CN111046336A - Polar coordinate Newton method load flow calculation method for jacobian matrix change - Google Patents

Polar coordinate Newton method load flow calculation method for jacobian matrix change Download PDF

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CN111046336A
CN111046336A CN201911383666.9A CN201911383666A CN111046336A CN 111046336 A CN111046336 A CN 111046336A CN 201911383666 A CN201911383666 A CN 201911383666A CN 111046336 A CN111046336 A CN 111046336A
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姚玉斌
王珏
肖炳旭
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Dalian Maritime University
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Abstract

The invention discloses a polar coordinate Newton method load flow calculation method for the change of a Jacobian matrix, which divides the load flow calculation into two stages, and adopts different methods to calculate the Jacobian matrix in different stages, which is called as a two-stage method. The iteration times of the 1 st stage are set, and practice shows that when the iteration times of the 1 st stage are set to be 2 times, the effect is optimal, and the running time of load flow calculation can be effectively reduced. The method adopts a two-stage method to perform load flow calculation, not only solves the problem of convergence of polar coordinate Newton method load flow calculation when a power system containing small impedance branches is analyzed, but also reduces the running time of the load flow calculation. When the load flow calculation is not converged by adopting the conventional polar coordinate Newton method, the method can reliably converge and has less operation time than that of the prior art. The method can effectively solve the problem of convergence of the power system containing the small-impedance branch circuit in the conventional polar coordinate Newton method load flow calculation analysis, and can also perform load flow calculation on the normal power system without adverse effect.

Description

Polar coordinate Newton method load flow calculation method for jacobian matrix change
Technical Field
The invention relates to a load flow calculation method of a power system with small impedance branches, in particular to a polar coordinate Newton method load flow calculation method with the change of a Jacobian matrix.
Background
The power flow calculation is a basic calculation for researching the steady-state operation of the power system, and determines the operation state of the whole power system according to the given operation condition and the network structure of the power system. The power flow calculation is also the basis of other analyses of the power system, and the power flow calculation is used in safety analysis, transient stability analysis and the like. The Newton method becomes the mainstream method of the current load flow calculation due to the advantages of reliable convergence, high calculation speed and moderate memory requirement. The Newton method is divided into two forms of polar coordinates and rectangular coordinates, and the Newton method load flow calculation of the two forms is widely applied to a power system.
In polar coordinate newton method load flow calculation, the voltage of the node i is expressed by adopting a polar coordinate as follows:
Figure BDA0002342929840000011
for a normal power network, polar coordinate Newton method load flow calculation has good convergence, but when a sick network with small impedance branches is encountered, the polar coordinate Newton method load flow calculation can be dispersed. The small impedance branch is ubiquitous in a power system, the small impedance branch can be divided into a small impedance line and a small impedance transformer branch, and the line can be regarded as a transformer with a transformation ratio of 1:1 on a mathematical model.
As shown in fig. 1, a conventional polar newton method load flow calculation method mainly includes the following steps:
A. inputting original data and an initialization voltage;
according to the characteristics of the nodes of the power system, the nodes of the power system are divided into 3 types by load flow calculation: the node with known active power and reactive power and unknown node voltage amplitude and voltage phase angle is injected into the node and is called as a PQ node; the node with known node injection active power and voltage amplitude and unknown node injection reactive power and voltage phase angle is called PV node; the node with known node voltage amplitude and voltage phase angle and unknown node injection active power and reactive power is called a balance node.
The voltage initialization adopts flat start, namely the voltage amplitudes of the PV node and the balance node are set values, and the voltage amplitude of the PQ node is 1.0; the voltage phase angles of all nodes take 0.0. The voltage phase angle is here in units of radians, other quantities take per unit values.
B. Forming a node admittance matrix;
C. setting an iteration count t to be 0;
D. calculating the node power and the node power unbalance amount to obtain the maximum unbalance amount delta Wmax；
The node power calculation formula is as follows:
Figure BDA0002342929840000021
in the formula, Pi、QiRespectively the active power and the reactive power of the node i; u shapei、UkThe voltage amplitudes of the node i and the node k are respectively; thetaik＝θi-θk，θiAnd thetakVoltage phase angles of the node i and the node k respectively; gik、BikAre respectively node admittance matrix elements YikThe real and imaginary parts of (c); n is the number of nodes.
