A new MVA sensitivity method for fast accurate contingency evaluation

Electrical Power and Energy Systems 38 (2012) 1–8

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A new MVA sensitivity method for fast accurate contingency evaluation Z.Q. Wu ⇑, Zhong Hao, Deng Yang Department of Electrical & Information Engineering, Hunan University, Changsha 410082, China

a r t i c l e

i n f o

Article history: Received 7 April 2010 Received in revised form 9 June 2011 Accepted 12 June 2011 Available online 8 February 2012 Keywords: Contingency evaluation Branch outage parameter Derivative MVA power sensitivity

a b s t r a c t A new fast algorithm of calculating node voltage and branch power flow under N  1 network topology is proposed based on MVA power sensitivity in this paper. The voltage of system under base network topology is calculated based on Newton power flow approach. The branch outage is represented by the branch outage parameter. The first order derivative of voltage with respect to branch outage parameter is calculated by using the convergent power flow equation of the base network topology, further the voltage derivative with respect to branch MVA power can easily be obtained. The voltage under N  1 network topology can be obtained by correcting the voltage at base network topology using the sensitivity. The advantage of the proposed approach is its high speed and high accuracy, especially for the branch MW power flow. The proposed sensitivity is justified theoretically. The proposed method can be used very quickly to obtain the accurate node voltage under different N  1 network topologies, and then branch power flow violation and voltage violation can be obtained. Tests on IEEE 14, 30, 118, 300 bus systems and Hunan power system in China verify the correctness and practical utility of the approach. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The analysis of the effects of hundreds of outages on line flows and bus voltages is required for the real-time security analysis [1– 3]. This challenges the solution method in terms of speed, accuracy. In a variety of studies such as branch violation, voltage violation and voltage stability assessment, post-contingency voltage is desired. The use of the full Newton power flow [4] for contingency analysis is computationally expensive. To speed up the computations, fast approximate power flow methods were proposed, such as the decoupled power flow [5,6] and the iterative linear AC power flow [7]. These methods use iterative procedures that take advantage of the usually loose coupling between real power and voltage magnitudes, and reactive power and voltage angles. In [8–14], single iteration methods are used to identify system limit violations. To achieve reasonable computational speed, these methods used an approximate [11,13] or partial system solution [8–10]. Methods have been developed which select and solve only the stressed portion of the network [12,14]. These methods work quite well for branch MW flow analysis. The evaluation for post-contingency voltage is more difficult. One of the most popular methods [11] uses a single iteration of the fast decoupled power flow based on compensation. It requires the calculation of the Q-mismatches for the entire network, even though the outage may significantly affect only a small portion. When the system is heavily loaded, the accuracy for only single ⇑ Corresponding author. E-mail address: [email protected] (Z.Q. Wu). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.06.035

iteration is not good enough. The local solution methods based on Y bus Gauss Seidel iterations [9,10] take advantage of the local nature of the majority of outages. However the method has the inherent disadvantage of Y bus Gauss Seidel iteration, the convergent speed is very slow. Another method for branch and voltage violation analysis [14] is proposed by taking into account the mostly local nature of the majority of outages without ignoring the potentially wide-spread effects. The method uses only incomplete single iteration of AC power flow, it has the same disadvantage as single AC iteration, the accuracy is not good for heavily loaded system. Distribution factors were introduced to evaluate the postcontingency system status [15]. The method uses sensitivities to estimate the post-contingency voltages. Distribution factors are computed using the decoupled power flow method. Real power distribution factor is regarded as providing a good trade-off between speed and accuracy. Voltage and reactive power distribution factors are not commonly used due to the large estimation errors [16]. In [17], under the assumption of P–h and Q–V decoupling and small angle differences, the reactive flow and voltage factors for branch outage are derived, although the estimation errors were decreased, it still remained large. As decoupling may not be valid for systems under stress conditions [18], the decoupling assumption was relaxed in [19] to derive voltage and reactive generation factors for line and generation outage studies. The voltage estimates were significantly improved, but the reactive flows gave extremely inaccurate results. By using sensitivity analysis, a segment linear sensitivity method is used to estimate the post-contingency voltage and reactive

