Analytical method for short-circuit analysis of shunt reactors

Abstract

The application discloses an analytic method for short-circuit analysis of a hollow shunt reactor and belongs to the technical field of power equipment analysis and diagnosis. The analytic method comprises the following steps: constructing a coupling inductance matrix of a winding of the hollow reactor; determining a short-circuit node set and a residual node set; performing block processing on the inductance matrix according to the residual node set and the short-circuit node set; calculating a reduced inductance matrix through Kron reduction operation; calculating a residual inductance value according to the reduced inductance matrix; calculating an inductance loss percentage according to a ratio of the residual inductance value to an initial inductance value, and classifying fault severity according to the inductance loss percentage. The application replaces time-consuming finite element simulation with analytic calculation through Kron reduction, compresses single fault analysis time from several hours to seconds, and realizes a complete closed loop from residual inductance calculation, fault current solution to operation and maintenance decision.

Description

Technical Field

[0001] This invention belongs to the field of power equipment analysis and diagnosis technology, and particularly relates to an analytical method for short-circuit analysis of air-core shunt reactors, applicable to shunt reactors in high-voltage and ultra-high-voltage transmission systems. Background Technology

[0002] In high-voltage and ultra-high-voltage transmission systems, air-core shunt reactors are key equipment for reactive power compensation and suppression of power frequency overvoltages. Due to their high operating voltage and large capacity, reactor windings may experience inter-turn or inter-layer short-circuit faults during long-term operation due to insulation aging, overvoltage impacts, or manufacturing defects. Once a short-circuit fault occurs, a huge circulating current will be generated in the fault circuit, leading to local overheating, further exacerbating insulation degradation, and even causing equipment burnout and system blackout. Therefore, accurately analyzing the impact of short-circuit faults on reactor parameters and calculating the fault current under short-circuit conditions is of paramount importance for the protection design and operation and maintenance of reactors.

[0003] Shortcomings of existing methods: (1) High computational cost of the finite element method (FEM) Traditional methods employ two-dimensional or three-dimensional finite element method (FEM) software to construct a detailed electromagnetic model of the reactor; short-circuit faults are simulated by applying zero-voltage boundary conditions to the faulty turns. While this method offers extremely high accuracy, simulating a large air-core reactor (which may contain dozens to hundreds of winding layers, each with dozens of turns) often takes several hours. When fault analysis needs to be performed turn-by-turn for various fault scenarios, the computational burden increases exponentially, making the FEM method practically unsustainable for many inter-turn faults.

[0004] (2) The analytical method failed to capture the interlayer coupling effect. Significant electromagnetic coupling exists between the winding layers of an air-core reactor, which can be described by a complete inductance matrix (including self-inductance and mutual inductance). When a short circuit occurs in a specific layer or number of turns, the current in the fault circuit depends not only on the self-inductance of that layer but is also strongly affected by the mutual inductance coupling with all other layers. Simplified series-parallel equivalent circuit models cannot accurately capture this complex multi-body coupling characteristic.

[0005] (3) Transient analysis tools such as ATP / EMTP have limited support for the size of the coupled matrix. Current transient analysis tools, such as ATP / EMTP, impose significant limitations on matrix size when modeling electromagnetic coupling. While these platforms enable rapid simulations and are sufficient to characterize interlayer coupling, they are limited by a fixed maximum matrix size (e.g., ATP-EMTP currently supports a maximum of 40 nodes). This limitation makes them unsuitable for complex systems such as shunt reactors, which often contain hundreds of layers and turns.

[0006] (4) Lack of a systematic progressive failure analysis methodology Short-circuit faults in reactors are usually not instantaneous failures, but rather develop gradually from single-turn short circuits to multi-turn short circuits, then to full-layer short circuits, and may even develop to the entire coil assembly. Existing technologies lack a systematic and computationally efficient method to simulate this gradual evolution process and accurately calculate the residual inductance and fault current at each stage.

[0007] (5) Fault current calculation ignores complex impedance characteristics Existing methods typically consider only the pure reactance (inductive impedance) to calculate the fault current, neglecting the winding resistance; or they use a simplified equivalent circuit model. For air-core reactors, although the resistance of each layer is relatively small, it is a non-negligible part of the fault loop impedance, especially when only a small number of turns are short-circuited, the resistance has a significant impact on the magnitude and phase angle of the fault current.

[0008] Therefore, it is urgent to design an analytical method for short-circuit analysis of air-core parallel reactors to solve the problems mentioned above. Summary of the Invention

[0009] The purpose of this invention is to provide an analytical method for short-circuit analysis of air-core parallel reactors. This method features matrix reduction through Kron reduction, replacing time-consuming finite element simulation with analytical calculation, thus improving computational efficiency to the second level. It accurately captures interlayer electromagnetic coupling and resistance effects through a complete inductance matrix and complex impedance. It overcomes the limitations of matrix size, supports layer / turn progressive fault analysis without node limitations, and combines quantitative grading standards to achieve a complete closed loop from numerical calculation to decision support. This invention solves the problems mentioned in the background art.

