Load change and load redistribution cooperative attack method oriented to power distribution network and defense method thereof

Abstract

The invention belongs to the technical field of electric digital data processing, and particularly relates to a load change and load redistribution cooperative attack method oriented to a power distribution network and a defense method thereof. According to the attack method, a double-layer Stackelberg game model between an attacker and a scheduler is constructed, the model is converted into a single-layer mixed integer linear programming problem by utilizing a KKT condition, efficient solving of an optimal cooperative attack strategy is achieved, a mathematical model of the attacker fuses double means of load change attack and load redistribution attack, and the optimal cooperative attack strategy is effectively solved. Physical layer disturbance is realized by directly tampering the real load of the node, and a scheduler is misguided to perceive by forging measurement data, so that collaborative attack of physical and information layers is realized under the same game framework, and the attack is more hidden and destructive; in order to cope with the attack method, a three-layer attack and defense model of soft open point defense is also introduced, the solving capability is stronger, the application range is wider, and the attack risk of the power distribution network can be more truly evaluated and effectively defended.

Description

Technical Field

[0001] This invention belongs to the field of electrical digital data processing technology, and in particular relates to a method for coordinated attack on load change and load redistribution in power distribution networks and a method for defense against such attacks. Background Technology

[0002] With the accelerated digitalization of power systems and the widespread deployment of new equipment such as distributed energy resources and controllable loads in distribution networks, the power system's reliance on information and communication networks is increasing. In these distribution systems, attackers can exploit vulnerabilities in the information layer to manipulate loads, inducing voltage out-of-bounds anomalies and causing system instability. Therefore, researching load attack strategies targeting distribution networks is of great significance for improving the resilience and security of power systems.

[0003] Load alteration attacks (LAA) are a typical form of cyberattack in power systems. These attacks manipulate remotely controllable loads (RCLs) to induce voltage instability and line overload. While LAAs directly disrupt the actual load, affecting system operation and inducing voltage fluctuations or power flow anomalies, the system can promptly detect load fluctuations and their associated voltage exceedance risks due to the direct manipulation of actual load values. This allows for mitigation through scheduling optimization and topology switching, effectively suppressing the attack's impact.

[0004] Load Redistribution Attack (LRA) is a typical perception layer attack. Attackers manipulate load measurement data to redistribute load values ​​across nodes without altering the total system load, thus misleading state estimation and scheduling decisions. LRAs are highly covert because they forge perception data to mislead state estimation and scheduling decisions by manipulating load measurement data. Since they do not change the actual physical load but only affect scheduling output, the attack itself is unlikely to cause serious voltage exceedances or system anomalies; it typically manifests only as low-risk anomalies such as increased generation costs and uneven power flow distribution.

[0005] In summary, both load change attacks and load redistribution attacks have certain limitations and are difficult to effectively break through the multiple defense mechanisms of the power system.

[0006] Therefore, proposing a collaborative attack method for load change and load redistribution in power distribution networks, as well as a defense method to deal with such attacks, is of great significance for ensuring the safe and stable operation of the power system.

[0007] A smart soft switch, or Soft Open Point (SOP), is a power electronics-based device widely used in power distribution networks to improve system flexibility and reliability. An SOP primarily consists of back-to-back voltage source converters (VSCs) that enable precise regulation of power flow between feeders, voltage support, and reactive power compensation. SOPs can address issues such as power fluctuations, localized overloads, and voltage quality problems caused by distributed energy resource integration.

[0008] In view of this, this invention proposes a method for coordinated attacks on load change and load redistribution in distribution networks to address the limitations of existing load change and load redistribution attacks in terms of concealment and destructiveness. Furthermore, considering the control capabilities of active devices such as smart soft switches (SOPs) in the system, a defense method is further constructed that integrates SOP response, attack decision-making, and scheduling feedback into a three-layer collaborative modeling framework. This method is used to suppress voltage risks caused by such coordinated attacks, achieve intervention and control against these attacks, and improve the overall defense capability of the system. Summary of the Invention

[0009] This invention aims to overcome at least one of the defects of the prior art and proposes a collaborative attack strategy for distribution networks, which combines load change attack (LAA) and load redistribution attack (LRA). By constructing a collaborative mechanism of physical disturbance and perceived tampering, it induces the dispatch system to generate infeasible generation schemes, causing the actual voltage of the power system to exceed the safe range, triggering voltage overshoot or even system collapse. The optimal joint attack strategy is solved by constructing a Stackelberg game optimization model. At the same time, in order to deal with the above-mentioned attack methods, this invention proposes a defense method with a three-layer game model.

[0010] The detailed technical solution of this invention is as follows: A coordinated attack method for load change and load redistribution in power distribution networks includes the following steps: S1. Based on the interaction between the attacker and the scheduler, a Stackelberg game model is constructed, in which the upper layer is the attacker model and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution collaborative attack method. S2. Using the Karush-Kuhn-Tucker conditions, the Big M method, and binary variables, the Stackelberg game model constructed in step S1 is simplified into a single-layer mixed integer programming model. The single-layer mixed integer programming model is then solved to obtain the optimal attack vector set.

[0011] Preferably, in the Stackelberg game model described in step S1, the attacker model's task is to determine the optimal attack vector injected into the original data to maximize the system voltage deviation; while the lower-level scheduler model, after receiving the optimal attack vector from the upper level, responds to the attack through power generation scheduling to minimize the voltage deviation and maintain stable system operation.

[0012] Preferably, the attacker model described in step S1 has the following objective function, which aims to maximize the deviation of the system voltage: (1) in, This refers to the active power load value that has been altered due to a load change attack. This refers to the reactive load value that has been altered due to a load change attack. This refers to the change in active power measurements caused by load redistribution attacks. This refers to the reactive power measurement value changed due to a load redistribution attack, where N represents the node. This refers to the square of the voltage reference value at node i. This refers to the actual squared voltage value of node i after it has been attacked. The attacker model has the following constraints: Power balance constraints: (2) (3) (4) (5) in, and These represent the active and reactive power flow measurements from node i to its child node k, respectively, which are altered by a combined attack of load change and load redistribution. and These represent the active and reactive power flow measurements from node j to node i, respectively, which are altered by a coordinated attack involving load change and load redistribution. Here, j is the parent node of i. and These represent the active and reactive loads at node i, respectively. and Representing nodes respectively i The active / reactive load values ​​were altered due to a load change attack. and Representing nodes respectively i The changes in active / reactive power measurements due to load redistribution attacks. and They are nodes iThe active / reactive power output of the distributed power source is given by N, where N is the set of nodes. Represents all branches connected to node i. The actual active power on Represents all branches connected to node i. The actual reactive power; and These represent the actual active power and reactive power flow values ​​after the coordinated attack, respectively. Formulas (6) to (11) are constraints on load change attacks and load redistribution attacks, as follows: (6) (7) (8) (9) =0, (10) (11) in, and These represent the original active and reactive loads of node i. and These are the proportional coefficients used to constrain attack strength. This represents the set of nodes that can be attacked in a load change attack. This represents the set of nodes that can be attacked in a load redistribution attack. Formulas (6) to (9) ensure that the attack amplitude does not exceed the parameter. and The allowed range is defined, thus limiting it to the allowed range of the true value, in order to maintain the stealth of the attack. Formulas (10) to (11) ensure that the attacker can only attack the given node.

[0013] The stealth constraints of LR attacks are as follows: , (12) in, and Represents a node i The changes in active and reactive power measurements due to load redistribution attacks; Formulas (13) to (14) are voltage constraints, as follows: (13) (14) in, Let be the square of the actual line voltage at node i. Let be the square of the actual voltage of line at node j. Let (i, j) be the resistance of the line. Let be the reactance of line (i, j), and the detailed calculation is given by formula (13). Formula (14) explicitly embeds the scheduler's optimal response, so the attacker can predict and utilize the scheduler's defense strategy, thereby forming the optimal attack scheme under the combined effect of physical disturbance and information tampering. This is the active power output vector of the distributed generator. This refers to the reactive power output vector of the distributed generator. The square of the voltage reference value at node i is represented. This represents the estimated squared voltage value at node i. This is the vector of the generator's output active power. This is the vector of reactive power output by the generator.

[0014] The objective function of the scheduler model described in step S1 is as follows: (15) The objective function formula (15) aims to minimize the sum of squared errors between the estimated squared voltage values ​​of all nodes in the system and their reference values. Represents the active power output vector of a distributed generator. This represents the reactive power output vector of a distributed generator, where N represents the set of nodes. This represents the square of the voltage reference value at node i. This represents the estimated squared voltage value of node i. The voltage estimate here is based on the load information that has been attacked and tampered with, and is not the actual operating voltage of the system. The scheduler model has the following constraints: Active and reactive power balance constraints at nodes: (16) (17) Voltage constraint: (18) Constraints on the generator: (19) (20) (twenty one) (twenty two) (twenty three) in, Let (i, j) be the resistance of the line. Let (i, j) be the reactance of the line. and Let represent the lower and upper bounds of the active power output vector of the distributed generator, respectively. and Let represent the lower and upper bounds of the reactive power output vector of the distributed generator, respectively. and To perform an LA attack, modify the sum of the active load and reactive power of all nodes. and This is the sum of the active and reactive power outputs of the distributed generator. and This is the sum of the original active and reactive loads of all nodes. Represents all generator nodes. Refers to the estimated squared voltage value of node j.

