Novel Techniques for Continuation Method to Calculate the Limit-Induced Bifurcation of the Power Flow Equation

8/6/2019 Novel Techniques for Continuation Method to Calculate the Limit-Induced Bifurcation of the Power Flow Equation

1/16

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Shanghai Jiaotong University]

On: 3 August 2010

Access details: Access Details: [subscription number 917206489]

Publisher Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-

41 Mortimer Street, London W1T 3JH, UK

Electric Power Components and SystemsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713399721

Novel Techniques for Continuation Method to Calculate the Limit-inducedBifurcation of the Power Flow EquationGuo-Yun Caoa; Chen Chenaa Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China

Online publication date: 28 June 2010

To cite this Article Cao, Guo-Yun and Chen, Chen(2010) 'Novel Techniques for Continuation Method to Calculate theLimit-induced Bifurcation of the Power Flow Equation', Electric Power Components and Systems, 38: 9, 1061 1075

To link to this Article: DOI: 10.1080/15325001003649435URL: http://dx.doi.org/10.1080/15325001003649435

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.
http://www.informaworld.com/smpp/title~content=t713399721http://dx.doi.org/10.1080/15325001003649435http://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://dx.doi.org/10.1080/15325001003649435http://www.informaworld.com/smpp/title~content=t713399721


2/16

Electric Power Components and Systems, 38:10611075, 2010

Copyright Taylor & Francis Group, LLC

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325001003649435

Novel Techniques for Continuation Methodto Calculate the Limit-induced Bifurcation

of the Power Flow Equation

GUO-YUN CAO1

and CHEN CHEN1

1Department of Electrical Engineering, Shanghai Jiao Tong University,

Shanghai, China

Abstract This article proposes an improved continuation method to calculate thelimit-induced bifurcation associated with the power flow equation due to the encounterwith reactive power limits by generators. One of the distinguishing features is that

the maximum loadability characteristics of limit-induced bifurcation are used foridentification so that the computation efforts can be minimized. Another distinctive

characteristic of the method is that, to ensure accuracy, the tangent vector of thecontinuation power flow equation at the reactive power limit encountering point is

used to differentiate between limit-induced bifurcation and saddle-node bifurcation,as the latter also represents maximum loadability. The proposed method also adopts

the conventional schemes of secant prediction, arc-length correction, and a new step-size control, i.e. , the convergence-dependent step-size control. Numerical results ofthe IEEE 118-bus system are used to illustrate the improvements of the proposedcontinuation method in comparison with the existing one.

Keywords continuation method, convergence-dependent step-size control, maximumloadability, power flow equation, reactive power limit, saddle-node and limit-inducedbifurcations, voltage stability

1. Introduction

Voltage stability has become a main concern for secure operation of stressed power

systems in recent years [14]. Theoretical studies showed that the small-disturbance

stability analysis of power system voltage stability can be based on the power flow

(PF) equation [58]. Among them, saddle-node bifurcation (SNB) and limit-induced

bifurcation (LIB) associated with the PF equation are two kinds of mechanisms that leadto voltage collapse [3, 4, 9, 10]. Numerical methods for bifurcation analysis have been

proposed and used to analyze and compute these two kinds of local bifurcations. Roughly

speaking, these computational methods can be classified into the following categories.

1) Direct method: The principle of this method is that the conditions for the bifur-

cations are formulated as a set of non-linear algebraic equations. These algebraic

equations can then be solved by iteration methods, such as the Newton method,

and thus, the bifurcation is obtained. Based on this idea, the point of collapse

Received 29 August 2009; accepted 16 December 2009.

Address correspondence to Dr. Guo-yun Cao, Department of Electrical Engineering, 800Dongchuan Road, Shanghai Jiao Tong University, Shanghai, 200240, China. E-mail: [email protected]

1061


3/16

1062 G.-Y. Cao and C. Chen

method to calculate the SNB in AC/DC power systems was investigated in

[11, 12].

2) Optimization method: The principle of this method is that the conditions for the

bifurcations are included as a constraint of an optimization problem in an either

explicit or implicit way, and then the bifurcation can be obtained by solving the

optimization problems [13, 14].

3) Continuation method: The idea of this method is that the equilibrium of the PF

equation is first calculated as the bifurcation parameter varies, thus obtaining the

equilibrium curve. Generally, only a sequence of discrete points on the equilibrium

curve, where the parameter is equal to some discrete values, can be obtained. Then

between each two successive points, it is necessary to detect whether some kind

of bifurcation occurs or not and to locate it within pre-specified accuracy if there

is one. Usually the continuation method uses the schemes of predictor, corrector,

and step-size control to trace the equilibrium curve [1526].