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power [20]. Good accuracy is achieved, but the method requires several times of sensitivity calculation. By using the optimal technique, partial network solution was proposed to estimate the post contingency voltage in [21,22]. However the method may have the convergent problems. Ref. [23] proposed a method using the sensitivity of system load margin with respect to branch power, via the sensitivity of system load margin with respect to branch parameter, to estimate the post contingency system load margin. With the development of the computer technology, the post-contingency voltage estimation is attracting more and more attention. However as mentioned above, the problem for the approximate network solution is that only single complete or incomplete iteration of the fast decoupled power flow is used to obtain the branch flow and voltage. For some heavily loaded power system, the analysis will lead to wrong conclusions. If more iterations of the fast decoupled power flow are adopted, partial re-factorization of the Jacobean matrix or some kind of compensation must be adopted, it will require computation time for large number of branch outages. Although a lot of improvements have been made, the distribution factor method still has the problem of bad accuracy. Because of the deregulation of the electricity industry, modern power system is loaded become more and more heavily. The fast decoupled power flow method does not converged very well sometime, which will seriously affect the reliability of security analysis software. A fast accurate algorithm of calculating the post-contingency voltage is proposed based on branch MVA sensitivity in this paper. The voltage under base network topology is calculated by using Newton–Raphson power flow approach. The branch outage is represented by a branch outage parameter. The derivative of voltage with respect to branch outage parameter is calculated by using the convergent power flow equation on the base network topology, and then the first order of node voltage derivative with respect to branch MVA power can be obtained easily. The post-contingency voltage and further the branch power flow can be obtained by correcting the voltage at base network topology using MVA sensitivity. The advantage of the proposed approach is its high speed, high accuracy, especially the branch MW power flow. The reason why the accuracy is high is that the proposed sensitivity has a very good linear relation. The proposed method can be used to replace the current methods for obtaining the accurate post-contingency node voltage very quickly, and further the branch violation and voltage violation can also be obtained. Tests on IEEE 14, 30, 118 and 300 bus systems and Hunan power system in China verify the correctness and practical utility of the approach.

For slack bus, the node voltage amplitude and phase angle are given. In (1)–(4), Gij and Bij are the real part and imaginary part of the admittance of branch i–j, respectively; ei and fi are the voltage real part and imaginary part of node i, respectively; Pis and Qis are the given real and reactive power of node i, respectively; Vi is the amplitude of node voltage; DPi and DQi are the mismatched real and reactive power respectively; DV 2i is the squared mismatched voltage. In rectangular coordinate, the correction equation for Newton method is as in (5).

2. Power flow equations for Newton method

    dDV @ DW ¼ ½J dlkm @ lkm

Assume the system consists of n nodes, among them there are p PQ nodes, n  p  1 PV nodes, one slack bus. In rectangular system, for the PQ node I, the following power flow equation can be obtained.

DPi ¼ Pis  ei

n n X X ðGij ej  Bij fj Þ  fi ðGij fj þ Bij ej Þ ¼ 0 j¼1

DQ i ¼ Q is  fi

ð5Þ

In Eq. (5), [DW] is the mismatched power column, [J] is the Jacobean matrix, [DV] is the updated node voltage change column. For PQ node,

½DW i  ¼ ½DPi DQ i t ;

½Jij  ¼

h

@ DP i @ej

@ DP i @fj

i frac@ DQ i @ej @ D@fQj i ; j ¼ 1; . . . ; n

For PV node,

½DW i  ¼ ½DPi DV 2i t ; ½Jij  ¼

h

@ DP i @ej

3.1. Branch outage representation As shown in Fig. 1, the branch outage can be represented by the branch admittance parameter l. l = 1 represents that the branch is in operation, l = 0 represents that the branch is out of operation. 3.2. Derivative of node voltage with respect to branch outage parameter The power flow solution is obtained for the base network topology. If single branch between k and m is out of operation, it can be represented by the branch outage parameter lkm = 0. After the power flow solution is converged, differentiating Eq. (5) with respect to the branch outage parameter lkm, Eq. (6) can be obtained.

DV 2i ¼ V 2is  ðe2i þ fi2 Þ ¼ 0 ði ¼ 1; 2; . . . ; n  p  1Þ

ð6Þ

Eq. (6) is the equation for the solution of first order hvoltage i derivative with respect to branch outage parameter lkm. @@lDW in km (6) can be determined by using (7).

ð1Þ ð2Þ

For PV node I, the following equations can be obtained.

j¼1

; j ¼ 1; . . . ; n

3. Proposed method for post-contingency voltage estimation

j¼1

n n X X ðGij ej  Bij fj Þ  fi ðGij fj þ Bij ej Þ ¼ 0

i

AC power flow could be solved for each contingency, followed by a check for limit violations and major shifts from the initial system conditions. However such an approach is not feasible for practical systems consisting of hundreds of buses. To overcome this computational barrier, this paper proposes an efficient and accurate method to obtain the post contingency node voltage.

ði ¼ 1; 2; . . . ; pÞ

DPi ¼ Pis  ei

2

@ DV @ DP i frac@ DV 2i @ej @fj i @fj

j¼1

n n X X ðGij ej  Bij fj Þ þ ei ðGij fj þ Bij ej Þ ¼ 0 j¼1

½J½DV ¼ ½DW

ð3Þ

j¼1

ð4Þ Fig. 1. Branch outage representation.