[0015] Preferably, step S2 specifically includes: S21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model are transformed into equivalent constraints. The equivalent constraints include the original feasibility conditions, gradient balance conditions, complementary relaxation conditions, and Lagrange multiplier nonnegativity. S22. Complementary relaxation conditions for linearized equivalent constraints based on the Big M method and binary variables; S23. Add the original feasibility conditions, gradient balance conditions, non-negativity of Lagrange multipliers, and complementary relaxation conditions obtained in step S21 and linearized in step S22 to the upper-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to the single-level mixed integer linear programming (MILP) problem, and solves the linear programming problem of the transformed single-level mixed integer programming model.

[0016] More preferably, the transformed equivalent constraints described in step S21 specifically include the following four categories: Original feasibility: All original constraints must be satisfied; the original constraints are the constraints contained in the scheduler model. Gradient equilibrium condition: (twenty four) (25) (26) (27) (28) in, , , , and These are the Lagrange multipliers of the equality constraints of formulas (16), (17), (18), (22), and (23), respectively; , , and These are the Lagrange multipliers of the inequality constraints of formulas (19) and (20) and must be non-negative. Represents the resistance of line (i, j). Indicates the standard voltage of node i. Represents the voltage at node i. The sum of the Grarange multipliers of formula (16) representing all child nodes of node i. The sum of the Lagrange multipliers of formula (17) representing all child nodes of node i, The sum of the Lagrange multipliers of formula (18) representing all child nodes of node i.

[0017] Complementary relaxation conditions: (29) (30) (31) (32) Nonnegativity of Lagrange multipliers: (33) This condition stipulates that all Lagrange multipliers associated with the inequality constraints must be greater than or equal to zero at the optimal solution. Represents all generator nodes; In a further preferred embodiment, step S22 specifically includes: The complementary relaxation conditions are achieved using the Big M method and binary variable linearization, as follows: (34) (35) (36) in, , , , All are Lagrange multipliers for inequalities, where M is a maximal number. and Each is a binary variable used for confirmation. and Has its minimum value been obtained? and If the minimum value is reached, the value is 1; otherwise, the value is 0. and Also a binary variable used for confirmation and Has its maximum value been obtained? and If the value is obtained, the value is 1; otherwise, the value is 0.

[0018] Preferably, in step S23, the original feasibility conditions, gradient balance conditions, Lagrange multiplier nonnegativity, and complementary relaxation conditions obtained in step S22 (after linearization) are added to the upper-level attacker model to obtain a single-level mixed-integer programming model. This completes the equivalent transformation from the original two-level game problem to a single-level mixed-integer linear programming (MILP) problem. The transformed single-level MILP problem is as follows:

[0019] The constraints include: Upper-level constraints: Formulas (1) to (13) Original feasibility conditions: Formulas (18)~(23) Gradient conditions: Equations (24) to (28) Complementary relaxation conditions: Equations (34)~(36) Nonnegativity of Lagrange multipliers in inequalities: Formula (33) Preferably, in step S23, the existing commercial solver CPLEX is used to solve the single-layer MILP problem to obtain the optimal attack vector.

[0020] This invention also provides a defense method against coordinated attacks involving load changes and load redistribution in power distribution networks, comprising the following steps: P1. Based on the interaction between the defender, the attacker, and the scheduler, a three-layer game model is constructed, in which the upper layer is the defender model, the middle layer is the attacker model, and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution collaborative attack method. P2. Using KKT conditions, the lower-middle level game model in the constructed three-level game model is simplified into a single-level mixed integer programming model, and the three-level game problem is decomposed into a main problem and sub-problem structure. P3. Use the Column-and-Constraint Generation method to solve the main problem and subproblems obtained in step P2 in order to obtain the optimal solution for defense.

[0021] Preferably, in the three-layer model, the upper-layer defender model minimizes the actual voltage deviation of the system by adjusting the active and reactive power output of the SOP; the middle-layer attacker model is tasked with determining the attack vector injected into the original data to maximize the system voltage deviation; and the lower-layer scheduler model, after receiving the optimal attack from the middle layer, responds to the attack through power generation scheduling to minimize the voltage deviation and maintain stable system operation.

[0022] A further preferred embodiment of the upper-layer defender model described in step P1 is as follows: The objective function is: (37) The actual active and reactive power constraints are: (38) (39) The actual voltage constraint is: (40) The network constraints for the operation of the power distribution system are: (41) (42) The SOCP expression constraints of the SOP operation model are: (43) (44) (45) (46) in, A collection of SOP equipment. For a single SOP, The loss of active power extracted and injected into the SOP; and These represent the active power and reactive power injected or extracted by the SOP at node i, respectively. The loss factor for SOP. For the capacity of SOP, and For SOP, extract the correlation coefficient between injected reactive power and capacity at node i; , These are vectors representing the active and reactive power injected into the SOP, respectively. , These are the weight coefficients of the objective function. and These represent the active and reactive power flowing from node i to node j, respectively. It is the maximum permissible apparent power capacity. This represents the actual voltage at node i. and These are the minimum and maximum values ​​of the normal voltage. The active power injected / extracted at node i for SOP. This refers to the loss at node i caused by the injection / extraction of active and reactive power during the SOP. The active power extracted by SOP is injected at node j. This represents the loss at node j caused by the injection / extraction of active and reactive power in the SOP.

[0023] The mid-level attacker model is shown below: The objective function is: (47) The constraints include: The constraints for actual active and reactive power and measured active and reactive power are as follows: (48) (49) (50) (51) The constraints for load alteration attacks and load redistribution attacks are: (52) (53) (54) (55) =0, (56) (57) The stealth constraints of LR attacks are as follows: , (58) The constraints on the actual voltage are: (59); The lower-level scheduler model is shown below: The objective function is: (60) The constraints include: The active and reactive power balance constraints are as follows: (61) (62) The voltage constraints are estimated as follows: (63) The upper and lower bound constraints of the generator are as follows: (64) (65) (66) The total power generation constraints are as follows: (67) (68)

[0024] Preferably, step P2 specifically includes: P21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model in the three-level game model are transformed into equivalent constraints. The equivalent constraints include the original feasibility, gradient balance condition, complementary relaxation condition, and Lagrange multiplier nonnegativity. P22. Complementary relaxation conditions obtained using the Big M method and binary variable linearization step P21: P23. Add the original feasibility conditions, gradient balance conditions, nonnegativity of Lagrange multipliers, and complementary relaxation conditions obtained after linearization in step P22 to the mid-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to a single-level mixed integer linear programming (MILP) problem. The transformed single-level MILP problem is the subproblem, and the model is shown below:

[0025] The constraints include: Mid-level constraints: Formulas (48)~(59) Original feasibility conditions: Formulas (63)~(68) Gradient conditions: Same as the attack method, formulas (24)~(28) Complementary relaxation conditions: Same as the attack method, formulas (34)~(36) Nonnegativity of Lagrange multipliers in inequalities: Same as attack method, formula (33) After the optimal attack vector is output from the subproblems, the main problem is constructed by a weighted combination of objective functions: The main question is as follows: (69) (70) The constraints for active power and reactive power are as follows: (71) (72) in, As decision variables, This is the vector of the generator's output active power. This is the vector of reactive power output by the generator. For the introduced weighting coefficients, This represents the actual voltage in round I. This represents the loss of the SOP in round I at node i.

[0026] In this main problem, decision variables are introduced. As an auxiliary variable, its value is not less than the sum of the system voltage deviation and the SOP loss, as detailed in formula (69). Given the attack vector, the optimal value of the main problem is determined. Therefore, The value of cannot exceed the optimal value of the objective function in the original three-level optimization problem. In other words, in the main problem... The optimal solution provides a lower bound, LB, for the objective function of the three-level optimization problem.

[0027] Preferably, the specific steps for solving step P3 are as follows: (1) Given the SOP vector SOP* and the upper limit of the number of iterations At the same time, set the iteration counter I=0; (2) Set the upper bound UB to positive infinity and the lower bound LB to negative infinity; (3) If I< If I < Return SOP*, end the process, and output SOP* as the final result; (4) Solve the subproblem with SOP* as the parameter to obtain the most unfavorable attack vector D* and its sum of squared voltage errors f* under the SOP decision. ; (5) Substituting D* into the principal problem, we obtain the new optimal solution SOP* and the objective value for the SOP decision. And update the lower bound LB in turn. Update the upper bound to ; (6) Determine whether the convergence condition is met. If the convergence condition is met, output SOP* as the final result; if the convergence condition is not met, execute... Repeat steps (3) to (6).

[0028] Compared with the prior art, the present invention has the following beneficial effects: (1) This invention addresses the limitations of existing load alteration attacks (LAA) and load redistribution attacks (LRA) in terms of concealment and destructiveness, and proposes a modeling and analysis method for coordinated load replacement and load redistribution attacks in distribution networks. By constructing a two-layer Stackelberg game model between the attacker and the scheduler, and using KKT conditions to transform it into a single-layer mixed-integer linear programming problem, the optimal coordinated attack strategy can be efficiently solved. The attacker's mathematical model integrates the dual means of load alteration attacks and load redistribution attacks. It not only achieves physical layer disturbance by directly tampering with the actual load of nodes, but also misleads the scheduler's perception by forging measurement data. Thus, it achieves coordinated attacks at the physical and information layers under the same game framework, making the attack more concealed and destructive.

