Adaptive resilient control method for microgrid byzantine attack based on auxiliary vector

Abstract

The application provides a kind of microgrid byzantine attack adaptive resilience control method based on auxiliary vector, it is related to microgrid technical field.The method first constructs microgrid system, and initializes the key parameters of microgrid system configuration;Based on the difference between normal distributed power supply and malicious distributed power supply in auxiliary vector convergence behavior, identify malicious distributed power supply, isolate the remaining normal distributed power supply corresponding auxiliary vector after malicious distributed power supply, and make it re-converge under new topology;Establish adaptive control framework dynamically adjusted according to the real-time risk level of microgrid system, set the adaptive step of frequency deviation correction of each distributed power supply, define the local evaluation function of each distributed power supply, calculate the gradient of local evaluation function based on auxiliary vector weighting, introduce momentum term to smooth filter the gradient update direction, continuously use auxiliary vector for weighted decision in the whole control process, and obtain the frequency value of each distributed power supply at next time.

Description

Technical Field

[0001] This invention relates to the field of microgrid technology, and in particular to an adaptive resilient control method for microgrids based on auxiliary vectors and Byzantine attack. Background Technology

[0002] Microgrids, as small-scale power systems, typically operate in islanded or grid-connected modes. Islanded microgrids generally employ three levels of control to achieve stable operation. Primary control ensures supply and demand balance within the islanded microgrid. Secondary control restores voltage and frequency deviations caused by primary control, returning the system to its rated operating point. Tertiary control focuses on the economic optimization of microgrid scheduling, minimizing operating costs or maximizing benefits. At the secondary control level of islanded microgrids, traditional centralized control methods suffer from low reliability and single-point-of-failure issues, making them ill-suited to the development needs of new microgrids with high-penetration renewable energy integration. Therefore, distributed secondary control of microgrid systems has received extensive research attention.

[0003] Distributed secondary control (DG) synchronizes the frequencies of distributed generation sources (DGs) within the system to the standard power frequency by sharing frequency information among them. However, the limited global information available to DG makes it vulnerable to cybersecurity threats. In islanded microgrids, not all DGs can be considered completely trustworthy; several malicious DGs may lurk within the system. These malicious DGs interfere with or even disrupt the consensus protocol in DG by sending forged or misleading information to neighboring DGs; such attacks are collectively known as Byzantine attacks. Byzantine attacks hinder the system from achieving normal frequency consistency and can significantly weaken the resilience of the microgrid, leading to cascading failures such as system instability, posing a serious threat to the safety and stability of power infrastructure. Therefore, researching the resilience control of islanded microgrids under Byzantine attacks is of great significance.

[0004] Current research on resilient control strategies for microgrid networks mainly focuses on spoofed data injection attacks or denial-of-service attacks, as well as hybrid attacks, resulting in relatively limited attack types. Byzantine attacks, however, are a more complex and dangerous type of attack in microgrid systems. Malicious data gates (DGs) can disrupt the consistency between DGs by sending contradictory, deceptive, or randomly erroneous data, making them difficult to detect by attack detection mechanisms. Currently, microgrids still face several challenges and shortcomings in resilient control against Byzantine attacks.

[0005] (1) Existing methods for identifying Byzantine DGs consider interference in the system and incorporate detection thresholds. Unreasonable thresholds can directly lead to missed or false detections of malicious DGs, causing the system's security control algorithm to fail. After Byzantine DGs are detected and isolated, the communication topology of the microgrid changes. Existing technologies typically require all normal DGs to re-run the complete consensus iteration process under the new topology to achieve consensus on state information again. This process requires multiple rounds of information exchange, consuming a large amount of communication resources and time, making it difficult to meet the real-time response requirements of microgrid frequency control.

[0006] (2) Existing anti-attack methods typically use the frequency values ​​provided by neighboring DGs for state updates. However, Byzantine attacks pollute the frequency values ​​sent by neighbors, causing normal nodes to calculate incorrect gradient directions based on erroneous information, thus distorting the gradient calculation direction. When malicious DGs are detected and removed, the number of remaining normal neighbors decreases, leading to an imbalance in gradient weights and causing the update direction to deviate from the global optimum. In addition, the inherent random noise in the communication link can cause severe oscillations in the gradient direction. Existing technologies fail to simultaneously address the systematic bias introduced by attacks and the random perturbations introduced by noise, severely reducing convergence accuracy.

[0007] (3) Existing methods do not consider the degree differences of DGs in the communication topology. DGs with higher degrees have a greater influence on information aggregation; while DGs with lower degrees have limited convergence speed due to insufficient information exchange. In addition, to ensure that the frequency state always operates within the feasible region, existing methods often perform projection operations on the updated state value in each iteration. The projection operation involves solving constraint optimization problems, which has high computational complexity. As the penetration rate of DGs in microgrid systems continues to increase, frequent projection operations will significantly increase the processor burden and occupy the resource quota of other control tasks. Summary of the Invention

[0008] The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing an adaptive elastic control method for microgrids based on auxiliary vectors to realize distributed elastic frequency control for microgrids under Byzantine attack.

[0009] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: On the one hand, this invention provides an adaptive resilient control method for microgrids based on auxiliary vectors in the face of Byzantine attacks, comprising: A microgrid system is constructed, and its key parameters are initialized and configured; the microgrid system has a total of n There are 10 distributed generation (DGs), including normal DGs and Byzantine DGs; each DG maintains a... n Auxiliary vector of dimension Auxiliary vector update mechanism based on microgrid system state design; Based on the difference in auxiliary vector convergence behavior between normal distributed generation (DGs) and malicious DGs, malicious DGs are identified, and after isolating malicious DGs, the auxiliary vectors corresponding to the remaining normal DGs are re-converged under the new topology. An adaptive control framework is established to dynamically adjust based on the real-time risk level of the microgrid system. An adaptive step size for frequency deviation correction of each distributed power source is set, a local evaluation function for each distributed power source is defined, the gradient of the local evaluation function is calculated based on auxiliary vector weighting, and a momentum term is introduced to smooth the gradient update direction, achieving an optimal trade-off between convergence speed and elasticity. Throughout the control process, auxiliary vectors are continuously used for weighted decision-making to obtain the frequency value of each distributed power source at the next moment.