Setting No. 1-m nodes as PQ nodes, No. m + 1-n-1 nodes as PV nodes, No. n nodes as balance nodes, and the calculation formula of the node power unbalance amount is as follows:
Figure BDA0002342929840000022
in the formula,. DELTA.Pi、ΔQiRespectively the active power unbalance amount and the reactive power unbalance amount of the node i; pis、QisRespectively giving an injection active power and an injection reactive power for the node i; m is the number of PQ nodes.
The balance nodes do not participate in iterative computation, and the power unbalance of the nodes does not need to be computed.
And calculating the maximum value of the absolute value in the power unbalance of each node, which is called as the maximum unbalance.
E. Judging the absolute value | Delta W of the maximum unbalance amountmaxWhether | is less than convergence precision ε; if the convergence precision is smaller than the convergence precision epsilon, executing a step I; otherwise, executing step F.
F. Forming a Jacobian matrix J;
the formula for calculating the elements of the jacobian matrix J (i ≠ J) is as follows:
Figure BDA0002342929840000023
Figure BDA0002342929840000024
Figure BDA0002342929840000025
Figure BDA0002342929840000031
in the formula (I), the compound is shown in the specification,
Figure BDA0002342929840000032
is the partial derivative of the active power unbalance of the node i to the voltage phase angle of the node j;
Figure BDA0002342929840000033
is the partial derivative of the active power unbalance of the node i to the voltage amplitude of the node j;
Figure BDA0002342929840000034
is the partial derivative of the reactive power unbalance of the node i to the voltage phase angle of the node j;
Figure BDA0002342929840000035
is the partial derivative of the reactive power unbalance of the node i to the voltage amplitude of the node j;
the formula for calculating the elements (i ═ J) of the jacobian matrix J is as follows:
Figure BDA0002342929840000036
Figure BDA0002342929840000037
Figure BDA0002342929840000038
Figure BDA0002342929840000039
in the formula (I), the compound is shown in the specification,
Figure BDA00023429298400000310
the partial derivative of the active power unbalance of the node i to the voltage phase angle of the node i is obtained;
Figure BDA00023429298400000311
the partial derivative of the active power unbalance of the node i to the voltage amplitude of the node i is obtained;
Figure BDA00023429298400000312
is the partial derivative of the reactive power unbalance of the node i to the voltage phase angle of the node i;
Figure BDA00023429298400000313
is the partial derivative of the amount of reactive power imbalance at node i to the voltage magnitude at node i.
Or by the following formula:
Figure BDA00023429298400000314
in the formula, Pi、QiAnd respectively calculating the active power and the reactive power of the node i according to the formula (1).
G. Solving a correction equation and correcting the node voltage amplitude U and the phase angle theta;
the basic equation of the load flow calculation is a nonlinear equation, and a successive linearization method is generally adopted for iterative solution. And (4) a correction equation (12) obtained through linearization is used for solving the correction quantity of the voltage amplitude and the phase angle.
Figure BDA0002342929840000041
Wherein J is a Jacobian matrix of (n + m-1) × (n + m-1) order, and H, N, M, L are four partitioned submatrices of the Jacobian matrix respectively, and the dimensions are (n-1) × (n-1) order, (n-1) × m order, mx (n-1) order, and mx order, respectively; Δ θ ═ Δ θ1,…,Δθn-1]TThe upper mark T represents transposition for the column vector of the phase angle correction quantity of the node voltage; delta U/U ═ delta U1/U1,…,ΔUm/Um]TDividing the node voltage amplitude correction quantity by the column vector of the node voltage amplitude; Δ P ═ Δ P1,…,ΔPn-1]TThe active power unbalance column vector of the node is obtained; Δ Q ═ Δ Q1,…,ΔQm]TIs a column vector of node reactive power unbalance.
The node voltage correction formula is as follows:
Figure BDA0002342929840000042
in the formula, superscript (t) represents the t iteration; delta UiAnd Δ θiThe voltage amplitude correction and the voltage phase angle correction of the node i are respectively.