[0029] (2) This invention introduces power measurement values ​​as key variables in attack and defense modeling, achieving unified modeling and analysis of physical layer load disturbances and information layer measurement tampering. By explicitly introducing the measured values ​​of active and reactive power of nodes into the attacker model, it can characterize the misleading effect of load redistribution attacks on the scheduling perception layer. At the same time, combined with the impact of load change attacks on actual power injection, the collaborative attack model can reflect both the changes in real system voltage and the erroneous response of the scheduler driven by false measurement data. This technical means ensures that the model can simultaneously characterize the entire link distortion process of "perception-response-operation", thereby achieving an accurate characterization of the concealment and destructiveness of collaborative attacks.

[0030] (3) This invention proposes a collaborative attack strategy that can mislead the dispatch system to generate infeasible power generation schemes and induce voltage overruns during operation, effectively breaking through the original perception and defense mechanisms of the power system and improving the concealment and harm of the attack effect.

[0031] (4) The method of this invention verifies the ability of cooperative attacks to undermine system voltage stability in multiple operating scenarios, demonstrating good adaptability and effectiveness under various load levels, attack amplitudes, and network scales. This research provides a modeling framework and attack examples for revealing the vulnerability of power systems in the context of complex cyber-physical coupling, and also provides a theoretical basis for the design of subsequent defense mechanisms. Based on this two-layer game model, this invention can accurately reveal the most dangerous attack methods under joint attacks, providing theoretical support for the design of power system defense mechanisms. This research on the joint load attack model and defense strategy for distribution networks is of great significance for improving the resilience and security of power systems.

[0032] (5) The defense method of the three-layer game model constructed in this invention is based on the perception information of the system's operating status and potential attacks. The upper layer adjusts the active and reactive power output of the soft-start SOP equipment to minimize voltage deviation and construct an active defense response. The middle layer attacker maximizes the voltage deviation within the system by comprehensively considering constraints such as load disturbance amplitude limits, active power balance constraints, and power synchronization adjustment requirements. The bottom layer defender performs power system scheduling based on the perception information after the attack, aiming to minimize the estimated voltage deviation and optimize the active and reactive power output strategies of distributed generators. Based on this three-layer game model, this invention can more accurately depict the complex interaction relationship between the attacker, defender, and scheduler, reveal the most destructive strategy combination under joint attacks, and provide theoretical support for constructing a power system defense mechanism, which is of great significance for improving the resilience and security of the power system.

[0033] (6) This invention also introduces a three-layer attack and defense framework for soft opening point defense, which has stronger solution capabilities and a wider range of applications, and can more realistically assess and effectively defend against the risk of power distribution network attacks. Attached Figure Description

[0034] Figure 1 This is a schematic diagram of the master-slave game between the attacker and the scheduler in Embodiment 1 of the present invention; Figure 2 This is a node voltage distribution diagram under different attack methods in Embodiment 1 of the present invention, wherein the LAA attack amplitude coefficient is... LRA attack amplitude coefficient ; Figure 3 The diagram shows the estimated voltage trend and the actual voltage trend of node 65 under different attack intensity parameters in the cooperative attack method of Embodiment 1 of the present invention. (a) is the estimated voltage trend and (b) is the actual voltage trend. Figure 4 The voltage distribution diagram is shown after executing the cooperative attack method of Embodiment 1 of the present invention in a 141-node system. Figure 5 This is a graph showing the voltage response trend of node 65 under the condition that the total load level varies from 93% to 107% in the cooperative attack method of Embodiment 1 of the present invention. Figure 6 This is a flowchart of the solution algorithm for Embodiment 2 of the present invention; Figure 7 This is a schematic diagram of the main problem and sub-problems of the three-layer game model in Embodiment 2 of the present invention; Figure 8 This is a comparison diagram of the voltage distribution of the IEEE 69-node power distribution system with and without the defense method enabled under the cooperative attack method in Embodiment 2 of the present invention. Detailed Implementation

[0035] To better illustrate the technical solution, design concept, and practical effects of the present invention, a detailed description will be provided below in conjunction with the accompanying drawings and specific embodiments. These embodiments are intended to aid in understanding the present invention and are not intended to limit the scope of protection of the present invention. For those skilled in the art, various adjustments, modifications, or equivalent substitutions can be made to these embodiments without departing from the spirit and core ideas of the present invention, and these should all be considered to fall within the scope of protection claimed by the present invention.

[0036] To verify the effectiveness and universality of the proposed coordinated attack model for load change and load redistribution in distribution networks, this invention uses the cplex solver in the MATLAB environment to model and simulate typical distribution network systems. The basic test system uses the IEEE 69-bus distribution network, and performs format conversion and power flow initialization based on MATPOWER data.

[0037] The attack method of this invention is based on the following: (1) A power distribution system can be abstracted as a directed connected graph G={ },in Represents a node. Represents the set of branches. Node set. ={1,2,..., } represents the bus in the system, numbered from 1 to N. Each branch connecting the bus can be represented as a node pair. ),in This represents the parent node of bus i. The set of all branches is defined as follows: ={( )| The system has a tree-like radial architecture, meaning that starting from the root node, every other node in the system can be reached through a unique, non-repeating path. Node 1 is designated as the root node, and every other node in the system has a parent node, which is the neighboring node closest to the root node. For any node k, the unique path from the root node to k is denoted as... The root node is not included, but k is included. The depth of a node is defined as the distance between that node and the root node, i.e., the number of branches.

[0038] (2) In this study, the Linearized Distribution Flow (LinDistFlow) model is used to model the power flow of the distribution system. This model simplifies the traditional branch power flow model by ignoring branch power losses, and is therefore widely used in power flow modeling of distribution systems. The power flow equations based on this model are shown below: (1) (2) (3) Formula (1) represents the node The active power balance relationship at the location, specifically, from its parent node Inflow node active power Subtract nodes Active load at the location , equal to from node The sum of active power flowing out to all its child nodes; Formula (2) represents the node Reactive power balance at the node; The reactive power balance at a given location is determined by the relationship between the following three factors: reactive power flowing in from the parent node... The reactive load of the node itself And the reactive power flowing from that node to its child nodes. Formula (3) represents the relationship between node voltages, where , These are the resistance and reactance parameters of the branch, respectively; (3) It should be assumed that the active and reactive power output capabilities of each distributed generation unit (DG) are limited to a specific range, as shown below: (4) (5) in and They represent the first i The minimum and maximum active and reactive power output of a distributed generation unit. Indicates the first i The active power that a distributed generation unit can output. Indicates the first i The reactive power that a distributed generation unit can output.

[0039] (4) In Load Changing Attack (LAA), attackers exploit potential information system vulnerabilities to turn controllable loads and other equipment into "zombie devices" and uniformly control their switching. This coordinated operation aims to disrupt the stability of the power system. In this scenario, the original LinDistFlow model formulas (1) and (2) can be rewritten as... (6) (7) in Represents the set of nodes that can be attacked. Attackers on the node Active / reactive load disturbances injected at the location, This represents the sum of active power flowing from node i to all its child nodes. This represents the sum of reactive power flowing from node i to all its child nodes;

[0040] (5) Load redistribution attack (LRA) is a typical spoofed data injection attack. Its essence lies in not changing the physical state of the actual load injection in the power system, but misleading the dispatch center's judgment of the system state by tampering with the load data at the measurement level, thereby causing incorrect generation dispatch or resource allocation. It is based on the following assumptions: the generator output measurement value cannot be tampered with by the attacker because there is a direct communication link between the control center and the power plant control room. Once the value is abnormal, it will be detected immediately, thus effectively detecting any malicious tampering. The power injection value on the bus without generators or loads cannot be tampered with because the measurement value of these buses is zero. The load measurement data is transmitted through the SCADA system, so it may be tampered with by the attacker. According to the LinDistFlow model, the node voltage is closely related to the reactive power flow in the power grid. In the load bus, the demand for reactive power is dynamic and its corresponding measurement equipment is widely distributed. Therefore, the attacker can use both active and reactive power measurement units to launch an attack on the load bus.

[0041] Based on the above assumptions, a load redistribution attack must satisfy the following constraints: First, since generator measurements cannot be tampered with, the modifications to active and reactive loads must satisfy the constraint that their sum is 0 to ensure power balance. (8) in, For the load bus set The active load modification of node i; To maintain the concealment of injecting false data, the active / reactive power measurement values ​​must be modified simultaneously: (9) (10) in, and The fake active / reactive load injected by the attacker. In a load redistribution attack, this represents the set of nodes that can be attacked. The forged data injected into each load node cannot deviate too much from its normal value to avoid attracting the attention of the scheduler. (11) (12) in, and The fake active / reactive load injected by the attacker. This represents the normal active load value when node i is operating normally. This represents the normal reactive load value when node i is operating normally. This represents a coefficient used to limit the degree of deviation from the intended scope of falsified data.