Note that continuation method has a good balance among the reliability, accuracy,and efficiency needed by an algorithm; thus, it is widely used in power system voltage

stability analysis.

It is known that the reactive power has an important influence on voltage stability;

also, generators with a voltage controller are fast and important reactive power sources.

Voltage instability may then arise when the reactive power maximum limit is encountered

by the generator (referred to as the limit point hereafter in this article), and its occurrence

of voltage instability in this case is called an LIB [4, 9]. In [25, 26] predictor and corrector

schemes were proposed to calculate the limit points in succession and identify the LIB.

The main idea of [25, 26] was that the tangent vector to the equilibrium curve of the

PF equation is first calculated, and then the step-size based on the tangent vector isused to identify the next generator that will first meet its reactive power limit. Some

problems might arise with the above prediction scheme. For example, if there are some

generators meeting their reactive power limits almost at the same time, then the obtained

step-size will be much smaller. Another problem is that this scheme cannot ensure that

the prediction always identified the correct generator, as indicated by [26]. Thus, whether

or not the generators in the system encounter their limits should still be examined after

solving the next point on the P-V curve.

Also, the condition for an LIB is derived in [26] and, to use it to check whether

a limit point is an LIB or not, two tangent vectors are calculated at each limit point

for two PF equations that are related to the ones before and after the limit encountered,

respectively.For the analysis and computation of an LIB of a PF equation due to an encounter with

reactive power limits by generators, [9] was the first to address the mechanism of voltage

collapse due to such an encounter, and the author related the collapse to transcritical

bifurcation. In [25], the predictor-corrector scheme was proposed to calculate the limit

points in succession; however, it did not provide a clear judgment as to whether or not a

limit point is the voltage collapse point. The differences of the limit points on the upper

branch, tip, and lower branch of the P-V curves were described in [26], and, hence, a

criterion to identify the LIB was proposed, thus improving the scheme.

First, this article describes how the LIB is a maximum loadability of the power system

based on the SNB phenomenon of the PF equation and discusses the differences betweenthe SNB and LIB. Then an improved continuation method is proposed to calculate the

LIB and differentiate it from the SNB. The advantages of the method are that the obtained


4/16

Techniques to Calculate Limit-induced Bifurcation 1063

result is accurate, the associated computation is simpler, and fewer computational efforts

are needed than the existent one [26]. This article does not address any more complex

phenomena, such as chaos in the power system dynamics [27, 28], rather it focuses on

the analysis and computation of the local bifurcations associated with the PF equation.

The article is organized as follows. Section 2 analyzes the similarity and differ-

ences between the SNB and LIB of PF equations. The proposed continuation method

is described in Section 3, and numerical case studies on the IEEE 118-bus system are

presented in Section 4. Conclusions are drawn in Section 5.

2. Comparisons Between PF Equation SNBs and LIBs

Consider the following PF equation that is dependent on a scalar bifurcation parameter:

f.x; / D 0; x 2 Rn; (1)

where x is the state variable for the equation, usually representing the unknown voltagemagnitudes and phase angles at the buses of the power system network, and is the

bifurcation parameter that represents the load and generation variations. Generally, it

appears in the PF equation in the following way:

PG D PG0 C PS;

PL D PL0 C PD;

QL D QL0 C QD ;

(2)

where PG , PL, and QL are the active power of the generation and the active andreactive power of the load, respectively. Their initial values are PG0 , PL0, and QL0,

and the direction of load and generation variations is represented by PS, QD , and QD,

respectively.

Voltage instability may result from the SNB where the PF Jacobian is singular

as the system becomes stressed. This bifurcation is shown in Figure 1(a), and it divides

the P-V curves into the upper and lower branches, which are depicted by the solid and

dashed lines, respectively. They represent the voltage stability and instability conditions,

respectively. Obviously, the SNB represents a maximum loadability constrained by the

PF equation.

When the reactive power limits of generators are taken into consideration, two

different situations for the limit encountered arise. One is that the limit point is theintersection of both the upper branches of the P-V curves associated with the PF equations

before and after the limit encountered. Such a limit point is called an ordinary limit point,

where it remains voltage stable; see Figure 1(b). The other is that the limit point is the

intersection of the upper and lower branch of the P-V curves associated with the above

two PF equations, respectively. This intersection is the so-called LIB shown in Figure 1(c).

