Alternating current optimal defense resource allocation method based on two-stage hybrid optimization

Abstract

The invention discloses an optimal communication defense resource allocation method based on two-stage hybrid optimization. According to the technical scheme, the method is characterized by comprising the steps that in the first stage, global configuration rough search is conducted, global exploration is conducted on the AC-ODRA problem under an alternating current model through a myxobacteria algorithm, the deployment position (discrete variable) of defense resources is rapidly determined, and a high-quality candidate solution is obtained; in the second stage, resource input fine adjustment is conducted, specifically, for the candidate solutions in the first stage, the deployment positions of the candidate solutions are fixed, and high-precision local optimization is conducted on the specific resource input amount (continuous variables) under the augmented Lagrange framework through an adaptive torque estimation optimization method of multi-starting-point random disturbance guided by instrument importance. According to the method, global search and local fine tuning are combined, the solving precision and robustness of the AC-ODRA problem are remarkably improved, a defense resource allocation scheme with lower cost can be obtained on the premise of guaranteeing the safety of a power grid, and the method has important theoretical value and application prospects.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of smart grid security, and particularly relates to a method for optimal defense resource allocation in a power network physical system to resist false data injection attacks on state estimation, and more particularly to an optimization solving technique for large-scale power grids under alternating current power models. BACKGROUND

[0002] With the deep integration of information communication technology and power systems, smart grids are increasingly facing network attack threats represented by false data injection attacks (FDIAs) while improving operational efficiency. FDIAs can mislead grid operators to make incorrect estimates of system state by maliciously tampering with measurement data, which may cause economic losses and even large-scale safety accidents.

[0003] As a proactive defense strategy, optimal defense resource allocation (ODRA) aims to determine an optimal defense resource (such as encryption modules, video surveillance, protective personnel, etc.) allocation strategy to minimize the overall defense resource investment while ensuring that any set of state variables cannot be tampered with by attackers (i.e., the minimum cost required to launch a successful attack on any subset of state variables should exceed the resources available to the attacker) to protect the power grid.

[0004] In the prior art, existing research has explored the ODRA problem in depth. For example, considering that the relationship between defense measures and attack costs often exhibits nonlinear characteristics in reality, a mixed integer nonlinear programming (MINLP) problem is constructed. Since this problem is difficult to solve with traditional commercial solvers, a slime mould algorithm with initialization pool (SMA-IP) is proposed by Xu et al. in 2025 in Optimal Defense Resource Allocation Considering Nonlinear Attack Cost in Power Systems, but this research is mainly conducted under a direct current (DC) model. However, the essence of evolutionary algorithms such as SMA is to perform global random search, and after determining the deployment location of defense resources (i.e., discrete variables), its ability to perform local fine optimization on continuous resource inputs (i.e., continuous variables) is relatively limited, which may result in the final calculated defense cost not being the theoretical minimum value, i.e., there is still some optimization space.

[0005] In addition, in order to extend the ODRA problem to more challenging large-scale power grid scenarios, a network partitioning-based strategy is proposed in the literature. Although this study mainly verifies its partitioning strategy under the DC model, it also prospectively gives the basic mathematical construction of the ODRA problem under the alternating current (AC) model, such as defining a binary structure matrix that can reflect the measurement-state dependency relationship under the AC model. However, this literature only uses the general SMA-IP algorithm to solve the sub-problems after partitioning, and does not design a special and high-precision solving method for the inherent and more complex nonlinear and high-coupling characteristics of the AC model, especially for the optimization of continuous variables.

[0006] In summary, the prior art has the following deficiencies: when dealing with the increasingly important AC-ODRA problem, there is a lack of a hybrid solving framework that can balance global optimal configuration and local accurate optimization, and relying only on a single-stage evolutionary algorithm may lead to insufficient optimality of the solution. Therefore, how to design a new optimization method to solve the complex AC-ODRA problem to achieve better optimality performance is a technical problem to be solved. SUMMARY

[0007] The purpose of the present application is to solve the technical problem of insufficient solving optimality and resource-saving defense resource allocation scheme caused by relying only on a single-stage evolutionary algorithm when the existing ODRA solving method is applied to a complex alternating current model, and to propose an alternating current optimal defense resource allocation method based on two-stage hybrid optimization.

