CN102354983A - Method for determining weak nodes in voltage stability based on matrix perturbation theory - Google Patents

Abstract

The invention provides a method for determining weak nodes in voltage stability based on a matrix perturbation theory. The method is characterized by comprising the following steps: constructing Jacobian matrix conditional number based on the matrix perturbation theory; constructing indexes of the weak nodes in the voltage stability based on sensitivity of the Jacobian matrix conditional number; and establishing load margin indexes of the nodes. The method provided by the invention has the advantages that the matrix perturbation theory is applied to voltage stability analysis of a power system, which opens a new approach for research and analysis of the voltage stability of the power system, overcomes defects of the traditional determination mode of the weak nodes in the static voltage stability and the defects of repeated power flow calculation in a continuation power flow (CPF) method for calculating load margin; and the calculation procedure is simple, the calculation speed is high and the engineering application value is high.

Description

A kind ofly confirm that based on the matrix perturbation theory voltage stablizes the weak node method

Technical field

The present invention is based on the matrix perturbation theory and confirms that voltage stablizes the weak node method, belongs to power system voltage stabilization prevention and control technology field.

Background technology

The interconnected big electrical network that with big unit, superhigh pressure, long distance power transmission is main feature is being when system brings great economic benefit; Changeable operational mode also makes it occur the fragility that is difficult to expect inevitably, and it is more outstanding to make that the power system voltage stabilization problem becomes.Intelligent grid environmental requirement dispatcher can be quick and precisely and is grasped the quantizating index of power grid security intuitively; This has higher requirement with regard to stablizing the research and analysis method for voltage, how to confirm fast and accurately that the weak node in the stable research of voltage has caused domestic and international experts and scholars' extensive concern.

In the power system voltage stabilization Journal of Sex Research, be defined as weak node with relatively easily losing the stable load bus of voltage in the system, its essence be exactly node can bear ability that load power changes relatively a little less than.At present, most continuous tide method and Sensitivity Analysis Method that adopt have clear concept, advantages such as easy to understand, but amount of calculation is bigger than normal, is inappropriate for stable analysis and the research of modern large-scale interconnected electric power system voltage.Though the Application of Singular Value Decomposition Method amount of calculation is less, the load margin index that it can't give egress can not realize the quantitative assessment to voltage stability.In the actual engineering, power system planning and operation department in the weak link, more pay close attention to the distance of current running status apart from critical point, i.e. stability margin in being concerned about network.If can when the fragility of electric power networks is made correct evaluation, can provide the load margin of weak node, will help power system operation department to take corresponding control measures early to guarantee the safe operation of electrical network.

Summary of the invention

The objective of the invention is to overcome existing power system voltage stabilization weak node and confirm the deficiency of method, provide a kind of have calculate simple, result accurately, be easy to realize and need not the repetition trend calculate and confirm that voltage stablizes the method for weak node.

Realize that the technical scheme that the present invention adopted is, a kind ofly confirm that based on the matrix perturbation theory voltage stablizes the weak node method, it is characterized in that it may further comprise the steps:

1) based on the structure of the Jacobian matrix conditional number of matrix perturbation theory

The equation that is used for electric power system tide calculating is:

JV＝W (1)

Wherein: J is used for the Jacobian matrix that system load flow calculates,

V representes the variation column vector [Δ θ Δ U/U] of node phase angle and voltage ^T,

W representes the variation column vector [Δ P Δ Q] of node active power and reactive power ^T

When system receives extraneous disturbance; System can meritoriously on the basis of present operating point redistribute with reactive power; Make system load flow reach a new operating point; Corresponding variation has also taken place in the element that this moment, total system was used for the Jacobian matrix of trend iteration; Power flow equation reaches balance at new operating point place, and there is following relation in system at new operating point place:

(J+ΔJ)(V+ΔV)＝(W+ΔW) (2)

Wherein: Δ J is that system is measured by the change of Jacobian matrix after the disturbance,

Δ V is that system is subjected to disturbance posterior nodal point voltage to change column vector,

Δ W is that system is subjected to disturbance posterior nodal point power to change column vector,

Make J '=J+ Δ J, then J ' is the Jacobian matrix of system at new operating point place power flow equation;

Establishing a capital corresponding to the Jacobian matrix J at two operating point places of trend convergence and J ' is non-singular matrix, and when system configuration did not change, J was identical with J ' dimension, can get according to the disturbance enclosure theorem of matrix inversion and equation group in the matrix perturbation theory:

$\frac{\left|\right|J^{\′-1}-J^{-1}\left|\right|}{\left|\right|J^{-1}\left|\right|}\≤\frac{\mathrm{\κ}}{1-\mathrm{\κ}{\left|\right|\mathrm{\ΔJ}\left|\right|}_2/{\left|\right|J\left|\right|}_2}\frac{{\left|\right|\mathrm{\ΔJ}\left|\right|}_2}{\left|\right|J\left|\right|}---\left(3\right)$

With

$\frac{\left|\right|\mathrm{\ΔW}\left|\right|}{\left|\right|W\left|\right|}\≤\frac{\mathrm{\κ}}{1-\mathrm{\κ}{\left|\right|\mathrm{\ΔJ}\left|\right|}_2/{\left|\right|J\left|\right|}_2}(\frac{{\left|\right|\mathrm{\ΔJ}\left|\right|}_2}{\left|\right|J\left|\right|}+\frac{\left|\right|\mathrm{\ΔV}\left|\right|}{\left|\right|V\left|\right|})---\left(4\right)$

Wherein:

κ＝||J|| ₂*||J ^-1|| ₂ (5)

Then, (5) formula can be defined the conditional number of Jacobian matrix according to the matrix perturbation theory;

Wherein: J is a Jacobian matrix in the power flow equation,

J ^-1Be the inverse matrix of J,

|| || ₂The 2-norm of representing matrix or vector;

2) stablize the structure of weak node index based on the voltage of Jacobian matrix conditional number sensitivity

State variable and control variables are respectively X=(x in the note system ₁, x ₂..., x _n) ^T, and Y=(y ₁, y ₂... y _n) ^T, n is the node number in the system,

By the differentiate rule of compound function, conditional number to the sensitivity of state variable does in the formula (5)

$\frac{\∂\mathrm{\κ}}{\∂x_i}=(\frac{\∂{\mathrm{\δ}}_{\max }}{\∂x_i}{\mathrm{\δ}}_{\min }-\frac{\∂{\mathrm{\δ}}_{\min }}{\∂x_i}{\mathrm{\δ}}_{\max })/{\mathrm{\δ}}_{\min }^2---\left(6\right)$

Wherein: δ _MaxAnd δ _MinBe the maximum and the minimum singular value of Jacobian matrix,

According to the differentiate rule of implicit function, singular value to the partial derivative of state variable in the system is:

$\frac{\∂\mathrm{\δ}}{\∂x_i}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{\∂\mathrm{\δ}}{\∂y_i}\frac{\∂y_i}{\∂x_i}(i=\mathrm{1,2},...,n)---\left(7\right)$

That is:

$\frac{\∂\mathrm{\δ}}{\∂Y}={\left[J^{-1}\right]}^T\frac{\∂\mathrm{\δ}}{\∂X}---\left(8\right)$

To sum up, can get the sensitivity degree of conditional number to control variables:

$\frac{\∂\mathrm{\κ}}{\∂y_i}={\left[J^{-1}\right]}^T(\frac{V_{\max }^T\frac{\∂J}{\∂x_i}{U}_{\max }}{V_{\max }^T{U}_{\max }}{\mathrm{\δ}}_{\min }-\frac{V_{\min }^T\frac{\∂J}{\∂x_i}{U}_{\min }}{V_{\min }^T{U}_{\min }}{\mathrm{\δ}}_{\max })/{\mathrm{\δ}}_{\min }^2---\left(9\right)$

Wherein: V _MaxAnd V _MinBe respectively δ _MaxAnd δ _MinCorresponding left singular vector,

U _MaxAnd U _MinBe respectively δ _MaxAnd δ _MinCorresponding right singular vector;

3) foundation of node load nargin index

The size of Jacobian matrix conditional number sensitivity has reflected the sensitivity that the stability of a system changes node power, and the definite condition number is a node load nargin index to the inverse of node control variable sensitivity, that is:

$L_p\left(i\right)=\frac{1}{\∂\mathrm{\τ}/\∂y_i}\×100\%---\left(10\right)$

The present invention is based on the matrix perturbation theory and confirm that it is to utilize the definite voltage of Jacobian matrix conditional number sensitivity realization to stablize the method for weak node that voltage is stablized the weak node method; Confirm that with traditional weak node method compares the advantage applies that has and exist in the large-scale electrical power system static electric voltage stability analysis in modern times: 1. Jacobian matrix conditional number and sensitivity mathematical derivation process thereof are rigorous, the explicit physical meaning of model; 2. under current running status, can confirm the weak node in the system, calculate node load nargin index simultaneously; 3. Jacobian matrix conditional number and Sensitivity calculation thereof are simple, and be consuming time few; The present invention is applied to the matrix perturbation theory in the power system voltage stabilization analysis; Opened up new approach for the research and analysis of power system voltage stabilization property, improved traditional static voltage stable in weak node confirm to repeat the defective that trend is calculated in the continuous tide method with calculated load nargin; It calculates simply, speed is fast, and practical applications is worth high.

