be modeled using GMM, as follows: (25) In the formula is the joint PDF of WT and PV; M is the number of Gaussian components; The weight of the m-th Gaussian component ×with its mean vector Covariance Matrix .

[0061] Then, we can give The cumulative distribution function (CDF): (26) Cumulative distribution function The core meaning is the probability that a random variable takes a value less than or equal to y, which is mathematically expressed as a probability density function: M represents the number of components that make up the normal distribution; This represents the weight of the m-th normal component; Let represent the normal cumulative distribution function of the i-th normal component at y.

[0062] Using equation (26), the equivalent transformation of the chance constraint is obtained: (27) in It is a set of nodes in a power distribution network. ; This represents the active power load of node i during time period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. These represent the active power outputs of DG (distributed energy source) and ESS (energy storage system) at time t, respectively. Represents the set of DG and ESS; express The inverse function can be calculated using the bisection method. This is the probability threshold.

[0063] Following this process, the recovery model is transformed into a deterministic mixed-integer linear programming (MILP) problem, which can be solved using commercial solvers.

[0064] Step 3: Design distributed recovery and boundary coordination strategies

[0065] The model is decomposed into distributed subproblems using the alternating direction multiplier method, a boundary variable compensation mechanism is introduced to enhance robustness under communication interruption, and dynamic recovery decision-making for multiple time periods is achieved based on rolling optimization.

[0066] Design a distributed dynamic recovery strategy based on ADMM decomposition.

[0067] The centralized recovery model is decomposed into multiple regional sub-problems using the alternating direction multiplier method. For the coupled boundary variables between sub-regions, consensus variables and Lagrange penalty terms are introduced to decouple global constraints, constructing a fully distributed optimization model. To improve the policy robustness during communication interruptions, a boundary variable compensation mechanism is designed, dynamically selecting optimized or preset compensation values ​​for information exchange based on the communication link status. Simultaneously, the tie-line switching logic is constrained to ensure the interconnected system meets radial operation requirements. At the dynamic recovery level, the first stage optimizes and determines the full-cycle network topology and energy storage scheduling plan, while the second stage employs rolling time-domain optimization, dynamically adjusting the output of distributed power sources and intelligent soft switches based on the latest renewable energy probability distribution to continuously optimize load recovery performance.

[0068] Step 3.1: Centralized DSR Model Decomposition

[0069] Step 3.1.1: Satisfy Subnet Radial Constraints: In this step, the physical connections between adjacent regions are isolated, and each subnet performs DSR independently to ensure radial configuration.

[0070] Step 3.1.2: Distributed Optimization of the Entire System: Using the subnet topology obtained in Step 1, a fully distributed model is developed to determine the inter-subnet reconstruction strategy and DER scheduling results. Furthermore, a boundary variable compensation mechanism is introduced to ensure the reliability of the recovery strategy in the event of a communication failure. For region a, it is represented as: (28) in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. Introducing consensus continuity and binary variables and To achieve model decomposition. and As dual variables; It is the set of adjacent regions of region a. This is the local objective function for region a, which typically includes the operating costs of that region (such as DER output costs and network loss costs), constraint penalty terms, etc., and is used to measure the economic efficiency and rationality of the operation of region a itself. Represented as It is the augmented Lagrange penalty coefficient. It is the penalty term coefficient of consistency constraints in distributed algorithms (such as ADMM), used to enhance the consistency of boundary variables between subnets; In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a.

[0071] A communication link failure prevented the control center from exchanging optimized boundary variables, causing the algorithm to fail to converge. Constraints were then added.

[0072] (29)

[0073] in For boundary decision variables; This indicates whether the communication link between adjacent areas a and b is valid. Then, the boundary variables for inter-regional exchange are determined through distributed optimization; if Then give the compensation value. The compensation strategy allows for the use of predetermined optimal boundary values ​​between adjacent regions without the need for information exchange. In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a.

[0074] In order to determine the switching decision of the communication line, two sets are defined: the set of feasible communication areas F and the set of unfeasible communication areas Y. The communication line between these two areas is subject to the following constraints.

[0075] (30)

[0076] in and Indicates the region a and B The state of the first line between them. It is a region a and B The set of all branches between. Indicates excluding branches The set of remaining branches. Equation (30) shows that region a and region B Only one connecting line is maintained between them. and Indicates the region a and B The status of the first line between them It is a region a and B The set of all branches between them. Indicates excluding branches The set of remaining branches, as shown in equation (30), indicates that region a and regionB Only one connecting line is required between them; Indicates the branch between regions a and B The on / off state variable (binary variable) takes a value of 1 to indicate that the branch is closed and connected, and a value of 0 to indicate that the branch is open. This indicates the relationship between a and B, excluding branches. The on / off state variables (binary variables) of other branches v, and their value rules are the same as those of other branches v. Consistent.