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! ! #t  " @ DW @ DP k @ DQ k @ DV 2k @ DP m @ DQ m @ DV 2m ;; ; ¼ ; ; or ; or @ lkm @ lkm @ lkm @ lkm @ lkm @ lkm @ lkm ð7Þ

It should be noticed that only four terms appear in (7). If the terminal node of the tripped branch is PV node, the reactive power at the node is not constrained. The node voltage at the PV node is not @ DV 2

2

changed with the branch tripping, therefore @l k and @@DlV m will be km km equal to 0. 3.3. Derivative of node voltage with respect to branch MVA power After the derivative of node voltage with respect to branch parameter is obtained, the derivative of node voltage with respect to branch MVA power can be obtained easily. By mathematical formulae manipulation, Eq. (8) can be obtained.

dV dV dSkm ¼ = dSkm dlkm dlkm

ð8Þ

where Skm is the MVA power of the faulted branch, which is equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2km þ Q 2km . The derivative of Skm can be expressed as in (9). dP km dQ km dSkm Pkm  dlkm þ Q km  dlkm ¼ dlkm Skm

ð9Þ

3.4. Post contingency voltage estimation By the voltage derivative obtained using (8) at the power flow convergent point for the base network topology, it is easily to calculate the post-contingency node voltage. Based on the derivative of node voltage with respect to the MVA power of the outage branch, the real and imaginary parts of postcontingency node voltage can be estimated by (10). N

V N1 ¼ V Ni þ i

dV i

N dSkm

 ðSkm Þ

ð10Þ

where Skm is the pre-contingency MVA power of the outage branch; V Ni is the pre contingency node voltage; V N1 is the post contingency i dV N

node voltage; dSNi is the node voltage sensitivity at the base network km

configuration obtained by (8).

branch admittance parameter; Vk and Vm are the complex voltage at node k and m, respectively; Ik and Im are the equivalent complex current (injection or withdrawing) at nodes k and m, respectively. Solution for (11), the node voltage can be expressed by (12).



Vk Vm



 ¼

Bk0 þ bkm  l; Bkm0  bkm  l

1

 

Bkm0  bkm  l; Bm0 þ bkm  l

Ik



ð12Þ

Im

Further manipulation for (12), if the equivalent node injection or withdraw current is considered constant before and after the branch is tripped, the node voltage real part or imaginary part at bus k and m can be expressed as in (13).

ek ¼

ak3 þ ak3  l ak1 þ ak2  l

ð13Þ

In (13), the parameters, ak1, ak2, ak3 and ak4 are determined by the network configuration and the system operation point, and they are different from to node to node. If the branch resistor is considered, similar expression to (13) can be obtained but it is more complex. By using the backward operation for the entire system LDU factorized admittance matrix, the relation between the voltage at any node and the branch admittance parameter is similar to (13), and again the parameters, ak1, ak2, ak3 and ak4 are different from node to node. When the branch admittance parameter l is changed, the node voltage and node injection (or withdraw) current will change, and therefore the true relation between the node voltage and the branch admittance parameter l is more complex than the expression in (13). The node voltage amplitude can be obtained by using the real part and the imaginary part of the node voltage. Obviously, the node voltage amplitude is a highly nonlinear function of the branch admittance parameter l. The simulation results verified the theoretical analysis results. Typical numerical curve between the node voltage and the branch admittance parameter is as shown in Fig. 2. In Fig. 2, the horizontal axis is the branch admittance parameter l, the vertical axis is the node voltage. If the first derivative of node voltage with respect to branch admittance parameter is used to estimate the post contingency voltage, the errors will be quite large. Although the piecewise linearity using the sensitivity of node voltage with respect to the parameter l can be used to estimate the post voltage, the computation time will be longer.

4. Justification for the proposed sensitivity 1.06

4.1. Nonlinear relation between node voltage and branch admittance parameter Considering N  1 contingency, if the terminal nodes of the tripped branch are the last two nodes, by using Gauss reduction, the node admittance equation for the last two nodes can be expressed as in (11) without taking into account the branch resistor.



Bk0 þ bkm  l; Bkm0  bkm  l Bkm0  bkm  l; Bm0 þ bkm  l



Vk Vm





¼

Ik Im



1.055

Node Voltage (pu)

In this section, the work is mainly focused on the justification of the proposed sensitivity method. It is to answer why the proposed sensitivity of node voltage with respect to pre contingency power flow is better than the sensitivity of node voltage with respect to the branch admittance parameter for the estimation of post voltage.

1.05 1.045 1.04 1.035 1.03

ð11Þ

In (11), Bk0 and Bm0 are the equivalent shunt susceptance at node k and m; Bkm0 is the equivalent susceptance between nodes k and m; bkm is the susceptance of the tripped branch; l is the

1.025 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. Typical curve of node voltage with respect to branch admittance parameter.

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Z.Q. Wu et al. / Electrical Power and Energy Systems 38 (2012) 1–8

4.2. Linear relation between node voltage and pre-contingency power flow of the tripped branch

1.06



Bk0 þ bkm ; Bkm0  bkm

"

Bkm0  bkm ; Bm0 þ bkm

V 0k V 0m

#

 ¼

Ik þ Ikm



Im  Ikm

ð14Þ

In (14), V 0k and V 0m are the post contingency node voltage at the terminals of the tripped branch; Ikm is the current through the branch from k to m. It can be obtained by using the power flow through the tripped branch and has the following expression.

