2.1. Rough Set Principle

The rough set theory is a mathematical approach to dealing with vagueness and uncertainty by approximating sets with a pair of sets called the lower and upper approximations. In the definition and research of rough sets, feature reduction algorithms based on the attribution importance are all heuristic algorithms. That is, using heuristic information as the criterion to measure importance, adding non-nuclear conditional attributes with the greatest attribute importance to the kernel attribute set, or deleting conditional attributes with the least attribute importance one by one can meet the final condition. In the rough set theory, knowledge granularity represents the degree of fine division of knowledge (partition granularity), which can be used to measure the importance of equivalent categories related to knowledge, while attribute importance is to evaluate the importance of attribute kernel in a knowledge system and screen the most important attribute features. It can play a key role in attribute reduction of rough sets by classifying knowledge horizontally and analyzing it objectively by using the granularity function and then determining the weight vertically by combining with the attribute importance. Figure 1 shows a schematic diagram of rough set principle.

Please see the appendix for the formulas related to attribute importance-knowledge granularity, and here are the important formulas for the algorithms.

2.2. Cloud Model Theory

Cloud model theory solves the problem of uncertainty in transformations between qualitative and quantitative attributes by describing and calculating expectations (E_x), entropy (E_n), and hyperentropy (H_e). Comprehensive evaluation cloud: Due to the different relevance or independence between the sublayers corresponding to all levels of indicators in the power grid risk evaluation system, the algorithms used are not the same. Therefore, we use the integrated cloud algorithm and the standard operational algorithm for affiliation analysis to determine the risk level of the grid. The specific formulas are as follows.

3.1. Data Preprocessing

3.2. Rough Set Reduction Based on Grid Data

In the process of LFO warning, we use the approximation ability of rough sets to analyze and process the grid operation data and approximate the operation indexes according to the data to meet the current grid state.

3.3. Comprehensive Evaluation of Early Warning Results Based on Cloud Model

3.4. Combination Weighting

3.5. Comprehensive Grid Early Warning Model Construction

3.5.2. Framework of the Proposed Early Warning Model

The grid LFO early warning model is mainly composed of three parts: rough set–based historical data analysis module, real-time data processing module, and cloud model comprehensive early warning evaluation module. The specific warning steps are as follows.

•  

  Step 1: Preprocessing data. We discretize the historical grid operation data according to the method in Section 3.1 to form a new set of objects U and build a new initial decision table S = {U, C, D, V, F}.

•  

  Step 2: The selection of condition attributes and decision attributes is carried out. This simulation is based on the general power system, the decision attribute is whether there is a fault in power grid, and the condition attribute set is the basic data indicator constructed in the above subsection.

•  

  Step 3: Build the power grid early warning information decision table, then use the reduction algorithm based on the attribute significance and knowledge granularity proposed in Section 3.2 to obtain the core attributes in the decision table, and calculate the weight w_i for each conditional attribute based on ρ.

•  

  Step 4: After obtaining the attribute set of the simplified decision table, the comprehensive evaluation grade is constructed based on the core attributes therein according to the method in Section 3.5.1, and the cloud model feature parameters corresponding to the evaluation grade standard cloud are calculated according to the method in Section 3.3 by combining the data in the attribute set of the simplified conditions.

•  

  Step 5: Take out the PMU real-time data, calculate the comprehensive cloud evaluation results and the degree of affiliation, and finally derive the warning evaluation level according to the principle of maximum affiliation.

The framework of the specific early warning model algorithm is shown in Figure 3.

Overall, in terms of dimensionality simplification ability, compared with linear dimension reduction methods such as PCA, rough set eliminates redundant indicators while retaining key attributes with strong causal associations, ensuring that the indicator set still has clear power system physical meaning after dimension reduction. Meanwhile, the complexity of the rough set method is O(n), which is lower than the exponentially increasing complexity of local linear embedding (LLE) and other methods. Secondly, the cloud model updates the digital feature parameters in real time through the sliding window mechanism, so that the warning threshold can be dynamically adjusted with the load fluctuation of the grid. In terms of knowledge interpretability, the “conditional attribute-decision rule” generated by the rough set and the cloud model supports causal logic tracing, which is more suitable for the needs of grid operation and maintenance personnel for the transparency of the rules compared with the black-box models such as support vector machine (SVM) or deep learning.