It can be seen that it also represents a maximum loadability, as the bifurcation parameter

begins to decrease its value along the lower branch of the incoming P-V curve. Note that

there may be more than one generator reaching their reactive power limits at the LIB.

Thus, both the SNB and the LIB can be identified by monitoring the parameter value

to check whether it experiences a maximum during the tracing of the P-V curve. Suchidentification just needs the computation of comparison, which is simple and efficient.

The following difference between the SNB and LIB can be used to differentiate them.


5/16


(a)

(b)

(c)

Figure 1. P-V curve at a load bus for different situations: (a) no limit encountered, (b) ordinary

limit point, and (c) limit-induced bifurcation.


6/16


That is, the PF Jacobian is singular at a SNB; thus, the tangent vector to the equilibrium

curve of the PF equation at the SNB will have a zero value for the parameter element,

while this is not the case for LIB. More details will be discussed in the following section.

3. Proposed Method

3.1. Selection for the Schemes of Prediction, Correction, and

Step-size Control

In the continuation method, the bifurcation parameter in Eq. (1) is also taken as a variable,

and the values of the state variables and parameter at each point of the equilibrium curve

are calculated by the correction. So, yD .x; / can be defined in Eq. (1), and it becomes:

f.y/ D 0; y 2 RnC1: (3)

The geometrical system in Eq. (3) defines a one-dimensional manifold (hyper-curve)

M in space RnC1. To trace this curve, the schemes of prediction, correction, and step-size

control are used repeatedly to generate a sequence of points on the curve. In this article,

the following schemes are chosen, which are discussed as follows.

1) Prediction of the next point. Assume that a point yj in the sequence has been

obtained. Then the approximate value QyjC1 for the next point can be obtained by

the prediction:

QyjC1 D yj C hj vj ; (4)

where hj and vj are the step-size and the normalized vector for the currentprediction supposed to be the j th step, respectively. The tangent prediction

implies that vj is the tangent vector of curve M at point yj ; i.e., it satisfies

Avj D 0; (5)

where A is the Jacobian of the derivative of f in Eq. (3) with respect to y at

point yj .

Another prediction scheme is the secant scheme, where the normalized vector

in Eq. (4) is determined by

vj Dyj yj1

kyj yj1k; (6)

where k k is the Euclidean norm of the vector.

Note that the secant prediction needs two previous points on the curve, while

the tangent prediction needs to solve Eq. (5). Thus, the second point may be

predicted from the initial operation point y0 D .x0; 0/ by tangent prediction and

other points by secant prediction in the continuation method.

Generally, the PF Jacobian at the initial operation point will not be singular,

and Appendix A proposes a method to solve for the tangent vector v0 at the

initial point in this case. It needs one decomposition of the rectangular matrixA

by LU factorization with column pivoting and one back-substitution for an upper

triangular matrix.


7/16


2) Arc-length scheme of correction to the accurate point. Having predicted the

approximate value for the next point, this point should be calculated within

a pre-specified accuracy. To this purpose an equation is added to Eq. (3) to

obtain

(f.y/ D 0

gj .y/ D 0; (7)

where j in gj .y/ represents the current correction.

Now by applying Newton method to Eq. (7), where the initial guess is

supplied by the predicted value QyjC1, the next point yjC1 can be obtained; see

Figure 1(a). Several schemes can be used to formulate gj .y/. To ensure the

robustness of the correction, the following is selected:

gj

.y/ D ky yj

k2

.hj

/2

: (8)

Geometrically, gj .y/ represents a circle whose center is at point yj and

radius is step-size hj ; see Figure 1(a). This is the arc-length correction. For ease

of reference, the three successive points yj1, yj , and yjC1 on the equilibrium

curve will be called the previous, current, and corrected (next) points, respectively,

hereafter.

3) Step-size control. Obviously, the step-size hj used in the prediction will have

an important influence on the convergence of the correction. To obtain a good

balance between the reliability and efficiency of a continuation method, some kind

of step-size control strategy needs to be incorporated in the method, especially

when the generator reactive power limit needs to be taken into consideration.

A rule-based step-size control was proposed and used to trace the equilibrium

curve without taking the generator reactive power limit into consideration in [19].

It is based on the decomposition of the equilibrium curve into three segments with

different characteristics. Then the prediction error or the gradient of the curve,

which express the different segments, are used to adapt the step-size control

during the continuation. However, as there is no clear knowledge on the division

of an equilibrium curve for a general power system beforehand, it is difficult to

specify the criterion constant used for step-size control, particularly when limit

point needs to be taken into consideration.