[0008] The purpose of the present application is achieved by the following technical solution: an alternating current optimal defense resource allocation method based on two-stage hybrid optimization, comprising the following steps:

[0009] Step one: global configuration coarse search stage: use the SMA-IP algorithm to perform global exploratory search on the entire solution space of the AC-ODRA problem. The main task of this stage is to solve the discrete binary variables in the problem, i.e., to determine where the defense resources should be deployed. The output of this step is a high-quality candidate solution, which includes a determined defense deployment location (the value of the binary decision variable vector φ) and an initial defense resource input quantity corresponding thereto (the initial value b initial ).

[0010] Step two, resource investment fine-tuning phase: fix the defense deployment position obtained in step one, and convert the original problem into a non-convex optimization sub-problem only about the continuous decision variable vector b. To solve this sub-problem with high precision, the present application adopts a multi-start random disturbance-based adaptive moment estimation (Multi-Start Adam) optimization method, which performs local fine-tuning optimization on the inner loop under the augmented Lagrangian double-loop optimization framework, and specifically includes the following sub-steps:

[0011] 2.1 First deterministic optimization: for the kth outer loop, when the first time (i.e., j = 1) of N inner loop optimizations is performed, a baseline fine-tuning solution is obtained by directly performing Adam optimization using the original starting point of the Adam optimizer.

[0012] 2.2 Multiple random disturbance optimization: in the subsequent N-1 inner loop optimizations (i.e., j = 2,..., N), non-uniform randomness guided by instrument importance is introduced to the starting point to generate N-1 new starting points; for each new starting point, Adam optimization based on the augmented Lagrangian framework is repeatedly performed.

[0013] 2.3 Optimal solution selection: from the baseline fine-tuning solution obtained in step 2.1 and the N-1 optimization results obtained in step 2.2, the solution with the minimum inner loop objective function value among all solutions is selected as the result of this inner loop and returned to the outer loop for updating the Lagrange multiplier.

[0014] The beneficial effects of the present application are as follows:

[0015] 1. Improved optimality of the solution: the present application combines the strong global search capability of evolutionary algorithms with the excellent local optimization capability of the Adam optimizer. This two-stage hybrid optimization approach can find more accurate solutions than single evolutionary algorithms, thereby significantly reducing the total cost of deploying defense resources while ensuring the same level of security.

[0016] 2. Enhanced robustness of the solution: the "multi-start + random disturbance" strategy guided by instrument importance used in the second stage effectively overcomes the defect of traditional gradient optimization methods that are prone to falling into poor local minima, greatly improving the stability and success rate of finding global or high-quality local optimal solutions.

[0017] 3. Filling the technical gap in the field: the present application first applies this advanced two-stage hybrid optimization framework, especially the multi-start random disturbance-based Adam optimizer, to solving complex AC-ODRA problems, providing a new and high-performance solution approach for the field and effectively addressing the precision bottleneck of existing technologies. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1A flow chart of an alternating current optimal defense resource allocation method based on two-stage mixed optimization is provided for an embodiment of the present application.

[0019] Figure 2 A second-stage augmented Lagrange-multiplier-multi-start Adam optimization implementation block diagram is provided for an embodiment of the present application. DETAILED DESCRIPTION

[0020] In order to make the above objectives, characteristics and advantages of the present application more apparent, specific embodiments of the present application are described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present application, and are not a limitation of the present application.

[0021] I. AC-ODRA problem formulation

[0022] First, a mathematical model of the AC-ODRA problem is established. In an alternating current power system with n state variables and m measurement instruments, the problem can be constructed as a mixed integer nonlinear programming (MINLP) problem as follows:

[0023] Minimize:

[0024] 1 T b

[0025] Subject to:

[0026] R-H * ·[φ1f1(b1),φ2f2(b2),v,φ m f m (b m )] T ≤0

[0027] 1 T φ-M≤0

[0028] b≥0,φ∈{0,1} m

[0029] Wherein:

[0030] 1 is an m-dimensional column vector with all elements being 1. 1 T denotes transposition of 1, so that it changes from a column vector to a row vector. Similarly, 0 is an n-dimensional column vector with all elements being 0.

