CN109818363B - Comprehensive safety correction method considering multiple rapid control means - Google Patents

Comprehensive safety correction method considering multiple rapid control means Download PDF

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CN109818363B
CN109818363B CN201910187721.0A CN201910187721A CN109818363B CN 109818363 B CN109818363 B CN 109818363B CN 201910187721 A CN201910187721 A CN 201910187721A CN 109818363 B CN109818363 B CN 109818363B
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林涛
毕如玉
陈汝斯
郭紫昱
周学明
徐遐龄
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Wuhan University WHU
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Abstract

The invention relates to a comprehensive safety correction method considering various rapid control means. Aiming at the requirement of reducing the control cost of a correction scheme, the comprehensive adoption of a plurality of rapid power flow control means in the safety correction is proposed: the controllable series compensation device compensation degree adjustment and the modulation of the multiterminal direct current power are switched to the transmission line, specifically include: 1) a safety correction optimization model of a variable topological structure power grid containing TCSC and MTDC based on mixed integer second-order cone programming is provided, and an accurate correction scheme can be obtained; 2) a heuristic line selection method based on branch on-off comprehensive influence indexes and an accelerated solving strategy of an induced objective function are provided, and solving efficiency is greatly improved. The invention has the beneficial effects that: the method can quickly obtain the accurate and low-control-cost safety correction scheme, and has important reference value for establishing the safety correction scheme on line.

Description

Comprehensive safety correction method considering multiple rapid control means
Technical Field
The invention belongs to the technical field of safety correction methods of alternating current and direct current hybrid power systems, and particularly relates to a comprehensive safety correction method taking various rapid control means into consideration based on a second-order cone.
Background
With the increase of the transmission capacity of the power grid, the power flow transfer caused by the faults of the alternating current lines or the direct current blocking and the like in the power grid is more serious, and even cascading faults can occur. And with the access of new energy power sources such as high-proportion wind power, photovoltaic and the like and the development of power markets, the operation mode of a power system becomes more variable. Under the background that the operation mode of the power system is complicated and changeable, if a corresponding safety correction scheme is still only made off-line for a typical operation mode, the correction scheme may have greater conservation. Therefore, it is necessary to obtain an accurate safety correction scheme through online calculation based on the current operation mode of the system.
Generally, the correction means of the safety correction scheme is to adjust the output of the generator and to cut off the load, with limited adjustment effect. And the adjustment of the generator cannot be effected quickly, and the adjustment process is long, which may increase the risk of cascading failure. It is therefore necessary to consider other rapid control means in the safety correction to reduce the amount of load shedding and shorten the recipe adjustment time. Further, safety correction schemes are usually established for fixed topologies, but main transmission lines are generally provided with devices such as circuit breakers, and conditions for changing the power grid topology are provided. In recent years, students have found that Transmission Switching (TS) is effective for tidal current regulation, and can significantly reduce the out-of-limit degree of a power grid, even eliminate the out-of-limit, so TS has been used as one of the means for safety correction in some practical power grids abroad.
After the TS is considered, the power grid topological structure is variable, the on-off state of each line becomes a binary variable of 0-1, and the complexity of a power flow model is increased remarkably. To simplify the calculation, the scholars usually adopt a linearization method, such as building a power flow optimization model considering TS based on a direct current power flow model. However, the power system itself is a highly nonlinear system, and especially in the case of a variable topology structure, the accuracy of the dc power flow model is difficult to guarantee, which may cause deviation in the adjustment effect of the obtained correction scheme. Therefore, it is necessary to consider constructing a corresponding safety correction optimization model based on the alternating current power flow model to obtain a more accurate safety correction scheme. However, for the nonlinear optimization model containing a large amount of 0-1 binary variables, the solving efficiency is generally low, and the online application is difficult. Therefore, a method for rapidly solving the model needs to be deeply researched, and support is provided for online application of comprehensive safety correction.
