Power system external impact vulnerability assessment method based on multi-target zero-sum game

Abstract

The invention discloses a power system external impact vulnerability assessment method based on a multi-target zero-sum game. The method comprises the following steps: establishing a power grid unit damage correlation propagation model according to a power grid structure and the types and number of actually included power grid units; on the basis of the power grid unit damage correlation propagation model, an attack and defense multi-target zero-sum game model is established according to optimization targets of attackers and defenders for different target functions and resource constraints of the attackers and the defenders; the attack and defense multi-target zero-sum game model is expressed as a double-layer mixed integer nonlinear optimization problem, and the problem is solved to obtain the minimum attack cost required by an external attacker to cause complete failure of the power system. According to the method, a multi-target attack and defense game model is established, and the minimum attack cost required by complete failure of the power system is solved from the perspective of attackers so as to reflect the vulnerability and security of a defense strategy of the power system on the overall level.

Description

Technical Field

[0001] The present invention belongs to the technical field of power system security defense, and in particular relates to a method for evaluating the external attack vulnerability of a power system based on multi-objective zero-sum game. Background Art

[0002] Assessing the external attack capability of power systems is of great significance to enhancing the ability of power systems to resist external malicious attacks. Current research work mainly starts from the perspective of defenders, takes the economic cost of external attacks on power systems as the objective function, and provides scheduling solutions or defense layout strategies to improve the security and defense capabilities of power systems by solving the attack and defense game problem, such as "Attack and defense game model based on power grid malicious attack damage analysis and its use method". However, in actual scenarios where there is an external attack threat to the power system, the attacker's goal is often to achieve the extreme situation of completely destroying the power system. In this scenario, how to accurately assess the external attack capability of the power system is an urgent problem to be solved. Summary of the invention

[0003] Purpose of the invention: In order to overcome the deficiencies in the prior art, a method for assessing the vulnerability of power systems to external attacks based on a multi-objective zero-sum game is provided. A multi-objective attack and defense game model is established to solve the minimum attack cost required to cause the complete failure of the power system from the attacker's perspective, so as to reflect the vulnerability and security of the power system defense strategy at the overall level.

[0004] Technical solution: To achieve the above-mentioned purpose, the present invention provides a method for evaluating the external attack vulnerability of a power system based on a multi-objective zero-sum game, comprising the following steps:

[0005] S1: A grid unit damage correlation propagation model is established based on the power grid structure and the types and numbers of grid units actually included, which is used to analyze and calculate the impact of any one or more grid units being damaged or disconnected from the grid on the grid connection status of other networks;

[0006] S2: Based on the grid unit damage correlation propagation model, an attack and defense multi-objective zero-sum game model is established according to the attacker and defender's optimization goals for different objective functions and their own resource constraints. This model is used to calculate the minimum cost that the attacker needs to pay to completely cause the power system to fail, while maintaining the resistance characteristics of the power system defender, thus forming a dynamic game process between the attacker and the defender.

[0007] S3: The attack-defense multi-objective zero-sum game model is expressed as a two-level mixed integer nonlinear optimization problem. The problem is solved to obtain the minimum attack cost required by an external attacker to cause the power system to fail completely.

[0008] Furthermore, the establishment of the power grid unit damage associated propagation model in step S1 includes:

[0009] The grid unit damage associated propagation model includes the associated grid-connected state constraints of four types of grid units: power plants, substations, buses and lines. The associated grid-connected state constraints consider the impact of the four types of grid units, power plants, substation buses and lines, on other grid units after being hit or disconnected from the grid. Two binary variables, hit state and grid-connected state, are established. The relationship between the grid-connected state of each type of grid unit and the hit state of the associated adjacent grid unit is as follows:

[0010]

[0011] Where i, j, k and h are the numbers of the power plant, transmission line, busbar and substation in the system respectively; is the grid-connected status of power plant i; The power plant i is in the state of being hit; is the grid connection status of the busbar associated with power plant i; is the grid-connected state of line j; is the hit state of line j; is the grid-connected status of the busbar at the starting point of line j; is the grid connection status of the bus at the end of line j; The bus k is in grid-connected state; The busbar k is in the struck state; The substation h is in the state of being struck; The grid-connected state of substation h; The substation h is in the state of being hit.

[0012] Furthermore, the attack-defense multi-objective zero-sum game model in step S2 includes two decision-maker models of an upper-level attacker model and a lower-level defender model, and a two-person game model consisting of two zero-sum game problems guided by two objective functions, namely, the attacker's attack cost and the defender's total loss.

[0013] Furthermore, the decision goal of the upper attacker model in step S2 is to minimize the attack cost e atk And maximize the defender's total loss f loss ,in,

[0014]

[0015] I, J, K and H are the total number of power plants, lines, busbars and substations in the power network, respectively; The attack cost required to cause damage to power plant i; The attack cost required to cause damage to line j; The attack cost required to cause the destruction of busbar k; is the attack cost required to cause damage to substation h; αk The importance of the load supplied by bus k; The load loss power of the bus k supplying power; The demand for power supply load for bus k; is the actual power supply load power of bus k; the attack cost can also be regarded as the defense capability of the grid unit. The attacker needs to pay at least an attack cost equal to the defense capability of the grid unit to destroy the network unit.