[0031] b=[b1,b2,...,b m ] T is an m-dimensional continuous decision variable vector, representing the amount of defense resources invested at each instrument. The optimization goal of the present application is to minimize the total investment, i.e.

[0032] φ = [φ1, φ2,..., φm]T m ] T is an m-dimensional binary decision variable vector. When φ i = 1, it means that defense resource is deployed on the ith meter; when φ i = 0, it means that no defense resource is deployed.

[0033] f i (b i ) is the attack cost function on the ith meter, which represents the cost paid by the attacker to break the meter when the defense resource input is b i . The function can be linear or nonlinear.

[0034] M is an integer, representing the upper limit of the number of meters on which the defense party can deploy defense measures, i.e., the defense budget.

[0035] R is an n-dimensional vector with all elements being R, R representing the upper limit of the total resources available to the attacker to launch attacks.

[0036] H * is an n x m binary structure matrix, which reflects the topological dependency relationship between the state variables of the AC model and the measurement meters. If the change of the jth state variable x j in the AC model of the power system affects the ith measurement meter z i , then the corresponding element in the matrix is 1; otherwise, it is 0.

[0037] The core meaning of the first constraint is that for any state variable, the total attack cost (i.e., the sum of the attack costs of all related meters) required by the attacker to successfully tamper with it must be greater than the upper limit of the resources R possessed by the attacker. Note that this constraint is expressed in vector form, which actually contains n sub-constraints.

[0038] The core meaning of the second constraint is that the upper limit of the number of meters on which the defense party can deploy defense measures is M.

[0039] The third constraint represents the basic properties of the decision variable.

[0040] II. Explanation of two-stage mixed optimization algorithm

[0041] The present application proposes a new method based on two-stage mixed optimization for solving the AC optimal defense resource allocation (AC-ODRA) problem. The overall process of the method is shown in Figure 1 , and the core is to decompose the complex mixed integer nonlinear programming problem into two related sub-stages to balance the globality and accuracy of the solution.

[0042] ​The first stage is a global configuration coarse search stage (step one in Figure 1 ), which aims to quickly lock the best combination of defense deployment locations by using the global search ability of evolutionary algorithms. The second stage is a resource input fine tuning stage (step two in Figure 1 ), which, based on the deployment locations determined in the previous stage, uses advanced local optimization algorithms to calculate the specific amount of resources to be invested at each location to achieve the lowest cost.

[0043] The specific implementation of each step will be described in detail below.

[0044] Step one: global configuration coarse search stage

[0045] Since the AC-ODRA problem is a highly nonlinear, non-convex MINLP problem, traditional solvers are difficult to effectively solve. Therefore, this step uses an advanced evolutionary algorithm, the slime mold algorithm with initialization pool (SMA-IP), to solve it. This algorithm has been proven in existing technology to have good performance in solving such problems. SMA-IP algorithm explores the solution space by simulating the foraging behavior of slime mold. Its individual code contains both binary variables φ and continuous variables b. Through its unique initialization strategy and evolution mechanism, the algorithm can effectively search within the complex feasible region that meets the constraints.

[0046] The purpose of this stage is not to directly obtain the final optimal solution, but to prepare for the fine tuning of the next stage. After sufficient iterative search, the SMA-IP algorithm outputs the best individual it has found, which is a high-quality candidate solution.

[0047] This candidate solution includes:

[0048] A certain, fixed defense deployment location - binary decision variable vector: φ * ;

[0049] A corresponding initial defense resource input amount - initial value of continuous decision variable vector b: b initial .

[0050] This candidate solution (φ * , b initial ) will be used as input for step two for further precise optimization.