Description of drawings

Fig. 1 IEEE57 node system structure chart.

Conditional number change curve when Fig. 2 IEEE57 node system load level increases.

Conditional number is to the idle sensitivity of each load bus under Fig. 3 IEEE57 node system two states.

Each node nargin index of Fig. 4 increases change curve with system loading.

Embodiment

Of the present inventionly a kind ofly confirm that based on the matrix perturbation theory voltage stablizes the weak node method, be used for the static voltage stability analysis of electric power system, may further comprise the steps:

1) based on the structure of the Jacobian matrix conditional number of matrix perturbation theory

The equation that is used for electric power system tide calculating is:

JV＝W (1)

Wherein: J is used for the Jacobian matrix that system load flow calculates,

V representes the variation column vector [Δ θ Δ U/U] of node phase angle and voltage ^T,

W representes the variation column vector [Δ P Δ Q] of node active power and reactive power ^T

When system received an extraneous disturbance (like the switching of load, the starting of motor etc.), system can redistribute meritorious and idle distribution on the basis of present operating point, make system load flow reach a new operating point.Corresponding variation has also taken place in the element that this moment, total system was used for the Jacobian matrix of trend iteration, and power flow equation reaches balance at new operating point place.There is following relation in system at new operating point place:

(J+ΔJ)(V+ΔV)＝(W+ΔW) (2)

Wherein: Δ J is that system is measured by the change of Jacobian matrix after the disturbance;

Δ V is that system is subjected to disturbance posterior nodal point voltage to change column vector;

Δ W is that system is subjected to disturbance posterior nodal point power to change column vector;

Make J '=J+ Δ J, then J ' is the Jacobian matrix of system at new operating point place power flow equation.

Establishing a capital corresponding to the Jacobian matrix J at two operating point places of trend convergence and J ' is non-singular matrix, and when system configuration did not change, J was identical with J ' dimension, can get according to the disturbance enclosure theorem of matrix inversion and equation group in the matrix perturbation theory:

$\frac{\left|\right|J^{\′-1}-J^{-1}\left|\right|}{\left|\right|J^{-1}\left|\right|}\≤\frac{\mathrm{\κ}}{1-\mathrm{\κ}{\left|\right|\mathrm{\ΔJ}\left|\right|}_2/{\left|\right|J\left|\right|}_2}\frac{{\left|\right|\mathrm{\ΔJ}\left|\right|}_2}{\left|\right|J\left|\right|}---\left(3\right)$

With

$\frac{\left|\right|\mathrm{\ΔW}\left|\right|}{\left|\right|W\left|\right|}\≤\frac{\mathrm{\κ}}{1-\mathrm{\κ}{\left|\right|\mathrm{\ΔJ}\left|\right|}_2/{\left|\right|J\left|\right|}_2}(\frac{{\left|\right|\mathrm{\ΔJ}\left|\right|}_2}{\left|\right|J\left|\right|}+\frac{\left|\right|\mathrm{\ΔV}\left|\right|}{\left|\right|V\left|\right|})---\left(4\right)$

Wherein:

κ＝||J|| ₂*||J ^-1|| ₂ (5)

Then, (5) formula can be defined the conditional number of Jacobian matrix according to the matrix perturbation theory;

Wherein: J is a Jacobian matrix in the power flow equation,

J ^-1Be the inverse matrix of J,

|| || ₂The 2-norm of representing matrix or vector;

2) stablize the structure of weak node index based on the voltage of Jacobian matrix conditional number sensitivity

State variable and control variables are respectively X=(x in the note system ₁, x ₂..., x _n) ^T, and Y=(y ₁, y ₂... y _n) ^T, n is the node number in the system,

By the differentiate rule of compound function, conditional number to the sensitivity of state variable does in the formula (5)