[0010] To achieve the above objectives, the analytical method for short-circuit analysis of hollow parallel reactors of the present invention provides the following specific technical solution: An analytical method for short-circuit analysis of air-core shunt reactors includes the following steps: S1. Constructing the air-core reactor winding Coupled inductor matrix Inductor matrix off-diagonal elements For nodes With nodes The mutual intuition between them, at this time Inductor matrix diagonal elements For nodes The self-awareness at this moment ; in, This represents the total number of winding nodes, where a node is either a winding layer node or an expanded winding turn node. S2. Determine the set of short-circuit nodes. and the set of remaining nodes ,in, Constituting all nodes, and It is an empty set; S3. Based on the remaining node set and short-circuit node set For inductor matrix Divide into blocks to obtain submatrices , , and ; in, The remaining nodes are the inductance submatrix. This is the mutual inductance submatrix between the remaining nodes and the short-circuited nodes. The inductance submatrix between short-circuited nodes, for transpose; S4. Calculate the reduced-order inductance matrix using Krone reduction operations. ,in for Moore-Penrose pseudo-inverse matrix; S5, based on the reduced-order inductance matrix Calculate the residual inductance value; S6. Calculate the inductance loss percentage based on the ratio of the residual inductance value to the initial inductance value, and classify the severity of the fault according to the inductance loss percentage.

[0011] Furthermore, this analytical method also includes an extended matrix construction step for inter-turn analysis of any coil layer, specifically: When inter-turn analysis of the target coil layer is required, replace the individual layer nodes of the target coil layer with Each independent turn node yields the expanded matrix dimension. ,in, The number of turns in this layer; the coil to which this target coil layer belongs is the target coil. The nodes in the expanded matrix are arranged in the following order: S11, the nodes of all layers of all coils preceding the target coil; S12, All turn nodes of the split layer in the target coil; S13, nodes of the remaining layers in the target coil, excluding the split layer; S14, the nodes of each layer of all coils after the target coil.

[0012] Furthermore, this analytical method also includes an adaptive inter-turn asymptotic short-circuit step: when At that time, each turn is added to the short-circuit node set S in turn, and a Kron reduction operation is performed once in each step; when At that time, turns 1 through 10 are added sequentially, then added in 5-turn increments, and finally the last turn is added in the last step. A Kron reduction operation is performed once in each step.

[0013] Furthermore, the residual inductance value The calculation method is as follows: for the reduced-order inductance matrix Summing all elements, i.e. .

[0014] Furthermore, the initial inductance value The calculation method is as follows: for the original complete inductance matrix Summing all elements, i.e.: .

[0015] Based on the initial inductance value and residual inductance value Calculate the percentage of inductor loss. The formula is .

[0016] Furthermore, the severity of the fault is classified according to the percentage of inductor loss as follows: A loss percentage of less than 10% indicates low risk. A loss percentage of 10% or less and less than 30% is considered medium risk. A loss percentage of 30% or less and less than 60% is considered high-risk. A loss percentage of ≥60% is considered a critical risk.

[0017] Furthermore, this analytical method also includes a short-circuit fault current calculation step: S21, According to the inductance matrix and node DC resistance vector Build Complex impedance matrix diagonal elements off-diagonal elements ; in, The imaginary unit, The system angular frequency, For nodes DC resistance; For nodes The self-awareness at this moment , For nodes With nodes The mutual intuition between them, at this time ; S22, According to rated voltage Total resistance of remaining nodes and residual resistance Calculate the normal operating current ; in, , This is the residual inductance value; S23. Based on the remaining node set and short-circuit node set For complex impedance matrix Divide the circuit into blocks and establish the short-circuit current equation. ; in, The impedance submatrix between short-circuit nodes. This is the coupling impedance submatrix between the short-circuited node and the remaining nodes. For length is The whole vector; S24. Solve the linear equations to obtain the fault current phasors of each short-circuit node. The maximum value of the fault current amplitude at all short-circuit nodes is taken as the short-circuit current. ; in, .

[0018] Furthermore, when the linear equation When the singularity or condition number exceeds a preset threshold, the least squares method is used to solve the problem. .

[0019] Furthermore, when performing inter-turn analysis, the DC resistance of each turn in the split layer is distributed in an equal manner: ; in, The DC resistance of the split layer. This represents the number of turns in this layer.