A new step-size control scheme is adopted, i.e., the convergence-dependentstep-size control, where the step-size is determined based on the convergence of

the correction. For example, if the correction needs one iteration to converge to the

solution, then the step-size can be increased for the next prediction and correction;

if two to four iterations are needed, the step-size can be kept unchanged; if five

to eight iterations are needed, the step-size may need to be decreased for the

next prediction and correction; if the correction does not converge after eight

iterations, the step-size should be decreased and the prediction (Eq. (4)) and

correction (Eq. (7)) redone to obtain the corrected one. It can be seen that

the step-size control is readily implemented and adaptive during the tracing

of the P-V curves. Note that the step-size needs to be confined between the

pre-specified minimum and maximum values for balance of the efficiency and

reliability.


8/16


3.2. LIB Detection

The previously selected schemes of prediction, correction, and step-size control are now

applied to trace the PF equation equilibrium curve and calculate the SNB or LIB. In

order to take the generator reactive power limit into consideration, the reactive power

outputs at each point are calculated for those PV buses specified generators. If, at apoint, there are no generators exceeding their reactive power limits, then the continuation

is able to continue. On the contrary, if there are some generators whose reactive power

outputs exceed the maximum limits (these generators will be referred to as the identified

generators or PV buses hereafter), then the PF equation f.y/ D 0 in Eq. (7) must be

changed in the following way and the correction redone by use of the corrected point

QyjC1, which was just obtained as the initial guess; see Figures 1(b) and 1(c).

The change of the PF equation in Eq. (7) is that all identified PV buses will be

changed into PQ buses; thus, the following equations are added to the PF equation

f.y/ D 0 in Eq. (7):

QG;i D Qlim;i ; (9)

where i represents all the identified generators. Also, the voltages at all these PV buses

will become unknown variables accordingly. For ease of reference, their voltages when

they are specified as PV buses are referred to as the pre-specified voltages hereafter. Note

that the total number of the identified generators will need to be confined to be equal to a

pre-specified number, for example, two to three to obtain a balance between the efficiency

and reliability before the PF is changed based on Eq. (9). Therefore, if the total number

exceeds the pre-specified number, the step-size will be decreased and the prediction and

correction redone.

Generally, the above techniques will be capable of handling both cases shown inFigures 1(a) and 1(b) before the continuation comes close to the LIB, and for both cases,

the parameter value at the corrected point will be larger than that at the current point.

Also, the voltage magnitudes at all identified PV buses will be equal to or smaller than

those pre-specified voltages.

Now suppose the continuation is approaching closer to an LIB. As a result, there arise

two possible situations for the arc-length correction scheme. One is that the correction

converges to point A shown in Figure 2(b), where the voltage at one of the identified

PV buses is greater than its originally pre-specified value. The other situation is that

the correction converges to point B shown in Figure 2(b), from which the bifurcation

parameter will begin to decrease.

Both situations shown in Figure 2(b) will be handled in the following way. For thefirst situation, decrease the step-size and redo the prediction and correction until there

is only one identified PV bus or more than one such bus at the newly corrected point.

(If there is more than one such bus, the newly corrected point is obtained by using a

pre-defined minimum value for the step-size, e.g., 104.) Moreover, it can be supposed

that the voltage at one of these identified PV buses is still greater than the pre-specified

value. Otherwise, the continuation method is continued. For ease of reference, the newly

corrected point is still noted as A, shown in Figure 2(b).

Similarly, for the second situation, the same work is done as in the first situation

decreasing the step-size and redoing the prediction and correctionto obtain the newly

corrected point so that there is only one or more than one identified generators reachingtheir reactive power limits at the corrected point. (If there is more than one, the corrected

point is obtained by using the minimum step-size.) Still, without loss of generality, it is


9/16


(a)

(b)

Figure 2. Tangent line: (a) at ordinary limit point and (b) at LIB.

assumed that the bifurcation parameter will begin to decrease from the corrected point,

noted as B shown in Figure 2(b).

Note that from the viewpoint of numerical computation, the occurrence of more than

one PV bus identified by using the minimum step-size stated above can indicate that there

are some generators meeting their reactive power limits at the same time.

