Distributed Parallel Solution Method and System for Optimal Power Flow

Abstract

This invention discloses a distributed parallel solution method and system for optimal power flow. The method includes the following steps: dividing a pre-defined undirected weighted graph based on a community detection algorithm to obtain several initial electrical communities; performing graph decomposition and pruning on each initial electrical community based on graph topology features to obtain several target electrical communities; each target electrical community iteratively solves its local OPF subproblem using the alternating direction multiplier method until the optimal solution corresponding to each target electrical community is obtained, generating the corresponding global optimal power flow result; wherein, in each iteration step, while independently solving for the optimal internal variables and optimal boundary variables corresponding to each target electrical community, the global consensus variable is also updated based on each optimal boundary variable; the penalty parameter is updated based on each optimal boundary variable, and the first Lagrange multiplier is adjusted based on the updated penalty parameter. The solution method proposed in this invention can effectively improve the solution efficiency.

Description

Technical Field

[0001] This invention relates to the field of power system optimal power flow calculation technology, and in particular to a distributed parallel solution method for optimal power flow based on community discovery and graph decomposition. Background Technology

[0002] Optimal Power Flow (OPF) is a core computational tool for the economical and safe operation of power systems. It optimizes generator output configuration to minimize generation costs or grid losses while meeting constraints such as generation output, node voltage, and line transmission limits. With the continuous expansion of the power grid and the integration of numerous renewable energy sources, traditional centralized OPF calculations face technical challenges such as excessive computational complexity, high memory consumption, and difficulty in adapting to distributed computing architectures. Distributed optimal power flow calculation has become an important research direction in this field, but the solution efficiency of existing distributed optimal power flow calculation methods still needs improvement. Summary of the Invention

[0003] The purpose of this invention is to solve the problem of low solution efficiency in existing distributed optimal power flow calculation methods, and to propose a parallel solution method for distributed optimal power flow.

[0004] To solve the above-mentioned technical problems, the present invention provides the following technical solution: Firstly, a method for parallel solving of distributed optimal power flow is provided, including the following steps: The pre-defined undirected weighted graph is divided based on the community detection algorithm to obtain several initial electrical communities; Based on graph topological features, graph decomposition and graph pruning are performed on each initial electrical community to obtain several target electrical communities; Each target electrical community solves its local OPF subproblem in parallel iteratively using the alternating direction multiplier method until the optimal solution for each target electrical community is obtained, generating the corresponding global optimal power flow result; One iteration step in which each target electrical community solves its local OPF subproblem in parallel includes the following steps: Based on the first global consensus variable, the first Lagrange multiplier and the penalty parameter, as well as the first internal variable and the first boundary variable corresponding to each target electrical community, the optimal internal variable and the optimal boundary variable corresponding to each target electrical community are solved independently. The optimal internal variable is used as the first internal variable corresponding to the next iteration step, and the optimal boundary variable is used as the first boundary variable corresponding to the next iteration step. A second global consensus variable is generated based on each optimal boundary variable, and it is used as the first global consensus variable in the next iteration step. The penalty parameters are updated based on the optimal boundary variables, and the first Lagrange multiplier is adjusted based on the updated penalty parameters to obtain the corresponding second Lagrange multiplier, which is then used as the first Lagrange multiplier for the next iteration step.

[0005] Existing technologies propose a method for solving AC optimal power flow based on distributed machine learning. This method first uses a community discovery algorithm (such as Louvain) to divide the power grid, and then independently trains a deep neural network (DNN) model in each divided region to predict the voltage amplitude and angle of that region. This method is essentially a data-driven approach, replacing the solution process of the physical model with a machine learning model. Its core lies in training the DNN with historical data to fit the mapping relationship from input (such as load) to output (such as voltage), but it lacks strict guarantees of physical constraints.

[0006] Existing technologies also propose solutions based on the alternating direction multiplier method in parallel, which are based on the divided power grid regions. However, these solutions do not perform deeper graph structure analysis and simplification on the sub-regions after partitioning. Furthermore, the penalty parameter update rules of ADMM are relatively fixed or based on simple residual feedback, lacking awareness of the topological characteristics of the power grid.

[0007] This application performs graph decomposition and graph pruning on the initial electrical communities obtained by the community detection algorithm, which can reduce dimensionality by utilizing the graph structure and improve the efficiency of subsequent solutions. During the solution process, only the boundary variables are exchanged, and the target electrical communities are coupled based on the boundary variables. Furthermore, the penalty parameters are updated based on the boundary variables to dynamically balance the convergence speed and numerical stability.

[0008] As one possible implementation method, the specific steps for updating the penalty parameter based on each optimal boundary variable are as follows: The boundary graph is updated based on the optimal boundary variables of each target electrical community, and graph features are extracted. The nodes of the boundary graph represent each target electrical community, the edges represent the connecting lines between each target electrical community, and the graph features include node degree, modularity, and boundary cut vertices. When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased; When the module degree is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced, and the second node degree threshold is less than or equal to the first node degree threshold. In all other cases, the penalty parameters remain unchanged.

[0009] Existing distributed OPF methods often employ fixed or simple residual feedback-based coordination parameters when dealing with inter-region coupling. This strategy ignores the differences in the graph roles of different boundary nodes within the power grid. For example, the impact on global convergence is clearly different between a cut point boundary connecting multiple regions and a single leaf node boundary; existing methods fail to perceive this graph structure information, leading to a one-size-fits-all coordination strategy that requires numerous iterations to converge at strongly coupled boundaries, resulting in low parallel efficiency.

[0010] This application utilizes the structural features (node ​​degree, cut vertices, modularity) of the boundary graph to drive parameter updates, which belongs to graph structure-driven methods. Compared with existing numerical-driven methods that update parameters based on numerical residuals, it can fundamentally reflect the importance and coupling strength of boundary nodes in the global topology, thereby accelerating convergence more effectively.

