Theoretical Studies of Active Power-Angle Sub-matrix for Power System Analysis

8/6/2019 Theoretical Studies of Active Power-Angle Sub-matrix for Power System Analysis

1/6

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(5): 16

DOI: 10.1007/s12204-008-0501-3

Theoretical Studies of Active Power/angle Sub-matrix for Power

System Analysis

CAO Guo-yun1 ( ), ZHANG Qing1 ( ), CHUNG T S2 ( ), CHEN Chen1 ( )(1. Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai 200240, China;2. Department of Electrical Engineering, Hong Kong Polytechnic University, Hong Kong, China)

Abstract: Properties of the active power/angle sub-matrix in the power flow Jacobian for power system anal-

ysis were studied. The sub-matrix is a dominant and irreducible matrix under very general conditions of power

systems, so that it is invertible. Also the necessary conditions for its singularity were given. These theoretical

results can be used to clarify the ambiguous understanding of the sub-matrix in current literature, and also

provide the theoretical foundations for the applications based on reduced p ower flow Jacobian. Numerical

simulation on the IEEE 118-bus power system is used to illustrate our results.

Key words: dominant matrix; irreducible matrix; power flow Jacobian; reduced power flow Jacobian; power

system analysis; V-Q sensitivity

CLC number: TM 711 Document code: A

Introduction

Voltage stability has become a main concern for se-cure operation of a stressed power system, and twokinds of approaches, i.e., the static and dynamic oneshave been developed to analyze, compute and designcontrols to maintain the system voltage stability[13].

The dynamic approach that uses the short- and/orlong-term simulation, can provide the time response of

a power system under a contingency. As there needsa large amount of computation for the simulation, thisapproach is usually used for the assessment of the ef-fectiveness of the protection and control designs for thevoltage stability, and the postmortem analysis for a par-ticular blackout caused by voltage instability[15]. Onthe other hand, the static approach, which is mainlybased on the power flow equation model of the powersystem, can not only determine whether the system isvoltage stable or not, but also calculate the sensitiv-ities that identify the voltage-weak buses, the impor-tant generators and transmission lines that have great

influence on the system voltage stability[415]. Suchinformation can direct us to design controls for voltagestability. Thus both the static and dynamic approacheshave their own advantages and disadvantages and theyare complementary for power system voltage stabilityanalysis and control.

Received date: 2007-06-26Foundation item: The National Natural Science Founda-

tion of China (No. 50307007)E-mail: [email protected]

1 Problem Formulation and LiteratureReview

1.1 Problem Formulation

According to the alternative current (AC) power sys-tem characteristics and the power system operationpractice, the idea of the V-Q sensitivity which is basedon the reduced power flow Jacobian is naturally pro-posed to analyze the voltage stability. The linear tech-

niques such as the singular value and eigenvalue decom-position are applied to the reduced Jacobian which linksthe incremental changes between the reactive power ofthe buses and the magnitude of their voltages [410].

Consider an electric power system with n + 1 nodes(buses) that do not count the ground bus, supposingthat nodes 1m are PQ buses, nodes m + 1n are PVbuses, and node n + 1 is the slack bus; then we canobtain the following standard PF equation:

Pis = Pi Ui

n+1j=1

Uj(Gij cos ij + Bij sin ij), (1a)

i = 1, 2, , n

Qis = Qi Ui

n+1j=1

Uj(Gij sin ij Bij cos ij), (1b)

i = 1, 2, , m

where, Pis = PGi PLDi and Qis = QGi QLDi arethe specified net injected active and reactive power; Piand Qi are defined as the active and reactive bus powerat node i, respectively. If the solution of Eq. (1) is ob-tained, then the linearized model of Eq. (1) around the


2/6

2 J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(5): 16

solution can be arrived:P

Q

= J

U

, (2)

where, P = [P1, P2, , Pn]T and Q =

[Q1, Q2, , Qm]T are the incremental changes in

the active and reactive bus power respectively; J =P PU

Q QU

is the power flow Jacobian.