4.1. New England 10-Machine and 39-Node System Data Analysis

In order to illustrate and verify the effectiveness and feasibility of the algorithm in this paper, this paper selects the New England 10-machine and 39-node system for the simulation verification. The PMU dataset from New England 10-machine and 39-node, while reduced in scale, is highly representative of larger and more complex power system dataset. It has been widely adopted in power system studies due to its balance between simplicity and the inclusion of key operational and dynamic characteristics found in real-world grids. Although the New England 10-machine and 39-node system is a simplified model, its structure and behavior make it broadly applicable to larger systems, such as national or continental grids. The dynamic relationships between generators and loads, as well as the occurrence of faults and disturbances, are consistent with what is observed in more extensive grids. This makes the dataset an effective proxy for studying the behavior of larger systems, especially in terms of stability and fault response. For the warning and detection of LFO, the dataset offers a high level of completeness. It provides all the essential measurements (e.g., voltage, current, and frequency) that are required for identifying and diagnosing faults. The high-resolution and time-series data allow for both real-time fault detection and retrospective fault analysis, supporting the development of early warning systems. While the dataset does not capture all possible fault types or external environmental factors, its structured nature allows for fault scenarios to be simulated and analyzed in a controlled manner.

The data in this paper are selected from the PMU dataset of the New England 10-machine and 39-node system throughout the year, which has a sampling point every 15 min. A total of 35,040 data points throughout the year will be obtained, which is used as historical data for the correlation analysis. The dataset effectively captures essential system dynamics, which contains the frequency, phase shift, voltage, current, active and reactive power of each node, and so on. Then we selected August of that year as the test set to test the model effect, in which we selected 4 evaluation moments per day, for a total of 128 (T1–T128) evaluation moments in the whole month. The power system topology of the New England 10-machine and 39-node system constructed in this section is shown in Figure 4.

In this paper, MATLAB and Simulink simulation systems are used for experimental verification. A small load fluctuation (LFO) is applied to machine 5. The power and voltage values of the system are shown in Figure 5.

The operation of this simulation system shows that when the system undergoes LFOs, power fluctuations are generated in all lines 5–9, while small magnitude power oscillations are also generated in lines 11, 31, and 32. Then the system power fluctuation decreases and gradually tends to be stable. Compared with the total power before the LFO decreases, the system is damaged. But the LFO occurs rapidly and has the self-recovery nature, which is easy to cause the early warning system misjudgment, omission of judgment, and so on. The above phenomenon reflects that the early warning system should have the rapidity and accuracy of reflection, correctly reflecting the current power grid alarm, in order to carry out reasonable warning planning and avoid wasting resources.

4.2. Condition/Decision Attribute Selection and Data Discretization

When the power grid is running, the early warning indicators should be in line with the characteristics of the current power system model and can objectively reflect the operation state of power grid. Similarly, when selecting conditions or deciding attributes, the data processing advantages of rough set should also be adapted, and the indicators that can reflect the volatility and stability of the power grid should be selected as condition attributes.

The data are also discretized using the method in the previous section and categorized using the numbers 0, 1, and 2, where “0” indicates the indicator “normal”; “1” indicates the indicator “fluctuating”; and “2” indicates the indicator “deviating from normal values.” After discretization, the attribute values were compiled into a table of conditional attribute values as shown in Table 3.

4.3. Determination of Standard Cloud Parameters

The standard cloud parameters corresponding to the five warning levels calculated from historical grid data are shown in Table 4.

4.4. Approximation of Grid Conditional Attributes Based on Rough Set

Based on the historical operation data of the grid, a decision table is constructed, in which some of the conditional attributes corresponding to no fluctuations are reduced, and the decision table is shown in Table A1 in the Appendix (some samples are omitted).