[0051] Step two: resource input fine tuning stage

[0052] The purpose of this stage is to optimize the continuous defense resource input amount b based on the defense deployment location φ * determined in the previous stage, to find the lowest cost feasible solution. The detailed process of this stage can be referred toFigure 2 .

[0053] 2.1 Sub-problem transformation. When the binary decision variable vector φ * is fixed, the original AC-ODRA problem is transformed into a non-linear, non-convex optimization sub-problem only about the continuous decision variable vector b. Its objective and constraints are the same as the model in Step 1, but φ has been treated as a constant φ * .

[0054] 2.2 Optimization framework: Augmented Lagrangian Method with double loops. To effectively handle the complex non-linear inequality constraints in this sub-problem, the Augmented Lagrangian Method is adopted as the core optimization framework. This method works collaboratively through an outer loop and an inner loop. The outer loop is responsible for dynamically adjusting the Lagrange multiplier vector λ and the penalty coefficient ρ to guide the solution to approach the feasible region; the inner loop is responsible for solving the optimal resource input under the given Lagrange multiplier vector λ and penalty coefficient ρ.

[0055] 2.2.1 Outer loop (Dual Update): The outer loop is the main process of the entire fine-tuning phase, responsible for iteratively updating the Lagrange multiplier vector λ and the penalty coefficient ρ.

[0056] 2.2.1.1 Initialization: At the beginning, the defense resource input b1 = b initial is initialized, the Lagrange multiplier vector λ is set to zero, and an initial penalty coefficient ρ1 (for example, ρ1 = 10) is set.

[0057] 2.2.1.2 Outer loop iteration (kth time):

[0058] ① Execute the inner loop:

[0059] Let the Lagrange multiplier vector be λ k = [λ 1,k , λ 2,k ,..., λ n,k ] T at the kth time of the outer loop. Take the current Lagrange multiplier vector λ k and the penalty coefficient ρ k as input, call the Adam optimizer of the inner loop, and solve the minimization problem of the following augmented Lagrangian function P(b) to obtain an updated defense resource input b k+1 :

[0060]

[0061] where g j(b) represents the degree of relaxation of the jth sub-constraint in the first constraint, which aims to protect the jth state variable from being tampered. Its specific mathematical expression is:

[0062]

[0063] where is the element in the jth row and ith column of the binary structure matrix H * . When g j (b) > 0, it indicates that the protection for the state variable j is insufficient, and the constraint is violated.

[0064] 2. Update the Lagrange multiplier vector λ:

[0065] After obtaining b k+1 , calculate its corresponding actual constraint relaxation value g j (b k+1 ). Then, for each sub-constraint (j = 1,..., n), update its corresponding Lagrange multiplier component λ j,k independently according to the following formula:

[0066] λ j,k+1 = max(0, λ j,k + ρ k · g j (b k+1 ))

[0067] The meaning of this update is that for those constraints that are still violated (i.e., g j (b k+1 ) > 0), their corresponding multiplier λ j,k+1 will increase, which is equivalent to increasing the "punishment level" for violating the constraint in the next inner loop, forcing the optimizer to find a solution that better satisfies the constraint.

[0068] 3. Update the penalty coefficient ρ:

[0069] To accelerate convergence, the penalty coefficient can be gradually increased:

[0070] ρ k+1 = γ · ρ k

[0071] where γ > 1 is the growth rate.

[0072] The outer loop continues until the change in the Lagrange multiplier is less than a set threshold or the maximum number of iterations is reached, and all constraints are satisfied. After the outer loop iteration ends, the final output b k+1 combined with φ * from step one, together constitute the final optimal defense scheme for the aforementioned AC-ODRA problem sought by the invention (φ *,b k+1 )。

[0073] 2.2.2 Inner loop (Primal Update): Multi-start Adam based subproblem solving

[0074] The goal of the inner loop is to find the b k that minimizes the augmented Lagrangian function under given λ k and ρ k+1 :

[0075]

[0076] Instead of using the standard gradient descent method, the present invention uses a more advanced Adam optimizer based on multi-start random perturbation to accomplish this task.