To obtain the relationship between Q and U fromEq. (2), we need to let P = 0, and so we can obtain

Q = JRU, (3)

where JR = [QU QP1 PU] is the reduced power

flow Jacobian. The diagonal elements in the inverse ofthe reduced power flow Jacobian are the V-Q sensitiv-ities related with the corresponding buses.

We can see that the formula to express the reduced

power flow Jacobian involves the inverse of the activepower/angle sub-matrix P. There exist two conceptsto treat the issue whether this sub-matrix is singularor non-singular in our current literature. One is to as-sume that it will be non-singular for the practical ap-plications, as pointed out in Ref. [10] that the case ofits singularity does not encounter us in our analysis forthe studied or practical power systems. The other con-cept is to relate its singularity with the loss of rotorangle stability of power systems. Note this idea comesas a generalization from the observation on the loss-less single-machine infinite bus (SMIB) system whenthe generator that is modeled by the classical model

sends the maximum active power to the infinite bus.Thus there needs not ambiguous but clear under-

standing of the properties of the sub-matrix to providea solid basis for the applications based on the reducedpower flow Jacobian. Further, if the inverse of the sub-matrix exists, then the Schurs formula can be used tolink the determinants between the PF Jacobian and thereduced Jacobian as

det(J) = det(P)det(JR). (4)

From Eq. (4) we know that the singularity of thepower flow Jacobian (i.e., a necessary condition for thesaddle-node bifurcation associated with the power flowequation) implies the singularity of the reduced Jaco-bian, thus the voltage collapse point in terms of small-signal voltage stability determined by the saddle-nodebifurcation associated with the power flow equation canalso be equivalently determined by the singularity ofthe reduced Jacobian. So it is mathematically rigorousto reduce Eq. (2) to Eq. (3) by letting P = 0, whileit was thought that the influence of the incrementalchanges of the active power on the incremental changesof the bus voltage is neglected when we use Eq. (3) toanalyze the voltage stability.

1.2 Literature Review

It should be noted that Ref. [15] also contained twotheorems (theorem 4 and 5) to discuss the properties ofthe sub-matrix. However there are some ambiguities inthe statements and the proofs of the theorems 4 and 5in Ref. [15].

Essentially theorem 4 in Ref. [15] studied the proper-ties of the sub-matrix P. We would indicate that therequirement by

Pjj

n

k=0,k=j

Pjk

> 0 (5)

with strict inequality for at least one row in P does notensure that P is irreducible. For example, the matrix

4 1 2

0 3 3

0 5 5

satisfies inequality (5) where the strict

inequality holds for the first row, however it is singularbecause of the linear dependence of its rows 2 and 3.

It seems that the condensation method does not ap-ply for the proofs of the theorems 4 and 5 in Ref. [15].For example in the proof of theorem 4 for the casen = 3, we cannot obtain that the expression of (q11 +q12 + q13)q33 + q13s is strictly positive under the con-dition that only the third row satisfies the strict in-equality (5) in Ref. [15], if we let q11 = q12 q13and q13 = 0. The similar defect exists for the caseof n = r, for example, if only the row r + 1 satis-fies the strict inequality condition but sj = 0 andqj,r+1

= 0, then we cannot derive the strictly positive-ness of qr+1,r+1(s

j ) + qj,r+1s

r+1 in Ref. [15].

Finally, in the proof of theorem 5 in Ref. [15] we can-not obtain that the determinant of nJ will equal tozero if one pivot appeared in the denominator changesits sign or becomes zero during the condensation, aswe know the determinant will not be continuous but beinfinite if the pivot equals to zero. Furthermore, thecondensation method is not applicable under this casebecause of the zero of the pivot entry.