[0077] Step 3.2: Dynamic Service Recovery Strategy

[0078] To address short-term fluctuations in renewable energy, a dynamic recovery strategy was adopted, which consists of two phases.

[0079] In the first stage, equation (28) is optimized to determine the system topology reconfiguration strategy and ESS scheduling results for the entire recovery cycle. This stage involves topology optimization to maximize connectivity between the power outage area and the local DER support area, thereby enhancing system resilience.

[0080] In the second stage, the spanning tree constraints (21)-(23) are related to the topology and ESS constraints. That is, equations (13)-(17) are removed, resulting in the following distributed service recovery model: (31) It is constrained by equations (2)-(12), (18)-(20), (27) and (29).

[0081] in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; Equation (31) is executed by the rolling horizon framework, which is dynamically reformulated as time progresses from t to t+1, incorporating the latest probability distribution results for renewable energy. The DSR strategy is dynamically adjusted during the optimization process, thereby improving its operational reliability.

[0082] At the same time, it is worth noting that the presence of extensive binary decision variables in equations (28) and (31) often leads to serious computational challenges, including inefficiency and non-convergence. These problems highlight the necessity of developing a feasible optimization algorithm.

[0083] Step 4: Propose an efficient distributed solution algorithm

[0084] We design an integer variable handling method that combines projection and relaxation iteration, and construct an induced acceleration objective function to improve the solution efficiency of mixed integer subproblems and ensure the convergence and computation speed of the distributed algorithm.

[0085] An improved distributed solution algorithm combining projection, relaxation iteration, and induced acceleration is proposed.

[0086] To efficiently handle a large number of integer variables in the model and ensure algorithm convergence, a "projection-relaxation iteration" mechanism is designed: First, the boundary integer variables are relaxed into continuous variables, and the projection operator is combined in the ADMM iteration to make them approximate 0 or 1; after the boundary variables converge and are fixed, the relaxation-iteration interactive solution is used for the internal integer variables, alternately optimizing its continuous relaxation model and the mixed integer linear programming sub-model after the boundary is fixed, until the convergence criterion is met; to accelerate the mixed integer linear programming solution process of each region's subproblem, the induction factor is calculated based on the current relaxed solution, and an objective function containing auxiliary induction terms is constructed to drive the integer variables to converge quickly to feasible integer solutions, thereby significantly improving the overall distributed computing efficiency.

[0087] Step 4.1: Optimize the framework design

[0088] The presence of binary variables related to operational control in both internal and boundary regions (e.g., node states, branch switching, and ESS operations) complicates the direct application of ADMM to solving DSR models. To address this, this paper proposes a novel optimization framework that can project and relax boundary variables.

[0089] This paper combines a method with the traditional ADMM to solve the convergence problem in the distributed DSR model. Based on this, induced acceleration techniques are employed to improve the computational efficiency of subproblems within the distributed optimization framework. The detailed implementation process is described below.

[0090] Step 4.2: Converging Projection and Iterative Process

[0091] The binary variables in equation (28) are of two types, namely To handle these binary variables, the different subproblems are formulated, and their optimal values ​​are determined sequentially.

[0092] Subproblem 1: Binary Variables The existence of boundary constraints hinders the smooth convergence of the ADMM iteration process. Therefore, an auxiliary continuous variable is introduced. Relax the Boolean constraints. Then, equation (28) can be modified using the following update strategy: (32a) (32b) (32c) (32d) (32e) In the formula, B represents the coupled network cluster; As a regularization factor; Indicates in The projection on the screen is used to round entries to 0 or 1. This is the penalty factor for ADMM. It refers to the update value of the intermediate variable at the boundary within the coupled network set B. and dual variables The mean of the sums.

[0093] In equation (32), the binary variables are relaxed to continuous variables, resulting in a convex formula that guarantees the effective convergence of ADMM. During the iteration process, a projection-based strategy is used to relax the variables. Push to Boolean values, as described in equation (32c). Then, the consensus variable... The optimized solution is projected onto the closest continuous solution. The binary value. Under the constraint of the augmented Lagrangian function, the convergence induction condition of the algorithm. This ensures that the optimal solution of ADMM is a binary solution.

[0094] In the ADMM step, gradually increase Beneficial for driving For stable convergence to Boolean values, the update strategy is as follows: (33) Where c is a constant; It is a conditional operator; and Let represent the original residual and the dual residual in iteration k, respectively. They converge to: (34) (35) in and This is the predetermined tolerance.

[0095] Subproblem 2: The solution to subproblem 1 fixes the boundary binary variables. Meanwhile, internal load switching variables The value must be a Boolean. This is because it's an integer variable. The existence of this model indicates that it belongs to a non-convex class, which leads to convergence problems in ADMM. Therefore, a solution strategy based on relaxation iteration is introduced.