[0077] 2.2.2.1 Adam optimizer: Adam is able to compute an adaptive learning rate for each decision variable (i.e., each b i ), thus achieving faster and more stable convergence in complex optimization surfaces.

[0078] Initialization: If the inner loop is in the first outer loop, the initial point b k,0 of the inner loop (i.e., b k,t-1 at t = 1) is initialized to the initial value of defense resource investment b1 of the first outer loop, otherwise it is initialized to the result b k of the last outer loop.

[0079] Inner loop iteration: At the t-th time step, its update equation is as follows:

[0080] ① Update the first moment (momentum term):

[0081]

[0082] where is the gradient of the objective function P(b) with respect to b.

[0083] ② Update the second moment (adaptive learning rate term):

[0084]

[0085] ③ Bias correction:

[0086]

[0087] ④ Update the parameter b:

[0088]

[0089] Among them, α is the basic learning rate, β1 and β2 are the decay rates (usually 0.9 and 0.999), and ∈ is a minimum value set to prevent division by zero (such as 10 -8 ). The inner loop continues until parameter b k,t The change in is less than the set threshold, or the maximum number of iterations is reached. After the inner loop iteration is completed, the final output b k,t As the result of the inner loop part of the outer loop b k+1 .

[0090] 2.2.2.2 Multi-starting point random perturbation strategy based on instrument importance

[0091] In order to further improve the solution quality of the inner loop and avoid falling into a bad local optimal solution, the present invention adopts a multi-start strategy for the above-mentioned Adam optimization process, and performs N independent adam optimizations on each starting point, and the other starting points except the original starting point are obtained by perturbing the original starting point. Different from the blind, uniform random perturbations in the prior art, the present invention proposes an adaptive perturbation method based on the importance of the instrument. The core idea of ​​this method is that not all defense positions are equally important, and the granularity of the disturbance exploration should be different. The resource input of instruments located in the key topological position of the power grid is more sensitive to the overall security of the system, and a more refined search should be carried out; while the instruments located in secondary positions can be explored on a larger scale. The specific implementation is as follows: Figure 2 As shown, it includes the following sub-steps:

[0092] ① First deterministic inner loop optimization: For the kth outer loop, when performing the first of N inner loop optimizations (i.e. j=1), directly use the original starting point b of the Adam optimizer start,1 =b k,0 (regardless of b k,0 Initialized to b1 or b k ). This ensures a complete exploration of the most direct optimization path, resulting in a baseline precision.

[0093] ② Inner loop optimization after multiple random perturbations: In the subsequent N-1 inner loop optimizations (i.e., j = 2, ..., N), non-uniform randomness guided by instrument importance is introduced to the starting point. This process includes:

[0094] (1) Calculate the importance of the instrument: First, according to the binary structure matrix H of the AC model * To quantify the importance of each instrument. Specifically, by * The matrix is ​​summed column-wise to obtain an m-dimensional instrument importance weight vector w:

[0095] w=sum(H* ,1) T

[0096] Here, sum(·,1) represents column-wise summation, and the result is a 1×m row vector, which is transposed. T Then we get an m×1 column vector w. The i-th element w in the vector i represents the number of state variables that the i-th instrument can affect, that is, the “defense coverage” of the instrument. i The larger the value, the greater the "defense coverage" of the instrument.

[0097] (2) Generate disturbance intensity guided by instrument importance: According to the instrument importance, for each effective decision variable b i (i.e. φ i * = 1) to generate a differentiated perturbation intensity. The basic principle is: the more important the instrument, the more subtle the perturbation (the smaller the intensity); the less important the instrument, the bolder the perturbation (the greater the intensity). This can be achieved by:

[0098] First, the instrument importance weight vector w is normalized to obtain the normalized weight vector w norm , so that its elements range in [0,1].