[0016] Furthermore, the decision goal of the lower-level defender model in step S2 is to maximize the attack cost e atk And minimize the defender's total loss f loss , affected by the cascading propagation effect caused by the upper-level attacker’s decision, the power system operation constraints that the defender needs to meet are:

[0017]

[0018] -θ max ≤θ k ≤θ max

[0019]

[0020] Among them, P i gen is the power generation capacity of power plant i; is the transmission power of line j; is the upper limit of power generation capacity of power plant i; θ k is the phase angle of busbar k; θ o(j) is the phase angle of the busbar at the starting point of line j; θ e(j) is the phase angle of the busbar at the end of line j; b j is the impedance of line j; is the upper limit of transmission power of line j; θ max is the maximum bus phase angle.

[0021] Furthermore, in the two-player game model in step S2, there are two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss. One of them is the attack cost e required by the attacker to attack the power system. atk , there is a min-max problem that is affected by the decisions of both the attacker and the defender:

[0022]

[0023] Second, the total loss of the defender's power system is f loss , there is a max-min problem 2 that is also affected by the decisions of the attacker and the defender:

[0024]

[0025] Where a is the set of attacker’s decision variables (x gen ,x line ,x bus ,x sub ,y gen ,y line ,y bus ,y sub );d is the set of defender’s decision variables (P gen ,P b ,θ,P load ,ΔP load ).

[0026] Furthermore, in step S2, in the two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss, for the defender, maximizing e atk and minimize f loss There is a significant consistency in the optimization objectives. This is because the defender can optimize the power flow to reduce the total loss of the power system after the attack to f loss If the minimum value is obtained, then compared with other non-optimized operation schemes, if the attacker wants to achieve the same attack effect, the minimum attack cost required by the attacker to deal with the defender's optimal power flow scheme is always the largest, which also achieves the goal of maximizing the attacker's attack cost. Therefore, under the condition of the joint model, the min-max problem of the two zero-sum game problems is simplified to:

[0027]

[0028] In the attack-defense multi-objective zero-sum game model, the attacker needs to completely fail the power system through external attacks, and the total loss of the power system will be foreseeably maintained at the maximum value, but this does not mean that the defender will stop the power flow optimization in the dynamic game process. The defender still needs to maintain the normal operation of the power system as much as possible. Combining the two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss, the max-min problem 2 is expressed as a constraint condition for a fixed game solution:

[0029]

[0030] in, It is the maximum loss when the power system fails completely.

[0031] Therefore, the attack-defense multi-objective zero-sum game model is mathematically a two-layer mixed integer nonlinear optimization problem.

[0032] Furthermore, the process of solving the problem in step S3 includes:

[0033] A1: Degrade the original problem into sub-problems;

[0034] The sub-problem in the method of reducing the original problem to sub-problems is to give the attacker an external attack cost, that is, the attacker's attack cost e atk is a single objective function zero-sum game under constant value conditions; for a given attacker's external attack cost, that is, the attacker's attack cost is fixed, the original multi-objective zero-sum game power system external attack vulnerability assessment model can be reduced to a sub-problem. The sub-problem is expressed as a zero-sum game model for solving the maximum loss that the attacker can cause to the power system under the premise that the attacker can pay a limited attack cost. It is a two-level mixed integer nonlinear optimization problem, and its objective function is

[0035]

[0036] Constraints:

[0037]

[0038]

[0039] -θ max ≤θ k ≤θ max

[0040]

[0041] The constraints include all the constraints in the grid unit damage association propagation model, the upper attacker model and the lower defender model, as well as the attack capability constraints caused by the limited attack cost of the attacker; among them, The maximum attack cost available to the attacker is a fixed value;

[0042] A2: Use the KKT condition by taking advantage of the strong duality of the underlying problem;

[0043] By using the strong duality of the lower-level problem of the subproblem and using the KKT condition, under the condition of the known attack decision of the upper-level attacker, the lower-level problem of the subproblem is a solvable linear optimization problem, and its dual problem can be derived. The lower-level problem and its dual problem must satisfy the strong duality condition. The KKT condition can be used to transform the double-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer nonlinear optimization problem.

[0044] Under the condition that the attack decision of the upper attacker is known, the lower problem is a solvable simple linear optimization problem, and the objective function is

[0045]

[0046] Constraints:

[0047]

[0048]

[0049] -θ max ≤θ k ≤θ max ,x k ,ξ k

[0050]

[0051] in, and are the grid-connected states of various grid units, and are constants especially when the attack decision of the upper attacker is known; ω k ,μ j ,γ i ,φ j , σ k ,κ k ,x k ,ξ k They are the dual variables corresponding to the constraints of the lower-level problem when solving the dual problem;

[0052] The dual problem can be deduced, and the objective function is

[0053]

[0054] Constraints:

[0055] ω con(i) +γ i ≤0,P i gen

[0056]

[0057] The lower-level problem is a linear problem whose objective function exists and is solvable in the feasible domain. The optimal solution of this problem and its dual problem must satisfy strong duality. The KKT condition can be used to transform the double-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer nonlinear optimization problem. For the lower-level problem of the subproblem, the specific constraint formula of the KKT condition is

[0058]

[0059] By using the dual problem and KKT condition to achieve the clamping effect on the lower-level objective function, the double-level mixed integer linear optimization problem of the subproblem is transformed into a single-level mixed integer nonlinear programming problem. The objective function is

[0060]

[0061] Constraints:

[0062]

[0063] -θ max ≤θ k ≤θ max

[0064]

