CN111614082B - Electric power system security domain boundary searching method based on Lagrange multiplier - Google Patents

Electric power system security domain boundary searching method based on Lagrange multiplier Download PDF

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CN111614082B
CN111614082B CN202010448662.0A CN202010448662A CN111614082B CN 111614082 B CN111614082 B CN 111614082B CN 202010448662 A CN202010448662 A CN 202010448662A CN 111614082 B CN111614082 B CN 111614082B
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CN111614082A (en
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姜涛
李雪
张琳玮
陈厚合
李国庆
张儒峰
王长江
李本新
于洋
王振浩
辛业春
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Northeast Electric Power University
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/10—Power transmission or distribution systems management focussing at grid-level, e.g. load flow analysis, node profile computation, meshed network optimisation, active network management or spinning reserve management
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
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Abstract

The invention discloses a method for searching a security domain boundary of an electric power system based on a Lagrange multiplier, which comprises the following steps: setting an initial power increasing direction, and establishing an SR critical point search optimization model; solving an SR critical point search optimization model by combining a displacement splitting preprocessor with local correction and a double conjugate gradient stabilization method to obtain an initial SR critical point; according to the characteristic that adjacent critical points on the SRB are in a limited neighborhood, SR critical point sets are searched through a boundary tracking algorithm and are sequentially connected to realize the search of the SRB.

Description

Electric power system security domain boundary searching method based on Lagrange multiplier
Technical Field
The invention relates to the field of static security domains, in particular to a method for searching a security domain boundary of an electric power system based on a Lagrange multiplier.
Background
With the increasing of the total capacity of the power system and the expanding of the network structure, the operation condition of the system becomes more and more tense, which brings a severe test to the safe and stable operation of the power grid[1]-[3]. And a static security domain (SR)[4]-[5]The method is an intuitive and effective power grid safety monitoring, defense and control method, and plays an extremely important role in the field of power system safety and stability analysis.
The static security domain of the power system refers to a set of all operating points which simultaneously satisfy a power flow equation and static security constraints in a power injection space[6]. According to different static safety and stability research objects of the power system, the power system SR can be divided into: a Thermal Security Region (TSR), a voltage security region (VCR), and a Voltage Stability Region (VSR). The current safe operation state of the system can be qualitatively evaluated by judging the relative position of the current operation point of the system and a Security Region Boundary (SRB), so that the search of the SRB is important for constructing the SR.
Hyperplane approximation[7]-[9]The method is a common method for searching the SRB of the power system, and the core idea of the method is to search a large number of SR critical points in a power injection space and approximate the SRB by the SR critical points. At present, the common search method for the SR critical point of the power system mainly has a limit calculation method[10]-[11]And Optimal Power Flow (OPF) method[12]. The limit calculation method is based on a power flow equation, carries out point-by-point iteration according to the determined power increasing direction, calculates the static safety margin of the system under the condition of meeting specific safety constraints, and evaluates the safety stability of the system through the static safety margin. The method comprises the steps of respectively establishing an SR critical point optimization model under a specific safety constraint condition according to different research objects by an OPF method, and obtaining the SR critical point by solving the optimization model. Both the limit calculation method and the OPF method search a series of SR critical point sets that satisfy certain safety constraints within the power injection space and highly approximate the SRB with these SR critical points.
However, in practical applications, the above method has the following disadvantages:
firstly, the SRB topological characteristics of different research objects are different, and the method is lack of universality; the continuous expansion of the system scale can increase the calculation burden of the method, so that the requirement for quickly constructing the SRB of the regional interconnection power system is difficult to meet; and thirdly, if all nodes and branches in the system are searched for corresponding SRBs to obtain the SRBs of the system, the calculated amount is huge and the applicability is poor. Therefore, the research on how to establish the SRB universal search model and quickly and accurately search the SRB according to the actual power system requirements has important significance on the online monitoring and evaluation of the safety and stability of the power system.
Reference to the literature
[1] Chenggang, Bayong, Zhao Jinquan, etc. evaluation of the probability of voltage stabilization of a large grid taking into account the randomness of load growth [ J ] protection and control of electric power systems, 2018,46(23):37-44.
[2] Ding Ming, Wang Wei Sheng, Wang Xiuli, etc. the influence of large-scale photovoltaic power generation on electric power systems is reviewed in [ J ]. Chinese Motor engineering Proc., 2014,34(1):1-14.
[3] Xue Yu Sheng, Lei xing, Xue Feng, etc. comments on the influence of wind power uncertainty on the power system [ J ]. Chinese Motor engineering reports, 2014,34 (29):5029-5040.
[4]Yang T,Yu Y.Steady-State Security Region-Based Voltage/Var Optimization Considering Power Injection Uncertainties in Distribution Grids[J].IEEE Transactions on Smart Grid,2019,10(3):2904-2911.
[5] Yangming, Chengming, Hanchong mountain, an effective static security domain method for real-time scheduling of an electric power system [ J ]. Chinese Motor engineering journal 2015,35(6): 1353-.
[6] Yu Shi Xin, electric power system security domain method review [ J ] Tianjin university bulletin, 2008,41(6): 635-.
[7] Wangfei, Yu Shi Xin, electric power system thermal stability security domain [ J ] based on wide area measurement system, proceedings of China Motor engineering, 2011,31(10):33-38.
[8] The power distribution system safety domain model [ J ] based on trend calculation, China Motor engineering report, 2017, 37(17):4941-4949.
[9]Yiwei Qiu,Hao Wu,Yongzhi Zhou.et al.Global Parametric Polynomial Approximation of Static Voltage Stability Region Boundaries[J].IEEE Transactions on Power Systems,2017,32(3):2362-2371.
[10] Lepeng, Suyin Sheng, Li construction, etc. are mostly security and calculation discussion of southern power grid technology in the mode of southern power grid, 2011,5(6):7-15.
[11] The strategy research on improving the robustness of continuous trend computing is carried out on Zhao Jinquan, Zenming, China Motor engineering journal, 2005,25(22):7-11.