[0096] During the relaxation phase of integer variables, Relaxation is continuous This forms a convex model, which is then solved using ADMM convergence.

[0097] (36)

[0098] It conforms to equations (2)-(20), (27) and (29).

[0099] and The update strategy is shown in equation (32). When solving equation (36), the boundary variables can be fixed to create a completely independent model for each region.

[0100] (37)

[0101] It conforms to equations (2)-(20) and (27).

[0102] in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; obtained in equation (37) After finding the optimal value, this value is kept fixed and used to reformulate equation (36). Equations (36) and (37) are optimized alternately until convergence is achieved. The optimal service recovery strategy is determined by solving subproblems 1 and 2 in sequence. The convergence criterion is: (38) in This summarizes the objectives for all regions in iteration z. This is the predetermined tolerance.

[0103] A rigorous proof of convergence for this method can be found in our previous work. Then, since there are no boundary binary variables, equation (31) in stage 2 can be solved by a relaxation iteration method.

[0104] Step 4.3: Acceleration Strategy

[0105] Acceleration Strategy: The computational efficiency of the algorithm largely depends on the solutions of the independent models (37), each of which is expressed as a MILP problem. To address this issue, an induced acceleration strategy is adopted to improve its computational performance. By adding an induced objective function, equation (37) is restated as follows: (39) It conforms to equations (2)-(20) and (27). in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. In the formula for The induced function is expressed as: (40) In the formula, This is the weighting factor. In equation (37) The relaxed solution is obtained where the binary variables are relaxed to continuous values ​​during the optimization process.

[0106] It can be observed that if ,So , exist Time is more than in The time is smaller. For minimization problems, It will tend towards 0. Similarly, if ,but Will be induced to 1, this is This trend. This method accelerates the solution process.

[0107] The entire algorithm program: provides a flowchart to illustrate the order of operations in the algorithm.

[0108] like Figure 2 As shown, step ① determines the radial topology of all independent subnets, and step ② determines the topology, DSR strategy, and dynamic service recovery scheme.

[0109] ①. Initialization phase: Initialize the relevant parameters of the Distributed Service Recovery (DSR) algorithm, including the upper limit of the number of iterations, the convergence error threshold, and the length of the rolling time domain; ②. Radial Topology Determination Step ①: Each sub-region network independently performs DSR calculations locally, and determines its own radial topology by solving the corresponding distributed optimization model; The original problem is broken down into two subproblems: Subproblem I: Determine the switching state of the boundary branch using the projection method (32); Subproblem II: By relaxing integer variables, construct the integer relaxation stage model (36) and the independent model solution stage (37) respectively, and iterate and update between the two stages; ③. Stage I Result Output: In Phase I, the network topology and static DSR strategy are determined, providing a foundation for subsequent dynamic recovery optimization. ④. Dynamic DSR Optimization Step ②: Based on the given stage I topology, the rolling time-domain framework is used to execute equation (31), and the relaxed-iterative method is combined to solve the distributed DSR optimization problem, thereby obtaining the dynamic DSR strategy of the system. ⑤. Acceleration Strategies and Iterative Updates: An acceleration strategy is introduced during the solution process to coordinate and update boundary variables and solutions to subproblems, thereby improving the efficiency and convergence speed of distributed solution. ⑥. Stage II Result Output: In Phase II, a dynamic DSR strategy that takes into account the characteristics of time evolution is obtained; ⑦. Termination Criterion: If the preset convergence condition is met, the calculation is terminated and the final optimization result is output; otherwise, return to step 2 and continue to perform iterative calculation.

[0110] This invention proposes a fully distributed dynamic recovery model for multi-regional distribution networks. This model improves recovery efficiency and system reliability by sharing limited boundary information within an opportunity-constrained rolling horizon framework, thereby facilitating comprehensive optimization of network reconfiguration and various cross-regional DER scheduling.

[0111] A distributed algorithm combining projection, iteration, and acceleration is employed to solve the D-DSR model. This improved version of the traditional ADMM enhances the convergence and computational efficiency of the large-scale distributed mixed-integer DSR model, while maintaining the adaptability of the resilient strategy in the event of network failures through a boundary variable compensation mechanism.

[0112] This embodiment employs two improved IEEE 33 bus and IEEE 123 bus systems to verify the effectiveness of the proposed method. Simulations were performed in Matlab 2018b using the Yalmip toolbox. The model was solved on a 64-bit computer with an Intel(R) i7-8500U CPU@1.8GHz and 8 GB of memory using the Cplex 12.7.1 optimizer. The IEEE 33 bus test system comprises three areas, where switches connecting to the upstream power grid are opened due to fault isolation, allowing for power restoration using local resources.

[0113] like Figure 3 As shown, the IEEE 33 bus system is divided into three zones, with one PV, one WT, four dg, one ESS unit, and one SOP connected across the different zones. Interconnection between zones utilizes four TSs. For fault isolation, switches between buses 0 and 1, 3 and 23, and 15 and 16 are open. Assume that after an extreme event, the power outage period is from 9:00 to 17:00.