[0010] Furthermore, the auxiliary vector Specifically, it is described as follows: ; in, Indicates the first i Auxiliary vectors of each power source Indicates the first i DG power supply i For the first j A neighbor's power supply DG j Trust weight, T Indicates vector transpose; The key parameters of the microgrid system include the auxiliary vector, frequency deviation, momentum, local risk value, and dynamic trade-off coefficient for each distributed power source; for the first... i One power source, with auxiliary vectors initialized to standard basis vectors: Frequency deviation initialized to For the frequency-feasible set, These are the minimum and maximum values ​​of the frequency deviation, respectively; the momentum is initialized to... Local risk values ​​are initialized to The dynamic tradeoff coefficients are initialized to... ,in, This is the initial value of the dynamic tradeoff coefficient.

[0011] Furthermore, the auxiliary vector update mechanism based on microgrid system state design is as follows: ; in, For the first i DG power supply i The set of neighboring power sources; For the first j A neighbor's power supply DGj For the first i DG power supply i The spatiotemporal consistency modulation factor; For the first i DG power supply i For the first j A neighbor's power supply DG j Communication weight; The spatiotemporal consistency modulation factor is shown in the following formula: ; in, For the first i DG power supply i The frequency deviation measured at time t; It is the first j A neighbor's power supply DG j Frequency deviation transmitted at time t; , , All are sensitivity coefficients greater than 0. The historical offset memory amount at time t is used to accumulate and represent the historical offset memory at time t. j A neighbor's power supply DG j For the first i DG power supply i The degree of persistent abnormality; The historical offset memory update rule is as follows: ; in, For memory factors, , All of them are weight parameters that are greater than 0.

[0012] Furthermore, the specific method for identifying malicious distributed generation (DGs) and, after isolating malicious DGs, making the auxiliary vectors corresponding to the remaining normal DGs converge again under the new topology is as follows: Definition of the first i DG power supply i For the first j A neighbor's power supply DG j The spatial deviation is: ; Define the first i DG power supply i The first-order time drift is: The second-order time drift is: ; Combining the spatial deviation and temporal drift of distributed power sources, construct the first... i DG power supply i For the first j A neighbor's power supply DG jContinuous risk score at time t As shown in the formula below: ; in, For smoothing coefficients; , , , All are weighted parameters that are greater than 0; Malicious distributed generation (DGs) are identified based on continuous risk scores; after identifying Byzantine-based malicious DGs, a set of normal distributed generation DGs is constructed. Among them, continuous risk score Not continuously judged as abnormal; when the j A neighbor's power supply DG j When a power source (DG) is identified as a Byzantine malicious power source, the malicious distributed power source (DG) is isolated. j This will be related to the malicious distributed power supply DG. j The communication weight of directly communicating distributed power sources is redistributed to the remaining normal neighboring distributed power sources; Using the already converged auxiliary vector under the original communication topology As the initial value, we obtain the first... i DG power supply i New auxiliary vector : ; in, To isolate the first j Distributed power supply (DG) j After that, distributed generation (DG) i With Distributed Power Generation (DG) k Communication weights between them.

[0013] Furthermore, the specific method for establishing the adaptive control framework that dynamically adjusts based on the real-time risk level of the microgrid system is as follows: Define the local risk perception function, the first i DG power supply i At any moment t The local risk value is the variance of the frequency values ​​of all received neighboring power source DGs: ; ; in, For the first i DG power supply i At any moment t The local risk value, For the first i DG power supply iThe number of neighboring distributed power sources; For the first i DG power supply i At the frequency of time t, For the first j A neighbor's power supply DG j At the frequency of time t, For the first i DG power supply i The arithmetic mean of all received frequencies at time t; Using exponentially weighted moving variance to achieve the first i DG power supply i At any moment t Smoothing estimate of local risk value: ; ; in, For the first i DG power supply i Exponentially weighted moving average of the receiving frequency; As a smoothing factor, it controls the degree to which historical information is retained; Through a distributed average consensus protocol, each distributed generation (DG) exchanges a smoothed estimate of its local risk value with its neighboring power sources. Without establishing a central node, each distributed generation (DG) approximates the average global risk value: ; in, For the first k The global risk value at the next iteration k This represents the number of iterations. For the first k During the nth iteration j A neighbor's power supply DG j The local risk value at time t; Define Lyapunov functions Measuring the deviation between the real-time frequency and the rated frequency of a microgrid system: ; in, This refers to the rated frequency value of the microgrid system. Define potential function Measuring the overall risk exposure of a microgrid system: ; in, This is a normalization constant, and its value is the typical variance of the microgrid system during normal operation. Construct the following drift-penalty term: ; in, For Lyapunov drift; This is the potential function drift; For the time t All historical information up to this point; For the conditional expectation operator, It is the set of dynamic trade-off coefficients for each distributed power source in a microgrid system; The drift-penalty term is an optimization objective used to minimize the deviation between the real-time frequency and the rated frequency of the distributed generation, while minimizing the overall risk exposure of the microgrid system. A dynamic trade-off is made between these two objectives; Lyapunov function Potential function The drift-penalty term constitutes the adaptive control framework of the entire microgrid system; Based on the adaptive control framework of microgrid systems, dynamic trade-off coefficients are set. Adaptive update based on global risk value: ; in, To adjust the speed coefficient; Risk threshold; and For dynamic trade-off coefficients The upper and lower bounds; when the global risk exceeds the threshold. Exponential growth shifts the control objective towards risk reduction; when overall risk decreases... Exponential decay leads to rapid convergence of the control objective.