[12] Tanshiqiang, a quick search method for static security domain boundaries of an electric power system, research [ D ]. northeast electric power university, 2019.
Disclosure of Invention
The invention provides a method for searching the boundary of a security domain of an electric power system based on a Lagrange multiplier, which not only inherits the characteristic of high-precision search of SR critical points of a traditional OPF model, but also avoids the calculation burden of searching the SR critical points by an OPF method, effectively reduces the time consumption for searching a single SR critical point, obviously improves the construction efficiency of an SRB (sequence-oriented binary search) and is described in detail in the following description:
a lagrangian multiplier based power system security domain boundary searching method, the method comprising:
setting an initial power increasing direction, and establishing an SR critical point search optimization model;
solving an SR critical point search optimization model by combining a displacement splitting preprocessor with local correction and a double conjugate gradient stabilization method to obtain an initial SR critical point;
according to the characteristic that adjacent critical points on the SRB are in a limited neighborhood, SR critical point sets are searched through a boundary tracking algorithm and are sequentially connected, so that the SRB search is realized.
Further, the power system security domain boundary comprises: a thermally stable security domain boundary, and a quiescent voltage security domain boundary;
the SR critical point search optimization model comprises: a thermal stability safety domain critical point searching optimization model and a static voltage safety domain critical point searching optimization model;
the SR critical points include: a thermally stable safety domain critical point, and a quiescent voltage safety domain critical point.
The thermal stability safety domain critical point search optimization model specifically comprises the following steps:
Figure BDA0002506721590000031
where λ is the static safety margin in the power increase direction d; x is the number of0A vector of initial state variables; y is0Injecting vectors for node power under a ground state; plThe active transmission capacity of the key branch l;
Figure BDA0002506721590000032
is the active transmission limit of branch l; f (x)0,y0) Is a conventional power flow equation of a power system.
Further, the solving of the stable security domain critical point search optimization model specifically includes:
(1) LMSS pretreatment is carried out on the Newton iterative correction equation set, and the specific process is as follows:
the coefficient matrix A is processed by block division, and the (1, 1) block matrix A is processed11Performing Hermitian splitting to obtain a Hermitian matrix H;
generate an LMSS preprocessor and note the preprocessor as psiLMSS；
Using LMSS preconditioner ΨLMSSPreprocessing the Newton iteration correction equation set to obtain an equivalent linear equation set of the Newton iteration correction equation set;
(2) solving an equivalent linear equation set by adopting a BICGSAB iterative method to obtain a vector eta to be solved; correcting the quantity x, lambda, mu and u to be solved; if the convergence accuracy is met, Newton iteration is finished, the quantities x, lambda, mu and u to be solved are output, and the calculation of the TSR critical point in the power increasing direction d is finished.
The static voltage safety domain critical point search optimization model specifically comprises the following steps:
Figure BDA0002506721590000041
in the formula, λ is a static safety margin of the system in the power increasing direction d; x is the number of0A vector of system initial state variables; y is0Injecting vectors for node power under a ground state; vkIs the voltage amplitude of node k;
Figure BDA0002506721590000042
is a set node k voltage magnitude limit.
Further, the solving of the search optimization model for the critical points of the static voltage safety domain specifically includes:
(1) and performing LMSS pretreatment on the other Newton iteration correction equation set, wherein the specific process is as follows:
the coefficient matrix B is processed by block division, and the (1, 1) block matrix B of the coefficient matrix B is processed11Performing Hermitian splitting to obtain a Hermitian matrix H;
generate an LMSS preprocessor and designate the preprocessor as ΨLMSSUsing LMSS preconditioner ΨLMSSPreprocessing the other Newton iteration correction equation set to obtain another equivalent linear equation set;
(2) solving the other linear equation set after the preprocessing by adopting a BICGSAB iterative method to obtain a vector y to be solved; correcting the quantity x, lambda, mu and u to be solved; if the convergence accuracy is met, Newton iteration is finished, the quantities x, lambda, mu and u to be solved are output, and the calculation of the VCR critical point in the power increasing direction d is finished.
The technical scheme provided by the invention has the beneficial effects that:
1. the method can effectively avoid the complex iterative optimization process of the OPF, thereby greatly reducing the time consumption for searching the critical points of the Thermal Stability Region Boundary (TSRB) and the static voltage region boundary (VCRB) of the power system and realizing the rapid construction of the SRB of the power system;
2. compared with the existing SRB construction method based on the OPF, the method not only greatly improves the construction efficiency, but also can ensure the accuracy of the static safety and stability evaluation of the system;
3. the method can be applied to SRB search of an actual power system, and compared with the existing OPF-based SRB construction method, the sensing capability of the voltage stability situation of the large power grid can be further improved.
Drawings
FIG. 1 is a flow chart of a Lagrange multiplier based generalized search method for security domain boundaries of an electrical power system;
fig. 2 is a schematic diagram of SRB search in a two-dimensional active injection space;
FIG. 3 is a diagram of an IEEE-14 node (Institute for Electrical and Electronic Engineers) test system;
FIG. 4 is a two-dimensional TSRB diagram with the IEEE-14 system coordinate axis as the load node;
FIG. 5 is a two-dimensional VCRB diagram with the IEEE-14 system coordinate axis as the load node.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention are described in further detail below.
In order to realize accurate and efficient construction of the SRB of the power system, the embodiment of the invention provides a power system security domain boundary searching method based on a Lagrange multiplier based on a traditional SR critical point searching optimization model.
Firstly, according to the characteristic that the static safety stability problem in an actual power system is caused by a limited number of key branches and nodes, the embodiment of the invention simplifies a conventional SR critical point general optimization model; then, a lagrangian multiplier method is introduced to solve the simplified SR critical point optimization model to directly obtain the SR critical point, wherein a Newton iteration correction equation set in the lagrangian multiplier method is solved by a method combining a displacement splitting (LMSS) preprocessor with local correction and a biconjugated gradient stabilized method (BICGSAB) in the embodiment of the invention, so that the calculation efficiency of the SR critical point is improved; and finally, according to the characteristic that adjacent critical points on the SRB are in a limited neighborhood, rapidly searching the SR critical point sets through a boundary tracking algorithm, and sequentially connecting the SR critical point sets to realize accurate and efficient search of the SRB.