[0020] On the other hand, this application also proposes a short-circuit analysis system for air-core shunt reactors, comprising: The inductor matrix loading module is used to load and verify data from an external data source. Coupled inductor matrix ; A DC resistance loading module is used to load the DC resistance value of each winding node; The short-circuit scheduling module is used to generate inter-layer progressive short-circuit scheduling schemes and / or inter-turn progressive short-circuit scheduling schemes; The Kron reduction calculation module is used to perform Kron reduction operations to eliminate short-circuited nodes; The residual inductance calculation module is used to calculate the equivalent residual inductance value based on the reduced-order inductance matrix. The fault current calculation module is used to construct the complex impedance matrix Z and solve for the short-circuit current. The fault classification module is used to classify the severity of faults based on the percentage of inductor loss. The results output module is used to output the analysis results of each short-circuit step.

[0021] This invention has the following advantages: By using Kron reduction to replace time-consuming finite element simulation with analytical calculation, the time for a single fault analysis is reduced from several hours to seconds. At the same time, it breaks through the 40-node scale limit of tools such as ATP-EMTP, supporting inter-layer and inter-turn progressive short-circuit analysis with unlimited number of nodes. By using complete inductance matrix and complex impedance modeling, it accurately captures strong electromagnetic coupling and resistance effects between layers. Combined with a four-level quantization classification standard, it realizes a complete closed loop from residual inductance calculation, fault current solution to operation and maintenance decision-making, and has significant advantages of high efficiency, accuracy, systematicity, robustness and clear physical meaning. Attached Figure Description

[0022] Figure 1 This is a flowchart illustrating the analytical method of the present invention; Figure 2 This is a schematic diagram of the relationship between progressive coil layer fault conditions and residual inductance according to the present invention. Figure 3 This is a schematic diagram of the relationship between the inter-turn progressive short-circuit fault condition and the residual inductance of the present invention. Detailed Implementation

[0023] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0024] Those skilled in the art will understand that although some embodiments herein include certain features included in other embodiments but not others, combinations of features from different embodiments are intended to be within the scope of the invention and form different embodiments. For example, in the claims, any of the claimed embodiments can be used in any combination.

[0025] The following is a reference to the appendix. Figure 1 To be continued Figure 3 The analytical method for short-circuit analysis of the hollow parallel reactor of the present invention is described.

[0026] Kron reduction, a classic method for reducing the order of network matrices, was first proposed by Gabriel Kron in the 1940s and initially applied to the equivalent calculation of power system networks. This method eliminates short-circuit nodes or internal nodes from the network equations through block matrix algebra operations, obtaining an equivalent reduced-order matrix. However, the systematic application of Kron reduction to the asymptotic short-circuit analysis of air-core reactors, and its combination with complex impedance matrices to achieve accurate fault current calculations, has not been reported in any published literature or patent records to date.

[0027] An analytical method for short-circuit analysis of air-core shunt reactors includes the following steps: S1. Constructing the air-core reactor winding Coupled inductor matrix Inductor matrix off-diagonal elements For nodes With nodes The mutual intuition between them, at this time Inductor matrix diagonal elements For nodes The self-awareness at this moment ; in, This represents the total number of winding nodes, where a node is either a winding layer node or an expanded winding turn node. For inter-layer analysis, each node corresponds to one winding layer. For example, a hollow reactor with 8 coils, each with 15 layers, has a total of inter-layer analysis nodes of [number missing]. =8×15=120, at this time, each node represents a winding layer, and the matrix dimension is 120×120.

[0028] For inter-turn analysis, replace the single layer node of the target coil layer with Each independent turn node yields the expanded matrix dimension. ,in, The number of turns in this layer is the target coil, and the coil to which this target coil layer belongs is the target coil. For example, if an inter-turn analysis is required for the first layer of the 8th coil (which has 45 turns in total), then one node in this layer is expanded into 45 independent turn nodes, while the nodes in the remaining 119 layers remain unchanged, expanding the total number of nodes to [number missing]. =120+45-1=164.

[0029] Node numbering uses a row-major order mapping rule, and the major order mapping rule uses row-first sorting to arrange the coils. ,layer For mapping to the global index, the total number of nodes ; in, The total number of coils, The number of layers for each coil.

[0030] Specifically, coil layer 1 corresponds to index 0, coil layer 2 corresponds to index 1, ..., coil layer 15 corresponds to index 14, coil layer 21 corresponds to index 15, and so on.

[0031] The nodes in the expanded matrix are arranged in the following order: S11, the nodes of all layers of all coils preceding the target coil; Specifically, all nodes of each layer of all coils preceding the target coil are placed at the front of the extended matrix, while maintaining the original row-major order. S12, All turn nodes of the split layer in the target coil; Specifically, the target coil is divided into layers. Each turn node (turn 1 to turn 2) Arranged in ascending order of turn number; S13, nodes of the remaining layers in the target coil, excluding the split layer; Specifically, the nodes of the remaining layers in the target coil, excluding the split layers, are arranged in their original order after the turn nodes; S14, the nodes of each layer of all coils after the target coil; Specifically, all nodes of each layer of all coils after the target coil are placed at the end of the extended matrix, while maintaining the original row-major order.