Disclosure of Invention
The invention provides a safety correction method considering various quick control means, which can quickly obtain an accurate safety correction scheme with low control cost. Aiming at the defects of high control cost and high conservation of a scheme formulated by a traditional method, three rapid power flow control means are proposed for reducing the load shedding amount: transmission line switching, TCSC (controllable Series compensator) compensation degree adjustment and Multi-terminal Direct Current (MTDC) power modulation. Aiming at the defect that a conventional linear optimization model may cause large deviation, in Order to obtain an accurate correction scheme, a safety correction optimization model of a variable topology structure power grid containing TCSC and MTDC is built based on a Mixed Integer Second Order Cone Programming (MISOCP) based on an alternating current power flow model. Aiming at the defect of low solving efficiency of a mixed integer programming model, in order to quickly solve the optimization model, a heuristic line selection method based on branch on-off comprehensive influence indexes and an accelerated solving strategy of an induced objective function are provided, so that the solving efficiency is greatly improved.
The technical problem of the invention is mainly solved by the following technical scheme:
a comprehensive safety correction method considering a plurality of rapid control means is characterized by comprising the following steps:
step 1, according to the current fault form, selecting the branch which is allowed to be disconnected in the optimization by adopting a heuristic line selection method based on the branch disconnection comprehensive influence index to form a set SL.
Step 2, according to the current fault form, adopting an accelerated solving strategy of an induced objective function, and according to a branch on-off comprehensive influence index IF l And constructing corresponding objective function additional terms to accelerate the solving speed of the optimization model.
Additional terms of the objective function are:
Figure BDA0001993400740000031
wherein the content of the first and second substances,
Figure BDA0001993400740000032
wherein SL is selected in step 1The set of branches that are allowed to be broken in the optimization. C v Is IF l Magnification/reduction factor of the index. This is to keep the additional terms of the objective function within reasonable bounds without affecting the optimization result. H l Is the open or closed state of the circuit breaker of line i. IF (intermediate frequency) circuit l,n Is to mix IF l After the indexes are sorted from small to large, the numerical value of the index at the nth position is ranked. C d Is the maximum difference between the allowed and original objective functions. NL denotes the maximum number of lines allowed to be disconnected in the correction scheme.
And 3, constructing a corresponding safety correction optimization model taking TS, TCSC and MTDC into consideration and adjusting aiming at the power grid which contains TCSC and MTDC and has a variable topological structure based on mixed integer second-order cone programming. The method specifically comprises decision variables, an objective function and constraint conditions.
And 4, solving the safety correction optimization model which is provided in the step 3 and takes various quick control means into account. The proposed model is a standard mixed integer second order cone programming mathematical model that can be solved by well-established commercial software such as CPLEX, GUROBI, MOSEK, etc.
In the above comprehensive safety correction method taking into account multiple fast control means, in step 1, the specific method of collecting SL includes:
calculating branch circuit disconnection comprehensive influence index IF of line l disconnection on all out-of-limit branch circuit flows l And sorting from small to large, selecting IF l The branches with the smallest indexes (less than 0 and the largest absolute values) are used as the branches to be disconnected, and a branch set SL allowing to be disconnected is constructed. IF (intermediate frequency) circuit l The index can reflect the comprehensive influence effect on the power flow of all the out-of-limit branches k after the line l is disconnected on the whole, and the calculation formula is as follows:
Figure BDA0001993400740000041
where OL is the set of out-of-limit legs. P l Is the power of line l. P k Is the power of line k whose power flow is out of limit.
Figure BDA0001993400740000042
Is the maximum power allowed by the line k with the power flow out of limit. D k,l The index is a branch Outage Distribution factor (LODF) index of the branch k when the branch l is in Outage state. When calculating LODF, the actual direction of the power flow needs to be taken as the positive direction.
In step 3, the comprehensive safety correction method taking into account a plurality of fast control means specifically includes the following steps:
and 3.1, the value of the decision variable of the safety correction optimization model determines the specific adjustment quantity of the corresponding adjustment measure in the finally obtained safety correction scheme. In the proposed model, decision variables are divided into a continuous type and a 0-1 binary type.
1) Continuous type variable: these variables correspond to devices that can be continuously adjusted. The method comprises the following specific steps: active and reactive power P generated by the generator of node i Gi ,Q Gi (ii) a Active and reactive power P injected into alternating current system by MTDC converter a CONVa 、Q CONVa (ii) a Intermediate variable K of degree of compensation of TCSC device installed on line ij between node i and node j TCij . Note that K is obtained by calculation TCij Then, the compensation k of the TCSC device is inversely solved according to the following formula c,ij As an adjustment amount for the TCSC in the calibration scheme.
k c,ij =K TCij /(1-K TCij )
2) Binary type 0-1 variable: these variables correspond to devices with only two states. The method specifically comprises the following steps: open or closed state H of circuit breaker of line ij ijH ij 1 denotes breaker closed, H ij 0 represents breaker open; cut-out or retention state L of the kth load i-k on node i di-kL di-k 1 denotes load retention, L di-k 0 indicates that the load is cut off.
