Correction Method of Sensitivity of Load Margin Respect to Branch Parameters Based on Thevenin Equivalent of Key Ports

Weiwei Miao, Hongjie Jia, Zeyin Dong Key laboratory of Smart Grid of Ministry of Education

Tianjin University Tianjin, China

niaoweiwei@tju.edu.cn, hjjia@tju.edu.cn

Abstract—Branch contingency ranking and screening methods based on the linear sensitivity of load margin respect to branch parameters are widely used in the power system security and stability assessment because of the rapid computation speed. However, due to the nonlinear nature of power system, linear estimate value may be erroneous, especially for some serious contingencies. In this paper, Thevenin equivalent of key ports and its fast modification as branch parameter varies is used to observe how the state variable impacted by the change of branch parameter. And then the relationship between the state variables and branch parameter is fitted by a set of quadratic functions. With the nonlinear information between the state variable and branch parameter, traditional linear sensitivity is corrected to get a more accurate estimate. As the branch parameter varies, the correction of Thevenin equivalent of key ports only needs a small amount of additional computation, and the calculation burden does not increase significantly with larger system, so this approach remains high computational efficiency. An IEEE 14-bus case is used to validate the proposed method in this paper.

Keywords- Branch Contingencies; Rank and Screen; Linear Sensitivity; Thevenin Equivalent

I. INTRODUCTION Severe contingencies identification and screening is

essential for power system stability assessment, with increasing levels of power demand, the incidence of serious contingency may cause the system become closer to the stability region boundaries, even worse, beyond the borders, meaning loss of stable equilibrium point. Therefore, the actual systems are often required to maintain stable and remain enough load margin when encountering any single branch outage, so called N-1 criterion. But for on-line monitoring and control, the huge number of branches may lead to very heavy computational burden, making it not realistic to analysis every contingency in detail. As a result, it is practical to firstly sort and screen contingencies to form critical credible contingency set for further analysis. Preliminary screening should be rapid, not very accurate, but should not miss any serious contingency.

Therefore, study of improving screening speed and accuracy follows mainly two ways:

1) One kind of approach is to simplify the voltage stability critical point calculating method. As derived in [3], a quadratic function could be used to describe the relationship between the state variables and load margin parameter around saddle-node bifurcation (SNB), and then a quadratic curve is used to fit the complete PV to quickly estimate the critical point. In [4] and

[6], continuous parameter of branch admittance is introduced, through which continuous method is used to trace the critical point of post-contingency directly from the critical point of pre-contingency system. [7] firstly used Damped Newton method to obtain a similar SNB point, and then fixed the voltage magnitude of a reference bus, thus the load margin parameter could be solved directly from parameterized power flow equations.

2) Another kind of approach is mainly based on linear sensitivity method to estimate the load margin. In [2], linear sensitivity of load margin respect to the branch parameter is derived. [5] used continuous power flow in branch parameter space to obtain the sensitivity at the virtual static stability critical point. Taking the advantages of highly calculating efficiency, methods based on sensitivity became the first choice in power system rapid contingency screening. However, due to the nonlinear nature of power system, using a linear estimation may be not accurate, especially for serious outages are quite erroneous. [2] also showed that the sensitivity of load margin to power injection is close to linear relationship, while the sensitivity of load margin to the branch parameter has strong nonlinearity. Thus scholars have proposed many methods to correct linear sensitivity to improve the estimate accuracy. [8] used a new sensitivity of load margin respect to branch power injection to estimate the load margin. Method in [9], which is also based on the linear sensitivity, exploited the nonlinear equations and linear optimization technique to improve estimation accuracy. [10] turned branch apparent power into two equivalent power source, and then used the fact that in-side power injection impacts little on load margin to fix the sensitivity.

In this paper, the basis of proposed method is still linear sensitivity. Nonlinear relationship between state variables of key nodes and branch parameter is obtained from rapid Thevenin equivalent modification of key ports when branch parameter varies, which will be then fitted by a set of quadratic functions. Finally the nonlinear relationship is taken into account to correct linear sensitivity estimation.

II. COMPUTATION AND CORRECTION OF SENSITIVITY OF LOAD MARGIN TO BRANCH PARAMETER

A. Computation of sensitivity of load margin to branch parameter

The model to parameterize branch admittance is shown in Fig.1. Continuous parameter μ is introduced for outage branch

The work was supported by Special Fund of the National Basic ResearchProgram of China (“973” Program, No.2009CB219701, No.2010CB234608),Tianjin Municipal Science and Technology Development Program of China(No. 09JCZDJC25000) and Research Fund for the Doctoral Program of Higher Education of China (No. 20090032110064).