[0032] By sequentially numbering the coils preceding the target coil, the turn nodes of the split layer are consecutively arranged at the original layer node positions of the target coil, with the remaining layers of the target coil following immediately after. The coils following the target coil are then sequentially numbered at the end. This order ensures the locality of matrix expansion, meaning that only a single node of the target layer needs to be replaced. The relative order of the remaining nodes remains unchanged, which facilitates matrix index management and subsequent block operations.

[0033] Taking an example of 8 coils, each with 15 layers, and the first layer of coil 8 being divided into 45 turns: Index 0-104: All layer nodes (105) from coil 1 to coil 7; Indexes 105-149: 45 turn nodes (turn 1 to turn 45) of the first layer of coil 8; Index 150-163: Layer nodes (14) of coil 8, layers 2 to 15.

[0034] This forms the extended matrix dimension. =105+45+14=164, which verifies... The correctness of the statement.

[0035] off-diagonal elements ,at this time : Represents a node With nodes Mutual inductance between them, when current is at the node During flow, at nodes The induced magnetic flux divided by the node The current in is According to the reciprocity theorem of electromagnetism, Therefore, the inductor matrix It is a symmetric matrix.

[0036] diagonal elements ,at this time : Represents a node Self-intuition, i.e., nodes When a unit current flows through the node, The magnetic flux induced in its own circuit.

[0037] Regarding the matrix acquisition method, the inductance matrix The elements can be obtained in any of the following ways: Finite element simulation: A detailed geometric model of the reactor is established using three-dimensional finite element software. Through static magnetic field or transient magnetic field simulation, each node is excited successively and the magnetic flux is calculated to obtain the complete inductance matrix.

[0038] Direct measurement: Before the reactor leaves the factory or during field testing, inductance measurement instruments are used to measure each layer / turn sequentially to construct an inductance matrix. For reactors already in operation, online / offline measurement techniques such as frequency response analysis can be used to obtain the inductance data.

[0039] Analytical calculation: Based on the geometric parameters of the reactor (coil diameter, height, number of turns, layer spacing, etc.), the mutual inductance between each layer / turn is calculated using classical electromagnetic formulas such as the Neumann formula.

[0040] Regarding matrix verification, in practice, due to measurement errors or simulation accuracy limitations, the obtained inductance matrix may exhibit slight asymmetry. Therefore, the matrix needs to be verified before performing subsequent block partitioning and reduction operations. Symmetry verification: Calculate the symmetry error When the symmetry error is less than a preset threshold (e.g., 0.1%), the matrix is ​​considered to meet the symmetry requirement; when it exceeds the threshold, symmetry processing is performed. .

[0041] Positive definiteness verification: Calculate the condition number of the matrix to evaluate whether the matrix is ​​positive definite. If the condition number is too large, it indicates that the matrix is ​​ill-conditioned, and it may be necessary to check the data source or perform preprocessing.

[0042] Full rank verification: Calculate the rank of the matrix to confirm. This ensures that the matrix is ​​full rank and that all nodes are linearly independent.

[0043] S2. Determine the set of short-circuit nodes. and the set of remaining nodes ,in, Constituting all nodes, and It is an empty set, meaning that every winding node is either in a short-circuit state (belonging to...). ), or be in a healthy state (belonging to There are no nodes that are both short-circuited and healthy, nor are there any nodes that do not belong to any category; Short-circuit node set : Defined as the set of all winding nodes that have experienced a short circuit; in actual fault scenarios, these nodes are short-circuited (considered as ideal short-circuit connections), and the voltage drop between the nodes is zero.

[0044] Remaining node set Defined as the set of all winding nodes that have not experienced a short circuit; these nodes maintain normal operating conditions and carry normal load current.

[0045] In progressive short-circuit analysis, the set S of short-circuit nodes is gradually expanded.

[0046] Layer progression: Starting from the initial layer, add one layer node at each step. =Initial state, =Level 1, = Layer 1, Layer 2, ..., until the terminal layer is reached.

[0047] Inter-turn asymptotic (adaptive step size strategy): Turn nodes are gradually added to the short-circuit node set according to the following adaptive step size strategy. : When T ≤ 10, each turn is added to the short-circuit node set in turn. Each step performs a Kron reduction operation once.

[0048] Right now: =1st turn, =1st turn, 2nd turn, =1st turn, 2nd turn, 3rd turn, ... =The first turn to the... Turns.

[0049] when At that time, turns 1 through 10 are added sequentially, then added in 5-turn increments, and finally the last turn is added in the last step. A Kron reduction operation is performed once in each step.

[0050] Turns 1 to 10: Add (turn 1, turn 2, turn 3, ..., turn 10) sequentially, performing a Kron reduction operation once per step; After the 10th turn: add turns in 5-turn increments (turn 15, turn 20, turn 25, turn 30, turn 35, turn 40, etc.), and perform a Kron reduction operation once per step; Final step: Add the last turn (turn) ), and perform a Kron reduction operation.