[0042] Example 1 This embodiment provides a method for coordinated attacks on load changes and load redistribution in power distribution networks, including the following steps: S1. Based on the interaction between the attacker and the scheduler, construct a Stackelberg game model, such as... Figure 1 As shown, the upper layer is the attacker model and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution coordinated attack method. Step S1 proposes a two-layer optimization model to design the most destructive attack scheme under given attack resource constraints. The attacker model's task is to determine the attack vector injected into the original data to maximize the system voltage deviation; while the lower-layer scheduler model, after receiving the optimal attack from the upper layer, responds to the system through power generation scheduling to minimize the voltage deviation and maintain stable system operation.

[0043] Specifically, the LinDistFlow model is used to characterize the operating state of the power system, and the upper-layer attacker model is shown below: The objective function is: (13) in, This refers to the active power load value that has been altered due to a load change attack. This refers to the reactive load value that has been altered due to a load change attack. This refers to the change in active power measurements caused by load redistribution attacks. This refers to the reactive power measurement value changed due to a load redistribution attack, where N represents the node. This refers to the square of the voltage reference value at node i. This refers to the actual squared voltage value of node i after it has been attacked. Its constraints include: Power balance constraints: (14) (15) (16) (17) in, and These represent the active and reactive power flow measurements from node i to its child node k, respectively, which are altered by a combined attack of load change and load redistribution. and These represent the active and reactive power flow measurements from node j to node i, respectively, which are altered by a coordinated attack involving load change and load redistribution. Here, j is the parent node of i. and These represent the active and reactive loads at node i, respectively. and Representing nodes respectively i The active / reactive load values ​​were altered due to a load change attack. and Representing nodes respectively i The changes in active / reactive power measurements due to load redistribution attacks. and They are nodes i The active / reactive power output of the distributed power source is given by N, where N is the set of nodes. Represents all branches connected to node i. The actual active power on Represents all branches connected to node i. The actual reactive power; and These represent the actual active power and reactive power flow values ​​after the coordinated attack, respectively. The constraints for load alteration attacks and load redistribution attacks are as follows: (18) (19) (20) (twenty one) =0, (twenty two) (twenty three) in, and These represent the original active and reactive loads of node i. and These are the proportional coefficients used to constrain attack strength. This represents the set of nodes that can be attacked in a load change attack. This represents the set of nodes that can be attacked in a load redistribution attack. Formulas (18) to (21) ensure that the attack amplitude does not exceed the parameter. and The defined allowable range restricts the attack to within the permissible range of the true value, thus maintaining the stealth of the attack. Formulas (22) to (23) ensure that the attacker can only attack the given node; The stealth constraints of LR attacks are as follows: (twenty four) in, and Represents a node i The changes in active and reactive power measurements due to load redistribution attacks; Voltage constraints are as follows: (25) (26) in, Let be the square of the actual line voltage at node i. Let be the square of the actual voltage of line at node j. Let (i, j) be the resistance of the line. Let (i, j) be the reactance of the line. This is the active power output vector of the distributed generator. This refers to the reactive power output vector of the distributed generator. This represents the estimated squared voltage value at node i.

[0044] In the attacker optimization problem formulas (13) to (26), the objective, as shown in formula (13), is to maximize the deviation of the system voltage. Furthermore, this study modifies the line overload model to characterize the mechanism of LR attacks under the LinDistFlow model. and Representing nodes respectively Active / reactive load values ​​altered by a load change attack; and Representing nodes respectively Due to changes in active / reactive power measurements caused by load redistribution attacks, system operators will execute safe and economical dispatch upon receiving these inputs, generating misleading power output. and As shown in formulas (14) and (16); since the actual active / reactive load in the system at this time is + and + The scheduling scheme will be implemented through the control system and applied to the real system, thereby generating the actual power of formulas (15) and (17). as well as Formula (25) demonstrates the constraint of actual power on actual voltage in the line; meanwhile, formulas (18)-(21) ensure that the attack amplitude will not exceed the parameters. and The defined range of restrictions is always kept within a specific range of its true value, thus ensuring the concealment of the attack; and through formulas (22) and (23), it is ensured that only the load of the attacking node can be modified; formula (24) is the concealment constraint of the LR attack; formula (26) explicitly embeds the scheduler's optimal response, so the attacker can predict and utilize the scheduler's defense strategy, thus forming the optimal attack scheme under the synergistic effect of physical disturbance and information tampering.

[0045] The lower-level problem represents the dispatcher's response strategy when faced with an attacker manipulating load measurements and actual load injection. The dispatcher optimizes the scheduling of controllable generation resources to bring the system voltage as close as possible to a given reference value. The lower-level dispatcher model is shown below: The objective function of the scheduler model is as follows: (27) The constraints include: Active and reactive power balance constraints at nodes: (28) (29) Voltage constraint: (30) Constraints on the generator: (31) (32) (33) (34) (35) in, Let (i, j) be the resistance of the line. Let (i, j) be the reactance of the line. and Let represent the lower and upper bounds of the active power output vector of the distributed generator, respectively. and Let represent the lower and upper bounds of the reactive power output vector of the distributed generator, respectively. and To perform an LA attack, modify the sum of the active load and reactive power of all nodes. and This is the sum of the active and reactive power outputs of the distributed generator. and This is the sum of the original active and reactive loads of all nodes. Represents all generator nodes. Refers to the estimated squared voltage value of node j.

[0046] The objective function formula (27) aims to minimize the sum of square errors between the estimated square voltage values ​​of all nodes in the system and their reference values. It should be noted that the voltage estimation here is based on the load information after being attacked and tampered with, rather than the actual operating voltage of the system. Formulas (28) and (29) respectively constrain the active and reactive power balance of the nodes. The load values ​​are calculated based on the data perceived by the scheduler. These data integrate the actual load changes and measurement disturbances implemented by the attacker. Formula (30) adopts the voltage drop expression in the LinDistFlow model. Its input is also based on the perceived power flow, rather than the actual power flow. This reflects the cognitive bias effect caused by the attack: the scheduler makes scheduling decisions based on erroneous information, trying to minimize the system voltage deviation, but may run counter to the real demand. Formulas (31) and (32) specify the upper and lower limits of generator output to ensure the physical feasibility of the scheduling scheme. Similarly, formula (33) ensures that only generator nodes can inject active / reactive power. Overall, the scheduler tries to minimize the perceived voltage deviation under the misled system observation, and the attacker uses this cognitive gap to induce scheduling direction, thereby generating a significant voltage deviation in the real system and achieving the attack objective.

[0047] The upper-layer attacker, considering constraints such as load disturbance amplitude limits, active power balance constraints, and power synchronization adjustment requirements, maximizes the voltage deviation within the system. The lower-layer dispatcher, based on the perceived information after the attack, performs power system dispatching with the goal of minimizing the estimated voltage deviation and optimizing the active and reactive power output strategies of distributed generators.

[0048] Since the lower layer is a convex optimization problem, this study adopts the Karush-Kuhn-Tucker (KKT) conditions to transform the objective function and constraints of the lower layer scheduler into a set of KKT conditions, which are then embedded as equivalent constraints into the upper layer attacker model to obtain a single-layer mixed integer programming model, and this model is then solved.

[0049] S2. Using the Karush-Kuhn-Tucker conditions, the Big M method, and binary variables, the Stackelberg game model constructed in step S1 is simplified into a single-layer mixed integer programming model. The single-layer mixed integer programming model is then solved to obtain the optimal attack vector set.

[0050] Step S2 is as follows: S21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model are transformed into equivalent constraints. The equivalent constraints include the original feasibility conditions, gradient balance conditions, complementary relaxation conditions, and Lagrange multiplier nonnegativity. Specifically, the KKT conditions are first-order necessary optimality conditions for constrained optimization problems, used to describe the relationship between the optimal solution and the constraints. The equivalent constraints after transformation described in step S21 mainly include the following four categories: Original feasibility: All original constraints must be satisfied, which are the constraints contained in the scheduler model; Gradient equilibrium condition: (37) (38) (39) (40) (41) in , , , and These are the Lagrange multipliers of the equality constraints of formulas (28), (29), (30), (34), and (35), respectively; , , and It is the Lagrange multiplier of the inequality constraint of formulas (31) and (32) and must be non-negative. Represents the resistance of line (i, j). Indicates the standard voltage of node i. Represents the voltage at node i. The sum of the Grarange multipliers of formula (28) representing all child nodes of node i. Let the sum of the Lagrange multipliers of formula (29) representing all child nodes of node i be given. The sum of the Lagrange multipliers of formula (30) for all child nodes of node i. Since formulas (28) to (30) reveal the implicit coupling relationship between generator output and node voltage, when deriving the KKT conditions, in addition to calculating the gradient of generator output variable, the gradient terms of node voltage and branch power must also be considered at the same time; formulas (37) to (39) correspond to the gradient balance constraints of voltage variable, active power and reactive power respectively, reflecting the coordination conditions between physical variables of the system; while formulas (40) and (41) further give the gradient conditions of generator output to ensure that the optimal coordinated response is achieved under voltage regulation and power dispatch.