[0011] In existing technologies, graph theory is often used only as a preprocessing tool and is no longer used after the region is partitioned. This application overcomes the bias of existing technologies by taking graph structure information (community density, boundary node roles) as core features and dynamically and adaptively feeding it back into the parameter adjustment of the parallel optimization algorithm (ADMM). The graph structure information of the power grid is integrated into the entire process of partitioning, simplification, and parallel coordination, forming a "three-in-one" collaborative design from data structure to computing architecture, realizing the deep integration and collaborative design of graph theory and physical optimization algorithm.

[0012] As one possible implementation, the step size of the alternating direction multiplier method is also updated based on the updated penalty parameters; The step size adjustment rule is as follows: ; Where, α t Let α be the current step size. t+1 For the updated step size, This refers to the updated penalty parameters.

[0013] As one possible implementation method: Identify the graph topology features of each initial electrical community, and decompose the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; The graph topological features include cut vertices, bridging edges, and radiating branches.

[0014] As one possible implementation method: The forward-backward substitution method is used to perform equivalent processing on each radiating branch, and the obtained equivalent complex power is injected into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community. The root node is the connection point between the radial branches and the core backbone subgraph.

[0015] This application uses the forward-backward substitution method to equate radiating branches to load injection into the core network. This is a mathematical equivalence, rather than a simple topology deletion. Through the design of equivalent pruning, this application can maintain the physical consistency of power balance while reducing dimensionality, thus ensuring the accuracy of subsequent OPF optimization.

[0016] As one possible implementation method: When the original residual is less than or equal to the preset original residual convergence threshold, and the dual residual is less than or equal to the preset dual residual convergence threshold, the iteration is determined to be converged. The original residual is used to measure the deviation between the global consensus variable and the boundary variable formed by the optimal boundary variables of all target electrical communities in the current iteration step; The dual residual is used to measure the degree of change of the global consensus variable in the current iteration step.

[0017] Secondly, a distributed parallel solution system for optimal power flow is proposed, including a community detection module, a graph decomposition module, a distributed OPF solution module, and a graph-aware coordinator: The community discovery module is used to divide a preset undirected weighted graph based on a community discovery algorithm to obtain several initial electrical communities. The graph decomposition module is used to perform graph decomposition and graph pruning on each initial electrical community based on graph topology features to obtain several target electrical communities; The distributed OPF solution module is used to solve the local OPF subproblem in parallel iteratively based on the alternating direction multiplier method for each target electrical community until the optimal solution corresponding to each target electrical community is obtained, and the corresponding global optimal power flow result is generated. Specifically: The distributed OPF solving module independently solves for the optimal internal variables and optimal boundary variables corresponding to each target electrical community based on the first global consensus variable, the first Lagrange multiplier and the penalty parameter, as well as the first internal variables and the first boundary variables corresponding to each target electrical community. The optimal internal variables are used as the first internal variables corresponding to the next iteration step, and the optimal boundary variables are used as the first boundary variables corresponding to the next iteration step. The graph-aware coordinator generates a second global consensus variable based on each optimal boundary variable, and uses it as the first global consensus variable for the next iteration step. The graph-aware coordinator updates the penalty parameters based on each optimal boundary variable, adjusts the first Lagrange multiplier based on the updated penalty parameters, obtains the corresponding second Lagrange multiplier, and uses it as the first Lagrange multiplier for the next iteration step.

[0018] As one possible implementation, the graph-aware coordinator includes a consensus computation unit and a boundary coordination unit, wherein the boundary coordination unit includes: The boundary graph maintenance unit is used to update the boundary graph and extract graph features based on the optimal boundary variables of each target electrical community. The nodes of the boundary graph represent each target electrical community, the edges represent the connecting lines between each target electrical community, and the graph features include node degree, modularity, and boundary cut vertices. The penalty parameter adjustment unit is used for: When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased; When the module degree is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced, and the second node degree threshold is less than or equal to the first node degree threshold. In all other cases, the penalty parameters remain unchanged.

[0019] As one possible implementation: the graph decomposition module includes a decomposition unit and a pruning unit; The decomposition unit is used to identify the graph topology features of each initial electrical community, and decomposes the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; the graph topology features include cut vertices, bridging edges and radial branches; The pruning unit is used to perform equivalent processing on each radiating branch using the forward-backward substitution method, and injects the obtained equivalent complex power into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community; the root node is the connection point between the radiating branch and the core backbone subgraph.

[0020] Thirdly, a computer-storable medium stores computer instructions, which, when invoked by a computer, are used to execute the distributed optimal power flow parallel solution method described in any of the preceding claims.

[0021] This invention, by adopting the above technical solutions, has significant technical effects: This invention performs graph decomposition and graph pruning on the initial electrical communities obtained by the community detection algorithm, which can reduce dimensionality by utilizing the graph structure and improve the efficiency of subsequent solutions. During the solution process, only the boundary variables are exchanged, and the target electrical communities are coupled based on the boundary variables. Furthermore, the penalty parameters are updated based on the boundary variables to dynamically balance the convergence speed and numerical stability. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 This is a flowchart illustrating the distributed optimal power flow parallel solution method of the present invention; Figure 2 This is a flowchart illustrating the process of partitioning a pre-defined undirected weighted graph based on a community detection algorithm. Figure 3 This is a schematic diagram of the power grid partitioning results based on a community detection algorithm in an example; Figure 4 This is a schematic diagram comparing the initial electrical community and the target electrical community obtained after graph decomposition and graph pruning in an example; Figure 5 This is a flowchart illustrating the process of graph-aware coordination based on the obtained optimal boundary variables. Figure 6 This is a schematic diagram of the distributed optimal power flow parallel solution system of the present invention. Detailed Implementation

[0024] The exemplary embodiments of this disclosure are described below with reference to the accompanying drawings, including various details of the embodiments to aid understanding, and should be considered merely exemplary. Therefore, those skilled in the art will recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this disclosure. In the following description, different "one embodiment" or "another embodiment" do not necessarily refer to the same embodiment. Furthermore, specific features, structures, or characteristics in one or more embodiments may be combined in any suitable form. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. Similarly, for clarity and brevity, descriptions of well-known functions and structures are omitted in the following description.