Example 1
According to different static safety and stability research objects of the electric power system, static safety domains (SR) of the electric power system can be divided into different types of safety domains, and effective evaluation and operation control of safety problems of different types of electric power systems can be realized by constructing various types of safety domains. For solving the problem of constructing a Thermal Stability Region (TSR) and a static voltage security region (VCR) of an electrical power system, an embodiment of the present invention provides a general search method for a Thermal Stability Region Boundary (TSRB) and a static voltage security region boundary (VCRB) of an electrical power system based on a lagrange multiplier, as shown in fig. 1, for convenience of description, the embodiment of the present invention refers to a TSRB and a VCRB as a Security Region Boundary (SRB) and refers to a TSR critical point and a VCR critical point as SR critical points, and the general search method includes the following steps:
101: setting an initial power increasing direction, and establishing an SR critical point search optimization model;
102: solving an SR critical point optimization model by adopting a new Lagrange multiplier method combining LMSS pretreatment and a BICGSAB iterative solution to obtain an initial SR critical point;
103: repeatedly changing the power increasing direction to the direction of reducing the power increasing direction angle by taking the initial SR critical point as a starting point, searching the SR critical point of the power system in the new power increasing direction by adopting an SR critical point search optimization model and a new Lagrange multiplier method, and mapping the SR critical point into a two-dimensional active injection space to obtain a new SR critical point;
104: if the power increase direction angle is less than or equal to 0 degrees, returning to the initial SR critical point;
105: repeatedly changing the power increasing direction to the direction of increasing the power increasing direction angle again by taking the initial SR critical point as a starting point, searching the SR critical point of the power system in the latest power increasing direction again, and mapping the SR critical point into a two-dimensional active power injection space to obtain a final SR critical point;
the searching process adopted in this step is the same as that in step 103, and is not described herein again.
106: and if the latest power increase direction angle is larger than or equal to 90 degrees, sequentially connecting all the final SR critical points to obtain the SRB of the two-dimensional active injection space of the power system.
In summary, in the embodiment of the present invention, a series of SR critical points are quickly searched by the new lagrangian multiplier method through the steps 101 to 106, so as to realize quick and accurate construction of the SRB of the power system.
Example 2
In this embodiment, taking TSR as an example, the scheme in embodiment 1 is further described below with reference to a specific calculation formula and fig. 2, and is described in detail in the following description:
201: the method comprises the following steps of establishing a TSR critical point optimization model by taking ground state power flow as a starting point and giving an initial power increasing direction d, wherein the specific process is as follows:
wherein
Figure BDA0002506721590000061
n＝2NL+NG(NLIs the total number of system load nodes, NGTotal number of system generator nodes).
For an actual power system, the thermal safety of the inter-region connection section is the key point of attention of operators in a power grid dispatching center, and the connection section is generally composed of a few key branches, so that the thermal safety of the key branches can be focused on, and the TSR of the system is further constructed. Aiming at the characteristic, for the thermal safety of a certain branch on the key section of the system, the TSRB critical point search optimization model corresponding to the TSR can be simplified as follows:
Figure BDA0002506721590000062
in the formula, λ is a static safety margin of the system in the power increasing direction d; x is a radical of a fluorine atom0A vector of system initial state variables; y is0Injecting vectors for node power under a ground state; plThe active transmission capacity of the key branch l;
Figure BDA0002506721590000071
the active transmission limit of branch i.
202: solving a TSR critical point optimization model shown in the formula (1) by adopting a new Lagrange multiplier method combining LMSS pretreatment and a BICGSAB iterative solution to obtain an initial TSR critical point 0, and mapping the obtained TSR critical point 0 to a two-dimensional active power injection space taking active power injection of i and j nodes as coordinate axes;
wherein the step 202 comprises:
1) to remove the absolute value of the second equality constraint in equation (1), the second equality constraint in equation (1) is transformed to yield:
Figure BDA0002506721590000072
wherein L islActive traffic P for branch llAnd active power transmission limit
Figure BDA0002506721590000073
The squared difference of (c).
2) Constructing a Lagrangian function for the optimization model in the formula (2), and converting the constraint problem into an unconstrained problem by means of a Lagrangian multiplier method to simplify the solution of the optimization model, wherein the constructed Lagrangian function is as follows:
L(x,λ,μ,u)＝-λ+μT·[f(x0,y0)-λd]+u·Ll (3)
in the formula, x is a system state variable, lambda is a system static safety margin, and mu and u are Lagrange multipliers; t is the transposed symbol.
3) Considering that the minimum value of the optimization problem has a necessary condition that the partial derivatives of the lagrangian function to all variables and multipliers are 0, therefore, the partial derivatives are respectively solved for the system state variable x, the system static safety margin λ and the lagrangian multipliers μ and u in the formula (3), and a set of equations shown in the formula (4) is obtained:
Figure BDA0002506721590000074
wherein J (x) is a trend Jacobian matrix;
Figure BDA0002506721590000081
is LlFor the partial derivatives of x, note
Figure BDA0002506721590000082
4) Iteratively solving the nonlinear equation set formula (4) by a Newton method, and firstly establishing a Newton iterative correction equation set as shown in a formula (5);
Figure BDA0002506721590000083
the modified equation set equation (5) is a set of linear equations of (2n +2) dimensions, in which,
Figure BDA0002506721590000084
Figure BDA0002506721590000085
the ith row and jth column elements are defined as:
Figure BDA0002506721590000086
let equation (5) be abbreviated: a η ═ b, where a is the coefficient matrix, η is the vector to be solved, and b is the known vector to the right of the equal sign.