H. D, enabling t to be t +1, and returning to the step D for next iteration;
I. output node and branch data.
For a normal power network, Newton method load flow calculation has good convergence, but when a sick network with small impedance branches is encountered, the Newton method load flow calculation can be diverged. The small impedance branch in the power system generally exists, the convergence is the most important index of the nonlinear problems of power flow calculation of the power system, and the solution of the equation cannot be obtained without convergence in the calculation. Therefore, the method has very important significance for improving the convergence of the polar coordinate Newton method load flow calculation for the power system with the small impedance branch.
The document 'a new method for solving the load flow of a system with small impedance branches' proposes to improve the Jacobian matrix calculated by the load flow through the conventional polar coordinate Newton methodAnd a method for calculating convergence of the power flow. The method adopts different methods to calculate the Jacobian matrix elements according to the type of the node connection branch, and the endpoint of the small impedance branch adopts a given value PisAnd QisAnd calculating the Jacobian elements, and calculating the Jacobian matrix elements by the end points of the normal impedance branches according to a conventional method, thereby effectively solving the divergence problem of the load flow calculation of the small impedance branch power system with the resistance of 0.
As shown in fig. 2, a polar coordinate newton method power flow calculation method proposed in the document "new method for solving power flow of system with small impedance branch" mainly includes the following steps:
A. inputting original data and an initialization voltage;
B. determining the branch type T connected with nodes at two ends according to the sizes of the branch resistance and the reactance;
the concrete steps of forming the branch type array connected with the nodes are as follows:
b1 reading branch data and small resistance threshold rminAnd a small reactance threshold xmin；
B2, clearing the branch type array T connected with the node;
b3, let d equal to 1;
b4, taking the numbers p and q of the first and last nodes of the branch d, a resistor r and a reactance x;
b5, judging whether r is less than or equal to rminAnd x is less than or equal to xminIf not, go to step B7;
b6, let Tp＝1，Tq＝1；
B7, let d be d + 1;
b8, judging whether d is larger than l branch number, if d is not larger than l, turning to step B4; otherwise, turning to the step C;
C. forming a node admittance matrix;
D. setting an iteration count t to be 0;
E. calculating the node power and the node power unbalance amount to obtain the maximum unbalance amount delta Wmax；
And calculating the node power according to the formula (1), and calculating the node power unbalance according to the formula (2).
Each is obtainedThe maximum value of the absolute value in the active power unbalance amount and the reactive power unbalance amount of the node is used as the maximum unbalance amount delta Wmax。
F. Judging the absolute value | Delta W of the maximum unbalance amountmaxWhether | is less than convergence precision ε; if the convergence precision is smaller than the convergence precision epsilon, executing a step J; otherwise, step G is performed.
G. Forming a Jacobian matrix J;
the jacobian matrix elements when i ≠ j are calculated as in equation (3) to equation (6).
The specific steps of calculating the jacobian matrix elements when i is j are as follows:
g1, let i equal 1;
g2, judging whether i is more than PQ node number m, if i is more than m, turning to step G5;
g3, determining whether T is satisfiediIf this condition is satisfied, the jacobian matrix element N is calculated as equation (14)iiAnd LiiGo to step G4, otherwise calculate Jacobian matrix element N according to equation (15)iiAnd LiiThen go to step G4;
Figure BDA0002342929840000061
in the formula, Pis、QisThe node i is given injected active power and injected reactive power, respectively.
Figure BDA0002342929840000062
G4 calculating the Jacobian matrix element M according to the formulaii；
Figure BDA0002342929840000063
G5 calculating the Jacobian matrix element H by the following formulaii；
Figure BDA0002342929840000064
G6, let i ═ i + 1;
g7, judging whether i is less than the number n of nodes, and turning to G2 if i is less than n; otherwise, turning to the step H;
H. solving a correction equation and correcting the node voltage amplitude U and the phase angle theta;
I. c, enabling t to be t +1, and returning to the step E for next iteration;
J. output node and branch data.
In step G of the above calculation method, when the jacobian matrix elements are calculated by using equations (3) to (6), if the constraint condition that i ≠ j is removed, when i ≠ j, it is obtained:
Figure BDA0002342929840000065
comparing equation (18) with equation (11), it can be seen that equation (18) is a part of equation (11), and node power P is used based on equation (18)iAnd QiThe formula (11) can be obtained by correcting.