[0025] Example 1: A distributed parallel solution method for optimal power flow, such as... Figure 1 As shown, it includes the following steps: S100. Construct an undirected weighted graph corresponding to the target power grid; The vertices of the undirected weighted graph are the power system nodes corresponding to the target power grid, and the edges are the transmission lines and transformers corresponding to the target power grid. The weights of the edges are based on electrical distance (such as the reciprocal of line impedance) or power transmission distribution factor.

[0026] In this embodiment, the bus, generator, and load in the power system are mapped to the vertices V of the graph, and the transmission lines and transformers are mapped to the edges E of the graph. An undirected weighted graph G of the power grid is constructed. The electrical coupling strength between nodes is quantified by the edge weights, so that the community detection algorithm can accurately characterize the electrical density between nodes and make the partitioning results highly consistent with the physical reality of the power grid. The specific calculation method for the undirected weighted graph G of the power grid is as follows: G=(V,E,w); Where V is the set of nodes (bus), E is the set of edges (line, transformer), and w is the edge weight; In this embodiment, the edge weight w ij Defined as the reciprocal of the electrical distance, those skilled in the art can select the line impedance modulus z according to actual needs. ij The reciprocal of or reactance x ij The reciprocal (applicable to high-voltage power grids) is calculated as follows: .

[0027] In this embodiment, grid topology parameters and real-time measurement data are obtained from the SCADA / EMS system. SCADA (Supervisory Control and Data Acquisition) is responsible for collecting real-time grid operation measurement data, while EMS (Energy Management System) stores the basic grid topology parameters. Specifically: Power grid topology parameters: These are the basic data that describe the physical structure of the power grid, including the basic parameters of equipment such as buses, generators, loads, transmission lines, and transformers in the power grid, as well as the connection relationships between these devices; Real-time measurement data: This refers to the dynamic operation data of the power grid, including the voltage, current, and power of each node, the output of each generator unit, the transmission power of each line, and the real-time power consumption of each load.

[0028] S200. Based on the community detection algorithm, the preset undirected weighted graph is divided to obtain several initial electrical communities; Those skilled in the art can initiate an OPF calculation task by timed or event-triggered method. When executing the OPF calculation task, the preset undirected weighted graph is partitioned based on the community detection algorithm. In this embodiment, a community detection algorithm is used to divide the undirected weighted graph into several electrically coupled sub-regions with weak inter-regional coupling, that is, to obtain several weakly coupled initial electrical communities. Those skilled in the art can select existing publicly available community detection algorithms to partition the undirected weighted graph according to actual needs. In this embodiment, the Louvain algorithm is used.

[0029] In this embodiment, modularity Q is used as the quality metric for community division, and the optimization objective is to maximize modularity Q to achieve automated identification of weakly coupled electrical zones in the power grid. The specific calculation method for modularity Q is as follows: ; Where m is the total edge weight; w ij k represents the edge weight. i c is the weighted degree of node i; i For the community to which node i belongs; when c i =c j When δ(ci,cj)=1, otherwise δ(cj)=1. i ,c j )=0; The specific calculation method for the total edge weight m is as follows: ; Weighted degree k i The specific calculation method is as follows: .

[0030] As an example, the Louvain algorithm is used for iterative optimization to achieve community partitioning, refer to... Figure 2 In each iteration step, the following steps are performed: S210, Modularity optimization stage; Traverse all nodes in the current power grid graph, calculate the modularity gain ΔQ for moving the target node from the current community to the corresponding neighbor community, and perform the node movement operation based on the modularity gain ΔQ. The current power grid graph is an undirected weighted graph or a power grid graph obtained from the previous iteration step; The specific calculation method for the modularity gain ΔQ is as follows: ; Where ∑in is the sum of edge weights within the community; ∑tot is the sum of edge weights associated with all nodes in the community; k i,in It is the sum of the edge weights between node i and nodes within the community; Execute node move operations that maximize the modularity gain ΔQ until there are no more node moves with ΔQ > 0; S220, Community Aggregation Phase: After all nodes with modularity gain greater than 0 have been moved, each community is merged into a supernode, and a new power grid is constructed. The nodes in the new power grid are the supernodes merged from the communities, and the edge weights between nodes are the sum of the edge weights between the original communities.

[0031] S230, Iteration termination judgment, until there is no room for improvement in modularity Q, the newly produced power grid graph is used as the graph topology decomposition of the initial electrical community based on electrical coupling characteristics, and each node in the graph topology decomposition is used as the corresponding initial electrical community.

[0032] Those skilled in the art can set their own iteration termination criteria, as long as they can identify that there is no room for improvement in the modularity Q. This embodiment does not limit such criteria.

[0033] Those skilled in the art can introduce adjustable parameters (such as resolution parameters) during the execution of the Louvain algorithm, and flexibly adjust the granularity of the power grid partition according to actual computing resources and accuracy requirements to adapt to different application scenarios.

[0034] As an example, taking a 118-node power grid system as the application object, the buses, generators, and loads in the system are modeled as vertices of the graph, and the lines and transformers are modeled as edges. The edge weight is defined as the reciprocal of the electrical distance (e.g., 1 / xij), constructing a weighted undirected graph reflecting the electrical coupling characteristics. Based on the above community detection algorithm, the modularity Q is iteratively optimized, and the 118-node system is finally automatically divided into 4 electrical communities (e.g., ...). Figure 3 As shown, different colors represent different communities. The nodes within each community are tightly coupled, and the communities are connected to each other only through a few tie lines, which realizes the weak coupling partitioning of the power grid. Different electrical communities are distinguished by different identifiers.