5) Solving a correction equation set formula (5) by adopting a method of combining a local correction displacement splitting (LMSS) preprocessor and a BICGSAB iterative solution to obtain correction quantities delta x, delta lambda, delta mu and delta u, wherein the specific solving process is as follows:
(1) LMSS pretreatment is carried out on the formula (5), and the specific process is as follows:
firstly, the coefficient matrix A is processed by block processingLet us order
Figure BDA0002506721590000087
Wherein the content of the first and second substances,
Figure BDA0002506721590000088
Figure BDA0002506721590000089
② a (1, 1) block matrix A to the coefficient matrix A11Hermitian (Hermitian) splitting is carried out to obtain a Hermitian matrix H, wherein,
Figure BDA00025067215900000810
generating LMSS preprocessor and marking the preprocessor as psiLMSSWherein, in the step (A),
Figure BDA0002506721590000091
wherein I is an identity matrix.
phi-LMSS preprocessorLMSSPreprocessing the equation (5), wherein the linear equation set equation (5) is equivalent to:
Figure BDA0002506721590000092
wherein, beta is an intermediate variable,
Figure BDA0002506721590000093
(2) solving the linear equation set formula (6) after the preprocessing by adopting a BICGSAB iterative method to obtain a vector eta to be solved, wherein the specific process is as follows:
given the initial value eta of the vector eta to be solved0Setting a residual threshold epsilon>0, calculating the initial residual amount r0＝b-Aη0Let us order
Figure BDA0002506721590000094
k＝1；
Wherein the content of the first and second substances,
Figure BDA0002506721590000095
to be capable of reacting with r0A set of mutually orthogonal bases, k being the number of iterations.
2 calculation of
Figure BDA0002506721590000096
If ρk-1If 0, the calculation fails; otherwise, entering the step (III);
where ρ isk-1Is an intermediate variable of the (k-1) th iteration, rk-1The residue of the (k-1) th iteration.
(iii) if k is 1, let pk＝rk-1(ii) a Otherwise let gammak-1＝(ζk-1/ωk-1)·(ρk-1/ρk-2)，pk＝rk-1+γk-1(pk-1-ωk-1υk-1)；
Wherein p iskSearch direction, ζ, for the kth iterationk-1、ωk-1、γk-1、υk-1All intermediate variables, rho, of the k-1 th iterationk-2Intermediate variable, p, for the k-2 th iterationk-1The search direction for the (k-1) th iteration;
fourthly, by
Figure BDA0002506721590000097
To obtain
Figure BDA0002506721590000098
Computing
Figure BDA0002506721590000099
sk＝rk-1-ζkυk；
Wherein the content of the first and second substances,
Figure BDA00025067215900000910
search direction, upsilon, for the kth iteration after LMSS preprocessingk、ζkAre all intermediate variables of the k-th iteration,skis the relative residual of the kth iteration.
Wu Rui | sk‖<Epsilon, order
Figure BDA00025067215900000911
Exiting the iteration process; otherwise, entering the step (sixthly);
wherein eta iskSolution of the quantity to be solved, eta, for the kth iterationk-1The solution of the required quantity obtained by the k-1 iteration is obtained.
Sixthly, by
Figure BDA00025067215900000912
To find out
Figure BDA00025067215900000913
Order to
Figure BDA00025067215900000914
ωk＝(sk，e)/(e，e)，
Figure BDA00025067215900000915
Figure BDA00025067215900000916
For the relative residual error of the kth iteration after LMSS preprocessing, e is the intermediate variable, ωkIs the intermediate variable of the kth iteration;
angles if etakIf the accuracy is met, the iteration process is exited; otherwise let rk＝sk-ωkAnd e, k is equal to k +1, the step III is returned until the solution eta is obtained, and the process is ended. Wherein r iskThe residue of the kth iteration.
6) Correcting the quantity x, lambda, mu and u to be obtained according to the formula (7);
Figure BDA0002506721590000101
7) setting Newton method convergence accuracy, substituting the corrected x, lambda, mu and u into formula (4), if formula (4) meets the convergence accuracy, then Newton iteration is finished, outputting the quantity x, lambda, mu and u to be solved, and calculating the TSR critical point in the power increasing direction d; otherwise, returning to the step 4), and repeating the Newton iteration process until x, lambda, mu and u meeting the convergence precision are obtained.
In specific implementation, the embodiment of the invention combines the BICGSAB iterative solution method with LMSS pretreatment and the non-precise Newton method to solve the formula (5), so as to improve the solving efficiency of the TSR critical point. The BICGSAB iterative solution judges when to stop iteration by comparing the current relative residual error with a preset threshold value, so the calculation precision can be controlled by modifying the threshold value. And the solution of the correction equation set formula (5) in the Newton cycle is to gradually approximate the solution of the nonlinear equation set formula (4), so that the solution precision of each cycle of the BICGSAB does not need to be too high, and convergence can be realized. Therefore, the embodiment of the present invention adopts the idea of the non-precise newton method to relax the inner loop solution precision, reduces the iteration times of the bicgsab iterative solution, and further improves the calculation efficiency of the algorithm, where the specific precision is set to 0.001, that is, the final error is less than 0.001, and the precision can also be set according to the needs in practical applications, and this is not described in detail in the embodiment of the present invention.
203: setting a power increasing angle delta alpha, and calculating a power increasing direction angle alpha corresponding to an initial power increasing direction d0Wherein
Figure BDA0002506721590000102
Figure BDA0002506721590000103
For the power increase component of node j in the power increase direction d,
Figure BDA0002506721590000104
is the power increase component of node i in the power increase direction d;
204: by adopting the research TSR critical point optimization model and solving method and combining with the SRB searching mode of boundary tracking, searching all T SR critical points, sequentially connecting the searched T SR critical points to obtain a two-dimensional T SRB, wherein the TSRB searching process of a two-dimensional active injection space is shown in the attached figure 2;
wherein the step 204 comprises:
1) let k be 1, k be the kth power system to be searched TSR critical point along the direction of decreasing the power increasing direction angle α;
2) along the direction of the angle alpha of the power increasing direction, calculating formulas according to the formulas (8) to (11) to obtain a new power increasing direction d under the alpha decreasing directionk，
Figure BDA0002506721590000105
Establishing a power increase direction dkOptimizing a model of the lower TSR critical point;
αk＝α0-k·Δα (8)
Figure BDA0002506721590000106
Figure BDA0002506721590000107
Figure BDA0002506721590000108
wherein alpha iskFor a new power increase direction dkCorresponding power increase angle, | d | is the modulo length of the initial power increase direction d,
Figure BDA0002506721590000111
in the direction of power increase dkThe power increase component of the intermediate node i,
Figure BDA0002506721590000112
in the direction of power increase dkThe power increase component of middle node j.