[0051] S3. Based on the remaining node set and short-circuit node set For inductor matrix Divide into blocks to obtain submatrices , , and ; in, The inductance submatrix between the remaining nodes (dimension) ), The mutual inductance submatrix between the remaining nodes and the short-circuited nodes (dimension) ), The inductance submatrix between short-circuit nodes (dimension) ), for transpose (dimension) ), This represents the number of remaining nodes. Let be the number of short-circuited nodes, and ; Taking a 120-layer reactor and a short circuit on the first layer as an example ( =1, =119): (119×119): The relationship between self-induction and mutual induction between layers 2 and 120; (119×1): Mutual inductance between layers 2 to 120 and layer 1 respectively; (1×119): Mutual inductance between layer 1 and layers 2 through 120 (for...) (transpose) (1×1): Self-perception of the first layer.

[0052] S4. Calculate the reduced-order inductance matrix using Krone reduction operations. ,in, for The Moore-Penrose pseudo-inverse matrix is ​​used to ensure numerical stability; In practical short-circuit analysis, the set of short-circuit nodes It may contain multiple electrically tightly coupled or even redundant nodes, leading to submatrix Singular or near-singular (e.g., extremely strong mutual inductance between multiple turn nodes within the same layer, making...) The determinant approaches zero. In this case, the ordinary inverse matrix does not exist or is numerically unstable, while the pseudo-inverse can provide a unique least-squares solution with minimum norm, ensuring that the reduction process is always executable.

[0053] even though It is not singular, but when its condition number is very large (ill-conditioned), direct inversion will amplify the input error, leading to a decline in the reduced value. Distortion and pseudo-inverse are achieved through singular value decomposition (SVD), which can truncate minimal singular values, suppress noise amplification, and thus improve the robustness of the entire algorithm.

[0054] The essence of Kron reduction is to substitute the current constraints at short-circuit nodes into the original system, eliminating the degrees of freedom of these nodes. The use of pseudo-inverse is equivalent to finding the optimal approximate solution in the constraint space, so that the reduced inductance matrix can still accurately reflect the electromagnetic coupling relationship between the remaining nodes, while avoiding non-physical solutions (such as negative inductance) caused by ill-conditioned matrices.

[0055] Specifically, reduced-order inductor matrix This represents the equivalent inductance matrix of the remaining windings after algebraic folding of the electromagnetic effects in the short-circuit loop (including the antimagnetic flux generated by the induced circulating current in the fault loop), which have been folded onto the remaining nodes.

[0056] S5, based on the reduced-order inductance matrix Calculate the residual inductance value ; Specifically, residual inductance value The calculation method is as follows: for the reduced-order inductance matrix Summing all elements, i.e. .

[0057] Residual inductance It represents the total equivalent inductance of the reactor as seen from the external terminals after a short circuit occurs. It is a key parameter for measuring the remaining performance of the reactor.

[0058] Furthermore, in the progressive short-circuit analysis, to characterize the marginal effect of each new short-circuit step on the residual inductance, the change in inductance between adjacent steps is also calculated. ,in, The residual inductance of the current step, The residual inductance from the previous step, and the initial inductance. Calculate the relative percentage change based on the baseline. This is used to reflect the instantaneous impact of the fault deterioration on the inductor parameters during this step. A negative value indicates that the inductance continues to decrease, and the larger the absolute value, the more severe the marginal impact of the new fault.

[0059] S6. Calculate the inductance loss percentage based on the ratio of the residual inductance value to the initial inductance value, and classify the severity of the fault according to the inductance loss percentage. Specifically, the initial inductance value The calculation method is as follows: for the original complete inductance matrix Summing all elements, i.e.: ; That is, the original inductance matrix constructed in step S1 ; Based on the initial inductance value and residual inductance value Calculate the percentage of inductor loss. The formula is .

[0060] The severity of the fault is classified according to the percentage of inductor loss as follows: A loss percentage of less than 10% indicates low risk. At this point, the inductance loss is relatively small, and the reactor can continue to operate for a short period of time, but monitoring should be strengthened. A loss percentage of 10% to 30% is considered medium risk, at which point the inductor loss is significant, and it is recommended to arrange a power outage for maintenance as soon as possible. A loss percentage of 30% to 60% indicates a high risk, as the inductor loss is severe and protective measures should be taken immediately to avoid prolonged operation. A loss percentage of ≥60% is considered a critical risk. At this point, the inductance loss is extremely severe, the reactor faces the risk of burning out, and it should be disconnected immediately.