[0051] Complementary relaxation conditions: (42) (43) (44) (45) Formulas (42) and (43) represent the complementary relaxation of the upper and lower limits of the generator's active power output constraints; formulas (44) and (45) represent the complementary relaxation of the upper and lower limits of the reactive power output constraints. If the constraint is in an "inactive" state, i.e., has not reached the upper or lower limits, the corresponding multiplier is zero; if the constraint is "active," i.e., has reached the upper limit, then the constraint function is zero, and the multiplier can be non-zero. Through this type of relationship, the system can automatically identify which variables have reached the boundary and impose "restrictions" on them, while which variables are freely adjustable and unconstrained.

[0052] Nonnegativity of Lagrange multipliers: (46) This condition stipulates that all Lagrange multipliers associated with the inequality constraints must be greater than or equal to zero at the optimal solution. Represents all generator nodes.

[0053] S22. Complementary relaxation conditions for linearized equivalent constraints based on the Big M method and binary variables; Specifically, the complementary conditions in the above formula (42) are nonlinear product terms and cannot be directly used in linear programming. They require the use of the Big M method + binary variable linearization complementary relaxation constraints.

[0054] (47) (48) (49) in, , , , All are Lagrange multipliers for inequalities, where M is a maximal number. and Each is a binary variable used for confirmation. and Has its minimum value been obtained? and If the minimum value is reached, the value is 1; otherwise, the value is 0. and Also a binary variable used for confirmation and Has its maximum value been obtained? and If the value is obtained, the value is 1; otherwise, the value is 0.

[0055] The complementary relaxation relationship of generator active power output involved in formulas (42) and (43) is rewritten as a linear inequality, forming formula (47); the complementary relationship of reactive power output involved in formulas (44) and (45) is also linearized, transforming into formula (48); by introducing The binary variable represents whether the constraint is active or not, and is used in conjunction with a sufficiently large constant M to control the linkage between the multiplier and the constraint function. All newly introduced binary variables satisfy the value range {0,1} of formula (49) to ensure validity.

[0056] S23. Add the original feasibility conditions, gradient balance conditions, non-negativity of Lagrange multipliers, and complementary relaxation conditions obtained in step S21 and linearized in step S22 to the upper-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to the single-level mixed integer linear programming (MILP) problem, and solves the linear programming problem of the transformed single-level mixed integer programming model.

[0057] Specifically, step S23 involves adding the original feasibility conditions, gradient balance conditions, Lagrange multiplier nonnegativity, and complementary relaxation conditions obtained in step S22 (after linearization) to the upper-level attacker model, resulting in a single-layer mixed-integer programming model. This completes the equivalent transformation from the original two-layer game problem to a single-layer mixed-integer linear programming (MILP) problem. The transformed single-layer MILP problem is as follows:

[0058] The constraints include: Upper-level constraints: Formulas (13) to (25) Original feasibility conditions: Formulas (30)~(35) Gradient conditions: Equations (37) to (41) Complementary relaxation conditions: Equations (47) to (49) Nonnegativity of Lagrange multipliers in inequalities: Formula (46) Specifically, in step S23, the existing commercial solver CPLEX is used to solve the single-layer MILP problem to obtain the optimal attack vector.

[0059] Figure 1This study demonstrates a master-slave game structure between an attacker and a scheduler. In this structure, the attacker, as the leader, pre-determines an optimal coordinated attack strategy based on a known scheduler response mechanism, aiming to maximize system voltage deviation. The scheduler, as the responder, receives perceived load data after interference from the attacker and adjusts the active and reactive power output of distributed generators (DGs) based on an optimization model, striving to minimize voltage deviation at each node, thus completing a scheduling response to the disturbance. This interaction mechanism reflects the attacker's deep deduction and guidance capabilities regarding scheduling strategies, and also reveals the potential threat of coordinated efforts between information domain misleading and physical domain intervention. Ultimately, the attacker selects the optimal attack vector under feasible constraints, inducing the scheduling system to generate infeasible power generation schemes and triggering systemic voltage instability during operation.

[0060] To comprehensively evaluate the effectiveness of collaborative attack strategies, the following four attack scenarios are designed for comparative analysis: Among them, LA attack parameters Set to 0.3, LR attack parameters are set to 0.5, the maximum active power output value of the generator is 0.2MW, minimum active power output value Maximum reactive power output value It is 0.2MVAr, the minimum reactive power output value. SOP capacity settings The reactive power factor at SOP is 0.2MW. and They are -0.8 and 0.8 respectively. Standard voltage The value is 1. The original load of the node and the line resistance and reactance are both set according to case 69 in Matpower. Safe voltage range. ~ The value ranges from 0.95 to 1.05.

[0061] No-attack scenario: The no-attack scenario serves as a benchmark to characterize the normal operation of the system under conditions of no malicious interference. In this scenario, the load data of all nodes is real and has not been tampered with. The scheduling center performs optimized scheduling based on the real-world perception information, and the output of distributed generators (DGs) is adjusted through the lower-level optimization model to minimize the voltage deviation of each node.

[0062] Load-only attack (LAA): Under the condition of normal distributed generator scheduling in the distribution network, the attacker only implements active / reactive disturbances on some remotely controllable load nodes, actually changing the node power injection value, and the voltage distribution state of the system after suffering such physical disturbances.

[0063] Load redistribution attack (LRA): Under the condition that the distribution network has normal distributed generator scheduling, the attacker only falsifies the measurement data of node load, misleading the scheduling system's perception of the grid status.

[0064] Jointly Implemented Collaborative Attack LAA+LRA: Under the condition that the distribution network has normal distributed generator scheduling, attackers simultaneously carry out load replacement and load redistribution attacks. By tampering with physical loads and load measurement data in synergy, they induce the dispatch center to generate infeasible power generation schemes and cause voltage instability during operation. This scenario combines physical disturbances and information misleading mechanisms, and is the most powerful and representative core verification scenario for attack effectiveness.

[0065] like Figure 2 This demonstrates the magnitude coefficient of LAA attacks. LRA attack amplitude coefficient The attack effects under different scenarios. The no-attack scenario is shown by the blue upper triangle in the figure: the node voltage is maintained in the range of 0.95–1.02 pu, the system runs stably, and there is no risk of exceeding the limit. The LA attack scenario is represented by the black square line in the diagram: due to direct disturbance of the physical load, the system's operating state changes significantly, with the voltage of multiple end nodes falling significantly below the safety threshold of 0.95 pu, indicating a strong voltage disruption capability. However, because this type of attack operates at the physical layer, its impact can be detected by the system during the scheduling phase. Once the system detects the anomaly, it can be mitigated through physical protection measures such as adding reactive power compensation equipment, adjusting distributed generation strategies, introducing topology switching, or installing smart switches. The LR attack scenario is represented by the green circle line in the diagram: the dashed line represents the voltage result obtained by the system scheduler based on the forged power data; the solid line represents the actual system voltage after the scheduling is put into operation. In the region between nodes 60 and 69, the dashed and solid curves show a significant deviation, indicating that the attacker successfully faked power injection information, misleading the system into believing it is operating normally. In reality, the actual voltage has decreased to some extent, but it still remains above the safety threshold of 0.95 pu, without causing system instability. Therefore, the LR attack has good concealment but lacks significant operational disturbance capability. The coordinated attack scenario is represented by the red diamond line in the diagram: In this scenario, although the voltage sensed by the dispatch system (i.e., the estimated voltage) remains within the normal range and does not reach the safety threshold, the actual voltage has severely exceeded the limits at multiple end nodes, falling far below 0.95 pu. This indicates that the attacker successfully misled the dispatch system by simultaneously manipulating sensed data and physical loads, inducing it to generate an infeasible dispatch scheme, ultimately leading to voltage instability during operation. Compared to implementing LAA or LRA alone, coordinated attacks are not only more destructive but also highly covert, significantly amplifying the attack effect and verifying the potential threat of coordinated mechanisms in power systems.

[0066] This embodiment also analyzes the impact of different attack strength parameters on the effectiveness of coordinated attacks, such as... Figure 3 As shown, Figure 3 (a) in the figure shows the estimated voltage trend of node 65 under different attack intensity parameters. Figure 3 (b) shows the actual voltage variation trend of node 65 under different attack strength parameters, where the parameters are... This represents the magnitude coefficient of the load redistribution attack LRA. This represents the magnitude coefficient of a load-change attack LAA, used to characterize the attacker's ability to control the system state under dual perturbations in the information and physical domains.

[0067] It can be observed that as the LRA attack coefficient increases... With the increase in the LAA attack coefficient, the estimated voltage at node 65 rose significantly, while the corresponding actual voltage decreased only slightly. This indicates that LAA focuses more on misleading the dispatch system's estimation of the grid state. Its essence lies in manipulating node load measurement data to induce the dispatch center to make incorrect generation dispatch decisions based on false perceptions, rather than directly creating drastic operational deviations. In contrast, with the increase in the LAA attack coefficient... The improvement in voltage estimation and actual voltage significantly reduced both, reflecting the stronger direct destructive power of this attack method on the physical system. This difference also explains the phenomenon that the actual voltage surface in the figure changes more steeply than the estimated voltage surface, reflecting the complementary mechanism of perception misdirection and physical perturbation in the coordinated attack: LRA disrupts perception and conceals anomalies, while LAA creates deviations and enhances damage. The two work together in the information and physical domains, significantly improving the stealth and destructiveness of the attack, exhibiting highly coordinated attack characteristics.