3.3. LIB Location

Now we know that there is a limit point in the neighborhood of the newly corrected

point, and this limit point might be an LIB, which should be checked. For this purpose,

the limit point is first calculated. It can be obtained by applying the scheme of correction

to the following equation: (f.y/ D 0

QG D Qlim; (10)

where f.y/ D 0 is the PF before the above-identified limit points encountered; i.e., it will

not be changed based on Eq. (9), even if some generators have exceeded their reactivepower limits. QG D Qlim implies that gj .y/ D 0 be substituted in Eq. (7) by making the

reactive power output of one identified generator equal to its maximum limit. In a word


10/16


system, Eq. (10) is obtained by changing one of the identified PV buses into a so-called

PQV bus [29], i.e., the active and reactive powers and the voltage magnitude is specified

for such a bus.

Note that only one identified PV bus was selected (even if there is more than one) to

change into a PQV bus, while other identified PV buses remain unchanged, as Eq. (10)

has the same number of unknown variables y as that of the equations it contains.

After obtaining the limit point by using the Newton method for Eq. (10), all identified

PV buses are changed into PQ buses based on Eq. (9) to obtain a new continuation PF

equation with the same form as Eq. (3) and to calculate a tangent vector at the limit point

for the new equation. Also, the tangent line can be drawn to the P-V curve for the new

equation at the limit point on the P-V plane; see Figures 2(a) and 2(b). Obviously, the

tangent line is the projection of the tangent vector onto the P-V plane. From Figure 2,

schemes can be developed to determine whether or not the limit point is an LIB. A

particular case is that the zero of parameter element in the tangent vector implies that

the limit point is the intersection of the upper branch with the SNB associated with the

PF equation after the limit is encountered; see Figure 3(a). It also appears to be an LIBas the intersection is usually transverse; thus, the qualitative property around the limit

point will have an abrupt change. Also, the condition for the zero parameter element in

the tangent vector is discussed in Appendix A. Otherwise, the following value can be

calculated:

vt;i =t ; (11)

where t is the parameter element in the vector, and vt;i is the voltage element in the

vector corresponding to the identified PV bus that was selected in Eq. (10). Now, if

the value calculated by Eq. (11) is positive, then the limit point is an LIB; otherwise, it

is not an LIB but an ordinary limit point, which is the case shown in Figure 3(b). If anLIB is obtained, then all points after it will be on the lower branch of the equilibrium

curve, and the continuation can be stopped if needed. On the contrary, the tangent vector

at the LIB just obtained can be used to predict the next point if the points on the lower

branch still need to be calculated. Note that only one computation of the tangent vector

(except the one needed to calculate the second point on the equilibrium curve) at a limit

point is needed to check whether it is an LIB in the whole continuation procedure.

Figure 3. Limit encountered: (a) instability due to LIB at SNB and (b) instability due to SNB

near SNB.


11/16


4. Numerical Studies and Discussions

4.1. Accuracy of the Proposed Method

The proposed method has been implemented in MATLAB (The MathWorks, Natick,

Massachusetts, USA) and successfully applied to numerically analyze several powersystem examples. For comparative studies, the numerical results are reported by applying

the proposed method in the IEEE 118-bus system [30], which is a benchmark system

studied by the methods in [25, 26]. Its accuracy is first demonstrated.

The contingency of load and generation increase from its initial operation condition

[30] is studied in the following way. The values of PS, QD, and QD in Eq. (2)

representing the direction of load and generation increase are equal to the initial operation

values ofPG0, PL0, and QL0, respectively. Thus, the parameter increases its value from

zero. The P-V curves at buses 10 and 8 obtained by the proposed method are shown in

Figure 4, where the upper and lower branches are represented by solid and dashed lines,

respectively. Note that these curves agree with those in [26]. So, the proposed method can

trace the equilibrium curve of the PF equation when the generator reactive power limits

are taken into consideration. Obviously, there is a maximum value for the bifurcation

parameter, which is at the right-most point of the equilibrium curve, and it must be

determined whether it is an LIB or SNB.

Then four successive points around the maximum point, which are the corrected

point by the method, are listed in Table 1, where the first column represents the number

of these points in the sequence of the points, the second column lists the parameter

values, and the last column lists the identified buses where the generators reactive power

outputs exceed or are equal to their limits as the exact value for each limit point has not

been calculated until now. It can obviously be seen that there is a maximum value for

the parameter between points 30 and 31 as the parameter value begins to decrease afterpoint 31; also, between points 30 and 31, generator 10 meets its reactive power limit.

Figure 4. P-V curves at buses 10 and 8.