$\frac{\∂\mathrm{\κ}}{\∂x_i}=(\frac{\∂{\mathrm{\δ}}_{\max }}{\∂x_i}{\mathrm{\δ}}_{\min }-\frac{\∂{\mathrm{\δ}}_{\min }}{\∂x_i}{\mathrm{\δ}}_{\max })/{\mathrm{\δ}}_{\min }^2---\left(6\right)$

Wherein: δ _MaxAnd δ _MinBe the maximum and the minimum singular value of Jacobian matrix,

According to the differentiate rule of implicit function, singular value to the partial derivative of state variable in the system is:

$\frac{\∂\mathrm{\δ}}{\∂x_i}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{\∂\mathrm{\δ}}{\∂y_i}\frac{\∂y_i}{\∂x_i}(i=\mathrm{1,2},...,n)---\left(7\right)$

That is:

$\frac{\∂\mathrm{\δ}}{\∂Y}={\left[J^{-1}\right]}^T\frac{\∂\mathrm{\δ}}{\∂X}---\left(8\right)$

To sum up, can get the sensitivity degree of conditional number to control variables:

$\frac{\∂\mathrm{\κ}}{\∂y_i}={\left[J^{-1}\right]}^T(\frac{V_{\max }^T\frac{\∂J}{\∂x_i}{U}_{\max }}{V_{\max }^T{U}_{\max }}{\mathrm{\δ}}_{\min }-\frac{V_{\min }^T\frac{\∂J}{\∂x_i}{U}_{\min }}{V_{\min }^T{U}_{\min }}{\mathrm{\δ}}_{\max })/{\mathrm{\δ}}_{\min }^2---\left(9\right)$

Wherein: V _MaxAnd V _MinBe respectively δ _MaxAnd δ _MinCorresponding left singular vector,

U _MaxAnd U _MinBe respectively δ _MaxAnd δ _MinCorresponding right singular vector;

3) foundation of node load nargin index

The size of Jacobian matrix conditional number sensitivity has reflected the sensitivity that the stability of a system changes node power, and the definite condition number is a node load nargin index to the inverse of node control variable sensitivity, that is:

$L_p\left(i\right)=\frac{1}{\∂\mathrm{\τ}/\∂y_i}\×100\%---\left(10\right)$

Can know by (10) formula, weak node in the system, the value of conditional number sensitivity is big more, and its load margin index is more little.When system was in voltage and stablizes critical condition, it is infinitely great that the conditional number sensitivity of each node is tending towards, and the nargin index will approach 0.

Instantiation:

It is that example is calculated and analyzed the validity of Jacobian matrix and conditional number thereof that the present invention comes verification system with the IEEE57 node system.IEEE57 node system structure chart as shown in Figure 1.

Under different trend modes, the conditional number of system is calculated the variation tendency of conditional number when being illustrated in figure 2 as the total system load, the number of times of k for increasing among the figure respectively with ratio 20% increase.

Can find out that in Fig. 2 the conditional number that different load level is corresponding is also inequality, and along with the increase of load level, conditional number increases gradually, when system was put near voltage collapse, the conditional number increasing degree was bigger.Aforementioned calculation result shows that the size of Jacobian matrix conditional number has reflected the voltage maintenance level of system to a certain extent, and the current operating point of its big more illustrative system of value is more near the voltage collapse point.

In the IEEE57 node system, as control variables, its node voltage is as state variable, respectively in the sensitivity of two kinds of load level computing system conditional numbers to the load bus control variables with the reactive power of each load bus.Its result of calculation as shown in Figure 3.The bigger part of nodes of conditional number Sensitirity va1ue has only drawn among Fig. 3.

State 1: the initial load state of system;

State 2: under the initial load state, all load buses increase power 10% with permanent power factor (PF).

In Fig. 3, the difference in size of sensitivity is comparatively remarkable, and it is more obvious that weak node and weakness zone seem.Sensitivity under the state 2 is greater than the sensitivity under the state 1 on the whole, and this is because the load level relative status 1 of state 2 wants high, compares state 1 stability of a system and worsens, and more approaches the critical condition of voltage collapse.But magnitude relationship does not change between each the node sensitivity under the two states.Explanation is under above-mentioned two kinds of trend modes, and the weak node of system and weakness zone do not change.As space is limited, only listed bigger part of nodes data and the ordering of system sensitivity in the table 1.

Weak node ordering under the table 1IEEE57 node system different conditions

Utilize the continuous tide method to calculate the meritorious nargin of each weak node, its result of calculation is as shown in table 2.