[0035] S300. Based on graph topology features, perform graph decomposition and graph pruning on each initial electrical community to obtain several target electrical communities; In this embodiment, based on the initial electrical community partitioning results, topological features such as internal cut points, bridging edges, and radial branches are identified, and the community is decomposed into a core backbone subgraph and multiple radial branch leaf graphs. The forward-backward substitution method is used to equate the radial branches to generalized loads injected into the root nodes of the core backbone subgraph. Pruning is performed through the mathematical equivalence of the radial branches to preserve the physical consistency of power balance. This embodiment cleverly decomposes the complex optimization problem that originally needed to be solved at the entire community dimension into two levels: core network optimization and branch analysis computation, by designing graph decomposition and graph pruning. While ensuring computational accuracy, it improves computational efficiency by reducing the dimensionality of the community network.

[0036] The specific steps are as follows: S310, Graph Decomposition; Identify the graph topology features of each initial electrical community, and decompose the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; The graph topological features include cut vertices, bridging edges, and radiating branches; All the radiating branches together form the radiating branch-leaf diagram corresponding to the initial electrical community; The initial electrical community, with the main trunk portion of the radial branches removed, is used as the core backbone subgraph; The resulting core backbone subgraph contains all the ring networks within the community and is the core object for subsequent OPF optimization; The resulting radial branch-leaf diagram is a passive feeder, which can be simplified equivalently through analytical calculation.

[0037] S320, Pruning diagram; The forward-backward substitution method is used to perform equivalent processing on each radiating branch, and the obtained equivalent complex power is injected into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community. The root node is the connection point between the radial branches and the core backbone subgraph. This embodiment, through graph pruning design, eliminates the need for detailed modeling of numerous nodes within the branches of the backbone network, thereby achieving dimensionality reduction in subsequent OPF calculations. Furthermore, unlike simple deletion, this embodiment maintains physical consistency of power balance through forward and backward substitution, ensuring calculation accuracy in subsequent OPF calculations.

[0038] Assuming the radiating branch consists of a root node r and several downstream nodes, the specific calculation steps for calculating the equivalent injected power using the forward-backward substitution method are as follows: S321. Forward process (from end to root): Calculate the active and reactive power flowing from each downstream node to the root node sequentially. Specifically: Active power P k The specific calculation method is as follows: P k =P k+1 +P loss,k +P load,k ; Among them, P k P1 represents the active power flowing from downstream node k to node k+1 (i.e., flowing towards the root direction). When k is a terminal node, P1 is the load power of that node itself. loss,k The active power loss on branch k (connecting node k and node k+1) is typically calculated based on the branch impedance and the power flowing through it; P load,k This represents the active power of the load connected at node k (or zero if the node has no load).

[0039] reactive power Q k The specific calculation method is as follows: Q k =Q k+1 +Q loss,k +Q load,k ; Among them, Qk Q1 represents the reactive power flowing from node k to node k+1 (i.e., flowing towards the root direction). When k is a terminal node, Q1 is the load power of that node itself. loss,k Q represents the reactive power loss on branch k (connecting node k and node k+1), which is typically calculated based on the branch impedance and the power flowing through it; load,k This represents the reactive power of the load connected at node k (or zero if the node has no load).

[0040] S322, Back-substitution process (from root to end): Calculate the voltage amplitude of each downstream node based on the root node voltage. Specifically: Voltage amplitude V k The specific calculation method is as follows: ; Among them, V k P represents the voltage amplitude (per unit or actual value) corresponding to downstream node k. k Q represents active power. k Reactive power; R k X is the resistance of branch k; k Let K be the reactance of branch k.

[0041] S323. Calculate the equivalent complex power. Based on the calculation results of forward and backward subtraction, obtain the equivalent injected complex power of the root node r of the radiating branch. Specifically: Equivalent complex power The specific calculation method is as follows: ; This represents the final calculated equivalent complex power at the root node r. For equivalent complex power The active component; For equivalent complex power The reactive component; j is the imaginary unit, which combines the active power P and reactive power Q in a complex domain for calculation, representing the orthogonal relationship between the two with a 90-degree phase difference.

[0042] S324. Utilizing the equivalent complex power corresponding to each radiating branch Replace the corresponding radiating branches to obtain the corresponding target electrical community; In this embodiment, after the equivalent pruning process, only the core backbone subgraph is retained as the object of subsequent OPF optimization. The radial branches do not need to participate in complex nonlinear optimization. The calculation is completed only through analytical formulas, which can reduce the number of nodes in the core optimization problem by 30% to 50%, and effectively reduce the optimization dimension.

[0043] Reference Figure 4As an example, a complex electrical community containing a ring network and multiple radial branches is taken as the processing object. The cut points and bridging edges inside the community are identified by graph decomposition technology. The radial branches are removed from the core backbone network. The equivalent complex power of each radial branch is calculated by forward and backward substitution method. The equivalent load is injected into the root node of the corresponding core backbone network.

[0044] In this embodiment, the interior of each initial electrical community obtained based on the community detection algorithm is finely deconstructed; Among them, the community detection algorithm can identify initially electrical communities with weak coupling, creating conditions for parallel computing; The graph decomposition design ensures the decoupling of the regions within the initial electrical community, providing an operational basis for dimensionality reduction in subsequent equivalent processing.

[0045] This embodiment uses the forward-backward substitution method to equate the radiating branches to loads injected into the core network. This equivalent treatment maintains the physical consistency of power balance, simplifying calculations while ensuring optimization accuracy.