[0114] A. Dynamic DSR Analysis under Multi-Source Coordination

[0115] 1) Advantages of Multi-Source Cooperative Service Recovery: This model enhances resilience by coordinating cross-regional network reconfiguration and der scheduling. Then, this section compares five cases: in Case 1, network reconfiguration is not included in the recovery model; in Cases 2 and 3, WT / PV and SOP are not involved in the recovery process. In Case 4, voltage deviation is not the optimization objective. The proposed model is evaluated in Case 5. The results are as follows... Figure 4 As shown.

[0116] The recovery results across five scenarios are shown in the table. In Case 1, nodes 16-18 and 23-25 ​​formed two electrical islands isolated from the overall system recovery process. In contrast, Case 5 employed a network reconfiguration strategy by shutting down switches TS3 and TS4, thereby establishing a connection to the system through Region 3. This strategy restored the islanded nodes in Region 2 while utilizing the regulation capabilities of the DG within the Region 1 island, ultimately increasing the system recovery level from 18.82 MWh to 19.22 MWh. Unlike Cases 2 and 3, the WT / PV and SOP units in Case 5 played a crucial role in providing active and reactive power support, which is essential for enhancing service recovery capabilities. Furthermore, except for Case 4, the voltage distribution remained within acceptable limits in all cases. These results clearly demonstrate that the model proposed in Case 5 can effectively coordinate various resources to enhance system resilience and mitigate voltage deviations.

[0117] B. Improve the performance of distributed algorithms

[0118] 1) Discussion on the convergence of the projection strategy: During topology reconstruction, the presence of binary variables at the boundaries hinders the direct application of distributed algorithms. This prompts us to use a projection strategy, and the impact of the regularization parameter is discussed.

[0119] from Figure 5 It can be seen that, When the binary boundary variables iterate in a fully relaxed state, the model converges to an infeasible solution. Increasing the regularization factor allows for successful convergence within a finite number of iterations. In this case, equation (32c) is used to implement a projection strategy to update the binary boundary variables, thus obtaining... and A feasible solution. However, as the regularization factor increases to... and The projection component becomes dominant, leading to convergence failure due to discretization effects. The results show that choosing an appropriate regularization factor is key to model convergence. Generally, we recommend a small initial value to activate the relaxation component in early iterations. When the binary boundary variables tend to stabilize, the regularization factor can be increased according to equation (33) to obtain a Boolean solution.

[0120] 2) The iterative ADMM method has better performance: To evaluate the feasibility of the proposed method, a comparative analysis was conducted with the standard ADMM method.

[0121] The standard ADMM, directly applied to mixed integer problems, failed to converge within 50 iterations, such as... Figure 6 In contrast, the proposed iterative method first solves a continuous relaxation problem, followed by a linear programming problem with fixed integer decision variables. By avoiding direct mixed-integer problem optimization, these two iterative problems converge precisely in 35 and 29 iterations, respectively. Furthermore, rigorous mathematical proofs guarantee the convergence of the two distributed problems and their iterative interactions.

[0122] C. Computational performance of the algorithm

[0123] To evaluate the computational performance of the algorithm, the Cplex optimizer and acceleration methods were used to address the repair problem, as described below.

[0124] The five cases in Section A were tested, and the results are summarized in Figure 7 The recovery process involves mixed-integer programming (MIP) because both network reconstruction in step 1 and load recovery optimization in step 2 require binary variables. Compared to directly solving the MIP model using a composite circuit optimizer, the accelerated method is computationally more efficient in most cases. Its slightly lower performance in a few specific cases is acceptable, as both methods are very fast in these situations. Notably, in scenarios with long solution times, the accelerated method improves computational speed by 8.8 to 80.0 times compared to the composite circuit optimizer. The results demonstrate that the accelerated method significantly improves the computational efficiency of the MIP model and is suitable for dynamic service recovery optimization.

[0125] This invention solves the following technical problems: 1. Solving the problem of difficult collaborative modeling of topology reconstruction and multi-source scheduling. To address the problem of highly coupled and difficult-to-unify modeling between network topology reconfiguration and various types of distributed energy sources (including dispatchable power sources, energy storage systems, and flexible smart switches) during distribution network service restoration, this invention constructs a unified restoration optimization model, incorporating network reconfiguration decisions and multi-source coordinated scheduling into the same optimization framework. Through boundary variable decoupling and consensus constraint design, the originally centralized and strongly coupled restoration model can be effectively decomposed, thereby realizing a distributed service restoration (DSR) model that can be solved in a distributed manner.

[0126] 2. Addressing the issue of unreliable recovery strategies due to short-term uncertainties in renewable energy.