[0014] Furthermore, the specific method for setting the adaptive step size for frequency deviation correction of each distributed power source, defining the local evaluation function of each distributed power source, calculating the gradient of the local evaluation function based on auxiliary vector weighting, and introducing a momentum term to smooth the gradient update direction is as follows: (1) Set the adaptive step size for frequency deviation correction of each distributed power source. This step size is a proportional coefficient applied to the frequency deviation correction amount. It determines the step size by which each distributed power source (DGs) adjusts its own frequency each time it receives the neighbor's frequency information: ; in, For the first i DG power supply i The step size for frequency deviation correction at time t. Used as the reference step size; The risk sensitivity coefficient controls the degree of risk decay on the step size; (2) Set an adaptive momentum coefficient that is coupled with the global risk level and the average confidence level of the auxiliary vector; The average confidence level of the auxiliary vector is shown in the following formula: ; in, The average confidence level of the auxiliary vector is used to measure DG. i The average level of trust in all of his / her neighbors; The adaptive momentum coefficient is coupled with the global risk level and the average confidence level of the auxiliary vector. As shown in the formula below: ; in, The reference momentum coefficient; For momentum risk gain; This is a trust level adjustment coefficient; and The upper and lower bounds of the momentum coefficient; the average confidence level of the auxiliary vector. As a moderating factor, the momentum coefficient reflects both the external risk environment and the internal trust status. (3) Define the local evaluation function of each distributed power source, calculate the gradient of the local evaluation function based on the auxiliary vector weighting, and introduce the momentum term to smooth the gradient update direction; Regarding the first i DG power supply i Define local evaluation function : ; The local evaluation function is calculated by weighting the trust weights in the auxiliary vector. gradient: ; in, It is a balance coefficient used to adjust the intensity of convergence towards the rated frequency; Introducing the momentum term Local evaluation function Smoothing filtering is applied to the gradient update direction: ; momentum term The essence is a local evaluation function The accumulation of historical gradient directions involves weighted fusion of the current gradient direction with the gradient directions from all past moments.

[0015] Furthermore, the specific method for continuously using auxiliary vectors for weighted decision-making throughout the control process to obtain the frequency values ​​of each distributed power source at the next moment is as follows: Dynamic consensus computation based on auxiliary vector weighting; constructing a normal distributed generation (DGs) set. Above, using the trust weights in the auxiliary vector, the trust-weighted average of the frequency deviations of all neighboring distributed sources is calculated for the i-th distributed source, thus achieving dynamic weighted consensus calculation: ; ; in, It is the trust-weighted average of the frequency deviations of all neighboring distributed power sources; For the first i DG power supply i For the first k A neighbor's power supply DG j Trust weight, For normalized trust weights; Combining weighted consistency with momentum correction direction generates the first... i The ideal frequency value of a distributed power source at the next moment: ; in, For the first i Distributed power supply (DG) i The ideal frequency value at the next moment; By integrating the two objectives of consensus negotiation and gradient optimization, the ideal frequency value for the next time step is generated; consensus negotiation is used to synchronize neighbor frequencies, and gradient optimization is used to drive the frequency to converge toward the nominal value. If the frequency is ideal In the frequency feasible set In addition, the ideal frequency value is projected onto the feasible frequency set to obtain the first... i The frequency value of the distributed power source at the next moment: ; If the ideal frequency value is within the feasible set, then directly use the ideal frequency value. Assign the frequency value to the i-th distributed source at the next time step: ; in, This is a projection operator used to project the ideal frequency value onto the feasible frequency set. superior.

[0016] On the other hand, this application proposes an electronic device comprising: one or more processors, and a memory for storing instructions that, when executed by the one or more processors, cause the one or more processors to perform the aforementioned microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors.

[0017] Thirdly, this application proposes a computer-readable storage medium storing executable instructions that, when executed, cause a processor to perform the aforementioned microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors.

[0018] Fourthly, this application proposes a computer program product, including a computer program or instructions that, when executed by a processor, implement the aforementioned microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors.

[0019] The beneficial effects of the above technical solutions are as follows: The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors provided by this invention (1) constructs a continuous risk score by integrating auxiliary vector space deviation, first-order time drift, and second-order time drift, which effectively avoids misjudgment and omission of Byzantine DGs caused by system disturbances. The final dynamically constructed set of normal distributed power DGs can be directly used for topology reconstruction and elastic frequency control. The one-step iterative fast reconvergence technology after topology change avoids the disadvantage of traditional methods requiring multiple rounds of consensus iteration, and significantly reduces the time delay and computational burden caused by attack isolation. (2) The combination of dynamic consistency based on auxiliary vector weighting and momentum term ensures that frequency control converges to the global optimal solution. All steps rely only on local information and neighbor interaction, without global coordination, and have comprehensive advantages such as no single point of failure, strong scalability, and suitability for embedded deployment. (3) A dynamic optimal balance between convergence speed and elasticity is achieved. The step size is coupled with the local risk and the number of neighbors of the DGs, effectively balancing the influence of nodes with different degrees, avoiding the overshoot risk of high-degree nodes, and improving the convergence speed of low-degree nodes. The momentum coefficient is coupled with the global risk and the self-trust value of the auxiliary vector. In addition, the on-demand projection mechanism significantly reduces the computational burden of the system. Attached Figure Description

[0020] Figure 1 The flowchart illustrates the adaptive resilient control method for microgrids based on auxiliary vectors in an embodiment of the present invention. Detailed Implementation

[0021] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0022] Example 1: In this embodiment, the microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors is as follows: Figure 1 As shown, it includes the following steps: Step 1: Construct the microgrid system and initialize the key parameters of the microgrid system.

[0023] Step 1.1: Define the common features of a microgrid system. n There are 10 distributed generation (DGs), including normal DGs and Byzantine DGs. Each DG maintains one... n Auxiliary vector of dimension Specifically, it is described as follows: ; in, Indicates the first i Auxiliary vectors of each power source Indicates the first i DG power supply i For the first j A neighbor's power supply DG j Trust weight, T This represents the transpose of a vector.

[0024] Step 1.2: Initialize the key parameters of the microgrid system; the key parameters of the microgrid system include the auxiliary vector, frequency deviation, momentum, local risk value and dynamic trade-off coefficient of each distributed power source; In this embodiment, for the first in the microgrid system i DG power supply i The auxiliary vectors are initialized to standard basis vectors: Frequency deviation initialized to , These are the minimum and maximum values ​​of the frequency deviation, respectively; the momentum is initialized to... Local risk values ​​are initialized to The dynamic tradeoff coefficients are initialized to... ,in, This is the initial value of the dynamic tradeoff coefficient.