[0046] S400: Each target electrical community solves its local OPF subproblem in parallel iteratively using the alternating direction multiplier method until the optimal solution for each target electrical community is obtained, generating the corresponding global optimal power flow result.

[0047] Specifically, the following steps are included: S410, Pre-construct OPF subproblems; In this embodiment, for each target electrical region after deep decomposition, internal variables and boundary variables are defined. With the goal of minimizing the power generation cost of the sub-region, a distributed OPF sub-problem is constructed by combining power flow balance constraints, equipment operation constraints and boundary consistency constraints. Based on the Alternating Direction Multiplier Method (ADMM), an augmented Lagrangian function incorporating constraints is constructed, and a three-round iterative rule is set for parallel solution of subproblems, global consensus variable update, and Lagrangian multiplier update, forming a complete distributed OPF computation model; S411, Objective function; For each target electrical region k (k=1,2,...,N, where N is the total number of sub-regions), define an internal variable x. k With boundary variable y k ; Among them, the internal variable x k y is a decision variable that belongs only to subregion k and is not coupled with other regions; the boundary variable is y. k The electrical quantity decision variables are shared by the target electrical region k and its adjacent regions.

[0048] The objective function is constructed with the goal of minimizing the power generation cost within the target electrical region k: ; Among them, f k (x k ) is the objective function of the target electrical region k, which is usually the power generation cost function (such as a quadratic cost function) within that region.

[0049] S412, Constraints; The constraints of this subproblem include power flow balance within the region, unit operation constraints, and consistency constraints coupled through boundary variables, specifically: Power flow balance constraints ensure that power injection and outflow at nodes within the region must be balanced. Specific constraints are as follows: g k (x k ,y k )=0; Unit operating constraints include generator output limits, node voltage amplitude limits, and line transmission capacity. Specific constraints are as follows: h k (x k ,y k )≤0; Boundary consistency constraints ensure that the variables at the boundary nodes of each sub-region remain consistent with the global consensus variables. The specific constraints are as follows: y k =z; Where z is a global consensus variable, requiring that the electrical quantities of each sub-region at the boundary node be consistent with the global consensus variable, so as to realize the boundary splicing between regions.

[0050] This embodiment couples the target electrical communities by constraining the boundary consistency.

[0051] S413. Construct the augmented Lagrange function; Based on the aforementioned distributed OPF subproblem, the alternating direction multiplier method (ADMM) is used to solve it, incorporating the constraints into the objective function to construct an augmented Lagrangian function, the specific function of which is as follows: ; Where, λ k These are Lagrange multipliers (dual variables), used to penalize the deviation between boundary variables and consensus variables; The penalty parameter (step size factor in ADMM) controls the penalty strength for boundary consistency; z is the global consensus variable, which is the reference value (such as voltage amplitude and phase angle) that all sub-regions reach a consensus at the boundary node. Let be the square of the L2 norm, representing the sum of squares of the deviations, used to construct the quadratic penalty term.

[0052] S420, Parallel solution; In practical applications, computing nodes are assigned to each target electrical community, and each computing node is responsible for solving the OPF subproblem for one target electrical community.

[0053] An iterative step in which each target electrical community solves its local OPF subproblem in parallel includes the following steps: S421. Each target electrical community independently solves for its corresponding optimal internal and optimal boundary variables by minimizing the augmented Lagrange function, based on the first global consensus variable, the first Lagrange multiplier, the first internal variable, and the first boundary variable. The optimal internal variable is used as the first internal variable of the next iteration step, and the optimal boundary variable is used as the first boundary variable of the next iteration step. The expression corresponding to the local OPF subproblem is as follows: ; To augment the Lagrangian function, i.e.: ; in: The current target electrical community is the kth community; x k Indicates the corresponding first internal variable; y k Indicate the corresponding first boundary variable; Z t Represents the corresponding first global consensus variable; Indicates the corresponding first Lagrange multiplier; This represents the optimal internal variable obtained from the solution, that is, the first internal variable corresponding to the next iteration step; This represents the optimal boundary variable obtained by solving the problem, that is, the first boundary variable corresponding to the next iteration step; f k (x k This is used to indicate the generation cost of the target electrical community k; This is the penalty parameter.

[0054] The first global consensus variable, the first Lagrange multiplier, the first internal variable, and the first boundary variable mentioned above are either initial values ​​or updated values ​​calculated in the previous iteration step.

[0055] S422. Generate a second global consensus variable based on the optimal boundary variables of all target electrical communities, and use the second global consensus variable as the first global consensus variable for the next iteration step; In this embodiment, the average value is calculated based on the optimal boundary variables of each target electrical community, and the resulting average value is used as the second global consensus variable.

[0056] As an example, after all computing nodes have completed the current iteration, an average calculation is performed based on all optimal boundary variables, using the following formula: ; Among them, z t+1 The second global consensus variable is the average of the optimal boundary variables of all target electrical communities.

[0057] S423. Based on the deviation between the optimal boundary variable corresponding to the target electrical community and the second global consensus variable, adjust the first Lagrange multiplier to obtain the corresponding second Lagrange multiplier, and use the second Lagrange multiplier as the first Lagrange multiplier for the next iteration step. The specific function for adjusting the first Lagrange multiplier is as follows: ; in, This is the corresponding second Lagrange multiplier, used to correct the boundary variable bias in the next iteration; This is the penalty parameter.

[0058] Penalty parameters Used to control the penalty strength of boundary consistency constraints; The larger the value, the stronger the penalty for boundary variable deviation, which may speed up the convergence speed, but too large a value may lead to numerical instability. As one possible implementation, after obtaining the optimal boundary variables, the penalty parameters are updated based on the optimal boundary variables, and the first Lagrange multiplier is adjusted based on the updated penalty parameters to obtain the corresponding second Lagrange multiplier. In this embodiment, the penalty parameter The step size α is updated simultaneously, and subsequent updates are based on the updated penalty parameters. Solve for the step size α, referring to Figure 5 The specific steps are as follows: T100. Update the boundary map and extract graph features based on the optimal boundary variables of each target electrical community; The boundary map G b The nodes represent the target electrical communities, and the edges represent the connecting lines between the target electrical communities, that is: G b =(V b E b ); Among them, V b Representing the target electrical communities, Eb Represents the connection lines between target electrical communities.