12/16


Table 1

Successive points around the maximum point

No. Parameter value Identified PV buses

28 0.98753 6529 1.08091 46

30 1.10562 4

31 1.10206 10

Based on the proposed technique, it can be determined whether or not the limit point

encountered by generator 10 is an LIB. Table 2 shows the computation results needed

for the determination. Note that the computation is executed after the method identified

that the bifurcation parameter experienced a maximum value. From these results, it can

be observed that the parameter value at this limit point becomes the maximum amongthose listed in Table 1. Furthermore, the value calculated by Eq. (11) is 2.86237/1; thus,

it can be determined that the limit point is an LIB. In [26], the same studies have been

reported, where the load and generation increases are expressed by

PG D PG0;

PL D PL0;

QL D QL0:

(12)

That is, the bifurcation parameter increases from 1, and the calculated LIB is 2.1007.The value calculated by the proposed method will be equal to 1C1:10991 if the increase

of load and generation expressed by Eq. (2) takes the same form as Eq. (12). Thus, the

proposed method will give a more accurate result as the LIB represents the maximum

loadability; in other words the method calculates the LIB without any loss of accuracy.

4.2. Efficiency of the Proposed Method

The efficiency of the proposed continuation method is now compared with other existent

ones. First, the computation times needed by the two cases of the proposed method for the

above studies, denoted as Case I and II, are listed in Table 3. The two cases correspond

that the maximum number of the identified PV generator discussed in Section 3.2, pre-specified as 2 and 3, respectively; all other constants pre-specified in the continuation

method are the same. These main constants are as follows: the initial step-size used is 0.5,

the minimum and maximum step-sizes are confined between 0.001 and 0.8. There must be

Table 2

Results of LIB location

Parameter value

Voltage element at bus 10

in the tangent vector

Parameter element in

the tangent vector

1.10991 2.86237 1


13/16


Table 3

Computation times for the proposed method

Computation time (sec)

Minimum Maximum Average

Proposed method (Case I) 1.282 1.306 1.292

Proposed method (Case II) 1.183 1.293 1.225

36 and 31 repetitions of prediction and correction in the continuation process for these two

cases to obtain the LIB. All of the computations are the same; i.e., the load and generation

increase from the initial operation conditions in the unified way expressed by Eq. (2).

The MATLAB program requires different times for each execution; thus, each cal-

culation is executed ten times, and the minimum, maximum, and average time neededare recorded. All of these computations are run on an IBM compatible computer with an

Intel Core DUO E6300 1.86-GHz CPU and 1.00-GB RAM.

It took 3.1 sec to calculate the LIB from the same initial operation condition of

the studied system in [26], where the MATLAB program was run on a computer with

Pentium III (450-MHz CPU); therefore, a direct comparison cannot be made with the

proposed method for the computation time, as they are run on computers with different

performances. However, the following comparisons on computational costs can be made

between these two methods.

First note that there were 30 limit points encountered until the LIB was obtained,

so 60 computations of tangent vectors were necessary to obtain the LIB, as two tangentvectors were needed at each limit point in [26]. Also, one or two iterations were needed

for the corrector to converge to the equilibrium on the P-V curve in the same reference.

Suppose that the computation effort for a tangent vector is almost the same as that

for an iteration in the corrector as both computations are concerned with the numerical

solution of the linear system; thus, a considerable number of computations can be saved

if the proposed method is used as two tangent vector computations are totally needed.

Obviously, the more limit points encountered by the continuation for a general power

system, the more computation efforts can be saved.

Second, [26] used the tangent prediction. By using this prediction (not the secant

prediction used by the proposed method), and if the maximum loadability characteristic

of LIB is exploited, then only one tangent vector is needed by prediction for each limitpoint; thus, half of the computations for the tangent vectors can be saved for the same

reference. This is because the proposed method does not need to calculate the tangent

vector for the PF equation, which is associated to the one before the limit is encountered.

A few more topics on the LIB due to the limit encounter in power systems will now

be addressed. First, the reactive power limits of the generators are kept as fixed values

in this article. However, the more accurate model of such a limit is the reactive power

capability curve used for generator modeling. In principle, the proposed techniques are

applicable under such conditions, where the complexity is that it costs more to check

whether or not a generator operation is within the capability curve.