[0025] The dynamic tradeoff coefficient determines which objective takes precedence between rapidly synchronizing the frequency to 50Hz and preventing runaway risks. A larger value indicates a greater emphasis on resilience in the microgrid system, while a smaller value indicates a greater emphasis on convergence speed. When the microgrid system is first started up, sufficient risk information has not yet been collected, so a neutral initial value is set. It is in a balanced, standby state. As the microgrid system operates, this value is dynamically adjusted using the exponential update formula in step 4.3. It automatically increases when the attack is severe, and automatically decreases when the attack risk is weak, as the microgrid system prioritizes convergence speed.

[0026] Step 2: Design an auxiliary vector update mechanism based on the state of the microgrid system.

[0027] The update rule for the auxiliary vector is designed as follows: ; in, For the first i DG power supply i The set of neighboring power sources; For the first j A neighbor's power supply DG j For the first i DG power supply i The spatiotemporal consistency modulation factor; For the first i DG power supply i For the first j A neighbor's power supply DG j Communication weights.

[0028] To suppress malicious frequency spoofing, auxiliary vector spoofing, and short-term noise disturbances, the spatiotemporal consistency modulation factor is shown in the following formula: ; in, For the first i DG power supply i The frequency deviation measured at time t; It is the first j A neighbor's power supply DG j Frequency deviation transmitted at time t; , , All are sensitivity coefficients greater than 0. The historical offset memory amount at time t is used to accumulate and represent the historical offset memory at time t. j A neighbor's power supply DG j For the first i DG power supply i The degree of continued abnormality.

[0029] Historical offset memory The update rules are as follows: ; in, For memory factors, , All of them are weight parameters that are greater than 0.

[0030] When the j A neighbor's power supply DG j Frequency information and auxiliary vectors simultaneously with the first i DG power supply i When maintaining consistency, When the value approaches 1, neighbor information is received normally, meaning that among normal distributed generation (DGs), the auxiliary vectors converge uniformly to the left eigenvector of the adjacency matrix of the communication topology graph composed of all DGs.j A neighbor's power supply DG j When continuously sending abnormal frequency information or forging auxiliary vectors, This will significantly reduce the impact of malicious information on local auxiliary vector updates, meaning that when a power source DG suffers a Byzantine attack, the evolution trajectory of its auxiliary vectors will deviate from the common convergence trajectory of normal auxiliary vectors.

[0031] Step 3: Based on the difference in auxiliary vector convergence behavior between normal distributed generation (DGs) and malicious DGs, accurately identify malicious DGs, and after isolating malicious DGs, quickly re-converge the auxiliary vectors corresponding to the remaining normal DGs under the new topology through single-step iteration.

[0032] Step 3.1: Under the condition of no attack and the communication topology remaining connected, the auxiliary vectors maintained by normal distributed generation (DGs) asymptotically converge to the left eigenvector of the adjacency matrix of the communication network under local interactive iteration. However, malicious DGs will cause the relevant auxiliary vectors to deviate from the normal evolution trajectory. Therefore, the anomaly detection of distributed generation can be achieved by the joint deviation of the auxiliary vectors in the spatial and temporal domains.

[0033] Definition of the first i DG power supply i For the first j A neighbor's power supply DG j The spatial deviation is: Spatial deviation is compared to the difference between the auxiliary vectors corresponding to different distributed sources at the current moment. The definition of the first... i DG power supply i The first-order time drift is: The second-order time drift is: The time drift quantity characterizes the instantaneous evolution intensity and inflection point features of the auxiliary vector. First-order time drift characterizes the instantaneous evolution intensity, while second-order time drift characterizes the inflection point features. The auxiliary vector evolution of normal distributed generation (DGs) is usually smooth and tends to be stable, while the auxiliary vector of malicious DGs exhibits significant drift or irregular fluctuations under continuous attacks.

[0034] Step 3.2: Combining the spatial deviation and temporal drift of the distributed power source, construct the first... i DG power supply i For the first j A neighbor's power supply DG j Continuous risk score at time t As shown in the formula below: ; in, For smoothing coefficients; , , , All are weighted parameters that are greater than 0.

[0035] Malicious distributed generation (DGs) are identified based on continuous risk scores rather than a single instantaneous difference. If the... j A neighbor's power supply DG j Continuous risk scores corresponding to multiple consecutive moments consistently higher than the first i DG power supply i The average risk score of the remaining neighboring power sources within the neighborhood will then be used to determine the risk score of the first power source. j A neighbor's power supply DG j Mark as a suspicious neighbor; if the first j A neighbor's power supply DG j If a property is consistently marked as suspicious from the perspective of more than half of its normal neighbors, then the first property is determined to be suspicious. j A neighbor's power supply DG j For Byzantine malicious power supply DGs.

[0036] Step 3.3: Identify Byzantine-based malicious power supply DGs, and then construct a set of normal distributed power supply DGs. Among them, continuous risk score It has not been consistently identified as abnormal.

[0037] Step 3.4: Isolating Byzantine malicious power sources (DGs) will cause changes in the communication topology between distributed generation (DGs). Continuing to use the dynamic update mechanism for auxiliary vectors in Step 2 will significantly increase the convergence time of the microgrid system. Therefore, when the... j A neighbor's power supply DG j When a power source (DG) is identified as a Byzantine malicious power source, the malicious distributed power source (DG) is isolated. j This will be related to the malicious distributed power supply DG. j The communication weights of directly communicating distributed power sources are redistributed to the remaining normal neighboring distributed power sources: ; in, To isolate the first j Distributed power supply (DG) j After that, distributed generation (DG) i With Distributed Power Generation (DG) j Communication weights between them. To isolate the first j Distributed power supply (DG) j Previously, distributed generation (DG) i With Distributed Power Generation (DG) j Communication weights between them. To isolate the first j Distributed power supply (DG) j Previously, distributed generation (DG) i With Distributed Power Generation (DG) k Communication weights between them. Distributed generation (DG) k Also for distributed generation (DG) i The neighbors. To isolate the first j Distributed power supply (DG) j After that, distributed generation (DG) i With Distributed Power Generation (DG) k Communication weights between them.