Ikm ¼ Ikm;r þ jIkm;i ¼

ðPkm ek þ Q km fk Þ þ jðPkm fk  Q km ek Þ V 2k

ð15Þ

Ikm;r ¼ Ikm;r;0 þ

Ikm;i

dIkm;i dIkm;i dIkm;i ¼ Ikm;i;0 þ  Dek þ  Dfk þ dek dfk dP km dIkm;i  DPkm þ  DQ km dQ km

Ikm;0;i ¼

Pkm;0  ek;0 þ Q km;0  fk;0 V 2k;0 Pkm;0  fk;0  Q km;0  ek;0 V 2k;0

ð16Þ

ð17Þ

ð18Þ ð19Þ

In (18) and (19), all variables are set at the pre contingency variable value.

dPkm dPkm dPkm dPkm DPkm ¼  De k þ  Dfk þ  Dem þ  Dfm ð20Þ dek dfk dem dfm dQ km dQ km dQ km dQ km DQ km ¼  De k þ  Dfk þ  Dem þ  Dfm ð21Þ dek dfk dem dfm According to (14), the node voltage changes due to contingency of branch from k to m can be approximately expressed as in (22).



Bk0 þ bkm ; Bkm0  bkm Bkm0  bkm ; Bm0 þ bkm



   Ikm DV k ¼ DV m Ikm

1.045 1.04 1.035 1.03

ð22Þ

From (15)–(22), it is very clear that the relation between the node voltage change and the pre contingency branch power flow is a very good linear relation. Because the node voltage change due to the branch tripping is not too great, the first order expansion for the voltage can be expressed as in (23).

0

5

10

15

20

25

30

S (MVA) Fig. 3. Typical curve of node voltage with respect to pre contingency power flow of the tripped branch.

V N1 ¼ i

In (16) and (17), Iij,r,0 and Iij,r,0 are the pre contingency branch current real part and the imaginary part of the tripped branch, which can be expressed by (18) and (19); DV i ¼ Dei þ jDfi is the node voltage difference between post contingency node voltage and the pre contingency node voltage; DSij ¼ DPij þ jDQ ij is the branch power flow difference between post contingency power flow and the pre contingency power flow, which can be expressed by (20) and (21). Due to the fact that the voltage change is not great, by power system analysis experience, Eqs. (20) and (21) are very good approximation for the branch power flow change caused by the compensation currents.

Ikm;0;r ¼

1.05

1.025

Generally speaking, the post contingency node voltage will not change too much compared with the pre contingency node voltage. According to the power system analysis experience, Eq. (15) can well be approximated by using first order expansion as expressed in (16) and (17).

dIkm;r dIkm;r dIkm;r  Dek þ  Df k þ dek dfk dPkm dIkm;r  DPkm þ  DQ km dQ km

Node Voltage (pu)

1.055

If the current compensation method is considered for the contingency, then Eq. (14) can be obtained.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ei;N  Dei;N þ fi;N  Dfi;N 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2i;N1 þ fi;N1  V Ni þ 2 e2i;N þ fi;N

ð23Þ

From (23) and (15)–(22), it is very clear than the post contingency voltage amplitude is a very good linear function of the pre contingency power flow of the tripped branch. Lots of numerical analysis results verified the theoretical analysis result. Typical curve of the node voltage with respect to the pre contingency power flow of the tripped branch is shown as in Fig. 3 where the horizontal axis is the pre contingency power flow of the tripped branch (in MVA). It is very clear that curve is a good linear curve. In the above analysis for the linear relation between the node voltage and the pre contingency power flow of the tripped branch, the first order Taylor expansion can be used. All the first order expansion used above can solidly be supported by the related power system analysis experiences. Therefore the proposed sensitivity of node voltage with respect to the pre contingency power flow of the tripped branch is better than the branch admittance parameter based sensitivity method for the estimation of post contingency voltage. It should be pointed out that although the post contingency voltage can be estimated by (15)–(23) principally, the solution for (15)–(23) need to re-factorize the revised Jacobean matrix for each contingency, which is very time consuming. 5. Advantage of the proposed method The proposed method is a power flow solution based contingency screening. Compared with the Y bus Gauss Seidel local solution method [11] which has the inherent disadvantage of slow convergent speed, by using the proposed method, the accurate voltage and branch power flow can be obtained very quickly. Compared with matrix inversion lemma compensation based AC power flow iteration [18], there is no need for solution of two columns of node impedance elements. Compared with the equivalent injection power compensation based AC power flow [16,17], there is no need to solve the four sensitivity which requires four time of fast forward and fast backward operation. Compared with partial re-factorization based AC load flow [19], it is obviously that the proposed method does not need partial re-factorization for the Jacobean matrix. For a given outage branch, if the MVA power based sensitivity is used, only one of linear equation solution in (6) is needed. For all branch outages, it is not necessary to re-factorize

