CN107863772B - High-order trajectory sensitivity calculation method based on generalized Galerkin - Google Patents
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- H—ELECTRICITY
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- H—ELECTRICITY
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Abstract
The invention discloses a high-order track sensitivity calculation method based on generalized Galerkin, which belongs to the field of stability analysis and control of a power system, improves the precision when the safety analysis and control of the power system are carried out by providing the high-order track sensitivity calculation method based on generalized Galerkin, takes high-order information into account, can more comprehensively reflect the influence degree of state response in the dynamic process of the system by certain control parameters or initial conditions, simultaneously uses the generalized Galerkin method when calculating the high-order track sensitivity, realizes the decoupling among projection equations by selecting a specific test basis function, and meets the requirement of quick calculation. The method is applicable to various complex dynamic processes of the power system, can comprehensively consider various conditions of the system during operation, has wide application range and high solving speed, and can provide a scientific and reasonable analysis scheme for the safety and stability analysis and control of the power system.
Description
Technical Field
The invention relates to the technical field of power systems, in particular to the field of stability analysis and control of power systems, and provides a high-order trajectory sensitivity calculation method based on generalized Galerkin.
Background
The power system is a complex nonlinear time-varying system. A large number of nonlinear elements exist in the power system, and when the nonlinear elements are controlled, the system state can undergo a complex change process; meanwhile, discrete events can be caused by actions of elements such as a relay protection device and an on-load tap changer, so that the state variable of the system jumps. Therefore, when analyzing the power system, it is necessary to understand the influence of various parameters affecting the dynamic process of the power system on the dynamic response of the system. The track sensitivity is an effective tool in the safety analysis of the power system, and the influence of the factors on the dynamic quality is quantitatively analyzed by researching the sensitivity of the dynamic response of the system to certain parameters or initial conditions and even the sensitivity of a system model. According to the sensitivity index of the system state to each control parameter, the safety performance of the system can be improved, and the stability margin or the economic index of the system can be improved. Therefore, the sensitivity method is widely applied to various fields of power systems.
However, the conventional trajectory sensitivity is the linear expansion of the system state with respect to the control parameter, and only the first-order term of the taylor expansion is retained, and the influence of the higher-order term is ignored, so that in the case of large change of the control parameter, the corresponding state response variation quantity is seriously deviated from the actual value, and the accuracy of the calculation result is greatly reduced. Therefore, it is necessary to introduce high-order trajectory sensitivity to compensate for the accuracy deficiency, where the high-order trajectory sensitivity is obtained by reserving taylor expansion high-order terms on the basis of the conventional trajectory sensitivity, that is, the solution of the high-order trajectory sensitivity can be stated as the solution of the expansion that reserves high-order terms for the variable to be studied from another perspective. According to the definition, the information of the high-order item relates to the high-order partial derivative of the state variable to the control parameter, and the calculation process is complicated, so that the high-order track sensitivity rapid calculation method based on the generalized Galerkin method is provided.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: a high-order track sensitivity calculation method based on generalized Galerkin is provided.
In order to solve the technical problem, the scheme of the invention comprises the following steps:
1. and establishing a mathematical model of the power system.
The system comprises a generator, an exciter and a regulating system thereof, a prime motor and a speed regulating system thereof, a load, a power network and the like. The various models are divided into a plurality of detailed and simple models, and in actual calculation, proper models can be selected according to different calculation contents and requirements.
Thus, the dynamic process of the power system can be described by a set of differential-algebraic equations:
wherein x is [ x ]1,x2,...,xn]TIs a state variable column vector, and is related to the motion of a generator rotor, Automatic Voltage Regulation (AVR), an excitation system and the like; y ═ y1,y2,...yr]TIs a column vector formed by node voltage amplitude and phase angle algebraic variables; u ═ u1,u2,...,us]TIs a control parameter column vector.
2. Constructing polynomial basis functions related to control parameters and setting a state variable x of the system to [ x [ ]1,x2,...,xn]TAnd the algebraic variable y ═ y1,y2,...yr]TExpressed by the set of polynomial basis functions, the linear combination of the set of polynomial basis functions containing the coefficients to be solved is the expansion we need to take into account the higher order term information.