[0059] From boundary map G b Extract core graph features, including node degree, modularity, and boundary cut vertices, specifically: Node degree d(v): The number of boundaries connecting this region to other regions; Modularity Q: Quality of global community partitioning; Boundary cut vertex: If removing a node would cause the boundary graph to become disconnected, then it is a cut vertex.

[0060] T200. Based on the graph features, dynamically adjust the penalty parameters of ADMM according to preset heuristic rules. and step size α; Heuristic rules include penalty parameter adjustment rules and step size adjustment rules; The specific rules for adjusting the penalty parameters are as follows: When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased. In this embodiment, we set... Those skilled in the art can set the node degree threshold according to actual needs. In this embodiment, the node degree threshold is 3. When the module degree Q is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced. In this embodiment, let... The second node degree threshold is less than or equal to the first node degree threshold; those skilled in the art can set the module degree threshold and node degree threshold according to actual needs. When none of the above conditions apply, the penalty parameter remains unchanged. In this embodiment, we set... That is, to maintain the current state.

[0061] in Indicates the current penalty parameter. This indicates the updated penalty parameter. This is the adjustment rate, typically set to 0.1~0.3, used to control the sensitivity of parameter updates; The specific rules for step size adjustment are as follows: ; Where α is the step size, used to update the step size factor of the Lagrange multipliers, and the penalty parameter. Coordination settings ensure the Lagrange multiplier λ k The update and penalty parameter adjustment match, at this time This can improve the stability of iteration.

[0062] In this embodiment, the boundary graph is designed so that nodes represent sub-regions and edges represent connecting lines between regions. The ADMM penalty parameters are dynamically adjusted based on the degree (number of connected target electrical regions) and module degree value of the nodes in the boundary graph.

[0063] For example, for the articulation point boundary connecting three regions, Set to a larger value (e.g., 10.0), and set the step size α to 0.1; for a single boundary, Set the step size to 1.0 and the step size α to 0.5. This strategy reduces the number of iterations.

[0064] T300, based on updated penalty parameters Update the first Lagrange multiplier corresponding to each target electrical community to obtain the corresponding second Lagrange multiplier.

[0065] T400, convergence judgment; To accurately determine the convergence state of the iteration, the original residual and the dual residual are calculated. The specific calculation method is as follows: ; Where, r t+1 The original residual corresponding to the current iteration step, y, measures the boundary variable obtained in the current iteration step. t+1 With global consensus variable z t+1 Deviation between; S t+1 The dual residual corresponding to the current iteration step measures the global consensus variable z in the current iteration step. t+1 The degree of change; y t+1 The boundary variables are formed by the optimal boundary variables of all target electrical communities in the current iteration step; Z t This is the first global consensus variable corresponding to the current iteration step; Z t+1 This is the second global consensus variable corresponding to the current iteration step; Penalty parameters.

[0066] This embodiment sets a dual residual convergence rule: when the original residual is less than or equal to a preset original residual convergence threshold, and the dual residual is less than or equal to a preset dual residual convergence threshold, iterative convergence is determined. The specific convergence condition is as follows: , ; in, The original residual convergence threshold, The threshold values ​​for the dual residual convergence are preset precision control parameters.

[0067] T500, if the convergence condition is not met, the updated penalty parameters will be applied. Step size α and Lagrange multiplier λ k The new parameters are distributed to each computing node, and each computing node uses the new parameters to solve the local OPF subproblem asynchronously and in parallel again. T600. If the convergence condition is met, the calculation is terminated, and the optimal solution for each target electrical region is obtained. T700 integrates the optimal solutions of each target electrical zone and the analytical calculation results of the corresponding radial branches equivalent loads to obtain the global optimal power flow results, and displays the results in a visualization manner.

[0068] In summary, a weighted graph is constructed based on electrical distance, and the adaptive electrical community partitioning of the power grid is achieved by optimizing the Louvain algorithm. The partitioning results are highly consistent with the physical characteristics of electrical coupling in the power grid, resulting in tight coupling within communities and connections between communities via only a few tie lines, significantly reducing the number of coordination iterations in the distributed algorithm. By decomposing the electrical community into a core backbone subgraph and a radial branch-leaf graph through graph decomposition, and using the forward-backward substitution method to achieve mathematical equivalence simplification of the radial branches, the number of nodes in the core optimization problem is reduced by 30% to 50% while maintaining the physical consistency of power balance, significantly improving computational efficiency. The ADMM penalty parameters and step size are dynamically adjusted using the graph topology features of the boundary nodes, avoiding the compatibility issues of splicing traditional coordination strategy techniques, thus improving the convergence speed of the algorithm by more than 30% and solving the technical problem of slow convergence of strongly coupled boundary iterations.

[0069] Example 2: A distributed parallel optimal power flow solution system, used to implement the solution method disclosed in the above examples, wherein, as... Figure 6 As shown, the system's layered architecture consists of a data layer (graph database and real-time data interface), a computation layer (including a community discovery module, a graph decomposition module, and a distributed OPF solution module), a coordination layer (graph perception coordinator), and an application layer (visualization and early warning). Specifically: The data layer is used to store the unweighted graph of the target power grid, equipment parameters, and real-time operating data in the form of attribute graphs. It provides efficient graph traversal and query interfaces and supports the operation of community discovery and graph decomposition algorithms.