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Z.Q. Wu et al. / Electrical Power and Energy Systems 38 (2012) 1–8

the Jacobean matrix repeatedly. Fast calculation speed and high accuracy are the advantages of the proposed method. Compared with the published methods, the proposed method has obvious advantage in terms of calculation speed and accuracy. The accuracy for the branch MW power flow estimation is extremely high. The reason why the proposed method has very good accuracy is that the MVA sensitivity has very good linear relation. 6. Numerical results The method presented has been applied to estimate the postcontingency voltages and the branch power flows in various test systems. In this paper, representative results on the IEEE 14, 30, 118, 300 test systems [24] and a practical system in China are presented to illustrate the capabilities of the proposed MVA sensitivity method in (10). Post-outage voltage magnitudes are calculated both with full power flow and with the proposed method. Voltage errors for bus voltage magnitudes are defined as

eV ¼ jV LF  V PF j

ð24Þ

where VLF and VPF are the bus voltage magnitudes calculated by full load flow and the proposed method; respectively. Post-outage MVAR flows are calculated with the voltage from both full power flow and the proposed method. Errors for branch reactive flows are defined as

eQ ¼ jQ LF  Q PF j

ð25Þ

where QLF and QPF are the branch reactive power flows calculated by the voltage from both full load flow and the proposed method; respectively. Post-outage MVA flows are calculated with the voltage from both full power flow and the proposed method. Errors for branch MVA flows are defined as

eMVA ¼ jMVALF  MVAPF j

ð26Þ

where MVALF and MVAPF are the branch MVA power flows calculated by the voltage from both full load flow and the proposed method; respectively. Post-outage MW flows are calculated with the voltage from both full power flow and the proposed method. Errors for branch MW flows are defined as

eMW ¼ jMW LF  MW PF j

ð27Þ

where MWLF and MWPF are the branch MW power flows calculated by the voltage from both full load flow and the proposed method; respectively. 6.1. IEEE-14 bus test system Base case load flow results for IEEE 14-bus test system are summarized in Table 1. Among 17 simulations, the results of two representative outages are reported. The first one is the outage of the line connected between bus-7 and bus-9. The second one is the outage of the transformer connected between bus-5 and bus-6. The post outage voltage magnitudes at all the busses and the reactive power flows through all the lines for these contingencies are illustrated in Tables 2–4, respectively. Note that only single side reactive power flows are reported in Tables 3 and 4 for the sake of simplicity. Errors for both voltage magnitudes and reactive power flows are quite small. The maximum voltage magnitude error for branch 7–9 contingency is almost the same as that reported in [21,22]. On the other hand, the average errors of voltage magnitude are far less than the reported in [20–22]. For example, the average voltage

Table 1 Base case control variables of IEEE-14 test system. Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Generation (p.u.)

Bus voltages

Active

Reactive

Magnitude (p.u.)

Phase (Degree)

232.39 40 0 0 0 0 0 0 0 0 0 0 0 0

16.549 43.557 25.075 0 0 12.731 0 17.623 0 0 0 0 0 0

1.06 1.045 1.01 1.0177 1.0195 1.07 1.0615 1.09 1.0559 1.051 1.0569 1.0552 1.0504 1.0355

0.0 4.9826 12.725 10.313 8.7739 14.221 13.36 13.36 14.939 15.097 14.791 15.076 15.156 16.034

Table 2 Voltage magnitudes for the outage of line 7–9 in IEEE-14 test system. Bus

VLF

VPF

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.0600 1.0450 1.0100 1.0169 1.0174 1.0700 1.0671 1.0900 1.0291 1.0282 1.0446 1.0535 1.0459 1.0179

1.0600 1.0450 1.0100 1.0172 1.0177 1.0700 1.0664 1.0900 1.0336 1.0320 1.0468 1.0544 1.0471 1.0210

0 0 0 0.0003 0.0003 0 0.0007 0 0.0044 0.0037 0.0021 0.0009 0.0011 0.0030

Table 3 MVAR flows for outage of line 7–9 in IEEE-14 test system. Branch

Base (MVAR)

Exact (MVAR)

Proposed (MVAR)

Errors (MVAR)

1–2 1–5 2–3 2–4 2–5 3–4 4–5 4–7 4–9 5–6 6–11 6–12 6–13 7–8 7–9 9–10 9–14 10–11 12–13 13–14

20.404 3.855 3.5602 1.5504 1.171 4.4731 15.824 9.6811 0.42761 12.471 3.5605 2.503 7.2166 17.163 5.7787 4.2191 3.61 1.6151 0.75396 1.7472

20.26 4.7337 3.6261 0.783 1.9917 5.1904 14.819 13.54 6.1238 13.171 5.9873 2.6448 8.578 13.9 0 2.4603 2.546 3.354 0.84668 2.9507

20.205 4.6321 3.6315 0.91317 1.8678 5.0582 14.834 13.081 5.4003 13.012 5.0974 2.5269 8.1186 15.435 0 3.3302 3.1032 2.4828 0.73042 2.3809