3) Taking TSR critical point k-1 as the initial point, and recording the system state variable and the system static safety margin at the critical point k-1 as x respectivelyk-1、λk-1X at the critical point k-1 of TSRk-1、λk-1And LagThe Langerian multiplier is used as an initial value of a Newton method for iterative solution of the kth TSR critical point along the direction of reducing the power increasing direction angle alpha, and Newton iteration is carried out until x meeting the convergence precision is obtainedk、λkThen the power increase direction dkSolving the TSR critical point based on the Lagrange multiplier;
4) mapping the searched TSR critical point k to a two-dimensional active power injection space to obtain a new TSR boundary point;
5) checking alpha, if alpha is less than or equal to 0 degrees, continuing to execute the step 6), otherwise, making k equal to k +1 and returning to the step 2);
6) returning to the initial TSR critical point 0;
7) let k 'be 1, k' be the kth power system to be searched TSR critical point along the direction of increasing power increasing direction angle α;
8) along the direction of increasing the power increasing direction angle alpha, calculating formulas according to the formulas (12) to (15) to obtain a new power increasing direction d in the alpha increasing directionk'，
Figure BDA0002506721590000113
Establishing a power increase direction dk'Optimizing a model of the lower TSR critical point;
αk′＝α0+k′·Δα (12)
Figure BDA0002506721590000114
Figure BDA0002506721590000115
Figure BDA0002506721590000116
wherein, k' is the TSR critical point of the kth power system to be searched along the direction of increasing the power increasing direction angle alpha, alphak'For a new power increase direction dk'Corresponding power increaseAngle, | d | is the mode length of the initial power increase direction d,
Figure BDA0002506721590000117
in the direction of power increase dk'The power increase component of the intermediate node i,
Figure BDA0002506721590000118
in the direction of power increase dk'The power increase component of middle node j. 9) Taking a TSR critical point k '-1 as an initial point, and recording a system state variable and a system static safety margin at the critical point k' -1 as x respectivelyk'-1、λk'-1X at the critical point k' -1 of TSRk'-1、λk'-1And taking the Lagrange multiplier as an initial value of a k' th TSR (transient time response) critical point in the increasing direction of the power increasing direction angle alpha by Newton method iteration, and performing Newton iteration until x meeting the convergence precision is obtainedk'、λk'Then the power increase direction dk'Solving the TSR critical point based on the Lagrange multiplier;
10) mapping the searched TSR critical point k' into a two-dimensional active power injection space to obtain a new TSR boundary point;
11) and checking alpha, if alpha is larger than or equal to 90 degrees, sequentially connecting all the obtained TSR boundary points to obtain the TSRB of the two-dimensional active injection space, ending the process, and otherwise, making k '═ k' +1 and returning to the step 8).
In summary, in the embodiment of the present invention, through the steps 201 to 204, not only is the complicated iterative optimization process of the OPF effectively avoided, but also the solution efficiency for searching a single TSR critical point is accelerated, the construction efficiency of the TSRB of the power system is significantly improved, and the method has practical application value for evaluating the thermal stability of the power system.
Example 3
The present embodiment takes a VCR as an example, and the scheme in embodiment 1 is further described below with reference to a specific calculation formula and fig. 2, which is described in detail as follows:
301: taking ground state power flow as a starting point, giving an initial power increasing direction d, and establishing a VCR critical point optimization model, wherein the specific process is as follows:
wherein
Figure BDA0002506721590000121
n＝2NL+NG(NLIs the total number of system load nodes, NGTotal number of system generator nodes).
In actual power system operation, the number of nodes which have a significant influence on the operation safety of the power grid due to the out-of-limit system voltage amplitude is very limited. Thus, in constructing a VCR for a power system, the VCR may be constructed with emphasis on these voltage safety master nodes, which have a serious impact on the operational safety of the system. Therefore, when the power system VCR is constructed, only the VCR formed by the leading nodes influencing the voltage safety of the system is concerned, and further, the VCRB critical point search optimization model of the single voltage safety leading node of the power system is obtained as follows:
Figure BDA0002506721590000122
in the formula, λ is a static safety margin of the system in the power increasing direction d; x is the number of0A vector of system initial state variables; y is0Injecting vectors for node power under the ground state; vkIs the voltage amplitude of node k;
Figure BDA0002506721590000123
is the set node k voltage magnitude limit.
302: solving a VCR critical point optimization model shown in a formula (16) by adopting a new Lagrange multiplier method with LMSS pretreatment and BICGSAB iterative solution to obtain an initial VCR critical point 0, and mapping the obtained VCR critical point 0 to a two-dimensional active power injection space with active power injection of i and j nodes as coordinate axes;
wherein the step 302 comprises:
1) transforming the second equality constraint in equation (16) yields:
Figure BDA0002506721590000131
2) constructing a Lagrangian function for the optimization model of the formula (17), and converting the constraint problem into an unconstrained problem by means of a Lagrangian multiplier method to simplify the solution of the optimization model, wherein the constructed Lagrangian function is as follows:
Figure BDA0002506721590000138
in the formula, x is a system state variable, lambda is a system static safety margin, and mu and u are Lagrange multipliers.