[0038] Only distributed generation (DGs) affected by topology changes have their weights updated, and the weight adjustment formula is purely locally calculated without global coordination. The weights originally assigned to the removed DGs identified as malicious Byzantine sources are proportionally redistributed to other neighboring DGs. Then, auxiliary vectors that have converged under the original communication topology are used. As an initial value, the i-th value can be obtained in just one iteration. i DG power supply i New auxiliary vector : ; Each distributed generation (DG) relies only on the changes of its neighbors and the converged auxiliary vectors under the original communication topology as initial values. It does not require knowledge of the entire network topology and only needs one iteration to complete the convergence of auxiliary vectors under the new topology. This significantly reduces communication and computational overhead, speeds up the convergence, and has good distributed characteristics.

[0039] Step 4: Establish an adaptive control framework that dynamically adjusts based on the real-time risk level of the microgrid system, set the adaptive step size for frequency deviation correction of each distributed power source, define the local evaluation function of each distributed power source, calculate the gradient of the local evaluation function based on auxiliary vector weighting, and introduce a momentum term to smooth the gradient update direction, achieving an optimal trade-off between convergence speed and elasticity, and continuously using auxiliary vectors for weighted decision-making throughout the control process to obtain the frequency value of each distributed power source at the next moment.

[0040] Step 4.1: Define the local risk perception function, the first... i DG power supply i At any moment t The local risk value is the variance of the frequency values ​​(including its own) of all received neighboring power source DGs: ; ; in, For the first i DG power supply i At any moment t The local risk value, For the first i DG power supply i The number of neighboring distributed power sources; For the first i DG power supply i At the frequency of time t, For the first j A neighbor's power supply DG j At the frequency of time t, For the first i DG power supply i The arithmetic mean of all received frequencies at time t; To reduce the impact of instantaneous fluctuations, an exponentially weighted moving variance is used to achieve the first... i DG power supply i At any moment t Smoothing estimate of local risk value: ; ; in, For the first i DG power supply i Exponentially weighted moving average of the receiving frequency; As a smoothing factor, it controls the degree to which historical information is retained.

[0041] Step 4.2: Local risk values ​​may be affected by local anomalies; therefore, global risk is used to measure the overall security status of the microgrid system. Through a distributed average consensus protocol, each distributed generation (DGs) exchanges smoothly estimated local risk values ​​with its neighboring power sources. Without establishing a central node, each distributed generation (DG) approximates the average global risk value: ; in, For the first k The global risk value at the next iteration k This represents the number of iterations. For the first k During the nth iteration j A neighbor's power supply DG j The local risk value at time t.

[0042] Step 4.3: Define the Lyapunov function Measuring the deviation between the real-time frequency and the rated frequency of a microgrid system: ; in, This refers to the rated frequency value of the microgrid system. Define potential function Measuring the overall risk exposure of a microgrid system: ; in, This is a normalization constant, and its value is taken as the typical variance of the microgrid system during normal operation.

[0043] Construction drift - penalty: ; in, For Lyapunov drift; This is the potential function drift; For the time t All historical information up to this point; For conditional expectation operators; It is the set of dynamic trade-off coefficients for each distributed power source in a microgrid system.

[0044] The drift-penalty term is an optimization objective used to quickly minimize the deviation between the real-time frequency and the rated frequency of distributed generation, while minimizing the overall risk exposure of the microgrid system. A dynamic trade-off is made between these two objectives; Lyapunov function Potential function The drift-penalty term constitutes the adaptive control framework of the entire microgrid system; Therefore, based on the adaptive control framework of this microgrid system, the first... i DG power supply i Dynamic trade-off coefficient Adaptive update based on global risk value: ; in, To adjust the speed coefficient; Risk threshold; and For dynamic trade-off coefficients The upper and lower bounds. When the global risk exceeds the threshold. Exponential growth shifts the control objective towards risk reduction; when overall risk decreases... Exponential decay leads to rapid convergence of the control objective.

[0045] calculate The first formula is responsible for dynamically adjusting based on the global risk value. The value of the second formula is responsible for the safety limit, ensuring... Not out of control. Specifically, in the first formula, when When the risk threshold is significantly exceeded, It will grow at an exponential rate, rapidly shifting the control objective from pursuing speed to mitigating risk. When When significantly below the risk threshold, It will decrease, pursuing speed. However, in extreme cases, it may output excessively large positive values ​​or values ​​close to zero. The first formula may cause the controller to completely freeze frequency adjustment due to excessive conservatism, while the second formula will cause the system to completely lose its risk mitigation awareness, thus leading to oscillation and instability. Therefore, the second formula is calculated based on the first formula. Values ​​are subject to saturation constraints, and upper and lower bound constraints are used to ensure... It always operates within a reasonable range that effectively suppresses risks without losing its basic convergence capability.

[0046] Step 4.4: Set the adaptive step size for frequency deviation correction of each distributed generation (DG). This step size is a proportional coefficient applied to the frequency deviation correction amount, which determines the step size by which each DG adjusts its own frequency after receiving frequency information from its neighbor. ; in, For the first i DG power supply i The step size for frequency deviation correction at time t. Used as the reference step size; The risk sensitivity coefficient controls the degree of risk decay on the step size; the higher the risk, the more exponentially the step size decays, avoiding the amplification of the attack impact due to large step sizes; the more neighboring distributed power sources there are, the smaller the step size, to prevent them from having too much influence and causing excessive impact on the communication network composed of distributed power sources (DGs) in the microgrid system.

[0047] Step 4.5: Set adaptive momentum coefficients that are coupled with the global risk level and the average confidence level of the auxiliary vector; The average confidence level of the auxiliary vector is shown in the following formula: ; in, The average confidence level of the auxiliary vector is used to measure DG. i The average level of trust in all of his / her neighbors.