According to the analysis of a document 'a new method for solving the load flow of a system with small impedance branches', the power of the end point of the small impedance branch calculated according to the initial voltage value set by flat start is too large, so that the load flow calculation by a polar coordinate Newton method is diverged, and if the node power P is not usediAnd QiModifying to use a given injection node power PisAnd QisThe correction can solve the problem of divergence of power flow calculation, but makes algorithm logic complex and operation time longer.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a polar coordinate newton method load flow calculation method with a change of a jacobian matrix, so as to improve the convergence rate of a small-impedance branch power system with a resistance of not 0 in analysis and reduce the operation time.
In order to achieve the above purpose, the power flow calculation of the invention is divided into two stages, and the jacobian matrix is calculated by adopting different methods in different stages, which is referred to as a two-stage method. The number of iterations in stage 1 is set, and in this stage, the jacobian matrix element when i is j is directly calculated by using formula (18) to ensure that the power flow calculation converges, and in the iteration in stage 2, the jacobian matrix element when i is j is calculated by using formula (11) to reduce the number of iterations. The iteration number of the 1 st stage is a set value, and practice shows that the effect is best when the iteration number of the 1 st stage is set to be 2, and the running time of load flow calculation can be effectively reduced, so the iteration number of the 1 st stage is set to be 2.
The technical scheme of the invention is as follows: a polar coordinate Newton method load flow calculation method for changing a Jacobian matrix comprises the following steps:
A. inputting original data and an initialization voltage;
B. forming a node admittance matrix;
C. setting an iteration count t to be 0;
D. calculating the node power and the node power unbalance amount to obtain the maximum unbalance amount delta Wmax；
And calculating the node power according to the formula (1), and calculating the node power unbalance according to the formula (2).
Calculating the maximum value of the absolute value of the active power unbalance and the reactive power unbalance of each node as the maximum unbalance delta Wmax。
E. Judging the absolute value | Delta W of the maximum unbalance amountmaxWhether | is less than convergence precision ε; if the convergence precision is smaller than the convergence precision epsilon, executing a step J; otherwise, executing step F;
F. forming an initial jacobian matrix;
the elements of the initial jacobian matrix are calculated as follows:
Figure BDA0002342929840000071
Figure BDA0002342929840000072
Figure BDA0002342929840000073
Figure BDA0002342929840000074
G. judging whether a condition that t is less than or equal to 1 is met, if the condition is met, turning to a step H, otherwise, calculating the Jacobian matrix element according to the formula (11), and turning to the step H;
H. solving a correction equation and correcting the node voltage amplitude U and the phase angle theta;
I. d, enabling t to be t +1, and returning to the step D for next iteration;
J. output node and branch data.
Compared with the prior art, the invention has the following beneficial effects:
1. the method adopts a two-stage method to perform load flow calculation, not only solves the problem of convergence of polar coordinate Newton method load flow calculation when a power system containing small impedance branches is analyzed, but also reduces the running time of the load flow calculation. When the load flow calculation is not converged by adopting the conventional polar coordinate Newton method, the method can reliably converge and has less operation time than that of the prior art.
2. The method can effectively solve the problem of convergence of the power system containing the small-impedance branch circuit in the conventional polar coordinate Newton method load flow calculation analysis, and can also perform load flow calculation on the normal power system without adverse effect.
3. Compared with the prior art, the method has simple logic, is particularly suitable for designing the load flow calculation method by using matrix operation of Matlab, and achieves the purposes of simplifying the design and improving the calculation speed.
Drawings
The invention is shown in figure 4. Wherein:
fig. 1 is a flow chart of a conventional polar newton method load flow calculation.
Fig. 2 is a flow chart of a polar newton method load flow calculation of the literature method.
FIG. 3 is a flow chart of a literature method for forming an array of branch types to which a node is connected.
Fig. 4 is a flow chart of polar newton method load flow calculation according to the present invention.
Detailed Description
The following is combined with the attached drawingsThe present invention is further illustrated. According to the flow chart of polar coordinate Newton method load flow calculation shown in FIG. 4, load flow calculation is performed on an actual large-scale power grid. The actual large power grid has 445 nodes, which contain a large number of small impedance branches. Wherein, 49 small impedance branches with x less than or equal to 0.001, 41 small impedance branches with x less than or equal to 0.0001, and 22 small impedance branches with x less than or equal to 0.00001. The small impedance branch l between the node 118 and the node 125, where the impedance value is smallest118-125X is 0.00000001. The convergence accuracy of the load flow calculation is 0.00001. In order to verify the convergence of the power system with the small-impedance branch circuit with the resistance different from 0 calculated by the invention, the small-impedance branch circuit l118-125、l60-122And l287-310The resistance of (d) was changed to 0.0001.