3) Respectively solving partial derivatives of the system state variable x, the system static safety margin lambda and the Lagrange multipliers mu and u in the formula (18) to obtain a group of equations shown in a formula (19):
Figure BDA0002506721590000132
wherein J (x) is a trend Jacobian matrix;
Figure BDA0002506721590000133
is composed of
Figure BDA0002506721590000134
For the partial derivatives of x, note
Figure BDA0002506721590000135
4) Iteratively solving the nonlinear equation set formula (19) by a Newton method, and firstly establishing a Newton iterative correction equation set as shown in a formula (20);
Figure BDA0002506721590000136
equation (20) is a set of (2n +2) -dimensional linear equations in which,
Figure BDA0002506721590000137
let equation (20) be abbreviated: and By is z, wherein, B is a coefficient matrix, y is a vector to be solved, and z is a known vector on the right side of an equal sign.
5) Solving a correction equation set formula (20) by adopting a method of combining a local correction displacement splitting (LMSS) preprocessor and a BICGSAB iterative solution to obtain correction quantities delta x, delta lambda, delta mu and delta u, wherein the specific solving process is as follows:
(1) the LMSS pretreatment is carried out on the formula (20), and the specific process is as follows:
firstly, the coefficient matrix B is processed by block processing, and the order is
Figure BDA0002506721590000141
Wherein the content of the first and second substances,
Figure BDA0002506721590000142
② a (1, 1) block matrix B of the coefficient matrix B11Hermitian (Hermitian) splitting is carried out to obtain a Hermitian matrix H, wherein,
Figure BDA0002506721590000143
generating LMSS preprocessor and marking the preprocessor as psiLMSS，
Figure BDA0002506721590000144
Wherein I is an identity matrix.
phi-LMSS preprocessorLMSSPreprocessing is performed on equation (20), and the system of linear equations is equivalent to:
Figure BDA0002506721590000145
wherein chi is an intermediate variable,
Figure BDA0002506721590000146
(2) solving the linear equation set formula (21) after the pretreatment by adopting a BICGSAB iterative method to obtain a vector y to be solved,
the specific process is as follows:
given the initial value y of the vector y to be solved0Setting a residual threshold epsilon>0, calculating the initial residual amount r0＝z-By0Let us order
Figure BDA0002506721590000147
k＝1；
Wherein the content of the first and second substances,
Figure BDA0002506721590000148
to be capable of reacting with r0A set of mutually orthogonal bases, k being the number of iterations.
2 calculation of
Figure BDA0002506721590000149
If ρk-1If 0, the calculation fails; otherwise, entering the step (III);
where ρ isk-1Is an intermediate variable of the (k-1) th iteration, rk-1The residue of the (k-1) th iteration.
(iii) if k is 1, let p bek＝rk-1(ii) a Otherwise let gammak-1＝(ζk-1/ωk-1)·(ρk-1/ρk-2)，pk＝rk-1+γk-1(pk-1-ωk-1υk-1)；
Wherein p iskSearch direction, ζ, for the kth iterationk-1、ωk-1、γk-1、υk-1All intermediate variables, rho, of the k-1 th iterationk-2Intermediate variable, p, for the k-2 th iterationk-1The search direction for the (k-1) th iteration.
Fourthly, by
Figure BDA00025067215900001410
To obtain
Figure BDA00025067215900001411
Computing
Figure BDA00025067215900001412
sk＝rk-1-ζkυk；
Wherein the content of the first and second substances,
Figure BDA0002506721590000151
search direction, upsilon, for kth iteration after LMSS preprocessingk、ζkIntermediate variables, s, for all k-th iterationskIs the relative residual of the kth iteration.
Wu Rui | sk‖<Epsilon, order
Figure BDA0002506721590000152
Exiting the iteration process; otherwise, entering the step (sixthly);
wherein, ykFor the solution of the quantity to be solved, y, obtained for the kth iterationk-1The solution of the required quantity obtained by the k-1 iteration is obtained.
Sixthly by
Figure BDA0002506721590000153
To obtain
Figure BDA0002506721590000154
Order to
Figure BDA0002506721590000155
ωk＝(sk，e)/(e，e)，
Figure BDA0002506721590000156
Wherein the content of the first and second substances,
Figure BDA0002506721590000157
for the relative residual error of the kth iteration after LMSS preprocessing, e is the intermediate variable, ωkIs the intermediate variable of the kth iteration.
Is given bykIf the accuracy is met, the iteration process is exited; otherwise let rk＝sk-ωke, k is k +1, return to step (c) until obtaining solution y, the procedure is over, wherein, r is k +1kThe residue of the kth iteration.
6) Correcting the quantity x, lambda, mu and u to be obtained according to the formula (22);
Figure BDA0002506721590000158
7) setting Newton method convergence accuracy, substituting the corrected x, lambda, mu and u into an equation (19), if the equation (19) meets the convergence accuracy, then Newton iteration is finished, outputting the quantity x, lambda, mu and u to be solved, and finishing the calculation of VCR critical points in the power increasing direction d; otherwise, returning to the step 4), and repeating the Newton iteration process until x, lambda, mu and u meeting the convergence precision are obtained.
303: setting a power increasing angle delta alpha, and calculating a power increasing direction angle alpha corresponding to an initial power increasing direction d0In which
Figure BDA0002506721590000159
Figure BDA00025067215900001510
For the power increase component of node j in the power increase direction d,
Figure BDA00025067215900001511
is the power increase component of node i in the power increase direction d;
304: the VCRB search process for searching all VCR critical points by using the VCR critical point optimization model and solution method in this study and combining with the SRB search mode of boundary tracking, and connecting the searched VCR critical points in sequence to obtain a two-dimensional VCRB, and the two-dimensional active power injection space is shown in fig. 2, and the specific process of step 304 is the same as step 204 in embodiment 2, which is not described herein again in the embodiments of the present invention.
In summary, in the embodiment of the present invention, the VCR critical points can be directly calculated through the steps 301 to 304, thereby effectively avoiding the complex iterative optimization process of the OPF, reducing the time consumption for searching a single VCR critical point, significantly improving the VCRB construction efficiency of the power system, and having practical application value for the static voltage safety evaluation of the power system.