[0068] To evaluate the applicability of the proposed cooperative attack strategy under different operating conditions, the following two simulation experiments were conducted: (1) Perform a coordinated attack in the IEEE 141-node system to verify its adaptability in large-scale distribution networks; (2) Under the condition that the total load level varies in the range of 93% to 107%, observe the voltage response trend of key nodes, such as node 65.

[0069] Figure 4 This demonstrates the voltage distribution after a coordinated attack on a 141-node system. It can be seen that the attacker successfully caused the voltage of multiple nodes to drop below the safe threshold of 0.95 pu, but the voltage sensed by the scheduling system was... Figure 4The blue line, however, remained within a safe range, exhibiting a clear "normal perception, abnormal operation" state. This indicates that the attacker coordinated the LRA and LAA to achieve a high degree of cooperation between information domain misdirection and physical domain disturbance, causing the scheduling system to mistakenly believe that the system was in a stable state, thus failing to trigger an effective response or adjustment, ultimately leading to a serious voltage out-of-range during operation.

[0070] Figure 5 Further analysis was conducted on the trends of the estimated and actual voltages of node 65 under different load levels. As the total load level gradually increased, the system operating pressure intensified, and the effect of coordinated attacks strengthened accordingly. This was manifested in the orange bars in the estimated voltage graph consistently being higher than the blue bars in the actual voltage graph, with the difference between the two gradually widening. At 107% load, the actual voltage of node 65 had dropped below 0.92 pu, while the estimated voltage of the scheduling system still deviated significantly from the actual voltage, failing to accurately reflect the true operating status of the system and severely weakening its ability to identify and respond to potential risks.

[0071] In summary, the collaborative attack modeling and solution method based on load change and load redistribution proposed in this invention systematically reveals the potential vulnerabilities of power systems in the context of the integration of the information and physical domains. By constructing a two-layer Stackelberg game model between the attacker and the scheduler, and using KKT conditions to transform it into an equivalent single-layer mixed-integer linear programming problem, efficient solutions for the optimal attack strategy are achieved. Simulation experiments cover typical scenarios such as multiple attack modes, multi-parameter disturbances, and multiple system scales, verifying that the proposed collaborative attack strategy can effectively mislead the scheduling system and induce voltage out-of-bounds instability under various operating conditions. The results show that compared with single attack forms, collaborative attacks achieve a significant improvement in the balance between concealment and destructiveness, possessing stronger misleading capabilities and a system risk amplification effect, providing a theoretical basis and model support for constructing a new distribution network attack defense mechanism.

[0072] Example 2 The defense method in this embodiment is also based on the following: The operational constraints of a back-to-back VSC structure's SOP can be formally modeled as follows: (1) (2) (3) (4) in, The loss calculation factor for SOP. This refers to the reactive power injected / absorbed by SOP into node i. This represents the maximum capacity of the SOP located at node i. , This represents the upper and lower bounds of reactive power injection / absorption at SOP and the correlation coefficient between these bounds and the maximum capacity of SOP.

[0073] Formula (1) represents the active power conservation constraint of SOP; Formula (2) represents the active power loss of the two VSCs in SOP; Formula (3) represents the reactive power limit of VSC; and Formula (4) represents the capacity constraint of SOP. Since the SOP operation model is non-convex, it is difficult to directly solve its optimal solution. Therefore, the second-order cone programming form SOCP will be introduced in the following for equivalent reconstruction.

[0074] This embodiment provides a defense method against coordinated attacks involving load changes and load redistribution in power distribution networks, including the following steps: P1. Based on the interaction between the defender, the attacker, and the scheduler, a three-layer game model is constructed, in which the upper layer is the defender model, the middle layer is the attacker model, and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution collaborative attack method. Step P1 proposes a three-layer optimization model to design the most destructive attack scheme and corresponding strategy under given attack resource constraints. The upper-layer SOP defender model minimizes the actual voltage deviation of the system by adjusting the active and reactive power output of the SOP; the middle-layer attacker model's task is to determine the attack vector injected into the original data to maximize the system voltage deviation; and the lower-layer scheduler model, after receiving the optimal attack from the middle layer, responds to the attack through power generation scheduling to minimize the voltage deviation and maintain stable system operation.

[0075] Specifically, the LinDistFlow model is used to characterize the operating state of the power system, and the upper-level defender model is shown below: The objective function is: (5) The actual active and reactive power constraints are: (6) (7) The actual voltage constraint is: (8) The network constraints for the operation of the power distribution system are: (9) (10) The SOCP expression constraints of the SOP operation model are: (11) (12) (13) (14) in, A collection of SOP equipment. For a single SOP, The loss of active power extracted and injected into the SOP; and These represent the active power and reactive power injected or extracted by the SOP at node i, respectively. The loss factor for SOP. For the capacity of SOP, and The correlation coefficient between the injected reactive power and the capacity is extracted at node i for SOP. , These are vectors representing the active and reactive power injected into the SOP, respectively. , These are the weight coefficients of the objective function. and These represent the active and reactive power flowing from node i to node j, respectively. It is the maximum permissible apparent power capacity. This represents the actual voltage at node i. and These are the minimum and maximum values ​​of the normal voltage. The active power injected / extracted at node i for SOP. This refers to the loss at node i caused by the injection / extraction of active and reactive power during the SOP. The active power extracted by SOP is injected at node j. This represents the loss at node j caused by the injection / extraction of active and reactive power in the SOP.

[0076] In the upper-level defender model, the SOP controller is responsible for determining the operating strategy of the soft-opening SOP to minimize the voltage squared error and mitigate the impact of coordinated attacks on the power system. By coordinating the active and reactive power injection and absorption of the SOP, the SOP controller aims to minimize the voltage squared error caused by the successful coordinated attack. Its objective function is shown in formula (5). Assuming that the SOP has successfully detected the existence of the load redistribution LR attack, formulas (6) and (7) describe the actual active and reactive power flow in the system; formula (8) is used to model the actual voltage level in the system; formulas (9) and (10) guarantee the network constraints for the operation of the distribution system. Since the SOP operating model formulas (1) and (4) are non-convex, this invention adopts the second-order cone programming SOCP reconstruction method to rewrite them equivalently. Formulas (11) and (14) are the SOCP expressions of the SOP operating model; The mid-level attacker model is shown below: The objective function is: (15) The constraints include: The constraints for actual active and reactive power and measured active and reactive power are as follows: (16) (17) (18) (19) The constraints for load alteration attacks and load redistribution attacks are: (20) (twenty one) (twenty two) (twenty three) =0, (twenty four) (25) The stealth constraints of LR attacks are as follows: , (26) The constraints on the actual voltage are: (27) In the attacker optimization problem formulas (15)-(27), the objective, as shown in formula (15), is to maximize the deviation of the system voltage. Furthermore, this study modifies the line overload model to characterize the mechanism of LR attacks under the LinDistFlow model. and Representing nodes respectively Active / reactive load values ​​altered by a load change attack; and Representing nodes respectively Due to changes in active / reactive power measurements caused by load redistribution attacks, system operators will execute safe and economical dispatch upon receiving these inputs, generating misleading power output. and As shown in formulas (16) and (18); since the actual active / reactive load in the system at this time is + and + The scheduling scheme will be implemented through the control system and applied to the real system, thereby generating the actual power of formulas (17) and (19). as well as Formula (27) demonstrates the constraint of actual power on actual voltage in the line; meanwhile, formulas (20)-(23) ensure that the attack amplitude will not exceed the parameters. and The defined limit range is always kept within a specific range of its true value, thereby ensuring the concealment of the attack; and the formulas (24) and (25) ensure that only the load of the attacking node can be modified; formula (26) is the concealment constraint of the LR attack. The lower-level problem represents the dispatcher's response strategy when faced with an attacker manipulating load measurements and actual load injection. The dispatcher optimizes the scheduling of controllable generation resources to bring the system voltage as close as possible to a given reference value. The model of the lower-level dispatcher is shown below: The objective function is: (28) The active and reactive power balance constraints are as follows: (29) (30) The voltage constraints are estimated as follows: (31) The upper and lower bound constraints of the generator are as follows: (32) (33) (34) The total power generation constraints are as follows: (35) (36) The objective function formula (28) aims to minimize the sum of square errors between the estimated square voltage values ​​of all nodes in the system and their reference values. It should be noted that the voltage estimation here is based on the load information after being attacked and tampered with, rather than the actual operating voltage of the system. Formulas (29) and (30) respectively constrain the active and reactive power balance of the nodes. The load values ​​are calculated based on the data perceived by the defender. These data integrate the actual load changes and measurement disturbances implemented by the attacker. Formula (31) adopts the voltage drop expression in the LinDistFlow model. Its input is also based on the perceived power flow, rather than the actual power flow. This reflects the cognitive bias effect caused by the attack: the defender makes scheduling decisions based on erroneous information, trying to minimize the system voltage deviation, but may run counter to the real demand. Formulas (32) and (33) specify the upper and lower limits of generator output to ensure the physical feasibility of the scheduling scheme. Similarly, formula (34) ensures that only generator nodes can inject active / reactive power. Formulas (35) and (36) are the total power generation constraints of the system.

[0077] P2. Using KKT conditions, the lower-middle level game model in the constructed three-level game model is simplified into a single-level mixed integer programming model, and the three-level game problem is decomposed into a main problem and sub-problem structure.