Step 3.2 the goal of the security correction optimization model is to keep as much load as possible. If a plurality of sets of schemes with the same load cutting amount exist, a scheme of adjusting the generator as few as possible is selected. The objective function of the security correction scheme is:
Figure BDA0001993400740000051
where Gen is the set of generators. Load is a set of loads. P Gi,0 The generator, which is the pre-fault node i, injects active power. n is a radical of an alkyl radical d Is the number of independent loads on each load node. P LDi-k Is the active power of the load i-k. M is a large penalty factor, such as 1000.
Considering the accelerated solution strategy for the induced objective function proposed in step 2, the objective function taken when solving the model is:
Figure BDA0001993400740000052
the objective function is only adopted during optimization, the objective function additional item is removed when the objective function of the safety correction scheme is finally calculated, and only the generator adjustment amount and the load shedding amount are considered.
Step 3.3 the constraints of the safety corrected optimization model are as follows:
Figure BDA0001993400740000053
Figure BDA0001993400740000054
Figure BDA0001993400740000055
Figure BDA0001993400740000056
Figure BDA0001993400740000057
Figure BDA0001993400740000058
Figure BDA0001993400740000059
Figure BDA00019934007400000510
Figure BDA00019934007400000511
Figure BDA00019934007400000512
Figure BDA00019934007400000513
Figure BDA00019934007400000514
P TCi,ij =T ij /x ij ,Q TCi,ij =(U i,ij -F ij )/x ij ,P TCj , ij =-T ij /x ij ,Q TCj , ij =(U j , ij -F ij )/x ij
Figure BDA00019934007400000515
Figure BDA0001993400740000061
Figure BDA0001993400740000062
Figure BDA0001993400740000063
Figure BDA0001993400740000064
Figure BDA0001993400740000065
Figure BDA0001993400740000066
Figure BDA0001993400740000067
Figure BDA0001993400740000068
Figure BDA0001993400740000069
Figure BDA00019934007400000610
Figure BDA00019934007400000611
G i ≥|P Gi -P Gi,0 |,G i ∈{0,1},∑G i ≤N G
sum(1-H ij )≤NL,H ij ∈{0,1}
wherein, P ij 、Q ij Is the active and reactive power flowing from node i to node j. g ij And b ij Is the conductance and susceptance of line ij. b i,ij 、b j , ij Is the equivalent pair-ground susceptance of the i, j sides of line ij. V i 、V j Is the voltage at node i, j. The variable plus a dash or an underline indicates the upper or lower limit of the variable, respectively. W i 、W j Is the square of the voltage at nodes i, j,
Figure BDA00019934007400000612
and
Figure BDA00019934007400000613
is an auxiliary variable defined to adapt to the TS. W Cij And W Sij Is an auxiliary variable associated with line ij. Theta i 、θ j Representing the voltage phase angle at nodes i, j, respectively. Theta.theta. ij Is the voltage phase angle difference of line ij.
Figure BDA00019934007400000614
Is a defined auxiliary variable. r is ij And x ij Is the resistance and reactance of line ij. P Gi 、Q Gi 、S Gi The generated active power, reactive power and apparent power of the node i. P LDi 、Q LDi Is the load active and reactive power of node i. δ (i) is the set of nodes adjacent to node i. η (i) is the set of nodes on the other end of the TCSC tributary with node i. P TCi,ij 、P TCj , ij After the TCSC is accessed, the equivalent slave nodes i and j inject the active power of the line ij. Q TCi,ij 、Q TCj,ij Is the equivalent reactive power injected into line ij from nodes i and j. T is ij 、Fi j 、U i,ij And U j , ij Is an auxiliary variable for the TCSC tributary. P CONVa 、Q CONVa 、S CONVa Is the power flowing from the dc converter a into the ac grid. P DCa Is to change from a DC networkThe power of current a. Beta is a a Is the loss factor of converter a. P ab,DC Is the power flowing from node a to b in the dc network. δ (a) represents the set of nodes b adjacent to node a in the dc network. P is ls,DC Representing the power loss of the dc link ab. W DCa Representing the square of the voltage at node a. R ab,DC Representing the resistance of the dc line ab. G i And whether the generator is adjusted or not is shown, the value of 1 shows that the generator on the node i is adjusted, and 0 shows that the generator is not adjusted. N is a radical of G Is the total number of generators allowed to be adjusted.