For s control parameters u ═ u1,u2,...,us]TIf the order of the high-order track sensitivity is set to N, then u is set for each control parameter i1, 2.. s, a set of polynomial basis functions (power series) of maximum order N can be constructed:
considering s control parameters, the overall polynomial basis function is the tensor product of each control parameter basis function:
in the formula (I), the compound is shown in the specification,
for the set of overall polynomial basis functions
The number of (2).
Thus, the state variables and the algebraic variables of the system can be expressed as linear combinations of the polynomial basis functions with the coefficients to be determined, i.e. N-th order taylor expansions of the state variables and the algebraic variables with respect to the control variables:
in the formula, x*And y*Representing approximations of a state variable x and an algebraic variable y,
and
the coefficients of the higher order expansion to be solved, i.e. the sensitivity of the corresponding variable to the j-th order trajectory of the control parameter, are used.
3. Substituting the N-order expansion containing undetermined coefficients into a differential-algebraic equation set representing the dynamic process of the power system, selecting a group of special test basis functions to perform projection calculation on the substituted differential-algebraic equation, eliminating a variable u, and obtaining a model containing only the coefficients to be determined
And
differential-algebraic equations of (c).
Firstly, x is*And y*Substituting the expression into the original system differential-algebraic equation system to obtain:
since the inner product projection operation can be expressed as:
therefore, the inner product projection operation is carried out on the substituted differential-algebraic equations and the selected special test basis function:
in the formula，
To satisfy a particular set of test basis functions:
1) when k is more than or equal to 1 and less than or equal to N-1,
2) when k is equal to N, the process is repeated,
∫uiΓkdu＝1,i≥0
by performing an inner product projection operation with the set of test basis functions, the benefits of this are: in each inner product projection operation, only (n + r) -order differential-algebraic equation consistent with the original equation order needs to be solved to obtain (n + r) corresponding coefficients, and then the obtained coefficients are used as known quantities to be substituted into the next inner product projection equation, namely, the inner product projection equation consistent with the property of gammakThe decoupling between equations containing the coefficient to be determined can be realized by performing inner product projection operation, and the sensitivity of the locus with the higher order to be determined is solved in sequence by using the back-substitution idea, so that the original (N + r) multiplied by N is prevented from being directly subjected to the original (N + r) multiplied by NbThe complex solving process of the high-dimensional and mutually coupled differential-algebraic equation system of the order accelerates the sensitivity to the track
And
and solving the speed.
4. Because the projection equation of the original differential-algebraic equation system is still the differential-algebraic equation system, the numerical integration method can be used for solving the high-order track sensitivity
And
and calculating an initial value of the high-order track sensitivity, and performing numerical integration on the projection equation by using an implicit trapezoidal method to solve the high-order track sensitivity at each moment.
The initial value of the undetermined coefficient is given by:
the simulation step length is given, numerical integration is carried out by adopting a trapezoidal integration formula, and the high-order track sensitivity of each moment can be obtained
And
the beneficial results of the invention are as follows: the high-order trajectory sensitivity calculation method based on the generalized Galerkin calculates high-order information, improves the precision of safety analysis and control of a power system, can reflect the influence degree of state response in the dynamic process of the system on certain control parameters or initial conditions more comprehensively, and meanwhile, the generalized Galerkin method is used for calculating the high-order trajectory sensitivity, decoupling among projection equations is realized by selecting specific test basis functions, and the requirement of quick calculation is met. The method is applicable to various complex dynamic processes of the power system, can comprehensively consider various conditions of the system during operation, has wide application range and high solving speed, and can provide a scientific and reasonable analysis scheme for the safety and stability analysis and control of the power system.
Drawings
Fig. 1 is a flowchart of a high-order trajectory sensitivity calculation method based on generalized Galerkin.
Fig. 2 is a single-line structure diagram of a 9-node system of the machine according to embodiment 3.
Fig. 3 is a time domain simulation result diagram of the relative angle between the rotors of the generator 1 and the generator 2 in the embodiment of the invention.
Detailed Description
The present invention is further illustrated by the following figures and specific examples, which are to be understood as illustrative only and not as limiting the scope of the invention, which is to be given the full breadth of the appended claims and any and all equivalent modifications thereof which may occur to those skilled in the art upon reading the present specification.