[0127] In response to the significant power output fluctuations and large prediction errors of intermittent renewable energy sources such as wind and solar power over short timescales, this invention introduces the concept of split-blob bar / chance constraint modeling in the recovery strategy formulation process. By dynamically updating uncertainty information through a rolling optimization mechanism, the recovery decision can make full use of renewable energy resources while ensuring confidence levels, effectively avoiding load recovery failures caused by power output deviations, thereby significantly improving the reliability of the recovery strategy and the resilience of system operation.

[0128] 3. Addressing the convergence problem of mixed-integer distributed models with a large number of binary variables.

[0129] To address the problem that binary decision variables such as network switching status, load recovery status, and equipment operation mode are widely present in recovery models, leading to strong non-convexity of distributed mixed integer optimization models and difficulty in direct convergence of traditional alternating direction multiplier methods, this invention designs a solution mechanism combining relaxation, iteration, and projection, and introduces an induced acceleration strategy to guide integer variables to quickly approach the optimal discrete solution. This significantly improves the convergence performance and computational efficiency of distributed recovery models while ensuring solution feasibility.

[0130] 4. Improve the robustness and feasibility of recovery strategies in communication failure environments.

[0131] To address the problem of potentially damaged communication links and unreliable information exchange in multi-regional power distribution networks under extreme events, this invention introduces a boundary variable compensation mechanism and communication state constraints into a distributed recovery framework. When communication is normal, it achieves inter-regional collaborative optimization. When communication is restricted or interrupted, it utilizes a preset compensation strategy to maintain the feasibility of recovery decisions and the radial structure of the system, thereby improving the robustness and engineering applicability of the recovery strategy under communication failure environments.

[0132] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Unless otherwise specified, an element defined by the phrase "comprising..." or "including..." does not exclude the presence of additional elements in the process, method, article, or terminal device that includes said element. Additionally, in this document, "greater than," "less than," "exceeding," etc., are understood to exclude the stated number; "above," "below," "within," etc., are understood to include the stated number.

[0133] Although the above embodiments have been described, those skilled in the art, once they understand the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the above descriptions are merely embodiments of the present invention and do not limit the scope of patent protection of the present invention. Any equivalent structural or procedural transformations made using the content of the present invention's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the scope of patent protection of the present invention.

Claims

1. A method for enhancing the resilience of a distribution network based on distributed dynamic recovery, characterized in that, Includes the following steps: Step 1: Obtain power distribution network operation data; Step 2: Construct a two-stage dynamic recovery model, establish a two-stage recovery model of topology reconstruction and rolling optimization, with the goal of maximizing load recovery, consider network constraints and resource operation constraints, and transform the probabilistic model into a deterministic mixed-integer linear programming problem through chance constraint transformation; Step 3: Design a distributed recovery and boundary coordination strategy. The model is decomposed into distributed subproblems using the alternating direction multiplier method. A boundary variable compensation mechanism is introduced to enhance robustness under communication interruption. Multi-time dynamic recovery decision is realized based on rolling optimization. Step 4; An efficient distributed solution algorithm is proposed, and an integer variable handling method combining projection and relaxation iteration is designed. An induced acceleration objective function is constructed to improve the solution efficiency of mixed integer subproblems and ensure the convergence and computation speed of the distributed algorithm.

2. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 1, characterized in that, The operational data in step 1 includes obtaining the resistance and reactance parameters of the distribution network branches, the active / reactive power data of the load, the probability distribution data of wind and solar power output, the access location and allowable upper and lower limits of distributed power sources, and the access location and port capacity of smart soft switches with energy storage.

3. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 1, characterized in that, Step 2 specifically includes constructing a two-stage recovery framework that includes topology reconfiguration and rolling optimization, and establishing a corresponding mathematical model; the model aims to maximize the load recovery amount and minimize the weighted sum of voltage deviations. Its constraints cover power balance equations considering line switching, branch voltage and capacity safety constraints, distributed power source operation constraints, energy storage system charging and discharging constraints, smart soft switching operation constraints, and network radial operation constraints based on spanning tree. To address the uncertainty of renewable energy output, a Gaussian mixture model is used to describe its randomness and correlation. The chance-constrained programming method is used to transform the probabilistic power balance constraints into a deterministic linear equivalent form, thereby transforming the original non-convex probabilistic problem into a deterministic mixed-integer linear programming problem.

4. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 1, characterized in that, Step 3 designs a distributed dynamic recovery strategy based on ADMM decomposition. The alternating direction multiplier method is used to decompose the centralized recovery model into multiple regional sub-problems. For the coupling boundary variables between sub-regions, consensus variables and Lagrange penalty terms are introduced to decouple global constraints, constructing a fully distributed optimization model. To improve the strategy robustness during communication interruptions, a boundary variable compensation mechanism is designed, dynamically selecting optimized or preset compensation values ​​for information exchange based on the communication link status. Simultaneously, the tie-line switching logic is constrained to ensure that the interconnected system meets the requirements for radial operation. At the dynamic recovery level, the first phase optimizes and determines the full-cycle network topology and energy storage scheduling plan, while the second phase adopts rolling time-domain optimization, which dynamically adjusts the output of distributed power sources and smart soft switches based on the latest renewable energy probability distribution to continuously optimize the load recovery effect.

5. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 1, characterized in that, Step 4 proposes an improved distributed solution algorithm combining projection, relaxation iteration, and induced acceleration. To efficiently handle a large number of integer variables in the model and ensure algorithm convergence, a "projection-relaxation iteration" mechanism is designed: First, the boundary integer variables are relaxed into continuous variables, and the projection operator is combined in the ADMM iteration to make them approximate 0 or 1; after the boundary variables converge and are fixed, the internal integer variables are solved by relaxation-iteration interaction, alternately optimizing their continuous relaxation model and the mixed integer linear programming sub-model after the boundary is fixed, until the convergence criterion is met; to accelerate the mixed integer linear programming solution process of each region's subproblem, the induced factor is calculated based on the current relaxed solution, and an objective function containing auxiliary induced terms is constructed to drive the integer variables to converge quickly to feasible integer solutions, thereby significantly improving the overall distributed computing efficiency.

6. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 1, characterized in that, Step 2 is described in detail below: Step 2.1: Resume framework construction and build a "two-stage" fully distributed dynamic DSR framework: Step 2.1.1 Topology Reconstruction Phase: Implement network reconstruction strategies by controlling traditional TSS to optimize the overall topology and restore connectivity to the power outage area; Step 2.1.2 Rolling Optimization Phase: The rolling optimization method is used to iteratively determine the multi-source scheduling decision based on the latest operating state of the system, with the aim of maximizing load recovery; Step 2.2: Establishing the recovery model; Step 2.2.1, Objective Function: The primary objective of the recovery model is to ensure power supply to the affected area. Voltage deviation is also a key indicator for evaluating system operating status. Therefore, a linearly weighted objective function is adopted, defined as: (1) in A set of time periods representing the power outage or restoration process. ; It is a set of nodes in a power distribution network. ; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active power load of node i during time period t; This represents the voltage amplitude at node i at time t; Indicates the upper and lower limits of the voltage amplitude; and Let be the weight coefficient, and satisfy... The first term in equation (1) is dedicated to maximizing load recovery, and the second term represents the voltage, the degree to which it deviates from the desired range. ; Step 2.2.2, Constraints: The constraints considered include power flow, DER operation, and network radiality, and their mathematical formulas are as follows: (2) (3) (4) (5) (6) (7) (8) (9) (10) in Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; This represents the distribution of active and reactive power flow between nodes during a period; This represents the active and reactive power output of the DG within period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the active and reactive loads of node i during time period t; This indicates the active and reactive power output of port SOP within period t; These represent the active power outputs of the energy storage system ESS, wind power WT, and photovoltaic PV at time t, respectively. This indicates the active and reactive power output of a distributed energy source (DER). This represents the voltage amplitude at node i at time t; These are the resistance and reactance of the branch circuit; This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates the upper limit of active and reactive power capacity of the branch; A large positive number, used for logical constraint linearization; Indicates the upper and lower limits of the voltage amplitude; Constraints (2)-(5) represent the power balance equations. Ignoring power loss, considering line switching, the branch voltage constraint is captured in equations (6)-(7), while the safety constraint is specified by equations (8)-(10). The operating constraints of distributed generator sets (DG), ESS, and flexible SOPs follow the following equation: DG should comply with power and capacity limits: (11) (12) in This represents the active and reactive power output of the DG within period t; This indicates the maximum active power output and apparent power capacity of the DG; ESS includes charge / discharge limits: (13) (14) (15) (16) (17) in The ESS is in discharge mode; while The ESS is in charging mode; This represents the active power released and stored in the ESS within period t; Indicates the maximum permissible charge and discharge power of the ESS; This indicates the state of charge of the ESS at a certain moment; This indicates the discharge and charging efficiency of the ESS. These represent the lower limit of the low-charge state and the upper limit of the high-charge state of the ESS, respectively. Indicates ESS capacity; SOP considers port power limitations: (18) (19) (20) in This indicates the active and reactive power output of port SOP within period t; Represents the set of nodes that connect soft openings; This indicates the maximum reactive power output at SOP. Indicates SOP capacity; To enhance system resilience, topology reconstruction is employed. Therefore, a spanning tree model is used to formulate the conditions for network radiality, as shown in equations (21)-(23). (21) (22) (23) in This indicates that a value of 1 represents a branch that closes within the period; otherwise, a value of 0. Indicates a branch group in a distribution network; It is a set of nodes in a power distribution network. ; Represents the set of root nodes; introduces binary variables. and To indicate different directions of trends; Describes the power flow state in a certain direction between nodes i and j within time period t, where 1 indicates that there is power flow in that direction and 0 indicates that there is no power flow. Step 2.2.3, Model Transformation: In order to address the uncertainty of regeneration, a chance-constrained transformation method is introduced, which enhances the solvability of the model through linearization; (24) in Represents the set of nodes that connect soft openings; Represents a set of DG, ESS, and WT / PV; These represent the active power outputs of distributed energy (DG), energy storage system (ESS), wind power (WT), and photovoltaic (PV) at time t, respectively. This represents the active power load of node i during time period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. This represents the upper limit of the active power output of WT / PV at node r at time t, that is, the maximum active power that the device can output during this period. This is the probability threshold; Equation (24) is a chance constraint, which means that the probability that all available generator outputs can meet the restored load demand must be greater than the confidence level. The original DSR model, i.e., equations (1)-(24), is a non-convex probability problem, requiring model transformation to ensure its solvability. The inherent randomness and correlation of wavelet transform and photovoltaic power output pose challenges to accurately representing them using standard probability distribution models. Assuming that the actual power output of renewable energy is determined by the predicted value, accompanied by superimposed prediction errors, the probability density function (PDF) of wavelet transform and photovoltaic transformation is modeled using GMM, as follows: (25) In the formula A combined PDF for WT and PV; M is the number of Gaussian components; The weight of the m-th Gaussian component ×with its mean vector Covariance Matrix ; So, given Cumulative Distribution Function (CDF): (26) Cumulative distribution function The core meaning is the probability that a random variable takes a value less than or equal to y, which is mathematically expressed as a probability density function: M represents the number of components that make up the normal distribution; This represents the weight of the m-th normal component; Let y represent the normal cumulative distribution function of the i-th normal component at point y; Using equation (26), the equivalent transformation of the chance constraint is obtained: (27) in It is a set of nodes in a power distribution network. ; This represents the active power load of node i during time period t; This means that if the load is restored at time t, node i is 1; otherwise, it is 0. Let represent the active power output of distributed energy source (DG) and energy storage system (ESS) at time t, respectively. Represents the set of DG and ESS; express The inverse function is calculated using the bisection method; This is the probability threshold; Following this process, the recovery model is transformed into a deterministic mixed-integer linear programming (MILP) problem, which is solved using a commercial solver.

7. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 6, characterized in that, Step 3 is described in detail below: Step 3.1: Centralized DSR Model Decomposition Step 3.1.1: Satisfy Subnet Radial Constraints: In this step, the physical connections between adjacent regions are isolated, and each subnet performs DSR independently to ensure radial configuration; Step 3.1.2: Distributed Optimization of the Entire System: Using the subnet topology obtained in Step 1, a fully distributed model is developed to determine the inter-subnet reconstruction strategy and DER scheduling results. Furthermore, a boundary variable compensation mechanism is introduced to ensure the reliability of the recovery strategy in the event of a communication failure. For area a, this is represented as: (28) in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. Introducing consensus continuity and binary variables and To achieve model decomposition, and As dual variables; It is the set of adjacent regions of region a; This is the local objective function for region a, which typically includes the operating costs and constraint penalty terms of that region, and is used to measure the economy and rationality of the operation of region a itself. Represented as It is the augmented Lagrange penalty coefficient. It is the penalty term coefficient for consistency constraints in distributed algorithms, used to enhance the consistency of boundary variables between subnets; In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a. A communication link failure prevented the control center from exchanging optimization boundary variables, causing the algorithm to fail to converge. Constraints were then added. (29) in For boundary decision variables; This indicates whether the communication link between adjacent areas a and b is valid. Then, the boundary variables for inter-regional exchange are determined through distributed optimization; if Then give the compensation value. The compensation strategy allows for the use of predetermined optimal boundary values ​​between adjacent regions without the need for information exchange. In mathematics, "a" is a universal quantifier that represents all adjacent regions b for region a. Meanwhile, to determine the switching decision for the communication line, two sets are defined: the set of feasible communication areas F and the set of unfeasible communication areas Y. The communication line between these two areas is subject to the following constraints. (30) in and Indicates the region a and B The status of the first line between them It is a region a and B The set of all branches between them. Indicates excluding branches The set of remaining branches, as shown in equation (30), indicates that region a and region B Only one connecting line is required between them; Indicates the branch between regions a and B The on / off state variable takes a value of 1 to indicate that the branch is closed and connected, and a value of 0 to indicate that the branch is open. This indicates the relationship between a and B, excluding branches. The on / off state variables of other branches v, and their value rules are the same as those of other branches v. Consistent; Step 3.2: Dynamic Service Recovery Strategy To address short-term fluctuations in renewable energy, a dynamic recovery strategy was adopted, which consists of two phases. In the first stage, Equation (28) is optimized to determine the system topology reconfiguration strategy and ESS scheduling results for the entire recovery cycle. This stage involves topology optimization to maximize the connectivity between the power outage area and the local DER support area, thereby enhancing system resilience. In the second stage, the spanning tree constraints (21)-(23) are related to the constraints of the topology and ESS, that is, equations (13)-(17) are removed, resulting in the following distributed service recovery model: (31) Constrained by equations (2)-(12), (18)-(20), (27) and (29); in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; Equation (31) is executed by the rolling horizon framework, which is dynamically reformulated as time progresses from t to t+1 and incorporates the latest probability distribution results of renewable energy. The DSR strategy is dynamically adjusted during the optimization process, thereby improving its operational reliability. The presence of a wide range of binary decision variables in equations (28) and (31) often leads to serious computational challenges, including inefficiency and non-convergence.