According to the comprehensive safety correction method considering various rapid control means, the TCSC, the MTDC and the TS are adjusted, the solving efficiency of the model is further improved, and the generation time of a safety correction scheme is shortened from two aspects of reducing the number of binary variables and guiding the solving direction. According to the method provided by the invention, a correction scheme with small load shedding amount and accuracy can be quickly obtained.
Drawings
Fig. 1 is a TS-considered branch power model provided by the present invention.
Fig. 2 is a standard IEEE 57 node wiring diagram provided by the present invention.
Fig. 3 is a schematic diagram of the MTDC location in the modified IEEE 57 node algorithm used for the algorithm verification provided by the present invention.
Detailed Description
The technical scheme of the invention is further specifically described by the following embodiments and data analysis.
An embodiment of a comprehensive security correction method taking into account a plurality of rapid control means:
on the basis of the standard IEEE 57 node calculation example shown in fig. 2, the multi-terminal dc transmission system and two TCSC devices shown in fig. 3 are added to verify the control effect of these fast control means. TCSC devices are installed on line 1 (node 9-node 13) and line 2 (node 9-node 11), respectively.
The line 15 and the line 18 are described as an example of a fault trip.
Based on the method proposed in step 1, selectingComprehensive influence index IF for selecting branch circuit to be cut off l The minimum 10 lines are lines 28, 14, 27, 13, 19, 66, 16, 23, 65 and line 4, and only 1 of the 10 lines is allowed to be disconnected during the optimization process.
In order to verify the adjusting effect of the three rapid correcting methods provided by the invention, a group of comparison scenes is set. Scenario 1 is the result based entirely on the method of the present invention, and scenario 2 is the result when three fast correction means, TS, TCSC and MTDC, are not considered. The resulting security correction schemes for scenario 1 and scenario 2 are shown in table 1.
Table 12 safety correction scheme under group scene
Figure BDA0001993400740000081
As can be seen from the above table, the objective function of the security correction scheme of scenario 1 is much smaller than scenario 2. This shows that the method provided by the invention can significantly reduce the control cost.
In order to verify that the method provided by the invention can effectively eliminate the out-of-limit, the power flow verification is carried out on the basis of Matpower software. The flow results obtained after implementing the safety correction scheme based on the proposed second order cone optimization model (denoted as SOCP in table 2) and based on the Matpower software (denoted as Matpower) are shown in table 2. Only the load flow results for the legs with out-of-limit load flow and TCSC legs are listed, limited to space.
TABLE 2 tidal flow verification results
Figure BDA0001993400740000082
Figure BDA0001993400740000091
As can be seen from table 2, the proposed method results in a solution that is effective in eliminating the out-of-limit. And the trend result is more consistent with the result of the standard trend algorithm, and the accuracy of the method is verified.
Further, in order to verify the effectiveness of the proposed heuristic route selection method (acceleration method 1 for short) based on branch-off comprehensive influence indexes and the accelerated solving strategy (acceleration method 2 for short) of the induced objective function, two sets of scenes are additionally arranged. Scenario 3 is based on scenario 1, and the two accelerated solution methods are removed. Scenario 3 is based on scenario 1, and the acceleration method 2 is removed, i.e. only the acceleration method 1 is adopted. The objective function and optimization time pairs for the three sets of scenarios are shown in table 3.
Table 3 verification of the effectiveness of the acceleration method
Figure BDA0001993400740000092
As can be seen from table 3, the heuristic line selection method based on the branch breaking comprehensive influence index and the accelerated solving strategy of the induced objective function can significantly improve the solving efficiency on the premise of ensuring the accuracy of the solving result.
In conclusion, the method provided by the invention can quickly obtain the accurate safety correction scheme with low control cost, and has important reference value for establishing the safety correction scheme on line.