Example 4
The feasibility verification of the solutions of examples 1, 2 and 3 is carried out below with reference to the specific examples, fig. 3, fig. 4, fig. 5 and tables 1, 2, 3, 4, 5 and 6, as described in detail below:
in this embodiment, the validity of the method is verified by searching the two-dimensional thermal stability security domain boundary TSRB and the static voltage security domain boundary VCRB of the IEEE-14 node system, and the IEEE-14 node testing system is shown in fig. 3. The power increase direction angular step Δ α set in this embodiment is 0.15rad, and the convergence accuracy of newton's method is 10-5The maximum number of iterations is 100; the convergence precision of the BICGSAB iterative method is 10-3The maximum number of iterations is 10000.
1) Construction of TSRB
And searching the TSRB of the branch 2-3 in a two-dimensional load active power injection space by taking the thermal stability of the branch 2-3 as a research object. In this scenario, the active power transmission limit of the branch 2-3 is 1.0p.u., and the critical load nodes affecting the thermal stability of this branch are 4 and 5. Therefore, the active injection of the load nodes 4 and 5 is taken as a coordinate axis, and the TSRB of the branch circuits 2-3 is searched in a two-dimensional active power injection space by adopting the method.
Setting the power increasing direction corresponding to the first TSR critical point as d by taking the ground state as a starting point2＝[2.3240,0.4000,0,1.9120,0.3040,0,0,0,0,0,0,0,0,0]TAngle of power increase direction α thereof20.1577rad, solving an optimization model formula (1) by adopting a TSR critical point optimization model provided by the method and introducing a Lagrange multiplier method, searching to obtain an initial TSR critical point, wherein the coordinates of the TSR critical point 2 are (1.3387,0.2128) corresponding to the TSR critical point 2 in the graph 4, and the static safety margin lambda of the system is lambda20.7001p.u. Searching the system TSR critical point along the direction of decreasing power increase direction angle alpha, by d2、α2And delta alpha can obtain a new power increase direction angle alpha10.0077rad, corresponding to the system power increasing direction d1＝[2.3240,0.4000,0,1.9360,0.0149,0,0,0,0,0,0,0,0,0]TWith the state variable x of the TSR critical point 22And system static safety margin lambda2As d1Calculating d by using the optimization model and the solving method provided by the method for the initial value of the lower TSR critical point1Corresponding to the TSR critical point 1, obtaining the coordinates of the TSR critical point 1 as (1.4705,0.0113), and obtaining the system static safety margin lambda10.7596p.u. By analogy, the obtained TSR critical point is taken as the starting point of the adjacent critical point to be solved, the state variable and the system static safety margin are taken as the initial values of the critical point to be solved, and the power increase direction angle alpha of the TSR critical point 00-0.1423rad, in this case α0<0, order alpha0When d is equal to 0, calculate d0＝[2.3240,0.4000,0,1.9360,0,0,0,0,0,0,0,0,0,0]TAnd the lower TSR critical point 0 stops searching the TSR critical point along the alpha reducing direction.
Returning to the initial TSR critical point 2 again, similar to the process, sequentially searching the critical points 3, 4, 5, …, 12 along the increasing direction of alpha by adopting the optimization model and the solving method provided by the method, wherein the power increasing direction angle of the TSR critical point 12 is alpha121.6577rad, at this time: alpha is alpha12>Pi/2, order alpha12Pi/2, calculating d12＝[2.3240,0.4000,0,0,1.9360,0,0,0,0,0,0,0,0,0]TAt the lower SNB point 12, the system ends the TSR critical point search in the increasing direction of α in the first quadrant. The details of each TSR critical point are shown in table 1. The TSR critical points searched in fig. 3 are sequentially connected, and the TSRB in the two-dimensional load active power injection space formed by the load nodes 4 and 5 can be obtained.
TABLE 1 information of each TSR critical point searched by the method
Figure BDA0002506721590000161
Figure BDA0002506721590000171
To verify the correctness of the method for searching the TSRB, fig. 4 further shows the TSRB searched by the OPF method, and it can be known from the comparison of fig. 4 that: for the same power increasing direction, the TSR critical point obtained by the method is almost coincident with the TSR critical point obtained by the OPF method, and the correctness of searching the TSRB by the method is verified.
2) Construction of VCRB
The voltage safety of the node 10 is used as a research object, the upper limit value of the voltage amplitude of the node 10 is 1.05p.u., and the lower limit value of the voltage amplitude of the node 10 is 0.95p.u., and the VCRB of the node 10 is searched in a two-dimensional load active power injection space formed by the load nodes 4 and 5 by adopting the optimization model and the solving method provided by the method.
Setting the power increasing direction d of the initial VCR critical point with the ground state as the starting point2＝[2.3240,0.4000,0,1.9120,0.3040,0,0,0,0,0,0,0,0,0]TAngle of power increase direction α thereof20.1577rad, the VCRB of node 10 is searched by the lagrange multiplier-based VCRB search method provided by the method, and the detailed information of each critical point on the VCR is shown in table 2 and the detailed information of each critical point on the VCR is shown in table 3, corresponding to the shaded part in fig. 5.
TABLE 2 searching for VCR upper boundary critical point information using the method
Figure BDA0002506721590000172
Figure BDA0002506721590000181
TABLE 3 searching for VCR lower boundary critical point information by the method
Figure BDA0002506721590000182
To verify the correctness of the VCRB searched by the method, fig. 5 further shows the VCRB searched by the OPF method, as can be seen from the comparison of fig. 5: for the same power increasing direction, the VCR critical point obtained by the method almost coincides with the VCR critical point obtained by the OPF method, and the accuracy of VCRB searching by the method is verified.
3) Comparison of computational efficiency
This section further compares the calculation time for constructing the above security domain boundary between the method and the OPF method (computer hardware configuration: processor intel (r) core (tm) m3-7Y30 CPU @1.00GHz, memory 4.00GB), and the results are shown in table 4.