This embodiment takes a 9-node power system network as an example. The system has 3 generators, 3 loads and 9 branches. The branch data and generator parameters are listed in tables 1 and 2, respectively, and the system power flow under normal operating conditions is shown in table 3, and the system frequency is 60 Hz.
TABLE 1 Branch data
TABLE 2 Generator data
TABLE 3 System flow under Normal operating conditions
1. And establishing a mathematical model of the power system.
The model of the synchronous generator is transient potential E'qAnd the constant model ignores the influence of the speed regulator and does not consider the action of the excitation voltage change and the rotor damping winding. It is a set of differential-algebraic equations:
equation of motion of the rotor:
stator winding voltage balance equation
The power network equation:
YU＝I
the meaning of each parameter in the above equation is:
ω — angular velocity of the generator (per unit value);
ωssynchronous rotational speed, when referenced to synchronous rotational speed, ωsHas a per unit value of 1;
FJ——F＝1/TJwherein T isJIs the generator inertia time constant;
delta-included angle between electromotive force phasor of the generator and the synchronous rotating shaft;
Pm-the mechanical power of the generator;
Pe-the electromagnetic power of the generator;
Ud、Uq-generator end longitudinal and transverse axis voltages;
Ra-generator stator loop resistance;
E′q-electromotive force of the transverse axis after transient reactance of the generator;
X′d-generator longitudinal axis transient reactance;
Id、Iq-generator longitudinal and transverse axis currents;
Xq-generator cross-axis synchronous reactance;
y-system node complex admittance matrix (Y)ii＝Gii+jBii,Yij＝Yji＝Gij+jBij)；
I-node injection current source vector
2. A polynomial basis function is constructed for the control parameters, and the state variables and algebraic variables of the system are represented by the set of polynomial basis functions, the coefficients of which are the corresponding higher order trajectory sensitivities.
Selecting an inertia time constant F of the generator 1J1Study of parameter F as a control parameterJ1The influence on state variables and algebraic variables in the system.
In order to avoid the occurrence of trigonometric functions in subsequent projection calculation, intermediate variables e and f are used for replacing the original delta, wherein the e and the f satisfy the following relation:
then the generator's rotor equation of motion becomes:
setting the order of the higher-order track sensitivity to 6, using parameter FJ1The linear combination of polynomials to represent state variables, expansions such as e, f can be written as follows: (expansions of other state variables and algebraic variables are analogous thereto)
In the formula
And
for the parameter F, the state variables e and F, respectivelyJ1I order track sensitivity.
3. Will contain the sensitivity of the track of undetermined high order
And
of 6-step expansion e*And f*Substituting into the rotor motion equation to selectAnd performing projection calculation on the substituted differential equation by using a group of special test basis functions, eliminating parameters, and obtaining the differential equation only containing the sensitivity sum of the high-order track to be solved.
The rotor motion equation can be obtained by substituting an expansion equation containing unknown high-order track sensitivity into the rotor motion equation: (substitution of algebraic equations is analogous thereto)
Selecting a specific test basis function, and then carrying out inner product projection operation on the set of differential-algebraic equations to obtain a set of differential-algebraic equations only containing the sensitivity of the high-order trajectory to be solved:
4. solving high-order trajectory sensitivity by numerical integration
The simulation step length is given to be 0.02s, the simulation time is given to be 2s, numerical integration is carried out by adopting a trapezoidal integration formula, and the value of each coefficient of the system variable high-order expansion at each moment can be obtained.
After the coefficients of the higher-order expansion are determined, the state locus of the state variable when the parameters take different initial values can be obtained, and the time domain simulation result of the relative angle between the rotors of the generator 1 and the generator 2 is shown in fig. 3.
The above description is only a preferred embodiment of the present invention, and all equivalent changes and modifications made in accordance with the claims of the present invention should be covered by the present invention.