The specific embodiments described herein are merely illustrative of the spirit of the invention. Various modifications or additions may be made to the described embodiments or alternatives may be employed by those skilled in the art without departing from the spirit or ambit of the invention as defined in the appended claims.

Claims (2)

1. A comprehensive safety correction method considering a plurality of rapid control means is characterized by comprising the following steps:
step 1, selecting branches which are allowed to be disconnected in optimization by adopting a heuristic line selection method based on branch disconnection comprehensive influence indexes according to a current fault form to form a set SL;
step 2, according to the current fault form, adopting an accelerated solving strategy of an induced objective function, and obtaining the rootAccording to branch on-off comprehensive influence index IF l Constructing corresponding objective function additional terms to accelerate the solving speed of the optimization model;
additional terms of the objective function are:
Figure FDA0003739312210000011
wherein the content of the first and second substances,
Figure FDA0003739312210000012
wherein SL is the set of branch selected in step 1 that is allowed to be disconnected in the optimization; c v Is IF l Magnification/reduction factor of the index; this is to make the additional term of the objective function in a reasonable range without affecting the optimization result; h l Is the open or closed state of the circuit breaker of line l; IF (intermediate frequency) circuit l,n Is to mix IF l After the indexes are sorted from small to large, arranging the numerical value of the index at the nth position; c d Is the maximum difference between the allowed and original objective functions; NL denotes the maximum number of lines allowed to be disconnected in the correction scheme;
step 3, constructing a corresponding safety correction optimization model taking TS, TCSC and MTDC into consideration and adjusting aiming at a power grid which contains TCSC and MTDC and has a variable topological structure based on mixed integer second-order cone programming; the method specifically comprises the steps of decision variables, objective functions and constraint conditions;
step 4, solving the safety correction optimization model considering various quick control means provided in the step 3;
in step 3, the specific method is as follows:
step 3.1, the value of the decision variable of the safety correction optimization model determines the specific adjustment quantity of the corresponding adjustment measure in the finally obtained safety correction scheme; in the proposed model, decision variables are divided into a continuous type and a binary type of 0-1;
1) continuous type variable: these variables correspond to devices that can be continuously adjusted; the method specifically comprises the following steps: sending of node iActive and reactive power P generated by motor Gi ,Q Gi (ii) a Active and reactive power P injected into alternating current system by MTDC converter a CONVa 、Q CONVa (ii) a Intermediate variable K of degree of compensation of TCSC device installed on line ij between node i and node j TCij (ii) a Note that K is obtained by calculation TCij Then, the compensation degree k of the TCSC device needs to be solved reversely according to the following formula c,ij As an adjustment amount of the TCSC in the calibration scheme;
k c,ij =K TCij /(1-K TCij )
2) binary variable 0-1: these variables correspond to devices with only two states; the method comprises the following specific steps: open or closed state H of circuit breaker for line ij ij ,H ij 1 denotes breaker closed, H ij 0 represents breaker open; cut-out or retention state L of the kth load i-k on node i di-k ,L di-k 1 denotes load retention, L di-k 0 indicates that the load is cut off;
step 3.2 the goal of the security correction optimization model is to keep as much load as possible; if a plurality of sets of schemes with the same load cutting amount exist, a scheme for regulating the generator as few as possible is selected; the objective function of the security correction scheme is:
Figure FDA0003739312210000021
wherein Gen is the set of generators; load is the set of loads; p Gi,0 The generator which is the node i before the fault injects active power; n is a radical of an alkyl radical d Is the number of independent loads on each load node; p is LDi-k Is the active power of load i-k; m is a large penalty factor;
considering the accelerated solution strategy for inducing the objective function proposed in step 2, the objective function adopted when solving the model is:
Figure FDA0003739312210000031