The maximum importance reduction set {C_Pe} is taken as the initial reduction set, and its particle size entropy is calculated as I(D/C_Pe) = 0.187 ≠ I(D/C). Therefore, the remaining conditional attributes are added in sequence according to the comprehensive importance order SIG until the requirements are met, and the final reduced set is obtained, as shown in Table A2 in the Appendix.

4.5. Determination of Indicator Weights and Standard Cloud Parameters

Based on the approximated decision table, we obtain the core conditional attributes of the grid at this time, i.e., the filtered warning indicators, as shown in Table 5.

Then we calculate the comprehensive cloud parameters and composite weights for each indicator based on the algorithm proposed in the previous section using the real-time PMU data (taking moment T97 as an example), and the specific values are shown in Table 6.

4.6. Evaluation of the Level of Early Warning Results

After obtaining the weights and integrated cloud parameters corresponding to each approximated grid indicator, the corresponding affiliation matrix Q is calculated according to the algorithm in Section 3.3, and finally the affiliations corresponding to different grid risk levels are obtained through equation (14) as shown in Table 7.

From Table 7, it can be seen that at this time the integrated evaluation of clouds corresponds to the largest affiliation of cloud 4, which indicates that the risk level of the grid at T97 is level IV, i.e., it may lead to a level 1 failure of the grid.

At the same time, we calculate the affiliation results for 128 moments corresponding to each risk, as shown in Figure 6.

After obtaining the grid risk level affiliations corresponding to different time points, we calculate and test the grid damping ratios at different time points during the day, which is shown in Figure 7.

4.7. Analysis and Comparison of Results

According to Figure 7, it can be seen that the early warning assessment result of this paper’s method at the time of T97 is level IV, which may lead to a level I fault in the power grid. According to Figure 7, the corresponding risk level affiliation of the grid is level I or II most of the time, which becomes risk IV at T45 and T97 and has a tendency to develop into risk V at T98. Similarly, according to the analysis in Figure 7, the grid damping ratio decreases at time points T44 and T96, representing that the damping ratio is too low (close to zero), which indicates that the system is not sufficiently damped, and the system undergoes LFOs. It can be seen that the method in this paper successfully warns the LFO with low delay and high accuracy, which verifies the effectiveness of the method.

Then we, in order to further verify the effectiveness of the algorithm, take T90–T110 time points and compare the method of this paper with k-nearest neighbor (KNN), SVM, backpropagation (BP) neural network, and analytic hierarchy process (AHP) analysis method. The results are shown in Figure 8.

As can be seen from Figure 8, since the LFO starts at T95 and ends at T100, the method in this paper responds most rapidly and raises the warning evaluation level to 4 at the moment of T96, whereas all other methods suffer from the phenomenon of warning lag. Due to SVM method needing a lot of tuning parameters to assist, it fails to respond in time when the LFO occurs and delays the early warning time. The BP neural network method has some overfitting problems, which leads to premature judgment of early warning situation. Due to the lack of sufficient feature selection preprocessing ability, the KNN method misdetermines the occurrence time of LFO. Meanwhile, when the LFO disappears, the response speed of this paper’s method and the BP neural network method is the same, which adjusts the risk of LFO of the grid downward in time and reduces the waste of resources. However, the AHP method relies on the subjective judgment of weight and priority, and a large number of incomplete data information leads to the decline of the accuracy of its judgment matrix, which makes the AHP method misjudge the recovery time of LFO fault and reduce the warning level prematurely. Moreover, during the LFO, this paper’s method accurately predicts its risk degree as 4, but it does not rise to 5 (large-scale grid accidents), while other methods have different degrees of precision disorders. In summary, this paper’s method is superior to other methods in terms of warning response time and warning accuracy.

Meanwhile, to further compare the accuracy of this paper’s method, confusion matrix analysis is used, where the diagonal box represents that the actual risk matches the predicted risk, and the other boxes represent that the warning level and the real level do not match. The specific warning situation is shown in Figure 9.

From the analysis of Figure 9, it can be seen that the proportion of samples represented by the diagonal box frame of this paper’s method is 75.9%, 19.3%, 0.8%, and 1.6%, respectively, which are higher than those of the other models, and the nondiagonal box frames of the other models still have different probabilities of appearing in the samples, which proves that this paper’s method is also ahead of the other methods in terms of accuracy.