For comparison, the following 4 comparison methods are adopted to perform load flow calculation on the actual large-scale power grid at the same time:
the conventional method comprises the following steps: a conventional polar coordinate Newton method load flow calculation method;
literature methods: a method proposed in a document 'a new method for solving a system trend containing small-impedance branches';
comparative method 1: setting the iteration times of the 1 st stage as 1 by adopting a two-stage method;
comparative method 2: the two-stage method is adopted, and the iteration number of the 1 st stage is set to be 3.
The results of the number of iterations are shown in table 1.
TABLE 1 iterative results of different power flow calculation methods
Method of producing a composite material Conventional methods Literature methods The invention Comparative method 1 Comparative method 2
Iteration result Non-convergence 5 times of convergence 5 times of convergence 5 times of convergence Convergence of 7 times
As can be seen from table 1, for the modified 445-node actual power system calculation example, the conventional polar coordinate newton method load flow calculation method does not converge, and the present invention, the literature method, and the two comparison methods can converge, but the iteration number of the present invention is less than that of the comparison method.
When the Matlab M file is adopted for programming, the running time of different load flow calculation methods is shown in a table 2, the unit of the running time is s, and the running time does not include the time for reading data, calculating branch power and outputting results.
TABLE 2 run times of different load flow calculation methods
Method of producing a composite material Conventional methods Literature methods The invention Comparative method 1 Comparative method 2
Run time(s) ----- 0.150251 0.117330 0.130783 0.150248
Table 2 does not record the run time of the conventional method since the conventional polar newton method load flow calculation method does not converge. As can be seen from Table 2, for the modified 445 node actual power system example, the present invention is faster than the literature method and the other two comparative methods, and the running time of the present invention is reduced by 21.9% compared with the literature method.
The present invention can be implemented using any programming language and programming environment, such as C language, C + +, FORTRAN, Delphi, Matlab, etc. The development environment may employ Visual C + +, Borland C + + Builder, Visual FORTRAN, and the like.
The present invention is not limited to the embodiment, and any equivalent idea or change within the technical scope of the present invention is to be regarded as the protection scope of the present invention.

Claims (1)

1. A polar coordinate Newton method load flow calculation method for changing of a Jacobian matrix is characterized in that: the method comprises the following steps:
A. inputting original data and an initialization voltage;
according to the characteristics of the nodes of the power system, the nodes of the power system are divided into 3 types by load flow calculation: the node with known active power and reactive power and unknown node voltage amplitude and voltage phase angle is injected into the node and is called as a PQ node; the node with known node injection active power and voltage amplitude and unknown node injection reactive power and voltage phase angle is called PV node; the node with known node voltage amplitude and voltage phase angle and unknown node injection active power and reactive power is called a balance node;
the voltage initialization adopts flat start, namely the voltage amplitudes of the PV node and the balance node are set values, and the voltage amplitude of the PQ node is 1.0; the voltage phase angles of all the nodes are 0.0; the unit of the voltage phase angle is radian, and other quantities adopt per unit values;
B. forming a node admittance matrix;
C. setting an iteration count t to be 0;
D. calculating the node power and the node power unbalance amount to obtain the maximum unbalance amount delta Wmax；
The node power calculation formula is as follows:
Figure FDA0002342929830000011
in the formula, Pi、QiRespectively the active power and the reactive power of the node i; u shapei、UkThe voltage amplitudes of the node i and the node k are respectively; thetaik＝θi-θk，θiAnd thetakVoltage phase angles of the node i and the node k respectively; gik、BikAre respectively node admittance matrix elements YikThe real and imaginary parts of (c); n is the number of nodes;
setting No. 1-m nodes as PQ nodes, No. m + 1-n-1 nodes as PV nodes, No. n nodes as balance nodes, and the calculation formula of the node power unbalance amount is as follows:
Figure FDA0002342929830000012
in the formula,. DELTA.Pi、ΔQiRespectively the active power unbalance amount and the reactive power unbalance amount of the node i; pis、QisRespectively giving an injection active power and an injection reactive power for the node i; m is the number of PQ nodes;
the balance nodes do not participate in iterative computation, and the power unbalance of the nodes does not need to be computed;
solving the maximum value of the absolute value in the active power unbalance and the reactive power unbalance of each node, which is called as the maximum unbalance;
E. judging the absolute value | Delta W of the maximum unbalance amountmaxWhether | is less than convergence precision ε; if the convergence precision is smaller than the convergence precision epsilon, executing a step J; otherwise, executing step F;
F. forming an initial jacobian matrix;
the elements of the initial jacobian matrix are calculated as follows:
Figure FDA0002342929830000021
Figure FDA0002342929830000022
Figure FDA0002342929830000023
Figure FDA0002342929830000024
in the formula (I), the compound is shown in the specification,
Figure FDA0002342929830000025
is the partial derivative of the active power unbalance of the node i to the voltage phase angle of the node j;
Figure FDA0002342929830000026
is the partial derivative of the active power unbalance of the node i to the voltage amplitude of the node j;
Figure FDA0002342929830000027
is the partial derivative of the reactive power unbalance of the node i to the voltage phase angle of the node j;
Figure FDA0002342929830000028
of the magnitude of the voltage at node j for the amount of reactive power imbalance at node iA partial derivative;
G. judging whether a condition that t is less than or equal to 1 is met, if the condition is met, turning to a step H, otherwise, calculating the elements of the Jacobian matrix according to the formula (7), and turning to the step H;
Figure FDA0002342929830000029
in the formula (I), the compound is shown in the specification,
Figure FDA0002342929830000031
the partial derivative of the active power unbalance of the node i to the voltage phase angle of the node i is obtained;
Figure FDA0002342929830000032
the partial derivative of the active power unbalance of the node i to the voltage amplitude of the node i is obtained;
Figure FDA0002342929830000033
is the partial derivative of the reactive power unbalance of the node i to the voltage phase angle of the node i;
Figure FDA0002342929830000034
is the partial derivative of the reactive power unbalance of the node i to the voltage amplitude of the node i;
H. solving a correction equation and correcting the node voltage amplitude U and the phase angle theta;
the correction equation is:
Figure FDA0002342929830000035
wherein J is a Jacobian matrix of (n + m-1) × (n + m-1) order, and H, N, M, L are four partitioned submatrices of the Jacobian matrix respectively, and the dimensions are (n-1) × (n-1) order, (n-1) × m order, mx (n-1) order, and mx order, respectively; Δ θ ═ Δ θ1,…,Δθn-1]TThe upper mark T represents transposition for the column vector of the phase angle correction quantity of the node voltage; delta U/U ═ delta U1/U1,…,ΔUm/Um]TDividing the node voltage amplitude correction quantity by the column vector of the node voltage amplitude; Δ P ═ Δ P1,…,ΔPn-1]TThe active power unbalance column vector of the node is obtained; Δ Q ═ Δ Q1,…,ΔQm]TThe node reactive power unbalance column vector is obtained;
the voltage correction formula is as follows:
Figure FDA0002342929830000036
in the formula, superscript (t) represents the t iteration; delta UiAnd Δ θiRespectively is the voltage amplitude correction and the voltage phase angle correction of the node i;
I. d, enabling t to be t +1, and returning to the step D for next iteration;
J. output node and branch data.
CN201911383666.9A 2019-12-27 2019-12-27 Polar coordinate Newton method load flow calculation method for jacobian matrix change Pending CN111046336A (en)

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CN106532711A (en) * 2016-11-30 2017-03-22 大连海事大学 Newton method power flow calculation method which changes Jacobi matrix with iteration and node types
CN106709243A (en) * 2016-12-09 2017-05-24 大连海事大学 Small-impedance branch network contained compensation-method polar coordinate Newton method tidal current calculation method

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CN106532711A (en) * 2016-11-30 2017-03-22 大连海事大学 Newton method power flow calculation method which changes Jacobi matrix with iteration and node types
CN106709243A (en) * 2016-12-09 2017-05-24 大连海事大学 Small-impedance branch network contained compensation-method polar coordinate Newton method tidal current calculation method

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