[0070] The community discovery module is used to divide a preset undirected weighted graph based on a community discovery algorithm to obtain several initial electrical communities. The graph decomposition module is used to perform graph decomposition and graph pruning on each initial electrical community based on graph topology features to obtain several target electrical communities; The distributed OPF solving module contains multiple parallel computing nodes, each responsible for solving the OPF subproblem in a sub-region. It supports distributed optimization algorithms such as ADMM. It is used to solve the local OPF subproblem in parallel iteratively based on the alternating direction multiplier method for each target electrical community until the optimal solution for each target electrical community is obtained, generating the corresponding global optimal power flow result. The graph-aware coordinator is used to collect boundary variables of each subproblem, adaptively update the Lagrange multipliers and penalty parameters based on graph topological features (boundary node roles, modularity, etc.), and issue new boundary values.

[0071] In existing technologies, parallel computing architectures (such as distributed computing nodes) are separated from the mathematical structure of the problem itself (graph topology). Computing nodes can only passively execute tasks; In this embodiment, the parallel computing architecture itself is defined by a graph structure. The division of computing nodes (each node is responsible for a sub-region) directly corresponds to the community in graph theory, and the working logic of the coordinator (updating the Lagrange multipliers) is also directly driven by graph theory features (boundary graph).

[0072] Since the task boundaries (which sub-region is responsible for) of the distributed solver module are dynamically generated by the outputs of the community discovery module and the graph decomposition module, if the graph structure is removed, the task allocation of computing nodes will lose its physical meaning and become random load balancing, failing to guarantee the weakest coupling between regions. The core logic of the coordinator (the graph topology-aware adaptive coordination strategy) is also defined by graph theory features. Removing graph features means the coordinator loses its awareness of graph features and can only adjust based on fixed parameters or simple residual feedback.

[0073] The distributed OPF solving module independently solves for the optimal internal variables and optimal boundary variables corresponding to each target electrical community based on the first global consensus variable, the first Lagrange multiplier and the penalty parameter, as well as the first internal variables and the first boundary variables corresponding to each target electrical community. The optimal internal variables are used as the first internal variables corresponding to the next iteration step, and the optimal boundary variables are used as the first boundary variables corresponding to the next iteration step. The graph-aware coordinator generates a second global consensus variable based on each optimal boundary variable, and uses it as the first global consensus variable for the next iteration step. The graph-aware coordinator updates the penalty parameters based on each optimal boundary variable, adjusts the first Lagrange multiplier based on the updated penalty parameters, obtains the corresponding second Lagrange multiplier, and uses it as the first Lagrange multiplier for the next iteration step.

[0074] As one possible implementation, the graph-aware coordinator includes a consensus computing unit and a boundary coordination unit; The consensus calculation unit is used to generate a second global consensus variable based on each optimal boundary variable, and use it as the first global consensus variable for the next iteration step. The boundary coordination unit includes: The boundary graph maintenance unit is used to update the boundary graph and extract graph features based on the optimal boundary variables of each target electrical community. The nodes of the boundary graph represent each target electrical community, the edges represent the connecting lines between each target electrical community, and the graph features include node degree, modularity, and boundary cut vertices. The penalty parameter adjustment unit is used for: When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased; When the module degree is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced, and the second node degree threshold is less than or equal to the first node degree threshold. In all other cases, the penalty parameters remain unchanged.

[0075] The Lagrange multiplier calculation unit is used to adjust the first Lagrange multiplier based on the updated penalty parameter to obtain the corresponding second Lagrange multiplier, and use it as the first Lagrange multiplier for the next iteration step.

[0076] As one possible implementation, the boundary coordination unit further includes a step size adjustment unit for updating the step size based on the updated penalty parameters.

[0077] As one possible implementation, the graph decomposition module includes a decomposition unit and a pruning unit; The decomposition unit is used to identify the graph topology features of each initial electrical community, and decomposes the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; the graph topology features include cut vertices, bridging edges and radial branches; The pruning unit is used to perform equivalent processing on each radiating branch using the forward-backward substitution method, and injects the obtained equivalent complex power into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community; the root node is the connection point between the radiating branch and the core backbone subgraph.

[0078] Those skilled in the art can also design human-computer interaction modules according to actual needs: to display partitioning results, iterative convergence process, final optimal power flow scheme (unit output, node voltage, network loss, etc.), and to provide early warning for critical boundaries or nodes exceeding limits.

[0079] Example 3. A computer-storable medium storing computer instructions, which, when invoked by a computer, are used to execute the distributed optimal power flow parallel solution method as described in any of the preceding embodiments.

[0080] As the device embodiment is basically similar to the method embodiment, the description is relatively simple, and relevant parts can be found in the description of the method embodiment.

[0081] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0082] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, apparatus, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0083] This invention is described with reference to flowchart illustrations and / or block diagrams of the method, terminal device (system), and computer program product according to the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0084] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0085] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0086] It should be noted that: The phrase "an embodiment" or "an embodiment" used in this specification means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. Therefore, the phrase "an embodiment" or "an embodiment" appearing in various places throughout the specification does not necessarily refer to the same embodiment.

[0087] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments.

[0088] Furthermore, it should be noted that the shapes and names of the parts and components described in the specific embodiments described in this specification may differ. All equivalent or simple variations made to the structure, features, and principles described in this patent concept are included within the protection scope of this patent. Those skilled in the art to which this invention pertains can make various modifications or additions to the described specific embodiments or use similar methods to replace them.