[0061] After the above Kron reduction operation is completed and the residual inductance value is obtained, Subsequently, the present invention also includes a short-circuit fault current calculation step: S21, According to the inductance matrix and node DC resistance vector Build Complex impedance matrix diagonal elements , representing nodes Self-impedance, off-diagonal elements , representing nodes With nodes Mutual impedance between them; in, The imaginary unit is used to distinguish nodes. , The system angular frequency, For nodes DC resistance; For nodes The self-awareness at this moment , For nodes With nodes The mutual intuition between them, at this time ; When performing inter-turn analysis, the DC resistance of each turn in the split layer is distributed in an equal manner: ; in, DC resistance at each turn node, The DC resistance of the split layer. This represents the number of turns in this layer.

[0062] S22, According to rated voltage Total resistance of remaining nodes and residual resistance Calculate the normal operating current ; in, , This is the residual inductance value. This represents the magnitude of the total impedance. S23. Based on the remaining node set and short-circuit node set For complex impedance matrix Divide the circuit into blocks and establish the short-circuit current equation. ; in, The impedance submatrix between short-circuit nodes has dimensions of , Let be the coupling impedance submatrix between the short-circuited node and the remaining nodes, with dimensions . , For length is The whole A vector with dimension 1 , For each short-circuit node fault current phasor; S24. Solve the linear equations to obtain the fault current phasors of each short-circuit node. The maximum value of the fault current amplitude at all short-circuit nodes is taken as the short-circuit current. ; in, for The inverse matrix, when Applicable when not singular. For the first Fault current phasor of each short-circuit node, , Belongs to the set of short-circuit nodes .

[0063] When linear equations When the singularity or condition number exceeds a preset threshold, the least squares method is used to solve the problem. ; in, This is a least squares solver whose objective is to find a... This minimizes the L2 norm (square root of the sum of squares) of the residual vector.

[0064] That is, if If the condition is singular or pathological, the system will automatically switch to least squares: .

[0065] Interlayer short-circuit analysis of series and parallel reactors Consider a 500 kV, 20 Mvar air-core shunt reactor with 8 coils, each coil containing 15 winding layers (120 layers in total), each layer consisting of 45 turns, all layers connected in series, and an inductance matrix. It is a symmetric positive definite matrix of 120×120, whose elements are obtained through finite element simulation or direct measurement.

[0066] Example 1, Layer Analysis Flow: S31. Load a 120×120 inductor matrix from an Excel file; verify the matrix dimension, calculate the rank = 120 (full rank), symmetry error < 0.1%, condition number, and perform symmetry processing; S32. Load DC resistance vector from Excel resistance worksheet (120 values, one for each level); S33. Establish a (coil, layer) index mapping: Coil 1, layer 1 is index 0, Coil 1, layer 2 is index 1, ..., Coil 8, layer 15 is index 119. Total number of nodes. =8 × 15 = 120; S34, Calculate the initial inductance = Sum of all 120×120 elements; S35. Configure analysis parameters: coil to be analyzed = 1, starting layer = 1, ending layer = 15, frequency = 60 Hz, rated voltage = 288675 V (for current calculation). S36. Generation layer short-circuit scheduling scheme: Step 1 is {Level 1}, Step 2 is {Level 1, Level 2}, ..., Step 15 is {Level 1 to Level 15}; S37. For each step, perform Kron reduction and calculate the residual inductance. Percentage of loss , resistance Normal operating current Short-circuit current Severity level; S38. Output: Progressive analysis results table (step number, fault tag, residual inductance) Percentage of loss , resistance Normal operating current Short-circuit current Inductance change Percentage change in inductance Table 1 (Severity Level).

[0067] Table 1 Table 1 shows the representative results of this series-parallel reactor in progressive short-circuit analysis. As the number of short-circuit layers increases, the residual inductance... Monotonically decreasing, percentage loss The severity of the fault increases progressively, from low risk to critical risk. Short-circuit current. This reflects the most severe circulating current in the fault layer.

[0068] See Figure 2 , Figure 2 The curves corresponding to Table 1 show the number of short-circuit layers on the horizontal axis and the residual inductance value on the vertical axis. The curve gradually decreases from the initial value of 11039.40 mH to 9355.76 mH as the number of layers increases. The decrease is more significant in the first 8 layers, and then tends to flatten out. As the number of short-circuit layers increases, the residual inductance decreases monotonically, the percentage of loss increases, and the severity of the fault gradually increases from low risk to critical risk.

[0069] Example 2: Turn Splitting Analysis Process S41. Construct extended index mapping: =120+45-1=164; S42, Node Assignment: Index 0-104 (all layer nodes of coil 1 to coil 7), Index 105-149 (turns 1-45 of layer 1 of coil 8), Index 150-163 (layer nodes of layer 2 to layer 15 of coil 8). S43, Distribute the turn resistors: (The DC resistance of the first layer of coil 8 is divided by 45). S44. Generate turn short-circuit scheduling scheme (adaptive): Analyze turns 1-10 one by one, then analyze every 5 turns (15, 20, 25, 30, 35, 40, 45) -- a total of 17 turn steps; S45. After the turn analysis is completed, generate the layer analysis scheduling scheme for layers 2 to 15 of coil 8; S46. Perform asymptotic Kron reduction for each step and calculate the residual inductance. Percentage of loss , resistance Normal operating current Short-circuit current Severity level; S38. Output: Progressive analysis results table (step number, fault tag, residual inductance) Percentage of loss , resistance Normal operating current Short-circuit current Inductance change Percentage change in inductance Table 2 (Severity Level).