0.0560 0.1016 0.0054 0.1297 0.1322 0.1322 0.0150 0.4640 0.7235 0.1590 0.8899 0.1179 0.4594 1.5350 0 0.8699 0.5572 0.8715 0.1163 0.5698

error and the standard deviation in this paper are 0.00079 and 0.00024, respectively; while the counterparts reported in (20) are 0.0009 and 0.0034, respectively; the results reported in (21) and (22) are 0.0038 and 0.0033, respectively. Reactive power flows errors are also very small compared with [20,22]. The maximum errors for branch 7–9 and branch 5–6 in this paper are only 1.53 MVAR and 3.57 MVAR, respectively; while the maximum errors for the outages of branch 7–9 and branch 5–6

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Z.Q. Wu et al. / Electrical Power and Energy Systems 38 (2012) 1–8

Table 4 MVAR flows for outage of line 5–6 in IEEE-14 test system. Branch

Exact (MVAR)

Proposed (MVAR)

Error (MVAR)

1–2 1–5 2–3 2–4 2–5 3–4 4–5 4–7 4–9 5–6 6–11 6–12 6–13 7–8 7–9 9–10 9–14 10–11 12–13 13–14

21.47 0.40905 3.3111 3.044 2.374 3.1387 11.795 9.146 0.614 0 14.14 4.3906 12.349 14.734 0.9298 3.5069 1.6106 10.147 2.7012 9.0419

20.691 0.71676 3.4468 4.6750 4.0389 1.4395 10.139 9.7975 1.201 0 10.562 3.7652 10.707 13.552 0.4979 1.7832 0.3778 7.9934 2.0337 6.6147

0.7790 1.1258 0.1357 1.6304 1.6645 1.6992 1.6560 0.6513 0.5874 0 3.5780 0.6254 1.6420 1.1820 0.4319 1.7237 1.2328 2.1536 0.6675 2.4272

in [22] are 3.02 and 8.12 MVAR, respectively. The maximum reactive power flow error for branch 5–6 outage reported in [20] is 6.0 MVAR. The average reactive power flow error and standard deviation in this paper are 0.31 MVAR and 0.21 MVAR, respectively; while the counterparts reported in [20] are 0.26 MVAR and 0.75 MVAR, respectively; the results reported in [22] are 0.8 MVAR and 1.1 MVAR, respectively. 6.2. IEEE-30 bus test system IEEE 30 bus test system data are downloaded from [24]. Among 36 simulations, the results of one representative outage are reported. The outage is the line connected between bus-4 and bus-6.

Table 5 Voltage magnitude for outage of line 4–6 in IEEE-30 test system. Bus

VLF

VPF

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.06 1.043 1.0225 1.0133 1.01 1.0086 1.0012 1.01 1.0478 1.0388 1.082 1.059 1.071 1.0439 1.0366 1.0405 1.0348 1.0248 1.0212 1.0247 1.0265 1.0271 1.0242 1.0168 1.0112 0.99338 1.0166 1.0055 0.99659 0.98503

1.06 1.043 1.0228 1.0137 1.01 1.0102 1.0025 1.01 1.0493 1.0401 1.082 1.0595 1.071 1.0445 1.0373 1.0413 1.036 1.0258 1.0222 1.0258 1.0278 1.0284 1.0252 1.018 1.0126 0.99484 1.0182 1.0074 0.99821 0.98663

0 0 0.0003 0.0004 0 0.0016 0.0012 0 0.0014 0.0013 0 0.00047 0 0.00057 0.00068 0.00077 0.0012 0.0010 0.0010 0.0011 0.0013 0.0013 0.0010 0.0012 0.0014 0.0015 0.0016 0.0019 0.0016 0.0016

The post outage voltage magnitudes at all the busses are illustrated in Table 5. Errors for voltage magnitudes are quite small. The maximum voltage magnitude error for branch 4–6 is 0.0019 (p.u.), while the maximum voltage magnitude error for the same branch outage reported in [21] is 0.0052 (p.u.). Obviously, the maximum error in this paper is smaller than the reported in [21]. The average error of voltage magnitude in this paper is 0.00079 (p.u.), while the average voltage magnitude error reported in [21] is 0.0038 (p.u.). Again, the average voltage magnitude error in this paper is less than that reported in [21]. Reactive power flows and active power flows errors, as presented in Tables 6 and 7, are also very small. According to Table 6, the maximum flow MVAR errors for branch 4–6 outage in IEEE 30-bus system is only 4.5 MVAR, the average error for reactive power flow is only 0.213 MVAR. The MW flows for branch 4–6 outage in IEEE 30 bus test system are presented in Table 7, the maximum error is only 0.43 MW, the average error of MW flows is very small, which is equal to 0.06 MW. This is the main advantage of the proposed method. 6.3. Error statistics for different systems In this section, the error statistics of IEEE-14, IEEE-30, IEEE-118, IEEE-300 and Hunan Power system in China (800 buses) are reported.