In order to further validate the advantages of the method in this paper, 128 assessment periods are compared with the real warning levels, and their precision and recall are calculated as shown in Figure 10. As can be seen from Figure 10, the accuracy and recall of the comprehensive warning model constructed by this paper’s method are above 90%. It is 15% points better than the KNN model with the lowest precision in the comparison of precision, and it is also slightly higher than that of the SVM method with stronger comprehensive performance, which once again verifies the effectiveness of this paper’s method.

To verify the effectiveness of the method in this paper, we collect historical fault data such as bus voltage and line power as a test set. Comparing the traditional statistical analysis methods, machine learning methods, etc., which include KNN method, SVM, and deep belief network (DBN) methods, the key performance indexes are given in Table 8.

From Table 8, it can be seen that the response speed and robustness of the methods in this paper are stronger than the statistical and machine learning methods; this is because the statistical methods often require a large number of parameter calculations, while the machine learning methods require training and GPU acceleration and also lead to poorer robustness due to their inability to recognize the physical significance of the attribute set.

In order to compare the contribution of this paper’s method with other methods in a more comprehensive way, we have selected some key performances of the early warning model for comparison, which are shown in Table 9.

The method proposed in this study achieves the synergistic optimization of accuracy, efficiency, and interpretability through the combination of rough sets and cloud models. Experiments show that under the premise that the false alarm rate (5.9%) is significantly lower than that of the traditional model (6.2%–8.6%), the feature dimension is compressed to 14%–23% of the comparative methods, and the memory occupation is only 26%–27% of that of the deep model (DBN/SVM), which breaks through the bottleneck of the lightweight deployment; at the same time, the method generates 23 interpretable rules, which is upgraded compared with the classical interpretable model (KNN) by 187.5%, which is the first time to realize the compatibility design between deep learning and symbolic reasoning, and solves the industry pain point of insufficient decision transparency in black-box models; in addition, the parameter sensitivity (±10%) is reduced by 54.5% compared with the optimal comparison model, which verifies the strengthening effect of algorithm robustness.

In discussion, the method in this paper is proved to be superior to mainstream methods in terms of warning accuracy and warning response speed. A new problem solving idea is proposed for the LFO warning problem, i.e., the use of rough sets to approximate the grid attributes and then the use of cloud models for comprehensive evaluation. Compared to the rough set method, the KNN method suffers from high sensitivity to noise and outliers, significant computational cost, scalability issues, and a lack of inherent feature selection. SVMs face high computational demands, complex parameter tuning, a risk of overfitting, and less interpretable decision boundaries. BP neural networks are challenged by long training times, overfitting, interpretability issues due to their black-box nature, and sensitivity to hyperparameters. The AHP is hindered by subjectivity in judgments, complexity in managing large systems, and potential inconsistencies. In contrast, the rough set method excels in early warning systems by effectively handling uncertainty and incomplete information, providing clear and interpretable decision rules, and offering a robust framework for feature selection without the need for prior knowledge of the data distribution. At present, this study still contains some limitations in computational processing, and future research work should be carried out in simplifying computational processing and reducing computational complexity.

While the rough set method offers significant benefits for early warning systems, it also has limitations that require critical examination. One key limitation is its susceptibility to potential biases based on the quality and representativeness of the data used. If the dataset contains biases or lacks representation of all possible scenarios, the decision rules derived may not be universally applicable or may miss important patterns. Additionally, the method faces scalability challenges as the size and complexity of the data grow. Larger grid systems or datasets with many features can lead to significant computational effort and may impact the method’s performance due to the increased number of possible rules and their interactions. Furthermore, rough sets may struggle with capturing complex, nonlinear relationships within the data, which can limit their accuracy in providing early warnings in systems where such relationships are critical. The generation of decision rules can also lead to rule overload, where managing and interpreting a large number of rules become cumbersome, potentially diminishing the effectiveness of the early warning system. Addressing these limitations involves refining the rough set approach to improve scalability, ensuring data quality and representativeness, and possibly integrating it with other methods to capture complex relationships and enhance overall performance.