[0065] ω con(i) +γ i ≤0

[0066] κ k ≤α i

[0067] -ω k +σ k +κ k ≤0

[0068]

[0069] Where D is the set of decision variables of the dual problem (ω,μ,κ,γ,φ, σ,χ,ξ);

[0070] A3: Logical expansion of nonlinear terms;

[0071] The logical expansion of nonlinear terms is to transform all high-order variable combinations caused by multiplication of decision variables into a form containing only first-order variable addition and subtraction combinations through mathematical methods, thereby transforming the single-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer linear optimization problem;

[0072] By mathematically converting all high-order variable combinations caused by multiplication of decision variables in the problem into a form containing only first-order variable addition and subtraction combinations:

[0073] Mode middle, are binary variables. or When it is 0, The value of must be 0, and when and When both are 1, The value of must be 1; therefore, according to the formula The equality constraints expressed can be expanded into inequality constraints:

[0074]

[0075] Through the pair The logical expansion of can achieve the same function as the original equality constraint, but avoids the situation of multiplying decision quantities. Similarly, the formula Perform logical expansion:

[0076]

[0077] Mode and Middle multiplication There are cases where the boundary is infinite, and there are continuous variables with unbounded range of values. Explain; since the decision variable μ j The range of values ​​of has no upper and lower bounds. Using the common method to replace the multiplication of binary variables and continuous variables will result in the situation where the variables are added or subtracted from the infinite boundary. In this case, it is better to set The corresponding expanded constraints are:

[0078]

[0079] In the formula, M is a maximum positive number; at this time, when When j -μ' j =μ j ;when When j -μ' j =0; multiplied with the item before expansion The operation logic is consistent; similarly,

[0080]

[0081] and can be transformed into:

[0082]

[0083]

[0084] After this step, the single-layer mixed integer nonlinear optimization problem of the subproblem is transformed into a single-layer mixed integer linear optimization problem;

[0085] A4: Transition from sub-problem to original problem;

[0086] After the subproblem is transformed into a single-layer mixed integer linear optimization problem, the double-layer mixed integer nonlinear optimization problem in the original multi-objective zero-sum game model is transformed into a single-layer mixed integer linear optimization problem by restoring the objective function and non-common constraints. Its objective function is

[0087]

[0088] Constraints:

[0089]

[0090]

[0091] -θ max ≤θ k ≤θ max

[0092]

[0093] ω con(i) +γ i ≤0

[0094] κ k ≤α i

[0095] -ω k +σ k +κ k ≤0

[0096]

[0097]

[0098] Wherein, T is the set of variables newly added by the logic expansion in step S3 (θ' o ,θ′ e ,μ' j ,γ′ i ,σ' k );

[0099] A5: Use the solver to solve the single-level mixed integer linear optimization problem in the subproblem model and the original problem model.

[0100] Furthermore, the solver in step A5 includes CPLEX and GUROBI solvers, and the CPLEX or GUROBI solver is called through the YALMIP toolbox on the MATLAB platform to solve the model.

[0101] In the present invention, the defender's power system grid structure and its defense resource allocation scheme are input into the power system external attack vulnerability assessment model based on multi-objective zero-sum game, which can solve the minimum attack cost required to cause the power system to fail completely, thereby reflecting the external attack capability of the case power system and comparing the external vulnerability of the power system under different defense strategies; the sub-problem can be used as an analysis model for solving the maximum loss that can be caused to the power system under the premise that the attacker can pay a limited attack cost, and participate in the analysis of the external security issues of the power system. The maximum attack cost paid by the attacker and the defender's power system grid structure and its defense resource allocation scheme are input into the sub-problem model, which can solve the maximum damage loss that can be caused to the power system under the condition of limited attack capability of the attacker.

[0102] Under specific conditions (such as the defense capability units of different power grid units in the power system defense resource allocation plan can be regarded as integers), the sub-problem of the present invention can achieve the same function as the original problem by gradually increasing the maximum attack cost provided by the input parameter attacker so that the output power system loss value is the same as the maximum value. Therefore, the sub-problem model can be used as a verification method for the original problem model. At the same time, the sub-problem model is not as good as the original problem model in terms of calculation speed and calculation accuracy when realizing this function, which can prove the superiority of the original problem model to a certain extent.

[0103] The present invention uses the power system external attack vulnerability assessment model based on multi-objective zero-sum game to conduct a vulnerability assessment method for different defense resource allocation schemes of the power system. The defender's power system grid structure and its defense resource allocation scheme are input into the power system external attack vulnerability assessment model based on multi-objective zero-sum game. The minimum attack cost required to cause the power system to fail completely can be solved, thereby reflecting the external attack capability of the case power system and comparing the external vulnerability of the power system under different defense strategies.

[0104] The subproblem of the present invention can be used as an analytical model to solve the maximum loss that can be caused to the power system under the premise that the attacker can pay a limited attack cost, and participate in the analysis of the external security issues of the power system. The maximum attack cost paid by the attacker and the defender's power system grid structure and its defense resource allocation plan are input into the subproblem model, so as to solve the maximum damage loss that can be caused to the power system under the condition of limited attack capability of the attacker.

[0105] The present invention utilizes a method for verifying the correctness of the results of the original problem model using a sub-problem model. The sub-problem model is input with completely identical grid parameters as the original problem and the output results of the original problem model. The output value of the sub-problem model is compared with the maximum loss value of the power system. If the values ​​are completely consistent, it proves that the solution to the original problem is correct.