[0048] Then, the adaptive momentum coefficient coupled with the global risk level and the average confidence level of the auxiliary vector is calculated. : ; in, The reference momentum coefficient; For momentum risk gain; This is a trust level adjustment coefficient; and This represents the upper and lower bounds of the momentum coefficient. As global risk increases, the momentum coefficient increases, strengthening smoothing filtering; when the average confidence level is high, the momentum coefficient decreases, accelerating response speed. The average confidence level of the auxiliary vector. As a moderating factor, the momentum coefficient reflects both the external risk environment and the internal trust status.

[0049] calculate The first formula dynamically calculates the theoretically optimal value of the momentum coefficient based on global risk and average confidence level. However, its calculation results may exceed a reasonable range in extreme scenarios. When the global risk is extremely high, the momentum coefficient may be too large, causing the system to rely almost entirely on historical gradients and lose its ability to respond to real-time changes. When the risk is extremely low and the confidence level is extremely high, the momentum coefficient may be too small or even negative, causing the system to be overly sensitive to noise and trigger frequency oscillations. Therefore, the upper and lower bound constraints of the second formula must be used to ensure that the momentum coefficient used in the end is always within a reasonable and numerically stable range.

[0050] Step 4.6: Define the local evaluation function for each distributed power source, calculate the gradient of the local evaluation function based on the auxiliary vector weighting, and introduce a momentum term to smooth the gradient update direction.

[0051] Regarding the first i DG power supply i Define local evaluation function : ; The local evaluation function is calculated by weighting the trust weights in the auxiliary vector. gradient: ; in, Local evaluation function gradient, It is a balance coefficient used to adjust the intensity of convergence towards the rated frequency.

[0052] Introducing the momentum term Local evaluation function Smoothing filtering is applied to the gradient update direction: ; momentum term The essence is a local evaluation function The accumulation of historical gradient directions involves a weighted fusion of the current gradient direction with the gradient directions from all past moments. Among these, the momentum coefficient... It controls the proportion of historical information retained. When communication noise causes the current gradient direction to oscillate violently, the momentum term uses the inertia of historical directions to act as a low-pass filter, preventing the frequency adjustment from fluctuating drastically due to a single abnormal gradient. When a Byzantine attack causes malicious distributed generation in the microgrid system, making the current gradient point in the wrong direction, the normal historical gradient direction accumulated before the Byzantine attack will have a considerable weight in the update, weakening the pulling force of the erroneous gradient. Under normal circumstances, when the gradient direction remains consistent, the momentum term will form an accumulating acceleration effect, driving the frequency to converge rapidly to the rated value.

[0053] Therefore, the momentum term It can be viewed as a direction filter with memory, which allows the frequency update direction to refer to both the current calculation results and the accumulation of historical experience, thereby achieving a balance between noise suppression, Byzantine attack resilience defense and convergence acceleration.

[0054] Trust weights in auxiliary vectors Directly used for local evaluation functions The gradient calculation, through the introduction of auxiliary vectors, ensures that neighbors with high trust levels contribute more to the gradient, while the contributions of neighbors with low trust levels are automatically suppressed. This compensates for the gradient weight bias introduced by isolating malicious Byzantine sources (DGs) and restores the unbiasedness of global gradient descent. The momentum term uses historical gradient directions to smooth the current update, suppressing control oscillations caused by communication noise.

[0055] Step 4.7: Dynamic consistency calculation based on auxiliary vector weighting.

[0056] In step 3.1, a normal distributed generation (DGs) set is constructed. Above, using the trust weights in the auxiliary vector for the first... i Each distributed power source calculates the trust-weighted average of the frequency deviations of all its neighboring distributed power sources to achieve dynamic weighted consensus calculation. ; ; in, It is the trust-weighted average of the frequency deviations of all neighboring distributed power sources, representing the first... i Distributed power supply (DG) i The adjustment direction is made to align it with that of trusted neighboring distributed power sources. For example, if the frequency of a neighboring distributed power source is higher than its own, then... A positive value indicates that the DGi frequency should be increased. For the first i DG power supply i For the first k A neighbor's power supply DG j Trust weight, ; To normalize the trust weights, neighbors with higher trust levels have greater say in consensus negotiation, while the influence of neighbors with lower trust levels is automatically reduced.

[0057] Step 4.8: Combine the weighted consistency with the momentum term to generate the first... i The ideal frequency value of a distributed power source at the next moment: ; in, For the first i Distributed power supply (DG) i The ideal frequency value at the next moment; By integrating the two objectives of consensus negotiation and gradient optimization, an ideal frequency value for the next time step is generated. Consensus negotiation is used to synchronize neighbor frequencies, while gradient optimization is used to drive the frequency to converge toward the predetermined value.

[0058] If the frequency is ideal In the frequency feasible set In addition, the ideal frequency value is projected onto the feasible frequency set to obtain the frequency value of the i-th distributed source at the next time step: ; If the ideal frequency value is within the feasible set, then directly use the ideal frequency value. Assign the frequency value to the i-th distributed source at the next time step: ; in, This is a projection operator used to project the ideal frequency value onto the feasible frequency set. The present invention performs projection only when the ideal frequency value exceeds the feasible region, avoiding the high computational burden of projection at every step, while ensuring that the frequency state of all distributed generation (DGs) ultimately remains within the safe operating range.

[0059] Example 2: This embodiment proposes an electronic device, including: one or more processors, and a memory for storing instructions, which, when executed by the one or more processors, cause the one or more processors to execute the aforementioned microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors.

[0060] The electronic device may be a mobile phone, computer, or tablet computer, etc., and includes a memory and a processor. The memory stores a computer program, which, when executed by the processor, implements the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors as described in the embodiments. It is understood that the electronic device may also include input / output (I / O) interfaces and communication components.

[0061] The processor is used to execute all or part of the steps in the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors as described in the above embodiments. The memory is used to store various types of data, which may include, for example, instructions for any application or method in the electronic device, as well as application-related data.