the objective function is only adopted during optimization, the objective function additional item is removed when the objective function of the safety correction scheme is finally calculated, and only the generator adjustment amount and the load shedding amount are considered;
step 3.3 the constraints of the safety corrected optimization model are as follows:
Figure FDA0003739312210000032
Figure FDA0003739312210000033
Figure FDA0003739312210000034
Figure FDA0003739312210000035
Figure FDA0003739312210000036
Figure FDA0003739312210000037
Figure FDA0003739312210000041
Figure FDA0003739312210000042
Figure FDA0003739312210000043
Figure FDA0003739312210000044
Figure FDA0003739312210000045
Figure FDA0003739312210000046
P TCi,ij =T ij /x ij ,Q TCi,ij =(U i,ij -F ij )/x ij ,P TCj,ij =-T ij /x ij ,Q TCj,ij =(U j,ij -F ij )/x ij
Figure FDA0003739312210000047
Figure FDA0003739312210000048
T ij >=W sij K TCij +W sij K TCij -H ij W sij K TCij ,
Figure FDA0003739312210000049
Figure FDA00037393122100000410
F ij >=W cij K TCij +W cij K TCij -H ij W cij K TCij ,
Figure FDA00037393122100000411
Figure FDA00037393122100000412
Figure FDA00037393122100000413
Figure FDA00037393122100000414
Figure FDA00037393122100000415
Figure FDA00037393122100000416
W DCa -W DCb =2R ab,DC P ab,DC -R ab,DC P ls,DC
Figure FDA00037393122100000417
Figure FDA00037393122100000418
G i ≥|P Gi -P Gi,0 |,G i ∈{0,1},∑G i ≤N G
sum(1-H ij )≤NL,H ij ∈{0,1}
wherein, P ij 、Q ij Is the active and reactive power flowing from node i to node j; g ij And b ij Is the conductance and susceptance of line ij; b i,ij 、b j,ij Is the equivalent pair-ground susceptance of the i-side, j-side of line ij; v i 、V j Is the voltage of node i, j; the variable plus a dash or an underline respectively represents the upper limit or the lower limit of the variable; w i 、W j Is the square of the voltage at nodes i, j,
Figure FDA00037393122100000419
and
Figure FDA00037393122100000420
is an auxiliary variable defined to adapt to TS; w Cij And W Sij Is an auxiliary variable associated with line ij; theta i 、θ j Respectively representing voltage phase angles of the nodes i and j; theta ij Is the voltage phase angle difference of line ij; theta.theta. ij r Is a defined auxiliary variable; r is a radical of hydrogen ij And x ij Is the resistance and reactance of line ij; p is Gi 、Q Gi 、S Gi Is the active, reactive and apparent power of the node i; p LDi 、Q LDi Is the load active and reactive power of node i; δ (i) is the set of nodes adjacent to node i; η (i) is the set of other end nodes with node i being one end of the TCSC tributary; p TCi,ij 、P TCj,ij After TCSC is accessed, equivalent slave nodes i and j inject active power of a line ij; q TCi,ij 、Q TCj,ij Is equivalent reactive power injected into line ij from nodes i and j; t is ij 、F ij 、U i,ij And U j,ij Is an auxiliary variable for the TCSC tributary; p is CONVa 、Q CONVa 、S CONVa Is the power flowing into the ac grid from the dc converter a; p DCa Is the power flowing into converter a from the dc network; beta is a beta a Is the loss factor of converter a; p ab,DC Is the power flowing from node a to b in the dc network; δ (a) represents a set of nodes b adjacent to node a in the dc network; p ls,DC Means for indicating straightPower loss of flow line ab; w DCa Represents the square of the voltage at node a; r ab,DC Represents the resistance of the dc line ab; g i Whether the generator is adjusted or not is shown, the value of the generator is 1, the generator on the node i is adjusted, and 0 is not adjusted; n is a radical of G Is the total number of generators allowed to be adjusted.
2. A comprehensive safety correction method in consideration of multiple fast control means according to claim 1, wherein in step 1, the specific method of forming the set SL comprises:
calculating branch circuit disconnection comprehensive influence index IF of line l disconnection on all out-of-limit branch circuit flows l And sorting from small to large, selecting IF l Several branches with indexes smaller than 0 and the largest absolute value are used as branches to be cut off, and a branch set SL allowing to be cut off is constructed; IF (intermediate frequency) circuit l The index can reflect the comprehensive influence effect on the power flow of all out-of-limit branches k after the line l is disconnected on the whole, and the calculation formula is as follows:
Figure FDA0003739312210000051
where OL is the set of out-of-limit legs; p l Is the power of line l; p is k Is the power of line k whose power flow is out of limit;
Figure FDA0003739312210000052
is the maximum power allowed by the line k with the power flow out of limit; d k,l The branch disconnection distribution factor index of the branch k is compared with the branch l disconnection distribution factor index; when calculating LODF, the actual direction of the power flow needs to be taken as the positive direction.
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