Second, the voltage collapse may result from the activation of the on-load tap changeron the transformer. Generally, the change of the tap position is handled in the same

manner as in the conventional PF. In more detail, the admittance of the network and,


14/16


accordingly, the PF Jacobian is first changed based on the new position of the tap changer;

the resultant PF equation is then solved by the iteration method where the initial value

is taken as the previous PF equation solution. If the convergence is obtained, then the

power system is thought to be voltage stable. Otherwise, the system is thought to lose

the equilibrium that will lead to voltage collapse. Generally, the short- and long-term

time domain simulation is performed to validate the result. Note that, although there is

also a loss of equilibrium associated with such discrete (discontinuous) controller, it is

not a local bifurcation phenomenon but a kind of global bifurcation, while both the SNB

and LIB are the local bifurcations.

5. Conclusions

This article has first demonstrated that the LIB, due to the reactive power limit encoun-

tered by the generator, is the maximum loadability based on the SNB phenomenon of

the PF equation, and then presented an improved continuation method to calculate the

LIB. In more detail, the maximum loadability characteristics of the LIB are used forits identification, and the tangent vector at the limit point is used for its location and

distinction from the SNB. The main advantages of these techniques are that the obtained

result is accurate, and the associated computations are simpler and cost effective by

comparison with the existent ones. These techniques with convergence-dependent step-

size control are implemented to obtain the improved method; thus, the equilibrium curve

for the PF equation can be traced and the SNB and LIB can be calculated in a unified

way. The effectiveness of the proposed method is illustrated in its numerical applications

in the IEEE 118-bus system.

AcknowledgmentWe wish to thank the anonymous referees for the comments and suggestions.

References

1. IEEE/CIGRE Joint Task Force on Stability Terms and Definitions, Definition and classification

of power system stability, IEEE Trans. Power Syst., Vol. 19, No. 2, pp. 13871401, May 2004.

2. Taylor, C. W., Power System Voltage Stability, New York: McGraw-Hill, 1994.

3. Cutsem, T. V., and Vournas, C., Voltage Stability of Electric Power Systems, Boston: Kluwer

Academic Publishers, 1998.

4. Power System Stability Subcommittee, Voltage stability assessment: Concepts, practices and

tools, IEEE Power Engineering Society, Catalog No. SP101PSS, 2002.

5. Sauer, P. W., and Pai, M. A., Power system steady-state stability and the load-flow Jacobian,

IEEE Trans. Power Systems, Vol. 5, No. 4, pp. 13741383, November 1990.

6. Dobson, I., The irrelevance of load dynamics for the loading margin to voltage collapse and

its sensitivities, Proceedings: Bulk Power System Voltage Phenomena IIIVoltage Stability,

Security and Control, L. H. Fink (Ed.), pp. 509518, Davos, Switzerland, 2226 August 1994.

7. Caizares, C. A., Conditions for saddle-node bifurcations in AC/DC power system, Intl. J.

Elect. Power Energy Syst., Vol. 17, No. 1, pp. 6168, 1995.

8. Bompard, E., Carpaneto, E., Chicco, G., and Napoli, R., A dynamic interpretation of the

load-flow Jacobian singularity for voltage stability analysis, Intl. J. Elect. Power Energy Syst.,

Vol. 18, No. 6, pp. 385395, 1996.

9. Dobson, I., Observations on the geometry of saddle-node bifurcations and voltage collapsein electric power systems, IEEE Trans. Circuits Syst. (I), Vol. 39, No. 3, pp. 240243, March

1992.


15/16


10. Dobson, I., and Lu, L., Voltage collapse precipitated by the immediate change in stability

when generator reactive power limits are encountered, IEEE Trans. Circuits Syst. (I), Vol. 39,

No. 9, pp. 762766, September 1992.

11. Caizares, C. A., Alvarado, F. L., DeMarco, C. L., Dobson, I., and Long, W. F., Point of

collapse methods applied to AC/DC systems, IEEE Trans. Power Syst., Vol. 7, No. 2, pp. 673

683, May 1992.12. Caizares, C. A., and Alvarado, F. L., Point of collapse and continuation method for large

AC/DC systems, IEEE Trans. Power Syst., Vol. 8, No. 1, pp. 18, February 1993.

13. Dobson, I., and Lu, L., New methods for computing a closest saddle-node bifurcation and

worst case load power margin for voltage collapse, IEEE Trans. Power Syst., Vol. 8, No. 3,

pp. 905913, August 1993.

14. Alvarado, F., Dobson, I., and Hu, Y., Computation of closest bifurcation in power systems,

IEEE Trans. Power Syst., Vol. 9, No. 2, pp. 918928, May 1994.

15. Seydel, R., Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, 2nd ed.,

New York: Springer-Verlag, 1994.