[0078] Since the lower layer is a convex optimization problem, this study adopts Karush-Kuhn-Tucker (KKT) conditions to transform the objective function and constraint relationship in the lower layer scheduler's optimization problem into a set of KKT conditions, and embeds them as equivalent constraints into the middle layer attacker model to solve the main problem and subproblems.

[0079] Step P2 specifically involves: P21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model in the three-level game model are transformed into equivalent constraints. The equivalent constraints include the original feasibility, gradient balance condition, complementary relaxation condition, and Lagrange multiplier nonnegativity. KKT conditions are first-order necessary optimality conditions for constrained optimization problems, used to describe the relationship between the optimal solution and the constraints; the transformed equivalent constraints mainly include the following four categories: Initial feasibility: All initial constraints must be satisfied; Gradient equilibrium condition: Same as formulas (37) to (41) in Example 1; Complementary relaxation conditions: Same as formulas (42) to (45) in Example 1; Nonnegativity of Lagrange multipliers: Same as formula (46) in Example 1; P22. Complementary relaxation conditions obtained using the Big M method and binary variable linearization step P21: The formula is the same as in Example 1 (47)~(49); P23. Add the original feasibility conditions, gradient balance conditions, nonnegativity of Lagrange multipliers, and complementary relaxation conditions obtained after linearization in step P22 to the mid-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to a single-level mixed integer linear programming (MILP) problem. The transformed single-level MILP problem is the subproblem, and the model is shown below:

[0080] The constraints include: Mid-level constraints: Formulas (16)~(27) Original feasibility conditions: Formulas (31)~(36) Gradient conditions: Same as formulas (37) to (41) in Example 1. Complementary relaxation conditions: Same as formulas (47) to (49) in Example 1. Nonnegativity of Lagrange multipliers in inequalities: Same as formula (46) in Example 1. Due to the second part of the defender's objective function formula (5) Since the decision variables in the subproblem are irrelevant, replacing the objective function (15) in the subproblem with the objective function formula (5) of the defender will not affect the optimal solution of the subproblem. Under the condition of a fixed SOP vector, the objective function value formula (5) of the defender corresponding to the optimal solution of the subproblem will not be less than the optimal objective value of the three-layer optimization problem. In other words, the optimal solution of the subproblem provides an upper bound estimate for the objective function formula (5) in the three-layer optimization problem.

[0081] The optimal attack vector D* in the subproblem output includes... , , and Based on this, the problem of the three-level game model can be reformulated as a mini-minimum two-level optimization problem; to solve this mini-minimum two-level optimization problem, we use a weighted combination of objective functions to construct the master problem, abbreviated as MP. (37) (38) The constraints for active power and reactive power are as follows: (39) (40) in, As decision variables, This is the vector of the generator's output active power. This is the vector of reactive power output by the generator. For the introduced weighting coefficients, This represents the actual voltage in round I. This represents the loss of the SOP in round I at node i.

[0082] In this main problem, decision variables are introduced. As an auxiliary variable, its value is not less than the sum of the system voltage deviation and the SOP loss, as detailed in formula (37). Given the attack vector, the optimal value of the main problem is determined. Therefore, The value of cannot exceed the optimal value of the objective function in the original three-level optimization problem. In other words, in the main problem... The optimal solution provides a lower bound, LB, for the objective function of the three-level optimization problem.

[0083] P3, such as Figure 6 As shown, the Column-and-Constraint Generation method is used to solve the main problem and subproblems obtained in step P2 to obtain the optimal solution for defense: (1) Given the SOP vector SOP* and the upper limit of the number of iterations At the same time, set the iteration counter I=0; (2) Set the upper bound UB to positive infinity and the lower bound LB to negative infinity; (3) If I< If I < Return SOP*, end the process, and output SOP* as the final result; (4) Using the current SOP* as parameters, solve the subproblem to obtain the most unfavorable attack vector D* and its sum of squared voltage errors f* under the SOP decision. ; (5) Substituting D* into the principal problem, we obtain the new optimal solution SOP* and the objective value for the SOP decision. And update the lower bound LB in turn. Update the upper bound to ; (6) Determine whether the convergence condition is met. If the convergence condition is met, output SOP* as the final result; if the convergence condition is not met, execute... Repeat steps (3) to (6).

[0084] Figure 7 This paper demonstrates the logical structure of the three-layer optimized defense framework proposed in this invention. The framework consists of a main problem and sub-problems, forming a recursive game-theoretic relationship. The main problem is led by the intelligent soft switch SOP controller, whose core objective is to proactively mitigate voltage exceedance issues caused by attacks by adjusting the SOP's operating strategy, ensuring that the system operating voltage remains within a safe range. In each iteration, the main problem optimizes its control variables based on the attack schemes provided by the sub-problems and the system response results, thereby improving the defense effectiveness. The sub-problems are jointly composed of the attacker and the scheduling response, and each contains two phases. In the first phase, the attacker, knowing the initial SOP configuration, generates a highly destructive attack strategy aimed at inducing a significant deviation in system voltage. In the second phase, the system scheduler adjusts based on perceived information, attempting to alleviate voltage deviations and maintain operational stability.

[0085] Figure 8 This paper demonstrates the voltage distribution of the IEEE 69-node power distribution system under a coordinated attack, comparing scenarios with and without the soft Standard Operating Procedure (SOP) defense mechanism enabled. In the scenario with SOP enabled, the actual voltage of all nodes remained above the safe threshold of 0.95 pu, with no voltage overflow observed. This indicates that SOP can identify spoofed measurement data and effectively regulate power flow to suppress voltage drops caused by coordinated load changes and load redistribution (LR) attacks. Compared to the scenario without SOP deployment, the introduction of SOP significantly alleviates the voltage sag problem and provides enhanced voltage support and regulation capabilities.

[0086] In summary, this invention proposes a three-layer collaborative attack and defense modeling and solution method for distribution networks, systematically revealing the dynamic game relationship between attackers, schedulers, and SOP controllers in a cyber-physical fusion environment. The constructed three-layer optimization framework covers the interaction process of attack strategy design, scheduling response optimization, and SOP proactive defense, achieving, for the first time, a unified modeling of the attack-response-defense chain in theory. By nesting the lower-layer system scheduling and the middle-layer attack strategy into the solution of the main problem, and performing modeling transformation and reconstruction based on KKT conditions, the original non-convex three-layer optimization problem is successfully transformed into a solvable master-slave iterative structure. Simulation experiments have been conducted in several typical distribution network scenarios, and the results show that the proposed three-layer attack and defense mechanism can not only effectively alleviate the voltage overrun risk caused by composite attacks, but also improve the system's adaptive control capability under extreme disturbances by dynamically adjusting the SOP output. This method demonstrates significant effects in improving system resilience, security, and proactive defense, providing theoretical support and modeling basis for building intelligent, proactively responsive distribution network security defenses in the future.

Claims

1. A method for coordinated attack on load change and load redistribution in distribution networks, characterized in that, Includes the following steps: S1. Based on the interaction between the attacker and the scheduler, a Stackelberg game model is constructed, in which the upper layer is the attacker model and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution collaborative attack method. S2. Using the Karush-Kuhn-Tucker conditions, the Big M method, and binary variables, the Stackelberg game model constructed in step S1 is simplified into a single-layer mixed integer programming model. The single-layer mixed integer programming model is then solved to obtain the optimal attack vector set.

2. The cooperative attack method according to claim 1, characterized in that, The attacker model described in step S1 has the following objective function: (1) in, This refers to the active power load value that has been altered due to a load change attack. This refers to the reactive load value that has been altered due to a load change attack. This refers to the change in active power measurements caused by load redistribution attacks. This refers to the reactive power measurement value changed due to a load redistribution attack, where N represents the node. This refers to the square of the voltage reference value at node i. This refers to the actual squared voltage value of node i after it has been attacked. The constraints of the attacker model include: Power balance constraints: (2) (3) (4) (5) in, and These represent the active and reactive power flow measurements from node i to its child node k, respectively, which are altered by a combined attack of load change and load redistribution. and These represent the active and reactive power flow measurements from node j to node i, respectively, which are altered by a coordinated attack involving load change and load redistribution. Here, j is the parent node of i. and These represent the active and reactive loads at node i, respectively. and Representing nodes respectively i The active / reactive load values ​​were altered due to a load change attack. and Representing nodes respectively i The changes in active / reactive power measurements due to load redistribution attacks. and They are nodes i The active / reactive power output of the distributed power source is given by N, where N is the set of nodes. Represents all branches connected to node i. The actual active power on Represents all branches connected to node i. The actual reactive power; and These represent the actual active power and reactive power flow values ​​after the coordinated attack, respectively. Formulas (6) to (11) are constraints on load change attacks and load redistribution attacks, as follows: (6) (7) (8) (9) =0, (10) (11) in, and These represent the original active and reactive loads of node i. and These are the proportional coefficients used to constrain attack strength. This represents the set of nodes that can be attacked in a load change attack. This represents the set of nodes that can be attacked in a load redistribution attack; formulas (6) to (9) ensure that the attack amplitude does not exceed the parameter. and The allowed range is defined, thus limiting it to the allowed range of the true value, so as to maintain the concealment of the attack; Formulas (10) to (11) ensure that the attacker can only attack the given node; The stealth constraints of LR attacks are as follows: ， (12) in, and Represents a node i The changes in active and reactive power measurements due to load redistribution attacks; Formulas (13) to (14) are voltage constraints, as follows: (13) (14) in, Let be the square of the actual line voltage at node i. Let be the square of the actual voltage of line at node j. Let (i, j) be the resistance of the line. Let (i, j) be the reactance of the line. This is the active power output vector of the distributed generator. This refers to the reactive power output vector of the distributed generator. The square of the voltage reference value at node i is represented. This represents the estimated squared voltage value at node i. This is the vector of the generator's output active power. This is the vector of reactive power output by the generator.