Claims (5)
1. A high-order track sensitivity calculation method based on generalized Galerkin is characterized by comprising the following steps:
step (1): establishing a mathematical model of the power system;
step (2): constructing a polynomial basis function related to the control parameter based on the data obtained in the step (1), and setting a state variable x of the system to [ x [1,x2,...,xn]TAnd the algebraic variable y ═ y1,y2,...yr]TThe polynomial basis functions are used for representing, and the linear combination of the polynomial basis functions containing the coefficients to be solved is the required expansion which takes the information of the high-order terms into account;
and (3): based on the polynomial basis function obtained in the step (2), substituting an N-order expansion containing undetermined coefficients into a differential-algebraic equation system representing the dynamic process of the power system, selecting a group of test basis functions to perform projection calculation on the substituted differential-algebraic equation, and obtaining a polynomial basis function containing only the coefficients to be solved
And
differential-algebraic equations of (a);
and (4): solving the high-order track sensitivity by adopting a numerical integration method according to the fact that the differential-algebraic equation containing the coefficient to be solved obtained in the step (3) is still a differential-algebraic equation system
And
2. the generalized Galerkin-based high-order trajectory sensitivity calculation method according to claim 1, wherein: the step (1) is specifically as follows: the model comprises a generator, an exciter and a regulating system thereof, a prime mover and a speed regulating system thereof, a load and a power network; selecting corresponding models according to different calculation contents and requirements during actual calculation;
the dynamic process of the power system is described by the following differential-algebraic equation:
wherein x is [ x ]1,x2,...,xn]TIs a state variable column vector; y ═ y1,y2,...yr]TIs a column vector formed by node voltage amplitude and phase angle algebraic variables; u ═ u1,u2,...,us]TIs a control parameter column vector.
3. The generalized Galerkin-based high-order trajectory sensitivity calculation method according to claim 1, wherein: the step (2) is specifically as follows: for s control parameters u ═ u1,u2,...,us]TIf the order of the high-order track sensitivity is set to N, then u is set for each control parameteri1, 2.. s, a set of polynomial power series of maximum order N can be constructed:
considering s control parameters, the overall polynomial basis function is the tensor product of each control parameter basis function:
in the formula (I), the compound is shown in the specification,
for the set of overall polynomial basis functions
The number of (2);
the state variables and algebraic variables of the system can be expressed as linear combinations of the polynomial basis functions with coefficients to be determined, i.e. N-th order taylor expansions of the state variables and algebraic variables of the system with respect to the control variables:
in the formula, x*And y*Representing approximations of a state variable x and an algebraic variable y,
and
the coefficients of the higher order expansion to be solved, i.e. the sensitivity of the corresponding variable to the j-th order trajectory of the control parameter, are used.
4. The generalized Galerkin-based high-order trajectory sensitivity calculation method according to claim 1, wherein: the step (3) is specifically as follows: firstly, x is*And y*Substituting the expression into the original system differential-algebraic equation system to obtain:
since the inner product projection operation can be expressed as:
therefore, the inner product projection operation is carried out on the substituted differential-algebraic equations and the selected special test basis function:
in the formula (I), the compound is shown in the specification,
to satisfy a particular set of test basis functions:
1) when k is more than or equal to 1 and less than or equal to N-1,
when k is equal to N, the process is repeated,
∫uiΓkdu＝1,i≥0。
5. the generalized Galerkin-based high-order trajectory sensitivity calculation method according to claim 1, wherein: the step (4) is specifically as follows: calculating an initial value of the high-order track sensitivity, and performing numerical integration on a projection equation by using an implicit trapezoidal method to solve the high-order track sensitivity at each moment;
the initial value of the undetermined coefficient is given by:
the simulation step length is given, numerical integration is carried out by adopting a trapezoidal integration formula, and the high-order track sensitivity of each moment can be obtained
And
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| CN101369002A (en) * | 2008-10-08 | 2009-02-18 | 东北电力大学 | Method for amending simulation parameter of generator by measured track and trace sensitivity |
| CN104036118A (en) * | 2014-05-14 | 2014-09-10 | 浙江大学 | Method for obtaining power system parallelization track sensitivity |
| CN106959618A (en) * | 2017-05-05 | 2017-07-18 | 国网山东省电力公司电力科学研究院 | A kind of voltage control method for coordinating for optimizing weight based on ladder |
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| CN101369002A (en) * | 2008-10-08 | 2009-02-18 | 东北电力大学 | Method for amending simulation parameter of generator by measured track and trace sensitivity |
| CN104036118A (en) * | 2014-05-14 | 2014-09-10 | 浙江大学 | Method for obtaining power system parallelization track sensitivity |
| CN106959618A (en) * | 2017-05-05 | 2017-07-18 | 国网山东省电力公司电力科学研究院 | A kind of voltage control method for coordinating for optimizing weight based on ladder |
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