TABLE 4 comparison of the calculation time for SRB construction by the method and OPF method
Figure BDA0002506721590000183
As can be seen from the comparison results in table 4, the calculation time for constructing each security domain boundary by the OPF method is much longer than that of the method, and the method can greatly improve the construction efficiency of the security domain boundary while ensuring the accuracy of the security domain boundary, and has higher calculation efficiency.
Table 5 further counts the total iteration count of the newton method and the OPF method for searching the security domain boundary when the method searches the security domain boundary. Compared with the OPF method, the method has the advantages that the iteration times for solving the SR critical point optimization model can be greatly reduced by adopting the Lagrange multiplier method, so that the solving efficiency of the SR critical point is improved, and the construction efficiency of the SRB is further improved.
TABLE 5 comparison of the number of iterations of the method and OPF method for constructing SRB
Figure BDA0002506721590000191
Table 6 further shows the total iteration number and the average iteration number of the bicgsab iteration method when the LMSS preprocessed bicgsab iteration method is used to solve the newton iteration correction equation set when the security domain boundary is searched by the method, as can be seen from table 6, the cumulative internal iteration number of the bicgsab iteration solution is less when the LMSS preprocessing method is used to process the coefficient matrix, which indicates that the proposed LMSS preprocessor has a better preprocessing effect.
TABLE 6 Total and average iterations of the BICGSAB iteration method
Figure BDA0002506721590000192
In conclusion, compared with the existing SRB construction method based on the OPF, the method not only inherits the characteristic of high-precision search of the SR critical point by the OPF, but also well avoids the calculation burden of searching the SR critical point by the OPF, effectively reduces the time consumption of searching a single SR critical point, and remarkably improves the construction efficiency of the SRB. In addition, along with the continuous expansion of the system scale, the dimension of an SR critical point optimization model to be solved is increased rapidly, the calculation amount and the calculation complexity of SRB construction are increased, the method combines the BICGSAB iterative solution with LMSS pretreatment and the non-precise Newton method to solve the Newton iterative correction equation set, the solution efficiency of the SR critical point of the regional interconnected large-scale power system is favorably accelerated, meanwhile, the calculation efficiency of the SR critical point is further improved by the aid of the GPU parallel technology, and the SRB construction efficiency is improved.
In the embodiment of the present invention, except for the specific description of the model of each device, the model of other devices is not limited, as long as the device can perform the above functions.
Those skilled in the art will appreciate that the drawings are only schematic illustrations of preferred embodiments, and the above-described embodiments of the present invention are merely provided for description and do not represent the merits of the embodiments.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (1)

1. A Lagrange multiplier based electric power system security domain boundary searching method is characterized by comprising the following steps:
setting an initial power increasing direction, and establishing a static security domain SR critical point search optimization model;
solving an SR critical point search optimization model by combining a displacement splitting LMSS preprocessor with local correction and a biconjugate gradient stabilization method BICGSAB to obtain an initial SR critical point;
according to the characteristic that adjacent critical points on the security domain boundary SRB are in a limited neighborhood, SR critical point sets are searched through a boundary tracking algorithm and are sequentially connected to realize the search of the SRB;
the SR critical point search optimization model comprises: a thermal stability safety domain critical point searching optimization model and a static voltage safety domain critical point searching optimization model; the power system security domain boundary comprises: a thermally stable security domain boundary, and a static voltage security domain boundary;
the thermal stability safety domain critical point search optimization model specifically comprises the following steps:
Figure FDA0003585471050000011
where λ is the static safety margin in the power increase direction d; x is the number of0A vector of initial state variables; y is0Injecting vectors for node power under a ground state; plThe active transmission capacity of branch l;
Figure FDA0003585471050000012
is the active transmission limit of branch l; f (x)0,y0) Is a conventional power flow equation of the power system;
the method for solving the TSR critical point search optimization model of the thermal stability security domain specifically comprises the following steps:
(1) LMSS pretreatment is carried out on the Newton iterative correction equation set, and the specific process is as follows:
the coefficient matrix A is processed in a block mode, and the (1, 1) block matrix A is processed11Performing Hermitian splitting to obtain a Hermitian matrix H;
generate an LMSS preprocessor and note the preprocessor as psiLMSS；
Using LMSS preconditioner ΨLMSSPreprocessing the Newton iteration correction equation set to obtain an equivalent linear equation of the Newton iteration correction equation setGroup (d);
(2) solving an equivalent linear equation set by adopting a BICGSAB iterative method to obtain a vector eta to be solved; correcting the quantity x, lambda, mu and u to be solved; if the convergence accuracy is met, Newton iteration is finished, the quantities x, lambda, mu and u to be solved are output, and the calculation of the TSR critical point in the power increasing direction d is finished;
the critical point searching optimization model of the static voltage security domain VCR is specifically as follows:
Figure FDA0003585471050000021
in the formula, λ is a static safety margin of the system in the power increasing direction d; x is the number of0A vector of initial state variables of the system; y is0Injecting vectors for node power under a ground state; vkIs the voltage amplitude of node k;
Figure FDA0003585471050000022
setting a node k voltage amplitude limit value; f (x)0,y0) Is a conventional power flow equation of the power system;
the method for solving the critical point search optimization model of the static voltage safety domain specifically comprises the following steps:
(1) and performing LMSS pretreatment on the other Newton iteration correction equation set, wherein the specific process is as follows:
the coefficient matrix B is processed by block division, and the (1, 1) block matrix B of the coefficient matrix B is processed11Performing Hermitian splitting to obtain a Hermitian matrix H;
generate an LMSS preprocessor and designate the preprocessor as ΨLMSSUsing LMSS preconditioner ΨLMSSPreprocessing the other Newton iteration correction equation set to obtain another equivalent linear equation set;
(2) solving the other linear equation set after the preprocessing by adopting a BICGSAB iterative method to obtain a vector y to be solved; correcting the quantity x, lambda, mu and u to be solved; if the convergence accuracy is met, Newton iteration is finished, the quantities x, lambda, mu and u to be solved are output, and the calculation of the VCR critical point in the power increasing direction d is finished;
the SR critical points include: a thermally stable region critical point, and a quiescent voltage region critical point.
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