This paper will demonstrate that the sub-matrix willbe a dominant and irreducible matrix under very gen-eral conditions for a connected AC power system, i.e.,

the phase angle difference between each two buses withdirect connection is less than the impedance angle ofthe branch that links the two buses directly. Now letus give a brief review on the dominant, and the irre-ducible/reducible matrix.

Definition 1 Let A be a square matrix of dimen-sion n with entries aij. Then A is said to be diagonallydominant by its row if

|aii|

nj=1,j=i

|aij | (6)


3/6

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(5): 16 3

for i from 1 to n. In addition A is said to be strictlydiagonally dominant if

|aii| >n

j=1,j=i

|aij | (7)

for i from 1 to n; and if Eq. (7) exists for at least one

i, then A is said to be weakly diagonally dominant.Definition 2 A nn matrixA is called a reducible

matrix if there exists a permutation matrix S such that

STAS=

A11 A12

0 A22

(8)

where A11 and A22 are r and n r square matrix.Essentially Eq. (8) means that we can re-number (re-order) the nodes associated with A so that it has theform of the right hand side of Eq. (8). A is an irre-ducible matrix if it is not a reducible matrix.

The following proposition[16] relates the relationship

between the non-singularity and dominance of a matrixand will provide the basis our following analysis.Proposition If a square matrix is strictly diago-

nally dominant or weakly diagonally dominant and alsoirreducible, then it is invertible.

2 Global View on Sub-matrix and In-vestigation of Sign of Its Off-diagonalElements

2.1 Global View on Sub-matrix

The off-diagonal and diagonal elements in P are

Hij =Pij

= UiUj(Gij sin ij Bij cos ij), i = j

(9a)

Hii =Pii

= BiiU2i Qi. (9b)

Substituting Qi defined in Eq. (1b) into Eq. (9b) andrearranging the expression we can obtain

Hii = (Hi1 + Hi2 + + Hi(i1) + Hi(i+1)

+ + Hin) Hi(n+1). (10)

So we can see that P is the upper-left sub-matrix ofP defined below by deleting its last row and column:

P =

H11 H12 H1i H1n H1(n+1)

H21 H22 H2i H2n H2(n+1)...

......

......

......

Hi1 Hi2 Hii Hin Hi(n+1)...

......

......

......

Hn1 Hn2 Hni Hnn Hn(n+1)

H(n+1)1 H(n+1)2 H(n+1)i H(n+1)n H(n+1)(n+1)

(11)

From Eqs. (10) and (11) we know that P is singularas the sum of its each row equals to zero, thus we cananticipate that P will be non-singular.

From Eqs. (9a) and (11) we know that P has thesame sparity as the admittance matrix of the powersystem, so the off-diagonal element in P between twonodes is zero if there is no direct connection betweenthem. Also we know that P is always of positional

symmetry.Equations (10) and (11) not only provide a global

view on the active power/angle sub-matrix, but alsocan be used to show that this sub-matrix will be dom-inant by its row if all of the off-diagonal elements inP corresponding to two buses with direct connectionhave the same sign such as all the values of such ele-ments being positive or negative.

2.2 Investigation of Sign of Off-diagonal Ele-

ment

We now investigate the off-diagonal elements between

two nodes with direct connection. For this purpose wefirst simplify the notation in Eq. (9), i.e., let ij = ,E = Ui, U = Uj, Gij = G and Bij = B, and then Hijcan be written as

Hij = EU(G sin B cos ) (12)

As indicated by Eq. (9a), the value of the off-diagonalelement is dependent on the series impedance parame-ter between the two corresponding nodes, thus a circuitwhich is shown in Fig. 1 can be introduced to evaluateHij where the two nodes are supposed to be connectedby the impedance Z = R + jX, and in this case themutual admittance of Eq. (12) is

G +jB = 1

R +jX.

To ensure the off-diagonal element to be negative, wewill require that the phase angle difference between


4/6


the two nodes should satisfy

< < , (13)

where = tg1(X/R) is the impedance angle.