[0106] Based on the above content, the scheme of the present invention is summarized as follows: The present invention provides a power system external attack vulnerability assessment model based on multi-objective zero-sum game and its linearization solution method. First, in order to characterize the power system attack process, a damage cascade propagation model is established according to the correlation between four types of network units in the power system; secondly, in order to maintain the dynamic resistance characteristics of the defender during the attack process, the contradictory game relationship between the external attacker and the scheduling defender for different objective functions is considered, and a multi-objective zero-sum game model is proposed; thereafter, in order to achieve an accurate solution to the model, the KKT condition and linear transformation are used, and on the basis of transforming and solving the sub-problems, the double-layer mixed integer nonlinear optimization problem in the original problem is successfully transformed into a mixed integer linear optimization problem; finally, the minimum attack cost required to cause the power system to fail completely can be solved by a conventional solver, thereby evaluating the power system's external attack capability. The method of the present invention can provide power grid defense dispatchers and defense plan makers with a theoretical model for analyzing the weak links of the power system and the effectiveness of different land and resource allocation plans under external attack scenarios, so as to formulate more effective defense measures, optimize the operating capacity of the power system under extreme scenarios, and provide a scientific basis for attack and defense simulations against the power system.

[0107] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0108] (1) The present invention starts from the perspective of an external attacker and aims to solve the attack cost required to completely fail the power system. Compared with the prior art, the calculation results of the multi-objective zero-sum game power system external attack vulnerability assessment model established by the present invention can reflect the overall attack resistance of the power system under different defense strategies.

[0109] (2) The power system external attack vulnerability assessment model based on multi-objective zero-sum game established in the present invention takes into account the completely opposite optimization goals of the decision makers of the game for the attacker's cost function and the defender's loss function, which is closer to the actual situation of the attack and defense game. The zero-sum game framework is used to establish the model, which enables the defender to always maintain the resistance characteristics in the process of solving the minimum attack cost required for the complete failure of the power system, thereby ensuring the dynamic game process under the external attack scenario of the power system.

[0110] (3) The present invention establishes a complex multi-objective zero-sum game model, which is mathematically a two-layer mixed integer nonlinear optimization problem. In view of the complexity of the problem, a corresponding mathematical method is proposed to solve the complex original problem by transforming and solving sub-problems, thereby transforming the original problem into a single-layer mixed integer linear optimization problem, so that the problem can use a general solver to calculate the global optimal solution.

[0111] (4) The present invention can help power grid dispatchers and homeland defense resource allocators analyze and compare the overall advantages and disadvantages of different defense strategies, so as to select the overall optimal defense plan under external attack scenarios.

[0112] (5) In addition to providing the minimum attack cost required to completely paralyze the power system, the output results of the present invention also provide the attacker's attack plan, that is, the target grid unit to be attacked and the attack intensity. Therefore, under the condition of obtaining the layout of the defense resources of the hostile forces or the national power system, the model of the present invention can also evaluate the vulnerability of the enemy's power system and provide the optimal attack path with the minimum attack cost required by our side. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 is a flow chart of the method of the present invention;

[0114] Figure 2 It is a calculation flow chart of the linearization solution method in the present invention;

[0115] Figure 3 This is the IEEE 30-node system grid structure diagram;

[0116] Figure 4 This is a diagram showing the attack situation when the power system fails under the condition of executing defense plan 1;

[0117] Figure 5 This is a diagram showing the attack situation when the power system fails under the condition of implementing defense plan two. DETAILED DESCRIPTION

[0118] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0119] Embodiment 1:

[0120] like Figure 1 As shown, the present invention provides a method for evaluating the external attack vulnerability of a power system based on a multi-objective zero-sum game, comprising the following steps:

[0121] S1: A grid unit damage correlation propagation model is established based on the power grid structure and the types and numbers of grid units actually included, which is used to analyze and calculate the impact of any one or more grid units being damaged or disconnected from the grid on the grid connection status of other networks;

[0122] The grid unit damage associated propagation model includes the associated grid-connected state constraints of four types of grid units: power plants, substations, buses and lines. The associated grid-connected state constraints consider the impact of the four types of grid units, power plants, substation buses and lines, on other grid units after being hit or disconnected from the grid. Two binary variables, hit state and grid-connected state, are established. The relationship between the grid-connected state of each type of grid unit and the hit state of the associated adjacent grid unit is as follows:

[0123]

[0124] Where i, j, k and h are the numbers of the power plant, transmission line, busbar and substation in the system respectively; is the grid-connected status of power plant i; The power plant i is in the state of being hit; is the grid connection status of the busbar associated with power plant i; is the grid-connected state of line j; is the hit state of line j; is the grid-connected status of the busbar at the starting point of line j; is the grid connection status of the bus at the end of line j; The bus k is in grid-connected state; The busbar k is in the struck state; The substation h is in the state of being struck; The grid-connected state of substation h; The substation h is in the state of being hit.

[0125] S2: Based on the grid unit damage correlation propagation model, an attack and defense multi-objective zero-sum game model is established according to the attacker and defender's optimization goals for different objective functions and their own resource constraints. This model is used to calculate the minimum cost that the attacker needs to pay to completely cause the power system to fail, while maintaining the resistance characteristics of the power system defender, thus forming a dynamic game process between the attacker and the defender.