16. Doedel, E., AUTO: Software for continuation and bifurcation problems in ordinary differential

equations, available at http://indy.cs.concordia.ca/auto/

17. Ajjarapu, V., and Christy, C., The continuation power flow: A tool for steady state voltage

stability analysis, IEEE Trans. Power Syst., Vol. 7, No. 1, pp. 416423, February 1992.

18. Chiang, H. D., Flueck, A. J., Shah, K. S., and Balu, N., CPFLOW: A practical tool for tracing

power system steady state stationary behavior due to load and generation variations, IEEE

Trans. Power Syst., Vol. 10, No. 2, pp. 623634, May 1995.

19. Mori, H., and Yamada, S., Continuation power flow with the nonlinear predictor of the

Lagranges polynomial interpolation formula, IEEE/PES Transmission and Distribution Con-

ference and Exhibition 2002 Asia Pacific, pp. 11331138, Yokohama, Japan, 610 October

2002.

20. Mori, H., and Kojima, T., Hybrid continuation power flow with linear-nonlinear predictor,

International Conference on Power System Technology (POWCON 2004), pp. 969974, Sin-

gapore, 2124 November 2004.21. Hiskens, I. A., and Davy, R. J., Exploring the power flow equation space boundary, IEEE

Trans. Power Syst., Vol. 16, No. 3, pp. 389395, August 2001.

22. Guoyun, C., Hill, D. J., and Hui, R., Continuation of local bifurcations for differential-

algebraic equations and its application to power systems, IEE Proc. Generat. Transm. Distrib.,

Vol. 152, No. 4, pp. 575580, July 2005.

23. Lerm, A. A. P., Caizares, C. A., and Silva, A. S., Multiparameter bifurcation analysis of the

south Brazilian power system, IEEE Trans. Power Syst., Vol. 18, No. 2, pp. 737746, May

2003.

24. Zhou, Y., and Ajjarapu, V., A fast algorithm for identification and tracing of voltage and

oscillatory stability margin boundaries, Proc. IEEE, Vol. 93, No. 5, pp. 934946, May

2005.

25. Hiskens, I. A., and Chakrabarti, B. B., Direct calculation of reactive power limit points, Intl.

J. Elect. Power Energy Syst., Vol. 18, No. 2, pp. 121129, 1996.

26. Yorino, N., Li, H.-Q., and Sasaki, H., A predictor/corrector scheme for obtaining Q-limit points

for power flow studies, IEEE Trans. Power Syst., Vol. 20, No. 1, pp. 130137, February 2005.

27. Lai, L. L., Liang, Z. Y., and Jiang, R. H., Chaotic phenomena in power systems, Proceedings

of the IEE International Conference on Control 1991, pp. 910914, Edinburgh, UK, 2528

March 1991.

28. Chiang, H. D., Liu, C. W., Varaiya, P. P., Wu, F. F., and Lauby, M. G., Chaos in a simple

power system, IEEE Trans. Power Syst., Vol. 8, No. 4, pp. 14071417, May 1993.

29. Kataoka, Y., and Shinoda, Y., Voltage stability limit of electric power system with generator

reactive power constraints considered, IEEE Trans. Power Syst., Vol. 20, No. 2, pp. 951962,

May 2005.30. Power system test case archive, available at http://www.ee.washington.edu/research/pstca/

index.html.


16/16


Appendix

To solve Eq. (5) for vj , A is first decomposed by LU factorization with column pivoting,

i.e.,

PA D LU; (A1)

where P is a permutation matrix, L is a unit lower triangular matrix, and U is an upper

triangular matrix. Moreover, the former two are square matrices, and the latter is an

n .n C 1/ matrix.

Now select the 1 n columns of U as a square and upper triangular matrix a and

its last column as a vector b, and solve the following triangular linear equation for z:

az D b: (A2)

Finally, append 1 to the row vector zT, normalize and transpose it to obtain the

tangent vectorvj

; thus,vj D normalize.zT; 1/T

can be obtained. By its substitution inEq. (5), it can be proven that it will satisfy Eq. (5). Note that the tangent vector has

a positive value for its parameter element; thus, the parameter will increase based on

the prediction in Eq. (4). In the above calculation, the triangular matrix a needs to be

invertible. On the contrary, if it is singular, it can be concluded that the PF Jacobian

is singular, and the parameter element in the tangent vector should be equal to zero.

The singularity can be decided by the value of the last diagonal element in a. If it is very

small, for example 106, then it is singular.

Documents

Novel Techniques for Continuation Method to Calculate the Limit-Induced Bifurcation of the Power Flow Equation