Nodej jX RE


5/6

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(5): 16 5

3.2 Necessary Conditions for Its Singularity

Theorem 2 (necessary conditions for singularity ofP) IfP associated with a connected power systembecome singular, then at least one of the off-diagonalelements between two nodes with direct connection inP will be zero or positive.

Proof by the principle of contradiction As-sume that none of the off-diagonal elements with directconnection is positive or zero, then all of such elementsmust be negative, and so the sub-matrix will be non-singular according to the above theorem, which is acontradiction. This completes the proof.

Remarks There was an assumption that the

singularity ofP implies the loss of the power systemangle stability. Through Theorem 2 we can find thatthe singularity means that there exists at least a phaseangle difference between two directly connected nodesexceeding its impedance angle. Thus its singularity is

related with the operation condition of the transmissionsystem. Though this condition will certainly influencethe rotor angle stability of the power system, the small-disturbance analysis of the rotor angle stability of powersystem should be based on the linearized differential-algebraic equation model of power system dynamics.Thus we can see that it is not true to determine therotor angel small-signal stability by singularity of thesub-matrix, which is generalized from the observationon single-machine infinite bus system. There wassome doubt that the decoupling between active powerand phase angles, and reactive power and voltage mag-

nitude in power system can be extended to the stressedloading conditions[3]. Theorem 1 and 2 give the quan-titative results. Generally speaking the decoupling willalways exist for power system where the phase angledifferences between each two nodes with direct connec-tion satisfies Eq. (13).

4 Case Studies

In this section we present numeric studies on IEEE118-bus power system, which is helpful to understandour theoretical studies from the point of view of numer-ical computation. The data of the system and initial

operation conditions are the same as those of Ref. [17].The studies are based on the continuation power flow

where the generation and loads are varied in a uniformway. On each point of the obtained P-V curves we cal-culate the least module eigenvalue and condition num-ber of P, and also we compute the value of its off-diagonal element that change its sign first.

We select the P-V curve at node 17 to be demon-strated in our results. The numerical results are shownin Fig. 3 where all the x coordinates are the uniform co-efficients for the increase of load and generation whose

initial values equal to 1. The solid lines represent theupper branch of the P-V curve, and the dashed ones thelower branch, and all the values such as the least moduleeigenvalue, the condition numbers and the off-diagonalelements which change their sign first are shown accord-ingly.

1.00

0.98

0.96

0.94

U/(pV)

0.2

0.1

0

0.1

0.2

4

2

0

2

4

6

8

10

Eigenvalue

Off-diagon

alelement

1 2 3 4

Load increase coefficient

(a) Voltage at node 17

1 2 3 4


(b) Least module eigenvalue of the matrix

1 2 3 4


(c) One off-diagonal element of the matrix

1 2 3 4


(d) Condition number of the matrix

14

12

10

8

6

4

2

0

Conditionnumber/(104)

Fig. 3 Numerical results for IEEE 118-bus system

We can see that P is well-conditioned on the upper


6/6


branch as its condition number has almost been con-stant. Also it is far from singularity as its least moduleeigenvalues are always positive. While on the lowerbranch of the P-V curves there exists a point wherethe sub-matrix becomes singular; also at this point theleast module eigenvalue of this sub-matrix is zero and

its condition number becomes much larger, and one ofits off-diagonal elements (the one between node 38 and65) has already changed its sign into a positive valuethrough zero before the occurrence of the singularity ofthe sub-matrix. The numerical studies conform to ourtheoretical results of Theorem 1 and Theorem 2.