[0126] The attack-defense multi-objective zero-sum game model includes two decision-maker models, an upper-level attacker model and a lower-level defender model, and a two-person game model consisting of two zero-sum game problems guided by two objective functions, the attacker's attack cost and the defender's total loss.

[0127] The decision goal of the upper attacker model is to minimize the attack cost e atk And maximize the defender's total loss f loss ,in,

[0128]

[0129] I, J, K and H are the total number of power plants, lines, busbars and substations in the power network, respectively; The attack cost required to cause damage to power plant i; The attack cost required to cause damage to line j; The attack cost required to cause the destruction of busbar k; is the attack cost required to cause damage to substation h; α k The importance of the load supplied by bus k; The load loss power of the bus k supplying power; The demand for power supply load for bus k; is the actual power supply load power of bus k; the attack cost can also be regarded as the defense capability of the grid unit. The attacker needs to pay at least an attack cost equal to the defense capability of the grid unit to destroy the network unit.

[0130] The decision-making goal of the lower-level defender model is to maximize the attack cost e atk And minimize the defender's total loss f loss , affected by the cascading propagation effect caused by the upper-level attacker’s decision, the power system operation constraints that the defender needs to meet are:

[0131]

[0132] -θ max ≤θ k ≤θ max

[0133]

[0134] Among them, P i gen is the power generation capacity of power plant i; is the transmission power of line j; is the upper limit of power generation capacity of power plant i; θ k is the phase angle of busbar k; θ o(j) is the phase angle of the busbar at the starting point of line j; θ e(j) is the phase angle of the busbar at the end of line j; b j is the impedance of line j; is the upper limit of transmission power of line j; θ max is the maximum bus phase angle.

[0135] In the two-player game model, there are two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss. One of them is the attack cost e required by the attacker to attack the power system. atk , there is a min-max problem that is affected by the decisions of both the attacker and the defender:

[0136]

[0137] Second, the total loss of the defender's power system is floss , there is a max-min problem 2 that is also affected by the decisions of the attacker and the defender:

[0138]

[0139] Where a is the set of attacker’s decision variables (x gen ,x line ,x bus ,x sub ,y gen ,y line ,y bus ,y sub );d is the set of defender’s decision variables (P gen ,P b ,θ,P load ,ΔP load ).

[0140] In two zero-sum game problems guided by two objective functions, the attacker's attack cost and the defender's total loss, for the defender, maximizing e atk and minimize f loss There is a significant consistency in the optimization objectives. This is because the defender can optimize the power flow to reduce the total loss of the power system after the attack to f loss If the minimum value is obtained, then compared with other non-optimized operation schemes, if the attacker wants to achieve the same attack effect, the minimum attack cost required by the attacker to deal with the defender's optimal power flow scheme is always the largest, which also achieves the goal of maximizing the attacker's attack cost. Therefore, under the condition of the joint model, the min-max problem of the two zero-sum game problems is simplified to:

[0141]

[0142] In the attack-defense multi-objective zero-sum game model, the attacker needs to completely fail the power system through external attacks, and the total loss of the power system will be foreseeably maintained at the maximum value, but this does not mean that the defender will stop the power flow optimization in the dynamic game process. The defender still needs to maintain the normal operation of the power system as much as possible. Combining the two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss, the max-min problem 2 is expressed as a constraint condition for a fixed game solution:

[0143]

[0144] in, It is the maximum loss when the power system fails completely.

[0145] Therefore, the attack-defense multi-objective zero-sum game model is mathematically a two-layer mixed integer nonlinear optimization problem.

[0146] S3: The attack-defense multi-objective zero-sum game model is expressed as a two-level mixed integer nonlinear optimization problem. The problem is solved to obtain the minimum attack cost required by an external attacker to cause the power system to fail completely.

[0147] Reference Figure 2 , the solution process includes:

[0148] A1: Degrade the original problem into sub-problems;

[0149] The sub-problem in the method of reducing the original problem to sub-problems is to give the attacker an external attack cost, that is, the attacker's attack cost e atk is a single objective function zero-sum game under constant value conditions; for a given attacker's external attack cost, that is, the attacker's attack cost is fixed, the original multi-objective zero-sum game power system external attack vulnerability assessment model can be reduced to a sub-problem. The sub-problem is expressed as a zero-sum game model for solving the maximum loss that the attacker can cause to the power system under the premise that the attacker can pay a limited attack cost. It is a two-level mixed integer nonlinear optimization problem, and its objective function is

[0150]

[0151] Constraints:

[0152]

[0153] -θ max ≤θ k ≤θ max

[0154]

[0155] The constraints include all the constraints in the grid unit damage association propagation model, the upper attacker model and the lower defender model, as well as the attack capability constraints caused by the limited attack cost of the attacker; among them, The maximum attack cost available to the attacker is a fixed value;

[0156] A2: Use the KKT condition by taking advantage of the strong duality of the underlying problem of the subproblem;

[0157] By using the strong duality of the lower-level problem of the subproblem and using the KKT condition, under the condition of the known attack decision of the upper-level attacker, the lower-level problem of the subproblem is a solvable linear optimization problem, and its dual problem can be derived. The lower-level problem and its dual problem must satisfy the strong duality condition. The KKT condition can be used to transform the double-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer nonlinear optimization problem.