978-1-4577-0547-2/12/$31.00 ©2012 IEEE

connecting bus f and bus t, while μ=1 means branch in service, and μ=0 for out of service. Branch admittance and susceptance are denoted as ys and yb respectively.

Figure 1. Model of paramterized branch admittance

Considering the continuous parameter of branch admittance, the characteristic equation of power system corresponding to the SNB point can be expressed as (1) - (3):

)3(01

)2(0),,()1(0),(),,( 0

=−

==−+=

νν

νxfxgbyxf

xT

μλμλμλ

Where, nR∈x is the state variable vector, which represents bus voltage magnitude and phase angle in polar coordinates.

1R∈λ is the load margin parameter along with specific power injection varying direction. 1),,( ×∈ nRμλxf is parameterized power flow equation, and 1),( ×∈ nRμxg is nodal power injection equation. nnR ×∈xf represents the Jacobian matrix of f, and v is the right eigenvector corresponding to the zero eigenvalue of fx.

For base case, once the critical point has been found, sensitivity information could be obtained by taking the first partial derivatives of (1) - (3), and then we get (4):

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔΔΔΔ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0///

0200

0νf

fνx

νfνf

ff

xxxx

x

μ

μλ

μλμμ

T

(4)

Where, 1, ×∈ nRμλ ff are the first derivatives of f to λ and μ, and nnR ×∈μxxx ff , are the derivatives of fx to x and μ.

In order to eliminate unnecessary sensitivity information, such as Δv/Δμ, transpose of vector ω is used to pre-multiply the second row of (4), which is the left eigenvector corresponding to zero eigenvalue of the Jacobian matrix. So (4) is simplified to (5):

⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡ΔΔΔΔ

⎥⎦

⎤⎢⎣

⎡νfω

fxνfω

ff

xxx

x

μ

μλ

μλμ

TT //

0 (5)

Similarly, ω could also be transposed to pre-multiply the first row of (5), then Δx/Δμ is omitted and (6) is derived:

λ

μμλfωfω

T

T−=ΔΔ / (6)

Equation (6) is the expression of linear sensitivity of load margin respect to parameter. By ignoring the state variables changes, Δλ/Δμ could be easily calculated. But for some serious contingencies, load margin estimation by (6) could be erroneous [9]. In order to get a more accurate estimate, some methods require the information of Δx/Δμ to correct linear estimation.

B. Correction of sensitivity of load margin to branch parameter Suppose the estimation of Δx/Δμ could be obtained before

hand, and then it is possible to fix Δλ/Δμ by (5). Denote the corrected vector of Δx/Δμ as nR∈Δα , and the corrected value of Δλ/Δμ as 1R∈Δβ . Introduce a mismatch function F(Δβ) as (7):

⎥⎦

⎤⎢⎣

⎡−

−−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Δ+ΔΔ

Δ+ΔΔ

⎥⎦

⎤⎢⎣

⎡=Δ

νfωfαx

νfωff

Fxxx

x

μ

μλ

βμλμβ TT 0

)( (7)

The least square error of (7) could be represented by (8):

αfαfανfωανfω

αffff

FF

xxxxxx

x

ΔΔ+ΔΔ+

ΔΔ+Δ=

ΔΔ=Δ

TTTT

TT

T

)()(

2

)()()(2 ββ

βββε

λλλ

(8) It can be seen from (8), ε is a convex quadratic function of

Δβ, which implies the solution of minimum ε can be easily calculated by (9):

λλ

λβff

αff xT

T Δ−=Δ (9)

Once αΔ is known, Δβ can be obtained by (9), so the estimate of αΔ is the key step. And we will estimate αΔ by Thevenin equivalent of key ports in the next section.

III. CORRECTION OF LINEAR SENSITIVITY BASED ON THEVENIN EQUIVALENT OF KEY PORTS

A. Relationship between the state variables and branch parameter through Thevenin equivalent In recent years, the Thevenin equivalent of load nodes has

been extensively studied in voltage stability analysis [12-13], which gives local indicators through simplification of the network in order to quickly determine the stability of system. In this paper, the Thevenin equivalent of key nodes is used to observe how branch parameter variation impacts the state variables of key nodes, and then this information is used for sensitivity correction. Take the IEEE 14-bus system for an example and Fig.2 gives the relationship between the branch parameter and the state variables of key nodes by Thevenin equivalent. In sub-graph 1a) of Fig.2, tag c2-5 means branch 2-5 is out of service and tag n10 means the state variables are obtained from the Thevenin equivalent of the port between

node 10 and the earth. So are the remaining sub-graphs. Tab.I gives the actual load margin, the estimated load margin by linear sensitivity and the corresponding estimated error of the four contingencies in Fig.2.