[0070] Table 2 Table 2 shows the representative results of this series-parallel reactor in the inter-turn progressive short-circuit analysis. As the number of short-circuit turns increases, the residual inductance... Overall decrease, but the rate of decline is slowing, percentage loss The failure severity increased from 1.78% to 3.52%, but remained at a low-risk level across all steps, indicating that the main risk of inter-turn short circuits lies in the large circulating current in the short-circuit loop rather than a decrease in inductance parameters; short-circuit current The peak current reached 12527.01 A (approximately 178 times the normal current) during the first short circuit. It then gradually decreased with increasing number of short-circuited turns, revealing that the current surge was most severe during single-turn or a few-turn short circuits. Table 2 uses an adaptive step-size strategy (step-by-step for the first 10 turns, then every 5 turns), compressing the analysis steps from 44 to 17, validating the balance between efficiency and accuracy of this strategy.

[0071] See Figure 3 , Figure 3 The graph corresponding to the data in Table 2 shows the horizontal axis as the number of short-circuited turns (1-45 turns) and the vertical axis as the corresponding residual inductance value. The curve starts at 10842.92 mH and gradually decreases to 10656.98 mH as the number of short-circuit turns increases. The decrease is more significant in the first 10 turns, and the curve tends to flatten out after 10 turns, reflecting the physical law that the electromagnetic parameters change most drastically in the early stage of inter-turn short circuit.

[0072] On the other hand, this application also proposes a short-circuit analysis system for air-core shunt reactors, comprising: The inductor matrix loading module is used to load and verify data from an external data source. Coupled inductor matrix ; Specifically, the inductor matrix loading module is the core module of the system's data entry point, responsible for loading the air-core reactor from external data sources (such as Excel files, CSV files, MATLAB binary files, or databases). Coupled inductor matrix The integrity and validity of the matrix are verified to ensure that all subsequent calculation steps are based on correct and reliable original data.

[0073] A DC resistance loading module is used to load the DC resistance value of each winding node; Specifically, the DC resistance loading module is responsible for loading the DC resistance value of each winding node (layer node or extended turn node), which is a complex impedance matrix. The construction provides real data DC resistance is a physical parameter that cannot be ignored in fault current calculation, as it directly affects the amplitude and phase of the short-circuit current.

[0074] The short-circuit scheduling module is used to generate inter-layer progressive short-circuit scheduling schemes and / or inter-turn progressive short-circuit scheduling schemes; Specifically, the short-circuit scheduling module is the core scheduler of the system's progressive analysis capability. It is responsible for automatically generating inter-layer progressive short-circuit scheduling schemes and / or inter-turn progressive short-circuit scheduling schemes based on the fault scenario parameters configured by the user. This module implements the progressive expansion strategy in step S2 above into a specific, iteratively executable sequence of short-circuit steps, driving the Kron reduction calculation module to execute sequentially.

[0075] The Kron reduction calculation module is used to perform Kron reduction operations to eliminate short-circuited nodes; Specifically, the Kron reduction calculation module is the core computing engine of the system, responsible for executing steps S3 and S4 above. This module receives the inductance matrix. and short-circuit node set As input, short-circuit nodes are eliminated through block matrix algebra operations, and the output is a reduced-order inductance matrix. .

[0076] The residual inductance calculation module is used to calculate the equivalent residual inductance value based on the reduced-order inductance matrix. Specifically, the residual inductance calculation module is responsible for calculating the reduced-order inductance matrix output by the Kron reduction calculation module. Calculate the equivalent residual inductance of the reactor after a short-circuit fault. And calculate the initial inductance value under normal conditions. .

[0077] The fault current calculation module is used to construct the complex impedance matrix Z and solve for the short-circuit current. Specifically, the fault current calculation module is responsible for constructing the complex impedance matrix. It also solves for the short-circuit fault current. Corresponding to the above short-circuit fault current calculation steps, this module is called after the Kron reduction calculation and residual inductance calculation are completed.

[0078] The fault classification module is used to classify the severity of faults based on the percentage of inductor loss. Specifically, the fault classification module is responsible for classifying the severity of the current fault state based on the percentage of inductor loss, corresponding to the specific implementation of step S6 above. This module will abstract the numerical calculation results ( This transforms into intuitive engineering judgment criteria, providing a direct basis for the decision-making of operation and maintenance personnel.