3. The cooperative attack method according to claim 1, characterized in that, The objective function of the scheduler model described in step S1 is as follows: (15) The objective function formula (15) aims to minimize the sum of squared errors between the estimated squared voltage values ​​of all nodes in the system and their reference values. Represents the active power output vector of a distributed generator. This represents the reactive power output vector of a distributed generator, where N represents the set of nodes. This represents the square of the voltage reference value at node i. This represents the estimated squared voltage value of node i. The voltage estimate here is based on the load information that has been attacked and tampered with, and is not the actual operating voltage of the system. The constraints of the scheduler model include: Active and reactive power balance constraints at nodes: (16) (17) Voltage constraint: (18) Constraints on the generator: (19) (20) (21) (22) (23) in, Let (i, j) be the resistance of the line. Let (i, j) be the reactance of the line. and Let represent the lower and upper bounds of the active power output vector of the distributed generator, respectively. and Let represent the lower and upper bounds of the reactive power output vector of the distributed generator, respectively. and To perform an LA attack, modify the sum of the active load and reactive power of all nodes. and This is the sum of the active and reactive power outputs of the distributed generator. and This is the sum of the original active and reactive loads of all nodes. Represents all generator nodes. Refers to the estimated squared voltage value of node j.

4. The cooperative attack method according to claim 1, characterized in that, Step S2 is as follows: S21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model are transformed into equivalent constraints. The equivalent constraints include the original feasibility conditions, gradient balance conditions, complementary relaxation conditions, and Lagrange multiplier nonnegativity. S22. Complementary relaxation conditions for linearized equivalent constraints based on the Big M method and binary variables; S23. Add the original feasibility conditions, gradient balance conditions, non-negativity of Lagrange multipliers, and complementary relaxation conditions obtained in step S21 and linearized in step S22 to the upper-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to a single-level mixed integer linear programming problem, and solves the linear programming problem of the transformed single-level mixed integer programming model.

5. The cooperative attack method according to claim 4, characterized in that, The transformed equivalence constraints described in step S21 specifically include the following four categories: Original feasibility: All original constraints must be satisfied; the original constraints are the constraints contained in the scheduler model. Gradient equilibrium condition: (24) (25) (26) (27) (28) in, , , , and These are the Lagrange multipliers of the equality constraints of formulas (16), (17), (18), (22), and (23), respectively; , , and These are the Lagrange multipliers of the inequality constraints of formulas (19) and (20) and must be non-negative. Represents the resistance of line (i, j). Indicates the standard voltage of node i. Represents the voltage at node i. The sum of the Grarange multipliers of formula (16) representing all child nodes of node i. The sum of the Lagrange multipliers of formula (17) representing all child nodes of node i, The sum of the Lagrange multipliers of formula (18) representing all child nodes of node i; Complementary relaxation conditions: (29) (30) (31) (32) Nonnegativity of Lagrange multipliers: (33) This condition stipulates that all Lagrange multipliers associated with the inequality constraints must be greater than or equal to zero at the optimal solution. Represents all generator nodes; Step S22 is as follows: The complementary relaxation conditions are achieved using the Big M method and binary variable linearization, as follows: (34) (35) (36) in, , , , All are Lagrange multipliers for inequalities, where M is a maximal number. and Each is a binary variable used for confirmation. and Has its minimum value been obtained? and If the minimum value is reached, the value is 1; otherwise, the value is 0. and Also a binary variable used for confirmation and Has its maximum value been obtained? and If the value is obtained, the value is 1; otherwise, the value is 0. The linear programming problem of the transformed single-layer mixed integer programming model described in step S23 is as follows: The constraints include: Upper-level constraints: Formulas (1) to (13) Original feasibility conditions: Formulas (18)~(23) Gradient conditions: Equations (24) to (28) Complementary relaxation conditions: Equations (34)~(36) Nonnegativity of Lagrange multipliers in inequalities: Equation (33).

6. A defense method against coordinated attacks involving load changes and load redistribution in power distribution networks, characterized in that: Includes the following steps: P1. Based on the interaction between the defender, the attacker, and the scheduler, a three-layer game model is constructed, in which the upper layer is the defender model, the middle layer is the attacker model, and the lower layer is the scheduler model. The attacker model is an attack model of the combined load change and load redistribution collaborative attack method. P2. Using KKT conditions, the lower-middle level game model in the constructed three-level game model is simplified into a single-level mixed integer programming model, and the three-level game problem is decomposed into a main problem and sub-problem structure. P3. Use the Column-and-Constraint Generation method to solve the main problem and subproblems obtained in step P2 in order to obtain the optimal solution for defense.

7. The defense method according to claim 6, characterized in that, The defender model described in step P1 is as follows: The objective function is: (37) The actual active and reactive power constraints are: (38) (39) The actual voltage constraint is: (40) The network constraints for the operation of the power distribution system are: (41) (42) The SOCP expression constraints of the SOP operation model are: (43) (44) (45) (46) in, A collection of SOP equipment. For a single SOP, The loss of active power extracted and injected into the SOP; and These represent the active power and reactive power injected or extracted by the SOP at node i, respectively. The loss factor for SOP. For the capacity of SOP, and For SOP, extract the correlation coefficient between injected reactive power and capacity at node i; , These are vectors representing the active and reactive power injected into the SOP, respectively. , These are the weight coefficients of the objective function. and These represent the active and reactive power flowing from node i to node j, respectively. It is the maximum permissible apparent power capacity. This represents the actual voltage at node i. and These are the minimum and maximum values ​​of the normal voltage. The active power injected / extracted at node i for SOP. This refers to the loss at node i caused by the injection / extraction of active and reactive power during the SOP. The active power extracted by SOP is injected at node j. This represents the loss at node j caused by the injection / extraction of active and reactive power in the SOP.

8. The defense method according to claim 7, characterized in that, The attacker model described in step P1 is as follows: The objective function is: (47) The constraints include: The constraints for actual active and reactive power and measured active and reactive power are as follows: (48) (49) (50) (51) The constraints for load alteration attacks and load redistribution attacks are: (52) (53) (54) (55) =0, (56) (57) The stealth constraints of LR attacks are as follows: ， (58) The constraints on the actual voltage are: (59)； The scheduler model is as follows: The objective function is: (60) The constraints include: The active and reactive power balance constraints are as follows: (61) (62) The voltage constraints are estimated as follows: (63) The upper and lower bound constraints of the generator are as follows: (64) (65) (66) The total power generation constraints are as follows: (67) (68)。 9. The defense method according to claim 6, characterized in that, Step P2 specifically involves: P21. Using the Karush-Kuhn-Tucker conditions, the objective function and constraints of the lower-level scheduler model in the three-level game model are transformed into equivalent constraints. The equivalent constraints include the original feasibility, gradient balance condition, complementary relaxation condition, and Lagrange multiplier nonnegativity. P22. Complementary relaxation conditions obtained using the Big M method and binary variable linearization step P21: P23. Add the original feasibility conditions, gradient balance conditions, nonnegativity of Lagrange multipliers, and complementary relaxation conditions obtained after linearization in step P22 to the mid-level attacker model to obtain a single-level mixed integer programming model. This completes the equivalent transformation from the original two-level game problem to a single-level mixed integer linear programming (MILP) problem. The transformed single-level MILP problem is the subproblem, and the model is shown below: After the optimal attack vector is output from the subproblems, the main problem is constructed by a weighted combination of objective functions: The main question is as follows: (69) (70) The constraints are: (71) (72) in, As decision variables, This is the vector of the generator's output active power. This is the vector of reactive power output by the generator. For the introduced weighting coefficients, Indicates the first Wheel actual voltage, Indicates the first The loss of the SOP of the wheel at node i.

10. The defense method according to claim 6, characterized in that, The specific steps for solving step P3 are as follows: (1) Given the SOP vector SOP* and the upper limit of the number of iterations At the same time, set the iteration counter I=0; (2) Set the upper bound UB to positive infinity and the lower bound LB to negative infinity; (3) If I< If I < Return SOP*, end the process, and output SOP* as the final result; (4) Solve the subproblem with SOP* as the parameter to obtain the most unfavorable attack vector D* and its sum of squared voltage errors f* under the SOP decision. ; (5) Substituting D* into the principal problem, we obtain the new optimal solution SOP* and the objective value for the SOP decision. And update the lower bound LB in turn. Update the upper bound to ; (6) Determine whether the convergence condition is met. If the convergence condition is met, output SOP* as the final result; if the convergence condition is not met, execute... Repeat steps (3) to (6).

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