The meritorious nargin of each weak node of table 2IEEE57 node system

Table 1 is weak node ranking results and the voltage margin index result of calculation thereof under two kinds of load methods of IEEE57 node system.Can find out that by table 1 data in system, its nargin index of node a little less than the relative thin shows that more near 0 its load margin is more little.Table 2 is two kinds of continuous tide result of calculations under the load method.Contrast table 1 and table 2 result, the nargin index under node 33 two states is respectively 0.0958 and 0.0592, and the continuous tide result of calculation of nargin is 19.38MW and 10.45MW under its two states simultaneously.The checkout result of other nodes has These characteristics equally, and characteristics show the degree of closeness that can cross good reflection system and critical condition based on the nargin index of Jacobian matrix conditional number sensitivity.

Be the characteristic of further checking system weak node, utilize the continuous tide method to calculate the meritorious nargin of each weak node, its result of calculation is as shown in table 3.

The meritorious nargin of each weak node of table 3IEEE57 node system

With the power of each load bus in the growth pattern increase system of constant power factor, change curve such as Fig. 4 of each the weak node load margin index in the table 3.

Can find that from above example the size of Jacobian matrix conditional number can reflect the distance of system apart from voltage collapse point preferably.Through the calculating checking to static load nargin of continuous tide method, declaration condition is counted sensitivity and can be accurately the weak node of system be sorted, and its nargin index has been reacted in the system each node current state accurately apart from the distance of voltage collapse critical point.

Claims (1)

1. confirm that based on the matrix perturbation theory voltage stablizes the weak node method for one kind, it is characterized in that it may further comprise the steps:

1) based on the structure of the Jacobian matrix conditional number of matrix perturbation theory

The equation that is used for electric power system tide calculating is:

JV＝W (1)

Wherein: J is used for the Jacobian matrix that system load flow calculates,

V representes the variation column vector [Δ θ Δ U/U] of node phase angle and voltage ^T,

W representes the variation column vector [Δ P Δ Q] of node active power and reactive power ^T

When system receives extraneous disturbance; System can meritoriously on the basis of present operating point redistribute with reactive power; Make system load flow reach a new operating point; Corresponding variation has also taken place in the element that this moment, total system was used for the Jacobian matrix of trend iteration; Power flow equation reaches balance at new operating point place, and there is following relation in system at new operating point place:

(J+ΔJ)(V+ΔV)＝(W+ΔW) (2)

Wherein: Δ J is that system is measured by the change of Jacobian matrix after the disturbance,

Δ V is that system is subjected to disturbance posterior nodal point voltage to change column vector,

Δ W is that system is subjected to disturbance posterior nodal point power to change column vector,

Make J '=J+ Δ J, then J ' is the Jacobian matrix of system at new operating point place power flow equation;

Establishing a capital corresponding to the Jacobian matrix J at two operating point places of trend convergence and J ' is non-singular matrix, and when system configuration did not change, J was identical with J ' dimension, can get according to the disturbance enclosure theorem of matrix inversion and equation group in the matrix perturbation theory:

With

Wherein:

κ＝||J|| ₂*||J ^-1|| ₂ (5)

Then, (5) formula can be defined the conditional number of Jacobian matrix according to the matrix perturbation theory;

Wherein: J is a Jacobian matrix in the power flow equation,

J ^-1Be the inverse matrix of J,

|| || ₂The 2-norm of representing matrix or vector;

2) stablize the structure of weak node index based on the voltage of Jacobian matrix conditional number sensitivity

State variable and control variables are respectively X=(x in the note system ₁, x ₂..., x _n) ^T, and Y=(y ₁, y ₂..., y _n) ^T, n is the node number in the system,

By the differentiate rule of compound function, conditional number to the sensitivity of state variable does in the formula (5)

Wherein: δ _MaxAnd δ _MinBe the maximum and the minimum singular value of Jacobian matrix,

According to the differentiate rule of implicit function, singular value to the partial derivative of state variable in the system is:

That is:

To sum up, can get the sensitivity degree of conditional number to control variables:

Wherein: V _MaxAnd V _MinBe respectively δ _MaxAnd δ _MinCorresponding left singular vector,

U _MaxAnd U _MinBe respectively δ _MaxAnd δ _MinCorresponding right singular vector;

3) foundation of node load nargin index

The size of Jacobian matrix conditional number sensitivity has reflected the sensitivity that the stability of a system changes node power, and the definite condition number is a node load nargin index to the inverse of node control variable sensitivity, that is:

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