5 Conclusion

We study the properties of the active power/anglesub-matrix in the power flow Jacobian for power sys-tem analysis. Through deducing the Eq. (10) and defin-

ing the Eq. (11), the particular structure of the activepower/angle sub-matrix is first revealed, and a lot ofuseful results can be obtained based on these equa-tions.The sufficient conditions for the non-singularity ofthe sub-matrix are given in Theorem 1, and we discussthat these conditions will be satisfied with by an induc-tively dominant AC power systems. On the other hand,the necessary conditions for the singularity of the sub-matrix can be obtained accordingly, which is stated byTheorem 2. Numerical results are used to demonstratethe theoretical results. These results not only can beused to clarify the ambiguities on the understanding ofthe properties of the sub-matrix, but also provide a rig-orous mathematical basis for the applications based onthe reduced PF Jacobian, which is physically meaning-ful and widely used in the static analysis of the reactivepower and voltage stability analysis problems in powersystems.

References

[1] Taylor C W. Power system voltage stability [M]. New

York: McGraw Hill, 1994.

[2] Kundur P. Power system stability and control [M]. New

York:McGraw Hill, 1994.

[3] Cutsem T V, Vournas C. Voltage stability of electric

power systems[M]. New York: Kluwer Academic Pub-

lishers, 1998.

[4] Morison G K, Gao B, Kundur P. Voltage stability anal-

ysis using static and dynamic approaches [J]. IEEE

Trans Power Systems, 1993, 8(2): 11591171.

[5] Gao B, Morison G K, Kundur P. Towards the develop-

ment of a systematic approach for voltage stability as-

sessment of large-scale power systems [J]. IEEE Trans

Power Systems, 1996, 11(3): 13141324.[6] Gao B, Morison G K, Kundur P. Voltage stability eval-

uation using modal analysis [J]. IEEE Trans Power

Systems, 1992, 7(4): 15291542.

[7] Lof P A, Smed T, Anderson G, et al. Fast calculation

of a voltage stability index [J]. IEEE Trans Power Sys-

tems, 1992, 7(1): 5464.

[8] Lof P A, Smed T, Anderson G, et al. Voltage stabil-

ity indices for stressed power systems [J]. IEEE Trans

Power Systems, 1993, 8(1): 326335.

[9] Mansour Y, Xu W, Alvarado F, et al. SVC placement

using critical modes of voltage instability [J]. IEEE

Trans Power Systems, 1994, 9(2): 757763.

[10] Canizares C A, de Souza A C Z, Quintana V H. Com-parison of performance indices for detection of proxim-

ity to voltage collapse [J]. IEEE Trans Power Systems,

1996, 11(3): 14411450.

[11] Begovic M, Phadke A. Control of voltage stability us-

ing sensitivity analysis [J]. IEEE Trans Power Systems,

1992, 7(1): 114123.

[12] Dobson I. Observations on the geometry of saddle-nodebifurcation and voltage collapse in electric power sys-

tems [J]. IEEE Trans Circuits and Systems (I), 1992,

39(3): 240243.

[13] Greene S, Dobson I, Alvarado F. Sensitivity of the

loading margin to voltage collapse with respect to ar-

bitrary parameters [J]. IEEE Trans Power Systems,

1992, 12(1): 262272.

[14] Capitanescu F, Cutsem T V. Unified sensitivity analy-

sis of unstable or low voltage caused by load increasesor contingencies [J]. IEEE Trans Power Systems, 2005,

20(1): 321329.

[15] Grijalva S, Sauer P W. A necessary condition for power

flow Jacobian singularity based on branch complex

flows [J]. IEEE Trans Circuits and Systems (I), 2005,

52(7): 14061413.

[16] Kolotilina LY. Nonsingularity/singularity for non-

strictly block diagonally dominant matrices [J]. Linear

Algebraic and Its Applications, 2003, 359: 133159.

[17] Power system test case archive [EB/OL]. http://

www.ee.washington.edu/research/pstca/index.html,

1999.

Theoretical Studies of Active Power-Angle Sub-matrix for Power System Analysis

Documents

Transcript of Theoretical Studies of Active Power-Angle Sub-matrix for Power System Analysis