Table 6 MVAR flows for the outage of branch 4–6 in IEEE-30 test system. Branch

Base (MVAR)

Exact (MVAR)

Proposed (MVAR)

Error (MVAR)

1–2 1–3 2–4 3–4 2–5 2–6 4–6 5–7 6–7 6–8 6–9 6–11 9–11 9–10 4–12 12–13 12–14 12–15 12–16 14–15 16–17 15–18 18–19 19–20 10–20 10–17 10–21 10–22 21–22 15–23 22–24 23–24 24–25 25–26 25–27 28–27 27–29 27–30 29–30 8–28 6–28

21.09 4.566 3.895 3.601 1.751 0.4399 16.35 11.69 2.975 8.145 8.173 0.161 15.71 5.914 14.24 10.49 2.402 6.796 3.354 0.6469 1.44 1.598 0.6189 2.791 3.707 4.424 10.01 4.601 1.428 2.912 3.063 1.248 2.016 2.367 0.368 5.034 1.669 1.663 0.6059 0.392 0.037

26.73 6.802 12.03 3.484 0.6674 5.746 0 6.242 1.332 11.83 7.881 0.6329 17.25 8.691 16.74 9.05 1.555 4.972 1.889 0.2796 0.4491 0.175 0.9815 4.459 5.257 6.626 10.3 4.801 1.119 1.124 3.583 0.6929 0.4498 2.368 1.934 6.209 1.672 1.667 0.607 0.4878 0.2212

26.95 6.606 11.88 3.3 0.125 6.605 0 6.044 1.562 16.34 7.879 0.689 17.51 8.949 16.72 8.68 1.511 4.799 1.703 0.323 0.635 0.074 1.083 4.562 5.362 6.821 10.33 4.818 1.102 1.009 3.617 0.808 0.367 2.373 2.022 6.319 1.676 1.671 0.608 1.223 0.427

0.22 0.20 0.15 0.18 0.54 0.85 0.0 0.20 0.23 4.5 0.00 0.06 0.26 0.26 0.02 0.37 0.04 0.17 0.19 0.04 0.21 0.10 0.10 0.10 0.10 0.20 0.03 0.02 0.02 0.11 0.04 0.12 0.08 0.00 0.07 0.11 0.00 0.00 0.00 0.73 0.21

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Z.Q. Wu et al. / Electrical Power and Energy Systems 38 (2012) 1–8 Table 7 MW flows for the outage of branch 4–6 in IEEE-30 test system.

Table 10 Error statistics for different load levels in Hunan Power System.

Branch

Base (MW)

Exact (MW)

Proposed (MW)

Error (MW)

1–2 1–3 2–4 3–4 2–5 2–6 4–6 5–7 6–7 6–8 6–9 6–11 9–11 9–10 4–12 12–13 12–14 12–15 12–16 14–15 16–17 15–18 18–19 19–20 10–20 10–17 10–21 10–22 21–22 15–23 22–24 23–24 24–25 25–26 25–27 28–27 27–29 27–30 29–30 8–28 6–28

173.2 87.73 43.62 82.21 82.39 60.33 72.15 14.77 38.12 29.57 27.72 15.84 0.0 27.72 44.21 0.0 7.859 17.9 7.249 1.585 3.695 6.02 2.781 6.724 9.022 5.327 15.79 7.619 1.825 5.04 5.74 1.808 1.203 3.545 4.758 18.06 6.19 7.092 3.704 0.5419 18.67

198.2 66.09 11.79 61.9 99.52 98.41 0 1.023 21.93 28.91 16.9 9.643 0.0 16.9 65.42 0.0 10.03 26.86 17.32 3.717 13.57 11.23 7.902 1.636 3.881 4.46 14.52 6.794 3.079 10.68 3.668 7.377 2.247 3.545 1.307 14.6 6.192 7.094 3.704 1.204 15.85

198.4 66.05 11.69 61.86 99.69 98.59 0 1.048 22 28.48 16.84 9.632 0.10 16.94 65.49 0.01 10.03 26.89 17.36 3.716 13.61 11.24 7.912 1.635 3.883 4.484 14.56 6.811 3.082 10.7 3.68 7.391 2.258 3.556 1.307 14.65 6.215 7.121 3.718 1.1 15.8

0.20 0.04 0.10 0.04 0.03 0.18 0.00 0.03 0.07 0.43 0.06 0.10 0.10 0.04 0.07 0.01 0.00 0.03 0.04 0.00 0.04 0.01 0.01 0.00 0.00 0.02 0.04 0.02 0.00 0.02 0.01 0.02 0.01 0.01 0.00 0.05 0.03 0.03 0.01 0.10 0.05

Table 8 Error statistics for different load levels in IEEE-14 test system. System stress level Voltage magnitude error (p.u.) Branch reactive flow error (MVAR) Branch MW power flow error

Mean St. dev. Mean St. dev. Mean St. dev.