8. The distribution network resilience enhancement method based on distributed dynamic recovery as described in claim 7, characterized in that, Step 4 is described in detail below: Step 4.1: Optimize the framework design The presence of binary variables related to the operational control of the interior and boundary regions complicates the direct application of ADMM to solving the DSR model. To address this, a novel optimization framework is proposed that can iterate the projection and relaxation of boundary variables. The method is combined with the traditional ADMM to solve the convergence problem in the distributed DSR model. On this basis, induced acceleration techniques are used to improve the computational efficiency of subproblems within the distributed optimization framework. Step 4.2: Converging Projection and Iterative Process The binary variables in equation (28) are of two types, namely To handle these binary variables, different subproblems are formulated and their optimal values ​​are determined sequentially. Subproblem 1: Binary Variables The existence of boundary constraints hinders the smooth convergence of the ADMM iteration process. To address this, an auxiliary continuous variable is introduced. Relax the Boolean constraints, and then modify equation (28) using the following update strategy: (32a) (32b) (32c) (32d) (32e) In the formula, B represents the coupled network cluster; As a regularization factor; Indicates in The projection on the screen is used to round entries to 0 or 1. The penalty factor for ADMM; It refers to the update value of the intermediate variable at the boundary within the coupled network set B. and dual variables The mean of the sums; In equation (32), the binary variables are relaxed to continuous variables, resulting in a convex formula that ensures the effective convergence of ADMM. During the iteration process, a projection-based strategy is used to relax the variables. Push to Boolean values, as described in equation (32c); then, apply the consensus variable. The optimized solution is projected onto the closest continuous solution. Given the binary value of the algorithm, under the constraint of the augmented Lagrangian function, the convergence induction condition of the algorithm is... This ensured The optimal solution is a binary solution; In the ADMM step, gradually increase Beneficial for driving For stable convergence to Boolean values, the update strategy is as follows: (33) Where c is a constant; It is a conditional operator; and Let the original residual and the dual residual be represented by , respectively, in iteration k. They converge to: (34) (35) in and For predetermined tolerance; Subproblem 2: The solution to subproblem 1 fixes the boundary binary variables. Meanwhile, internal load switching variables The requirement is for a Boolean value, since it is an integer variable. The existence of this model indicates that it belongs to a non-convex class, which leads to a convergence problem in ADMM. To address this, a solution strategy based on relaxation iteration is introduced. During the relaxation phase of integer variables, Relaxation is continuous This forms a convex model, which is then solved using ADMM with convergence. (36) It conforms to equations (2)-(20), (27) and (29); and The update strategy is shown in equation (32). When solving equation (36), the boundary variables are fixed and a completely independent model is developed for each region. (37) It conforms to equations (2)-(20) and (27); in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. ; obtained in equation (37) After finding the optimal value, this value is kept fixed and used to reformulate equation (36); equations (36) and (37) are optimized alternately until convergence is achieved. The optimal service recovery strategy is determined by solving subproblems 1 and 2 in sequence. The convergence criterion is: (38) in To summarize the objectives for all regions in iteration z. For predetermined tolerance; Then, since there are no boundary binary variables, equation (31) in stage 2 is solved by the relaxation iteration method; Step 4.3: Acceleration Strategy Acceleration Strategy: The computational efficiency of the algorithm largely depends on the solutions of the independent models (37). Each model is expressed as a MILP problem. To address this issue, an induced acceleration strategy is adopted to improve its computational performance. By adding an induced objective function, equation (37) is restated as follows: (39) It conforms to equations (2)-(20) and (27); in and These are continuous binary decision variables within region a, which can be divided into boundary decision variables. In the formula for The induced function is expressed as: (40) In the formula, As a weighting factor, In equation (37) The relaxed solution, in which the binary variables are relaxed to continuous values ​​during the optimization process; if ,So , exist Time is more than in The time is smaller; for minimization problems, It will tend towards 0; similarly, if ,but Will be induced to 1, this is This trend indicates that this method accelerates the solution process.

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