Claims

1. A distributed parallel solution method for optimal power flow, characterized in that, Includes the following steps: The pre-defined undirected weighted graph is divided based on a community detection algorithm to obtain several initial electrical communities; Based on graph topological features, graph decomposition and graph pruning are performed on each initial electrical community to obtain several target electrical communities; Each target electrical community solves its local OPF subproblem in parallel iteratively using the alternating direction multiplier method until the optimal solution for each target electrical community is obtained, generating the corresponding global optimal power flow result; One iteration step in which each target electrical community solves its local OPF subproblem in parallel includes the following steps: Based on the first global consensus variable, the first Lagrange multiplier and the penalty parameter, as well as the first internal variable and the first boundary variable corresponding to each target electrical community, the optimal internal variable and the optimal boundary variable corresponding to each target electrical community are solved independently. The optimal internal variable is used as the first internal variable corresponding to the next iteration step, and the optimal boundary variable is used as the first boundary variable corresponding to the next iteration step. A second global consensus variable is generated based on each optimal boundary variable, and it is used as the first global consensus variable in the next iteration step. The penalty parameters are updated based on each optimal boundary variable, and the first Lagrange multiplier is adjusted based on the updated penalty parameters to obtain the corresponding second Lagrange multiplier, which is then used as the first Lagrange multiplier for the next iteration step. The specific steps for updating the penalty parameter based on each optimal boundary variable are as follows: The boundary graph is updated based on the optimal boundary variables of each target electrical community, and graph features are extracted. The nodes of the boundary graph represent each target electrical community, the edges represent the connecting lines between each target electrical community, and the graph features include node degree, modularity, and boundary cut vertices. When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased; When the module degree is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced, and the second node degree threshold is less than or equal to the first node degree threshold. In all other cases, the penalty parameters remain unchanged.

2. The distributed optimal power flow parallel solution method according to claim 1, characterized in that, The step size of the alternating direction multiplier method is also updated based on the updated penalty parameters; The step size adjustment rule is as follows: ； Where, α t Let α be the current step size. t+1 For the updated step size, ρ t+1 This refers to the updated penalty parameters.

3. A distributed optimal power flow parallel solution method according to any one of claims 1-2, characterized in that: Identify the graph topology features of each initial electrical community, and decompose the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; The graph topological features include cut vertices, bridging edges, and radiating branches.

4. The distributed optimal power flow parallel solution method according to claim 3, characterized in that: The forward-backward substitution method is used to perform equivalent processing on each radiating branch, and the obtained equivalent complex power is injected into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community. The root node is the connection point between the radial branches and the core backbone subgraph.

5. The distributed optimal power flow parallel solution method according to claim 4, characterized in that: When the original residual is less than or equal to the preset original residual convergence threshold, and the dual residual is less than or equal to the preset dual residual convergence threshold, the iteration is determined to be converged. The original residual is used to measure the deviation between the global consensus variable and the boundary variable formed by the optimal boundary variables of all target electrical communities in the current iteration step; The dual residual is used to measure the degree of change of the global consensus variable in the current iteration step.

6. A distributed parallel solution system for optimal power flow, characterized in that, It includes a community detection module, a graph decomposition module, a distributed OPF solver module, and a graph-aware coordinator: The community discovery module is used to divide a preset undirected weighted graph based on a community discovery algorithm to obtain several initial electrical communities. The graph decomposition module is used to perform graph decomposition and graph pruning on each initial electrical community based on graph topological features to obtain several target electrical communities; The distributed OPF solution module is used to solve the local OPF subproblem in parallel iteratively based on the alternating direction multiplier method for each target electrical community until the optimal solution corresponding to each target electrical community is obtained, and the corresponding global optimal power flow result is generated. Specifically: The distributed OPF solving module independently solves for the optimal internal variables and optimal boundary variables corresponding to each target electrical community based on the first global consensus variable, the first Lagrange multiplier and the penalty parameter, as well as the first internal variables and the first boundary variables corresponding to each target electrical community. The optimal internal variables are used as the first internal variables corresponding to the next iteration step, and the optimal boundary variables are used as the first boundary variables corresponding to the next iteration step. The graph-aware coordinator generates a second global consensus variable based on each optimal boundary variable, and uses it as the first global consensus variable for the next iteration step. The graph-aware coordinator updates the penalty parameters based on each optimal boundary variable, adjusts the first Lagrange multiplier based on the updated penalty parameters, obtains the corresponding second Lagrange multiplier, and uses it as the first Lagrange multiplier for the next iteration step. The graph-aware coordinator includes a consensus computation unit and a boundary coordination unit, wherein the boundary coordination unit includes: The boundary graph maintenance unit is used to update the boundary graph and extract graph features based on the optimal boundary variables of each target electrical community. The nodes of the boundary graph represent each target electrical community, the edges represent the connecting lines between each target electrical community, and the graph features include node degree, modularity, and boundary cut vertices. The penalty parameter adjustment unit is used for: When a boundary cut point exists or the node degree is greater than or equal to a preset first node degree threshold, the penalty parameter is increased; When the module degree is greater than or equal to a preset module degree threshold, and the node degree is less than a preset second node degree threshold, the penalty parameter is reduced, and the second node degree threshold is less than or equal to the first node degree threshold. In all other cases, the penalty parameters remain unchanged.

7. A distributed parallel optimal power flow solution system according to claim 6, characterized in that: The graph decomposition module includes a decomposition unit and a pruning unit; The decomposition unit is used to identify the graph topology features of each initial electrical community, and decomposes the corresponding initial electrical community into a core backbone subgraph and a radial branch-leaf graph based on the obtained graph topology features; the radial branch-leaf graph includes one or more radial branches; the graph topology features include cut vertices, bridging edges and radial branches; The pruning unit is used to perform equivalent processing on each radiating branch using the forward-backward substitution method, and injects the obtained equivalent complex power into the corresponding root node of the core backbone subgraph to obtain the corresponding target electrical community; the root node is the connection point between the radiating branch and the core backbone subgraph.

8. A computer-storable medium, characterized in that, The computer storage medium stores computer instructions, which, when invoked by the computer, are used to execute the distributed optimal power flow parallel solution method as described in any one of claims 1 to 5.

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