Figure 2. Relationhsip between the branch parameter and the state variables of key nodes

TABLE I. ACTUAL AND ESTIMATED LOAD MARGIIN

Branch Outage

Load Margin

Actual Value Estimated Value Estimated Error

c2-5 1.8904 1.9051 0.78%

c6-12 1.8437 1.9001 3.06%

c12-13 1.8891 1.9105 1.13%

c6-13 1.1396 1.7571 54.19%

Combined Fig.2 and Tab.I, it can be seen, for the non-severe contingencies, like c2-5, c6-12, c12-13, the state variables of key nodes reflect almost linear relationship with branch parameter, such as sub-graph 1a), 1b) to sub-graph 3a), 3b) in Fig.2 and the first three rows in Tab.I. However, the nonlinear relationship between the state variables of key nodes and branch parameter becomes significant after a serious branch outage occurs, such as sub-graph 4a) and 4b) in Fig.2. This phenomenon is similar to the PV curve whose critical point has strong nonlinearity. In fact, for severe contingency, this is due to becoming closer to the virtual critical point in branch parameter space. In this situation, linear estimate will be inaccurate. As shown in Tab.I, the estimate error for branch 6-

13 outage goes up to 54.19%. Thus, the nonlinear information reflected by Fig.2 and Tab.I can be used for sensitivity correction.

B. Rapid Thevenin equivalent modification as branch parameter varies Denote Y as the system nodal admittance matrix and Z as

the nodal impedance matrix. ]ee[eM lkm ,,,= represents the node-port correlation matrix, where, n

i R∈e means an all zero column vector except the i-th element is 1. M represents key nodes m, k, l and the earth forms the key ports. Multi-port Thevenin equivalent is shown in Fig.3:

Figure 3. Thevenin Equivalent of multi-port

In case of μ=1, Thevenin equivalent of key ports can be expressed by (10) and (11) [11]:

ZMMZ Teq = (10)

0VMV Teq = (11)

As the admittance of branch f-t changes from ys to μys, that is equivalent to remove (1-μ) ys from branch. Then we denote (1-μ)ys as yc, the correction of system admittance matrix becomes:

Tcy MMYY ′′−=′ (12)

Where, ][ ,tfeM =′ is the node-branch correlation matrix, n

tf R∈,e means an all zero column vector except the f-th element is 1 and the t-th element is minus 1.

With μ changes, the modified Thevenin equivalent is represented by (13) and (14):

T

seq

TTc

TT

Tc

T

Tc

T

TTeq

ay

y

y

y

ccZ

ZMMMZMMZMZMM

MZMMZMMZZM

MMMYM

MYMMZMZ

T

1

1

1

1

1

))1(

1(

)/1(

))/1((

)(

)()(

−

−

−

−

−

+−

−−=

′′′+−′−=

′′′+−′−=

′′−=

′=′=′

μ

μ

(13)

0VMcV

IZMMZMV

IZMMZZM

IYMVMV

T

seq

Tc

Teq

Tc

T

TTeq

ay

ay

ay

′+−

−−=

′+−′−=

′+−′−=

′=′=′

−

−

−

−

1

1

1

1

))1(

1(

)/1(

))/1((

)()(

μ

μ

(14)

Where, MZMc ′= T , MZM ′′= Ta

Equation (13) and (14) is called the modification formula of Thevenin equivalent when the branch parameter varies.

Here are tips that should be noticed for the modification of Thevenin equivalent when μ changes:

• Thevenin equivalent now becomes a tool for voltage stability assessment which will be executed as online task. So taking the advantage of existing results would be efficient;

• For the same branch outage, even μ varies, c and a in (13) and (14) only needs calculating once. What is more, because of the Sparsity of M and M', it is only necessary to obtain the specific element in the matrix Z. Above all, the calculation burden was reduced and will not increase significantly with larger system.

• Selection of key nodes should satisfy the following principles:

a. Nodes connected to branches with heavy active or reactive power flow;

b. Obvious weak nodes of the investigated system.

C. Sensitivity correction based on quadratic fitting As mentioned above, for the same contingency, when μ

changes, c and a only need calculating once. That is to say, given a value of μ, the state variables can be obtained quickly. Therefore, as μ varies in [0,1], it is possible to obtain the state variables quickly to fit a quadratic function.