[0062] The processor can be implemented as an Application Specific Integrated Circuit (ASIC), Digital Signal Processor (DSP), Programmable Logic Device (PLD), Field Programmable Gate Array (FPGA), controller, microcontroller, microprocessor, or other electronic components, and is used to execute the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors described in the above embodiments.

[0063] Example 3: This embodiment proposes a computer-readable storage medium that stores executable instructions. When these instructions are executed, if they are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium.

[0064] The computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors described in the various embodiments of this application.

[0065] The aforementioned storage media include: flash memory, hard disks, multimedia cards, card-type memory (e.g., SD (Secure Digital Memory Card) or DX (Memory Data Register, MDR) memory), random access memory (RAM), static random-access memory (SRAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), programmable read-only memory (PROM), magnetic storage, disks, optical discs, servers, APP (Application) application stores, and other media capable of storing program verification codes. These media store computer programs, which, when executed by a processor, can implement the various steps of the aforementioned microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors.

[0066] Example 4: This embodiment proposes a computer program product, including a computer program or instructions, which, when executed by a processor, implements the aforementioned microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors.

[0067] Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or part of the technical solution, can be embodied in the form of a computer program product.

[0068] The various embodiments in this application are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0069] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the present invention.

Claims

1. A microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors, characterized in that, include: Construct a microgrid system and initialize the key parameters of the microgrid system; The microgrid system has a total of n There are 10 distributed generation (DGs), including normal DGs and Byzantine DGs; each DG maintains a... n Auxiliary vector of dimension Auxiliary vector update mechanism based on microgrid system state design; Based on the difference in auxiliary vector convergence behavior between normal distributed generation (DGs) and malicious DGs, malicious DGs are identified, and after isolating malicious DGs, the auxiliary vectors corresponding to the remaining normal DGs are re-converged under the new topology. An adaptive control framework is established to dynamically adjust based on the real-time risk level of the microgrid system. An adaptive step size for frequency deviation correction of each distributed power source is set, a local evaluation function for each distributed power source is defined, the gradient of the local evaluation function is calculated based on auxiliary vector weighting, and a momentum term is introduced to smooth the gradient update direction. The auxiliary vector is continuously used for weighted decision-making throughout the control process to obtain the frequency value of each distributed power source at the next moment.

2. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 1, characterized in that, The auxiliary vector Specifically, it is described as follows: ； in, Indicates the first i Auxiliary vectors of each power source Indicates the first i DG power supply i For the first j A neighbor's power supply DG j Trust weight, T Indicates vector transpose; The key parameters of the microgrid system include the auxiliary vector, frequency deviation, momentum, local risk value, and dynamic trade-off coefficient for each distributed power source; for the first... i One power source, with auxiliary vectors initialized to standard basis vectors: Frequency deviation initialized to For the frequency-feasible set, These are the minimum and maximum values ​​of the frequency deviation, respectively; the momentum is initialized to... Local risk values ​​are initialized to The dynamic tradeoff coefficients are initialized to... ,in, This is the initial value of the dynamic tradeoff coefficient.

3. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 2, characterized in that, The auxiliary vector update mechanism based on microgrid system state design is as follows: ； in, For the first i DG power supply i The set of neighboring power sources; For the first j A neighbor's power supply DG j For the first i DG power supply i The spatiotemporal consistency modulation factor; For the first i DG power supply i For the first j A neighbor's power supply DG j Communication weight; The spatiotemporal consistency modulation factor is shown in the following formula: ； in, For the first i DG power supply i The frequency deviation measured at time t; It is the first j A neighbor's power supply DG j Frequency deviation transmitted at time t; , , All are sensitivity coefficients greater than 0. The historical offset memory amount at time t is used to accumulate and represent the historical offset memory at time t. j A neighbor's power supply DG j For the first i DG power supply i The degree of persistent abnormality; The historical offset memory update rule is as follows: ； in, For memory factors, , All of them are weight parameters that are greater than 0.

4. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 3, characterized in that, The specific method for identifying malicious distributed generation (DGs) and, after isolating malicious DGs, making the auxiliary vectors corresponding to the remaining normal DGs converge again under the new topology is as follows: Definition of the first i DG power supply i For the first j A neighbor's power supply DG j The spatial deviation is: ; Define the first i DG power supply i The first-order time drift is: The second-order time drift is: ; Combining the spatial deviation and temporal drift of distributed power sources, construct the first... i DG power supply i For the first j A neighbor's power supply DG j Continuous risk score at time t As shown in the formula below: ； in, For smoothing coefficients; , , , All are weighted parameters that are greater than 0; Malicious distributed generation (DGs) are identified based on continuous risk scores; after identifying Byzantine-based malicious DGs, a set of normal distributed generation DGs is constructed. Among them, continuous risk score Not continuously judged as abnormal; when the j A neighbor's power supply DG j When a power source (DG) is identified as a Byzantine malicious power source, the malicious distributed power source (DG) is isolated. j This will be related to the malicious distributed power supply DG. j The communication weights of directly communicating distributed power sources are redistributed to the remaining normal neighboring distributed power sources; the auxiliary vectors that have converged under the original communication topology are utilized. As the initial value, we obtain the first... i DG power supply i New auxiliary vector : ； in, To isolate the first j Distributed power supply (DG) j After that, distributed generation (DG) i With Distributed Power Generation (DG) k Communication weights between them.

5. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 4, characterized in that, The specific method for establishing an adaptive control framework that dynamically adjusts based on the real-time risk level of the microgrid system is as follows: Define the local risk perception function, the first i DG power supply i At any moment t The local risk value is the variance of the frequency values ​​of all received neighboring power source DGs: ； ； in, For the first i DG power supply i At any moment t The local risk value, For the first i DG power supply i The number of neighboring distributed power sources; For the first i DG power supply i At the frequency of time t, For the first j A neighbor's power supply DG j At the frequency of time t, For the first i DG power supply i The arithmetic mean of all received frequencies at time t; Using exponentially weighted moving variance to achieve the first i DG power supply i At any moment t Smoothing estimate of local risk value: ； ； in, For the first i DG power supply i Exponentially weighted moving average of the receiving frequency; As a smoothing factor, it controls the degree to which historical information is retained; Through a distributed average consensus protocol, each distributed generation (DG) exchanges a smoothed estimate of its local risk value with its neighboring power sources. Without establishing a central node, each distributed generation (DG) approximates the average global risk value: ； in, For the first k The global risk value at the next iteration k This represents the number of iterations. For the first k During the nth iteration j A neighbor's power supply DG j The local risk value at time t; Define Lyapunov functions Measuring the deviation between the real-time frequency and the rated frequency of a microgrid system: ； in, This refers to the rated frequency value of the microgrid system. Define potential function Measuring the overall risk exposure of a microgrid system: ； in, This is a normalization constant, and its value is the typical variance of the microgrid system during normal operation. Construct the following drift-penalty term: ； in, For Lyapunov drift; This is the potential function drift; For the time t All historical information up to this point; For conditional expectation operators; It is the set of dynamic trade-off coefficients for each distributed power source in a microgrid system; The drift-penalty term is an optimization objective used to minimize the deviation between the real-time frequency and the rated frequency of the distributed generation, while minimizing the overall risk exposure of the microgrid system. A dynamic trade-off is made between these two objectives; Lyapunov function Potential function The drift-penalty term constitutes the adaptive control framework of the entire microgrid system; Based on the adaptive control framework of microgrid systems, dynamic trade-off coefficients are set. Adaptive update based on global risk value: ； in, To adjust the speed coefficient; Risk threshold; and For dynamic trade-off coefficients The upper and lower bounds; when the global risk exceeds the threshold. Exponential growth shifts the control objective towards risk reduction; when overall risk decreases... Exponential decay leads to rapid convergence of the control objective.

6. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 5, characterized in that, The specific method for setting the adaptive step size for frequency deviation correction of each distributed power source, defining the local evaluation function of each distributed power source, calculating the gradient of the local evaluation function based on auxiliary vector weighting, and introducing a momentum term to smooth the gradient update direction is as follows: (1) Set the adaptive step size for frequency deviation correction of each distributed power source. This step size is a proportional coefficient applied to the frequency deviation correction amount. It determines the step size by which each distributed power source (DGs) adjusts its own frequency each time it receives the neighbor's frequency information: ； in, For the first i DG power supply i The step size for frequency deviation correction at time t. Used as the reference step size; The risk sensitivity coefficient controls the degree of risk decay on the step size; (2) Set an adaptive momentum coefficient that is coupled with the global risk level and the average confidence level of the auxiliary vector; The average confidence level of the auxiliary vector is shown in the following formula: ； in, The average confidence level of the auxiliary vector is used to measure DG. i The average level of trust in all of his / her neighbors; The adaptive momentum coefficient is coupled with the global risk level and the average confidence level of the auxiliary vector. As shown in the formula below: ； in, The reference momentum coefficient; For momentum risk gain; This is a trust level adjustment coefficient; and The upper and lower bounds of the momentum coefficient; the average confidence level of the auxiliary vector. As a moderating factor, the momentum coefficient reflects both the external risk environment and the internal trust status. (3) Define the local evaluation function of each distributed power source, calculate the gradient of the local evaluation function based on the auxiliary vector weighting, and introduce the momentum term to smooth the gradient update direction; Regarding the first i DG power supply i Define local evaluation function : ； The local evaluation function is calculated by weighting the trust weights in the auxiliary vector. gradient: ； in, It is a balance coefficient used to adjust the intensity of convergence towards the rated frequency; Introducing the momentum term Local evaluation function Smoothing filtering is applied to the gradient update direction: ； momentum term The essence is a local evaluation function The accumulation of historical gradient directions involves weighted fusion of the current gradient direction with the gradient directions from all past moments.

7. The microgrid Byzantine attack adaptive elastic control method based on auxiliary vectors according to claim 6, characterized in that, The specific method for continuously using auxiliary vectors for weighted decision-making throughout the entire control process to obtain the frequency values ​​of each distributed power source at the next moment is as follows: Dynamic consensus computation based on auxiliary vector weighting; constructing a normal distributed generation (DGs) set. Above, using the trust weights in the auxiliary vector, the trust-weighted average of the frequency deviations of all neighboring distributed sources is calculated for the i-th distributed source, thus achieving dynamic weighted consensus calculation: ； ； in, It is the trust-weighted average of the frequency deviations of all neighboring distributed power sources; For the first i DG power supply i For the first k A neighbor's power supply DG j Trust weight, For normalized trust weights; Combining weighted consistency with momentum correction direction generates the first... i The ideal frequency value of a distributed power source at the next moment: ； in, For the first i Distributed power supply (DG) i The ideal frequency value at the next moment; By integrating the two objectives of consensus negotiation and gradient optimization, the ideal frequency value for the next time step is generated; consensus negotiation is used to synchronize neighbor frequencies, and gradient optimization is used to drive the frequency to converge toward the nominal value. If the frequency is ideal In the frequency feasible set In addition, the ideal frequency value is projected onto the feasible frequency set to obtain the first... i The frequency value of the distributed power source at the next moment: ； If the ideal frequency value is within the feasible set, then directly use the ideal frequency value. Assign the frequency value to the i-th distributed source at the next time step: ； in, This is a projection operator used to project the ideal frequency value onto the feasible frequency set. superior.

8. An electronic device for executing the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors for any one of claims 1-7, characterized in that, include: One or more processors, and a memory for storing instructions that, when executed by the one or more processors, cause the one or more processors to perform the aforementioned auxiliary vector-based microgrid Byzantine attack adaptive resilient control method.

9. A computer-readable storage medium storing executable instructions for performing the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors as described in any one of claims 1-7, characterized in that, When the instruction is executed, it causes the processor to perform the adaptive resilient control method for microgrids based on auxiliary vectors and Byzantine attacks.

10. A computer program product for executing the microgrid Byzantine attack adaptive resilient control method based on auxiliary vectors as described in any one of claims 1-7, characterized in that, This includes a computer program or instructions that, when executed by a processor, implement the aforementioned adaptive resilient control method for microgrids based on auxiliary vectors and Byzantine attacks.

Images