[0158] Under the condition that the attack decision of the upper attacker is known, the lower problem is a solvable simple linear optimization problem, and the objective function is

[0159]

[0160] Constraints:

[0161]

[0162] in, and are the grid-connected states of various grid units, and are constants especially when the attack decision of the upper attacker is known; ω k ,μ j ,γ i ,φ j , σ k ,κ k ,x k ,ξ k They are the dual variables corresponding to the constraints of the lower-level problem when solving the dual problem;

[0163] The dual problem can be deduced, and the objective function is

[0164]

[0165] Constraints:

[0166] ω con(i) +γ i ≤0,P i gen

[0167]

[0168] The lower-level problem is a linear problem whose objective function exists and is solvable in the feasible domain. The optimal solution of this problem and its dual problem must satisfy strong duality. The KKT condition can be used to transform the double-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer nonlinear optimization problem. For the lower-level problem of the subproblem, the specific constraint formula of the KKT condition is

[0169]

[0170] By using the dual problem and KKT condition to achieve the clamping effect on the lower-level objective function, the double-level mixed integer linear optimization problem of the subproblem is transformed into a single-level mixed integer nonlinear programming problem. The objective function is

[0171]

[0172] Constraints:

[0173]

[0174]

[0175] -θ max ≤θ k ≤θ max

[0176]

[0177] ω con(i) +γ i ≤0

[0178] κ k ≤α i

[0179] -ω k +σ k +κ k ≤0

[0180]

[0181] Where D is the set of decision variables of the dual problem (ω,μ,κ,γ,φ, σ,χ,ξ);

[0182] A3: Logical expansion of nonlinear terms;

[0183] The logical expansion of nonlinear terms is to transform all high-order variable combinations caused by multiplication of decision variables into a form containing only first-order variable addition and subtraction combinations through mathematical methods, thereby transforming the single-level mixed integer nonlinear optimization problem of the subproblem into a single-level mixed integer linear optimization problem;

[0184] By mathematically converting all high-order variable combinations caused by multiplication of decision variables in the problem into a form containing only first-order variable addition and subtraction combinations:

[0185] Mode middle, are binary variables. or When it is 0, The value of must be 0, and when and When both are 1, The value of must be 1; therefore, according to the formula The equality constraints expressed can be expanded into inequality constraints:

[0186]

[0187] Through the pair The logical expansion of can achieve the same function as the original equality constraint, but avoids the situation of multiplying decision quantities. Similarly, the formula Perform logical expansion:

[0188]

[0189] Mode and Middle multiplication There are cases where the boundary is infinite, and there are continuous variables with unbounded range of values. Explain; since the decision variable μ j The range of values ​​of has no upper and lower bounds. Using the common method to replace the multiplication of binary variables and continuous variables will result in the situation where the variables are added or subtracted from the infinite boundary. In this case, it is better to set The corresponding expanded constraints are:

[0190]

[0191] In the formula, M is a maximum positive number; at this time, when When j -μ' j =μ j ;when When j -μ' j =0; multiplied with the item before expansion The operation logic is consistent; similarly,

[0192]

[0193] and can be transformed into:

[0194]

[0195] After this step, the single-layer mixed integer nonlinear optimization problem of the subproblem is transformed into a single-layer mixed integer linear optimization problem;

[0196] A4: Transition from sub-problem to original problem;

[0197] After the subproblem is transformed into a single-layer mixed integer linear optimization problem, the double-layer mixed integer nonlinear optimization problem in the original multi-objective zero-sum game model is transformed into a single-layer mixed integer linear optimization problem by restoring the objective function and non-common constraints. Its objective function is

[0198]

[0199] Constraints:

[0200]

[0201]

[0202] -θ max ≤θ k ≤θ max

[0203]

[0204] ω con(i) +γ i ≤0

[0205] κ k ≤α i

[0206] -ω k +σ k +κ k ≤0

[0207]

[0208]

[0209] Wherein, T is the set of variables newly added by the logic expansion in step S3 (θ' o ,θ′ e ,μ' j ,γ i ',σ' k );

[0210] A5: Use CPLEX or GUROBI solver to solve the subproblem model and the single-layer mixed integer linear optimization problem in the original problem model. Use the MATLAB platform YALMIP toolbox to call CPLEX or GUROBI solver to solve the model.

[0211] Embodiment 2:

[0212] The correctness of the results of the original problem model is verified by using the sub-problem model in the power system external attack vulnerability assessment model based on multi-objective zero-sum game. The specific method is as follows: the minimum attack cost value e solved by the original problem model is converted into atk The maximum attack cost of the attacker in the subproblem model parameter Input, when the power system grid structure of the original problem and the sub-problem analysis and calculation is the same, the output result of the sub-problem f loss The maximum loss value of the power system Keep it consistent, otherwise, it means that there are human operational errors in the process of model implementation.

[0213] In order to verify the effectiveness and practical effect of the present invention, two test examples are used for illustration, as follows: Test Example 1:

[0214] The external attack vulnerability assessment method of the power system based on multi-objective zero-sum game provided by the present invention is now used to calculate and analyze the external attack vulnerability of the improved IEEE30-node system case under different defense resource allocation strategies.

[0215] The grid structure of the IEEE 30-node system case is as follows: Figure 3 As shown, the system parameters are from the MATPOWER7.0 software package, and the power parameters of the power plant are modified in this test example, as shown in Tables 1 to 4.