50%

100%

120%

0.00030 0.00004 0.22 0.16 0.11836 0.34851

0.00079 0.00024 0.31 0.21 0.13201 0.22063

0.0011 0.0006 0.35 0.28 0.17117 0.29046

50%

100%

120%

0.00045 0.00043 0.14876 0.35869 0.07995 0.51626

0.00113 0.00364 0.21303 0.62127 0.09783 0.39871

0.00183 0.00768 0.31895 1.19905 0.15583 0.66427

Table 9 Error statistics for different load levels in IEEE-30 test system. System stress level Voltage error (p.u.) Branch reactive flow error (MVAR) Branch MW power flow error

Mean St. dev. Mean St. dev. Mean St. dev.

Effects of system stress levels on the accuracy of the computation are also analyzed. The real and reactive powers are changed in the same ratio with respect to the base-case powers at all the buses. The error statistics for three different load levels are illustrated in Tables 8 and 9. It can be observed from the table that there is not a significant increase in the errors for higher system stress levels. From Table 8, the error statistics for both voltage

Tripped branch

Maximum branch MVA error (MVA)

MVA flow of the branch with maximum MVA error

Average branch MVA flow error

1 7 11 57 82 118 139

0.37 1.34 0.40 1.08 0.33 19.53 1.42

626.19 270.51 103.36 421.74 192.18 143.77 110.44

0.002969 0.014638 0.003793 0.018039 0.005059 0.354100 0.007219

Table 11 Computation time and voltage magnitude errors for several test systems. Test system

Number of the selected contingency branch

IEEE-14 bus IEEE-30 bus IEEE-118 bus IEEE-300 bus Hunan power system (800 buses)

17 36 177 300 282

CPU time for single contingency (s) A

B

_ 0.016 0.281 2.015 6.660

_ 0.0025 0.0135 0.0705 0.249

Mean error (p.u.)

St. dev.

0.00079 0.00113 0.00018 0.00043 0.000426

0.000238 0.003643 0.00044 0.00111 0.0000015

A: full Newton method. B: proposed method.

magnitude and branch active, reactive power flow in IEEE 14-bus test system under different stress levels are smaller than that reported in [20,22]. For example, the mean voltage magnitude errors of IEEE 14-bus system under different stress levels in this paper are 0.0003, 0.00079 and 0.0011, respectively. While the counterparts reported in [22] are 0.0039, 0.0047 and 0.0058, respectively. The mean branch reactive power flow errors of IEEE 14-bus system under different stress levels in this paper are 0.22, 0.31 and 0.35, respectively; while the counterparts reported in [22] are 0.8, 1.8 and 2.3, respectively. The mean branch active power flow errors of IEEE 14-bus system under different stress levels in this paper are 0.118, 0.132 and 0.171 MW, respectively. From Table 9, the error statistics for both voltage magnitude and branch active, reactive power flow in IEEE 30-bus system under different stress levels are also very small. Another practical power system in China is analyzed. The system has more than 800 buses. Due to the paper length limitation, only seven contingencies are reported. The maximum branch MVA flow errors and the average branch MVA flow errors for different contingencies are listed in Table 10. In the table, the column of ‘Tripped branch’ represents the contingency branch; the column of ‘Maximum branch MVA error’ represents the maximum branch MVA error related to the selected contingency branch; the column of ‘MVA flow of the branch with maximum MVA error’ represents MVA flow of the branch with maximum MVA error related to the selected contingency branch; the column of ‘Average branch MVA flow error’ represents the average branch MVA flow error in the system, related to the selected contingency branch. According to the table, the error is very small. The accuracy of the proposed method should be the best among the current widely used commercial software. Therefore the proposed method can be used for the practical system online security analysis. The average computation time per branch outage, average voltage magnitude errors and standard deviation are presented in Table 11. According to the table, the CPU time of the proposed method and the full Newton method are presented. The greater

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Z.Q. Wu et al. / Electrical Power and Energy Systems 38 (2012) 1–8

is the system, the more efficient is the proposed method. The reason why the proposed method has very fast computation speed is that the proposed method needs only one fast forward and backward operation for a given branch outage. As for the accuracy, the mean voltage magnitude error and standard deviation are also smaller than the results reported in [21] for the tested standard IEEE systems. It should be pointed out that all simulations in this paper are performed on a Pentium IV 2.6-GHz personal computer. Principally the proposed method can be combined with the idea of partial network solution. In this case only partial node voltage derivatives are required, therefore the computation will speed up, but the accuracy will not seriously be affected.

7. Conclusions Based on the Newton power flow solution under base network topology, the voltage derivative with respect to branch outage parameter is obtained. Further the node voltage derivative with respect to branch MVA power is derived. By using sensitivity analysis, the post-contingency node voltage is obtained and further the branch power flows are determined. The proposed method is better than the methods suggested by the literatures in terms of accuracy and computation time. Therefore fast speed of calculation and high accuracy, especially for the branch MW power flow, are the advantages of the proposed method. The proposed method can be used to replace the currently used method for online security analysis. The greater is the system, the more efficient is the proposed method. The reason why the accuracy is high is that the proposed sensitivity has a very good linear relation. Theoretical analysis justified the proposed sensitivity method. The proposed method can be used to obtain the post-contingency entire network detailed operation status and it is very helpful for the online security analysis and voltage stability assessment of the large scale power system.

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