[0099] Then, the instrument importance-guided perturbation level vector p is calculated noise , whose component p noise,i The calculation formula is:

[0100] p noise,i =r base ×(1-w norm,i )

[0101] Among them, r base is a base perturbation level (e.g. 0.1), w norm,i It is w norm Thus, if the importance of the i-th instrument is higher, the corresponding disturbance level p noise,i The smaller it is.

[0102] (3) Apply perturbation to generate a new starting point: At the beginning of the jth optimization (j>1), a new starting point b is generated. start :

[0103] b start,j =max(0,b k,0 ⊙(1+p noise ⊙p rand )⊙φ * )

[0104] Where ⊙ represents the element-by-element multiplication of the vector, prand is a random vector uniformly distributed in the interval [-1, 1] and of the same dimension as b k,0 is a random vector uniformly distributed in the interval [-1, 1] and of the same dimension as b * By multiplying with φ

[0105] ③Optimal solution selection: from the benchmark fine-tuning solution obtained by the first deterministic inner loop optimization and the N-1 optimization results obtained by the inner loop optimization after multiple random disturbances, the solution with the minimum inner loop objective function value P(b) among all solutions is selected as the inner loop result b k+1 and returned to the outer loop for updating the Lagrange multiplier.

[0106] The above only describes the preferred embodiments of the present application, although the present application has been disclosed as above with the preferred embodiments, however, is not intended to limit the present application. Any person skilled in the art, without departing from the scope of the technical scheme of the present application, can use the above disclosed methods and technical contents to make many possible changes and modifications to the technical scheme of the present application, or modify equivalent embodiments. Therefore, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present application, without departing from the content of the technical scheme of the present application, still belongs to the scope of protection of the technical scheme of the present application.

Claims

1. A method for optimal communication defense resource allocation based on two-stage hybrid optimization, characterized in that: include: Global configuration rough search phase: Using the slime mold algorithm, a global exploratory search is performed on the entire solution space of the optimal defense resource allocation problem under the communication model, outputting candidate solutions. The candidate solutions include a determined defense deployment location represented by a binary decision variable vector and a corresponding initial defense resource investment represented by a continuous decision variable vector. Resource investment fine-tuning stage: The defense deployment positions obtained in the global configuration rough search stage are fixed, and the original problem is transformed into a nonlinear, non-convex optimization subproblem involving only a continuous decision variable vector. An adaptive torque estimation optimization method based on multi-starting point random perturbations is used to locally tune the defense resource investment in the inner loop within an augmented Lagrangian framework with a double-layer loop. The local optimal solution is returned to the outer loop for updating the Lagrangian multiplier. After the outer loop iterates, the optimal defense resource allocation plan is obtained.

2. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 1 is characterized in that: The augmented Lagrangian framework works in tandem with an outer loop and an inner loop. The outer loop is responsible for dynamically adjusting the Lagrangian multiplier and penalty coefficient to guide the solution to approach the feasible domain; the inner loop is responsible for solving the optimal defense resource investment under current conditions under given Lagrangian multipliers and penalty coefficients.

3. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 1 is characterized in that: The inner loop of the augmented Lagrangian framework is specifically: First deterministic inner loop optimization: During the first of N inner loop optimizations, Adam optimization is performed directly using the original starting point of the Adam optimizer to obtain a baseline precision adjustment. Inner-loop optimization with multiple random perturbations: In the subsequent N-1 inner-loop optimizations, non-uniform randomness guided by instrument importance is introduced to the starting point to generate N-1 new starting points. For each new starting point, Adam optimization based on the augmented Lagrangian framework is repeatedly performed to obtain N-1 solutions. Optimal solution selection: From the N-1 solutions obtained by the benchmark precision adjustment obtained by the first deterministic optimization and multiple random perturbation optimizations, the solution with the smallest inner loop objective function value is selected as the result of this inner loop and returned to the outer loop.

4. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 3 is characterized in that: If this inner loop is in the first outer loop, the starting point of the inner loop is initialized to the initial value of the defense resource input of the first outer loop; otherwise, it is initialized to the result of the previous outer loop.

5. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 3 is characterized in that: The inner loop optimization of the multiple random perturbations is specifically as follows: First, the importance of each instrument is quantified according to the binary structure matrix of the AC model, that is, the binary structure matrix is ​​summed column-wise to obtain the instrument importance weight vector; Then, a differentiated disturbance intensity is generated for each effective decision variable according to the importance of the instrument to obtain the disturbance level vector. The disturbance principle is that the more important the instrument, the smaller the disturbance intensity, and the less important the instrument, the greater the disturbance intensity. Finally, a new starting point is generated by combining the random vector, the perturbation level vector, and the defense deployment position obtained in the global configuration rough search phase.

6. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 5 is characterized in that: The calculation formula for the disturbance horizontal component of the i-th instrument is: p noise,i =r base ×(1-w norm,i ) Among them, r base is a basic perturbation level, w norm,i is the i-th element value of the instrument importance weight vector with the element range normalized to [0,1].

7. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 5 is characterized in that: For the starting point b k,0 The new starting point b after introducing non-uniform randomness guided by instrument importance start The calculation formula is: b start =max(0,b k,0 ⊙(1+p noise ⊙p rand )⊙φ * ) Where ⊙ represents the element-by-element multiplication of the vector, p rand is an element uniformly distributed in the interval [-1,1] and is the same as b k,0 Random vector of the same dimension, p noise is the disturbance level vector, φ * It is the defense deployment position obtained in the global configuration rough search phase.

8. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 1 is characterized in that: The outer loop of the augmented Lagrangian framework is specifically: Initializing the defense resource input amount to the initial defense resource input amount obtained in the global configuration rough search phase; initializing the Lagrange multiplier and the penalty coefficient; During each outer loop iteration, the current Lagrangian multiplier and penalty coefficient are first used as input to call the Adam optimizer in the inner loop to solve the minimization problem of the augmented Lagrangian function to obtain the updated defense resource input. Then, based on the updated defense resource input, the actual constraint relaxation value is calculated and the Lagrangian multiplier is updated. Finally, the penalty coefficient is increased. The outer loop continues until the change in the Lagrange multiplier is less than the set threshold or the maximum number of iterations is reached, and all constraints are satisfied. After the outer loop iteration ends, the final output of the defense resource input is combined with the defense deployment position obtained by the global configuration coarse search stage to form the optimal defense resource allocation plan.

9. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 1 is characterized in that: In an AC power system with n state variables and m measuring instruments, the optimal defense resource allocation problem under the AC model is formulated as a mixed integer nonlinear programming problem, i.e., minimizing 1 T b, and satisfy the following three constraints: R-H * ·[φ1f1(b1),φ2f2(b2),...,φ m f m (b m )] T ≤0 1 T φ-M≤0 b≥0,φ∈{0,1} m Where b=[b1,b2,...,b m ] T is an m-dimensional continuous decision variable vector, representing the amount of defense resources invested in each instrument; φ=[φ1,φ2,...,φ m ] T is an m-dimensional binary decision variable vector, when φ i =1 means deploying defense resources on the i-th meter. i = 0, no deployment; f i (b i ) is the attack cost function on the i-th meter, which means that when the defense resource investment is b i The cost that the attacker needs to pay to break through the meter when ; M represents the upper limit of the number of meters that the defender can deploy defense measures; R is an n-dimensional vector whose elements are all R, and R represents the upper limit of the total resources that the attacker can use to launch an attack; H * It is an n×m binary structure matrix that reflects the topological dependency between the state variables of the power grid and the measuring instruments under the AC model; the first constraint contains n sub-constraints.

10. The AC optimal defense resource allocation method based on two-stage hybrid optimization according to claim 9 is characterized in that: In the kth outer loop, the Lagrange multiplier vector λ is given k =[λ 1,k ,λ 2,k ,...,λ n,k ] T and penalty coefficient ρ k The goal of the inner loop is to find the amount of defense resources b that minimizes the augmented Lagrangian function P(b) k+1 : Among them, g j (b) represents the relaxation degree of the j-th sub-constraint in the first constraint, which is intended to protect the j-th state variable from being tampered with.

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