[0216] Table 1 Power plant parameters

[0217]

[0218] Table 2 Transmission line parameters

[0219]

[0220]

[0221] Table 3 Bus power supply parameters

[0222]

[0223] Table 4 Substation related parameters

[0224]

[0225] In order to facilitate experimental calculation and result comparison, two land resource allocation schemes with different focuses but the same total amount of defense resources are designed in the experiment. The busbars supporting the substation are not allocated defense resources and are shared with the substation where they are located:

[0226] Option 1: Allocate 5 defense units to each power plant and substation, 3 defense units to each independent bus, and 2 defense units to each transmission line. The total number of defense units used is 208 p.u.;

[0227] Option 2: Allocate 2 defense units to each power plant, transmission line and substation, and 4 defense units to each independent bus, with a total of 208 p.u. of defense units used.

[0228] With the help of the present invention, the minimum attack cost required for the power system to completely fail due to external attack under the above different defense schemes and the attacker's attack scheme are calculated as shown in Table 5.

[0229] Table 5 Calculation data of power system vulnerability analysis under different defense schemes

[0230]

[0231] It can be seen from Table 5 that, under the condition of consuming the same amount of defense resources, defense resource allocation scheme 1 enables the power system to withstand a higher level of external attacks, and its vulnerability to attacks is relatively low.

[0232] Figure 4 and Figure 5 It further provides grid attack images when the power system fails completely due to an attack under two types of defense schemes. Figure 4 and Figure 5 Comparing the attack situations of the two schemes and their defense schemes, it can be seen that defense scheme 1 focuses on the protection of power plants and substations. Therefore, the attacker's attack scheme focuses on attacking the busbars and lines, achieving the same effect as attacking the adjacent power plant. When defense scheme 2 relatively improves the defense capability of the busbar, the attacker chooses to directly attack the power plant.

[0233] Test Example 2:

[0234] Now, the minimum attack cost of the attacker solved by the original problem model in Test Example 1 is verified by using the neutron problem of the present invention. The input power system parameters are also the improved IEEE 30-node system case. Two experiments are set in this test example.

[0235] Experiment 1:

[0236] Using the same power system parameters and grid structure as in Test Case 1, the attacker's available attack cost is set to the attacker's minimum attack cost in the calculation results of Case 1 in Test Case 1. The calculation results are shown in Table 6.

[0237] Table 6 The maximum loss that can be caused to the power system

[0238]

[0239] Comparing Table 5 and Table 6, it can be seen that when the attacker invests the minimum attack cost solved by the original problem, the maximum loss can be caused to the power system, that is, the power system fails completely.

[0240] Experiment 2:

[0241] Based on Experiment 1, the attacker’s available attack cost is modified to 14 and 16, and the experiment is repeated. The calculation results are shown in Table 7.

[0242] Table 7 and The maximum loss that can be caused to the power system

[0243]

[0244] It can be seen from Table 7 that for the first defense resource allocation scheme, when the attacker's available attack cost is less than 15 p.u., the damage loss caused to the power system fails to reach the maximum value; when the attacker's available attack cost is greater than 15 p.u., the damage loss caused to the power system remains at the maximum value. This shows that the minimum attack cost required to cause the complete failure of the case power system through external attack is 15 p.u., which is consistent with the calculation results of the power system external attack vulnerability assessment model based on multi-objective zero-sum game in this invention.

Claims

1. A method for assessing the vulnerability of power system to external attack based on multi-objective zero-sum game, characterized in that: The steps include: S1: Establish a grid unit damage correlation propagation model based on the power grid structure and the types and numbers of grid units actually included; S2: Based on the grid unit damage correlation propagation model, an attack and defense multi-objective zero-sum game model is established according to the attacker and defender's optimization goals for different objective functions and their own resource constraints; S3: The attack-defense multi-objective zero-sum game model is expressed as a two-level mixed integer nonlinear optimization problem. The problem is solved to obtain the minimum attack cost required by an external attacker to cause the power system to fail completely.

2. According to claim 1, a method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game, characterized in that: The establishment of the grid unit damage associated propagation model in step S1 includes: The associated grid-connected state constraint considers the impact of the four types of grid units, namely power plants, substation buses and lines, on other grid units after being hit or disconnected from the grid. Two binary variables, hit state and grid-connected state, are established. The relationship between the grid-connected state of each type of grid unit and the hit state of the associated adjacent grid unit is as follows: Where i, j, k and h are the numbers of the power plant, transmission line, busbar and substation in the system respectively; is the grid-connected status of power plant i; The power plant i is in the state of being hit; is the grid connection status of the busbar associated with power plant i; is the grid-connected state of line j; is the hit state of line j; is the grid-connected status of the busbar at the starting point of line j; is the grid-connected status of the bus at the end of line j; The bus k is in grid-connected state; The busbar k is in the struck state; The substation h is in the state of being struck; The grid-connected state of substation h; The substation h is in the state of being hit.

3. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 1 is characterized in that: The attack-defense multi-objective zero-sum game model in step S2 includes two decision-maker models, an upper attacker model and a lower defender model, and a two-person game model consisting of two zero-sum game problems guided by two objective functions, the attacker's attack cost and the defender's total loss.

4. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 3 is characterized in that: The decision goal of the upper attacker model in step S2 is to minimize the attack cost e atk And maximize the defender's total loss f loss ,in, I, J, K and H are the total number of power plants, lines, busbars and substations in the power network, respectively; The attack cost required to cause damage to power plant i; The attack cost required to cause damage to line j; The attack cost required to cause the destruction of busbar k; is the attack cost required to cause damage to substation h; α k The importance of the load supplied by bus k; The load loss power of the bus k supplying power; The demand for power supply load for bus k; is the actual power supply load power of bus k; the attack cost can also be regarded as the defense capability of the grid unit. The attacker needs to pay at least an attack cost equal to the defense capability of the grid unit to destroy the network unit.

5. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 3 is characterized in that: The decision goal of the lower-level defender model in step S2 is to maximize the attack cost e atk And minimize the defender's total loss f loss , affected by the cascading propagation effect caused by the upper-level attacker’s decision, the power system operation constraints that the defender needs to meet are: Among them, P i gen is the power generation capacity of power plant i; is the transmission power of line j; is the upper limit of power generation capacity of power plant i; θ k is the phase angle of busbar k; θ o(j) is the phase angle of the busbar at the starting point of line j; θ e(j) is the phase angle of the busbar at the end of line j; b j is the impedance of line j; is the upper limit of transmission power of line j; θ max is the maximum bus phase angle.

6. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 3 is characterized in that: In the two-player game model in step S2, there are two zero-sum game problems guided by the two objective functions of the attacker's attack cost and the defender's total loss. One of them is the attack cost e required by the attacker to attack the power system. atk , there is a min-max problem that is affected by the decisions of both the attacker and the defender: Second, the total loss of the defender's power system is f loss , there is a max-min problem 2 that is also affected by the decisions of the attacker and the defender: Where a is the set of attacker’s decision variables (x gen ,x line ,x bus ,x sub ,y gen ,y line ,y bus ,y sub );d is the set of defender’s decision variables (P gen ,P b ,θ,P load ,ΔP load ).

7. A method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 6, characterized in that: In the two zero-sum game problems in step S2, the min-max problem 1 is simplified to: The second max-min problem is expressed as a constraint on a fixed game solution: in, It is the maximum loss when the power system fails completely.

8. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 7 is characterized in that: The process of solving the problem in step S3 includes: A1: Degrade the original problem into sub-problems; The subproblem is formulated as a zero-sum game model for solving the maximum loss that the attacker can cause to the power system under the premise that the attacker can pay a limited attack cost. It is a two-level mixed integer nonlinear optimization problem, and its objective function is Constraints: in, is the maximum attack cost available to the attacker, which is a certain value; A2: Use the KKT condition by taking advantage of the strong duality of the underlying problem; Under the condition that the attack decision of the upper attacker is known, the lower problem is a solvable simple linear optimization problem, and the objective function is Constraints: -θ max ≤θ k ≤θ max ,x k ,x k in, and are the grid-connected states of various grid units, and are constants especially when the attack decision of the upper attacker is known; ω k ,μ j ,γ i ,φ j , σ k ,κ k ,x k ,ξ k They are the dual variables corresponding to the constraints of the lower-level problem when solving the dual problem; The dual problem can be deduced, and the objective function is Constraints: The KKT condition is used to transform the double-layer mixed integer nonlinear optimization problem of the subproblem into a single-layer mixed integer nonlinear optimization problem. For the lower-layer problem of the subproblem, the specific constraint formula of the KKT condition is: By using the dual problem and KKT condition to achieve the clamping effect on the lower-level objective function, the double-level mixed integer linear optimization problem of the subproblem is transformed into a single-level mixed integer nonlinear programming problem. The objective function is Constraints: -θ max ≤θ k ≤θ max oh con(i) +g i ≤0 k k ≤α i -oh k +s k +k k ≤0 Where D is the set of decision variables for the dual problem A3: Logical expansion of nonlinear terms; By mathematically converting all high-order variable combinations caused by multiplication of decision variables in the problem into a form containing only first-order variable addition and subtraction combinations: Mode middle, are binary variables. or When it is 0, The value of must be 0, and when and When both are 1, The value of must be 1; therefore, according to the formula The equality constraints expressed can be expanded into inequality constraints: Pair Perform logical expansion: Mode and Middle multiplication There are cases where the boundary is infinite, and there are continuous multiplication terms with unbounded range of continuous variables. To explain; The corresponding expanded constraints are: In the formula, M is a maximum positive number; at this time, when When j -μ' j =μ j ;when When j -μ' j =0; multiplied with the item before expansion The operation logic is consistent; similarly, and can be transformed into: After this step, the single-layer mixed integer nonlinear optimization problem of the subproblem is transformed into a single-layer mixed integer linear optimization problem; A4: Transition from sub-problem to original problem; After the subproblem is transformed into a single-layer mixed integer linear optimization problem, the double-layer mixed integer nonlinear optimization problem in the original multi-objective zero-sum game model is transformed into a single-layer mixed integer linear optimization problem by restoring the objective function and non-common constraints. Its objective function is Constraints: -θ max ≤θ k ≤θ max oh con(i) +g i ≤0 k k ≤α i -oh k +s k +k k ≤0 Wherein, T is the set of variables newly added by the logic expansion in step S3 (θ' o ,θ' e ,μ' j ,γ' i ,σ' k ); A5: Use the solver to solve the single-level mixed integer linear optimization problem in the subproblem model and the original problem model.

9. The method for assessing the external attack vulnerability of a power system based on multi-objective zero-sum game according to claim 8 is characterized in that: The solver in step A5 includes CPLEX and GUROBI solvers, and the CPLEX or GUROBI solver is called through the YALMIP toolbox on the MATLAB platform to solve the model.

Images