322

1 hhhx ++= μμc (15)

Where， lc R∈x is the concerned state variables, which is a

subset of x, l is the number of concerned state variables, lR∈321 ,, hhh are the coefficients of quadratic fitting.

Then the correction cαΔ could be described by (16):

)1(222

)()(

1

2121

1

μμ

μμ

μμ

μ

−=−−+=

ΔΔ

−ΔΔ

=Δ=

hhhhh

xxα cc

c

(16)

Fulfilled αΔ with cαΔ , then Δβ could be calculated by (17):

)1(

)1(2)( 1

μ

μμβ

λλ

λ

λλ

λ

−=

−−=

Δ−=Δ

p

T

T

T

T

ffhff

ffαff xx

(17)

Where,λλ

λ

ffhff x

T

T

p 12−= .

From (16)-(17), we knows that αΔ and Δβ are both functions of μ, that is to say sensitivity correction value of state variable and load margin to μ varies when μ changes. The relationship between them could be illustrated by Fig.4, the first derivates of state variable to μ is linear to μ, so αΔ is linear to μ, then Δβ can be obtained by a linear transformation of αΔ . Therefore, the accumulated correction could be represented by the shaded area in Fig.4, as expressed by (18):

λλ

λμμ

μμβμμβλ

ffhff x

T

T

i

n

iic

pdp

d

11

0

1

01

2)1(

)()(

−==−=

Δ=ΔΔ=Δ

∫

∫∑=

(18)

Figure 4. Illustration of αΔ andΔβ

IV. EXAMPLES

Figure 5. IEEE 14-bus test system

In this paper, IEEE 14-bus system is used to verify the effectiveness of this method, whose single-line diagram is shown in Fig.5. Load margin of base case is 1.9256. In Tab.I, load margin obtained by continuous power flow, estimated load margin by linear sensitivity and estimated error for

branch outage c2-5, c6-12, c12-13 and c6-13 is listed. As can be seen from Tab.I, for a serious contingency c6-13, the linear sensitivity estimated error reaches up to 54.19%, so the sensitivity correction of this contingency will be mainly concerned. Based on the key nodes selecting principle, key nodes for Thevenin equivalent in this example will be 6,7,9,10,11 and 13.

Figure 6. Relationship between the branch parameter of c6-13 and voltage

magnitude of key nodes

TABLE II. QUADRATIC FITTING OF THE NONLINEAR RELATIONSHIP BETWEEN THE STATE VARIABLES AND THE BRANCH PARAMETER

Key Nodes h1 h2 h3

6 0.0150 -0.0276 1.0832

7 -0.0939 0.1738 0.7029

9 -0.0867 0.1625 0.5166

10 -0.0686 0.1280 0.5099

11 -0.0369 0.0697 0.7564

13 -0.3725 0.7124 0.5419

TABLE III. ACTUAL, ESTIMATED AND CORRECTED LOAD MARGIN

Branch Outage

Load Margin Actual Value

Estimated Value

Estimated Error

Corrected Value

Corrected Value

c2-5 1.8904 1.9051 0.78% 1.8931 0.14%

c6-12 1.8437 1.9001 3.06% 1.8281 -0.85%

c12-13 1.8891 1.9105 1.13% 1.8805 -0.46%

c6-13 1.1396 1.7571 54.19% 1.0610 -6.90%

Fig.6 shows the nonlinear relationship between state variables and branch 6-13 parameter through Thevenin equivalent of key nodes and fast modification. Tab.II gives the coefficients of quadratic fitting. Load margin correction value calculated by (18) is -0.6961, so the modified estimated load margin will be 1.0610, whose estimated error decreases to

minus 6.90% compared with linear estimate error of 54.19%. The estimate accuracy is significantly improved by this method. Tab.III also lists the correction result of three remaining branch outage in Tab.I, it can be seen that for non-severe contingency, this method can also improve the estimate accuracy.

V. CONCLUSION This paper presents a sensitivity correction method based

on Thevenin equivalent of key nodes. As the branch parameter changes, rapid modification of Thevenin equivalent of key nodes could quickly obtain the nonlinear relationship between the state variables and the branch parameter. Then the nonlinear relationship is fitted by quadratic functions to derive the sensitivity correction formula. IEEE 14-bus test system is used to validate the proposed method and test results shows that this method could improve estimate accuracy for both severe and non-severe contingency.

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