[0079] The results output module is used to output the analysis results of each short-circuit step.

[0080] Specifically, the results output module is the system's data outlet, responsible for outputting the calculation results of each module to the user in a structured and visual form, supporting multiple output formats and display methods.

[0081] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. An analytical method for short-circuit analysis of a hollow shunt reactor, characterized in that, Includes the following steps: S1. Constructing the air-core reactor winding Coupled inductor matrix Inductor matrix off-diagonal elements For nodes With nodes The mutual intuition between them, at this time Inductor matrix diagonal elements For nodes The self-awareness at this moment ; in, This represents the total number of winding nodes, where a node is either a winding layer node or an expanded winding turn node. When performing inter-layer analysis, the nodes are winding layer nodes, and each node corresponds to one winding layer; When inter-turn analysis of the target coil layer is required, replace the individual layer nodes of the target coil layer with Each independent turn node yields the expanded matrix dimension. ,in, The number of turns in this layer; the coil to which this target coil layer belongs is the target coil. The nodes in the expanded matrix are arranged in the following order: S11, the nodes of all layers of all coils preceding the target coil; S12, All turn nodes of the split layer in the target coil; S13, nodes of the remaining layers in the target coil, excluding the split layer; S14, the nodes of each layer of all coils after the target coil; S2. Determine the set of short-circuit nodes. and the set of remaining nodes ,in, Constituting all nodes, and It is an empty set; When performing inter-layer analysis, starting from the initial layer, one layer node is added to the short-circuit node set at each step. until the termination layer; When performing inter-turn analysis, the adaptive inter-turn asymptotic short-circuit procedure is as follows: when When the value is ≤10, each turn is added to the short-circuit node set in turn. Each step performs a Kron reduction operation once; when When the value is >10, the first to tenth turns are added sequentially, then the turns are added in 5-turn increments, and the last turn is added in the last step. A Kron reduction operation is performed once in each step. S3. Based on the remaining node set and short-circuit node set For inductor matrix Divide into blocks to obtain submatrices , , and ; in, The remaining nodes are the inductance submatrix. This is the mutual inductance submatrix between the remaining nodes and the short-circuited nodes. The inductance submatrix between short-circuit nodes, for transpose; S4. Calculate the reduced-order inductance matrix using Krone reduction operations. ,in for Moore-Penrose pseudo-inverse matrix; S5, based on the reduced-order inductance matrix Calculate the residual inductance value. The calculation method is as follows: for the reduced-order inductance matrix Summing all elements, i.e. ; S6. Calculate the inductance loss percentage based on the ratio of the residual inductance value to the initial inductance value, and classify the severity of the fault according to the inductance loss percentage.

2. The analytical method for short-circuit analysis of hollow parallel reactors according to claim 1, characterized in that, Initial inductance value The calculation method is as follows: for the original complete inductance matrix Summing all elements, i.e.: ; Based on the initial inductance value and residual inductance value Calculate the percentage of inductor loss. The formula is .

3. The analytical method for short-circuit analysis of hollow parallel reactors according to claim 2, characterized in that, The severity of the fault is classified according to the percentage of inductor loss as follows: A loss percentage of less than 10% indicates low risk. A loss percentage of 10% or less and less than 30% is considered medium risk. A loss percentage of 30% or less and less than 60% is considered high-risk. A loss percentage of ≥60% is considered a critical risk.

4. The analytical method for short-circuit analysis of hollow parallel reactors according to claim 1, characterized in that, It also includes the steps for calculating short-circuit fault current: S21, According to the inductance matrix and node DC resistance vector Build Complex impedance matrix diagonal elements off-diagonal elements ; in, The imaginary unit, The system angular frequency, For nodes DC resistance; For nodes The self-awareness at this moment , For nodes With nodes The mutual intuition between them, at this time ; S22, According to rated voltage Total resistance of remaining nodes and residual resistance Calculate the normal operating current ; in, , This is the residual inductance value; S23. Based on the remaining node set and short-circuit node set For complex impedance matrix Divide the circuit into blocks and establish the short-circuit current equation. ; in, The impedance submatrix between short-circuit nodes. This is the coupling impedance submatrix between the short-circuited node and the remaining nodes. For length is The whole vector; S24. Solve the linear equations to obtain the fault current phasors of each short-circuit node. The maximum value of the fault current amplitude at all short-circuit nodes is taken as the short-circuit current. ; in, .

5. The analytical method for short-circuit analysis of a hollow parallel reactor according to claim 4, characterized in that, When linear equations When the singularity or condition number exceeds a preset threshold, the least squares method is used to solve the problem. .

6. The analytical method for short-circuit analysis of a hollow parallel reactor according to claim 4, characterized in that, When performing inter-turn analysis, the DC resistance of each turn in the split layer is distributed in an equal manner: ; in, The DC resistance of the split layer. This represents the number of turns in this layer.

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