ch063- power systems

Grigsby, L.L., Hanson, A.P., Schlueter, R.A., Alemadi, N. Power SystemsThe Electrical Engineering HandbookEd. Richard C. DorfBoca Raton: CRC Press LLC, 2000

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63Power Systems

63.1 Power System AnalysisIntroduction Types of Power System Analyses The Power Flow Problem Formulation of the Bus Admittance Matrix Example Formulation of the Power Flow Equations P-V Buses Bus Classifications Generalized Power Flow Development Solution Methods Component Power Flows

63.2 Voltage InstabilityVoltage Stability Overview Voltage Stability Models and Simulation Tools Kinds, Classes, and Agents of Voltage Instability Proximity to Voltage Instability Future Research

.1 Power System Analysis

troduction

e equivalent circuit parameters of many power system components are described in Chapters 61, 64, and. The interconnection of the different elements allows development of an overall power system model. Thetem model provides the basis for computational simulation of the system performance under a wide varietyprojected operating conditions. Additionally, post mortem studies, performed after system disturbancesequipment failures, often provide valuable insight into contributing system conditions. The different typespower system analyses are discussed below; the type of analysis performed depends on the conditions to beessed.

pes of Power System Analyses

wer Flow Analysis

wer systems typically operate under slowly changing conditions which can be analyzed using steady statealysis. Further, transmission systems operate under balanced or near balanced conditions allowing per phasealysis to be used with a high degree of confidence in the solution. Power flow analysis provides the startingint for many other analyses. For example, the small signal and transient stability effects of a given disturbance dramatically affected by the pre-disturbance operating conditions of the power system. (A disturbanceulting in instability under heavily loaded system conditions may not have any adverse effects under lightlyded conditions.) Additionally, fault analysis and transient analysis can also be impacted by the pre-distur-

nce operating point of a power system (although, they are usually affected much less than transient stabilityd small signal stability analysis).

ult Analysis

lt analysis refers to power system analysis under severely unbalanced conditions. (Such conditions includewned or open conductors.) Fault analysis assesses the system behavior under the high current and/or severely

. Grigsby and P. Hansonburn University

A. Schlueter and Alemadihigan State University

2

unbalanced conditions typical during faults. The results of fault analyses are used to size and apply systemprotective devices (breakers, relays, etc.) Fault analysis is discussed in more detail in Section 61.5.

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ansient Stability Analysis

nsient stability analysis, unlike the analyses previously discussed, assesses the systems performance over ariod of time. The system model for transient stability analysis typically includes not only the transmissiontwork parameters, but also the dynamics data for the generators. Transient stability analysis is most oftend to determine if individual generating units will maintain synchronism with the power system followingisturbance (typically a fault).

tended Stability Analysis

tended stability analysis deals with system stability beyond the generating units first swing. In addition to generator data required for transient stability analysis, extended stability analysis requires excitation system,ed governor, and prime mover dynamic data. Often, extended stability analysis will also include dynamics

ta for control devices such as tap changing transformers, switched capacitors, and relays. The addition ofse elements to the system model complicates the analysis, but provides comprehensive simulation of nearlymajor system components and controls. Extended stability analyses complement small signal stability

alyses by verifying the existence of persistent oscillations and establishing the magnitudes of power and/orltage oscillations.

all Signal Stability Analysis

all signal stability assesses the stability of the power system when subjected to small perturbations. Smallnal stability uses a linearized model of the power system which includes generator, prime mover, and controlvice dynamics data. The system of nonlinear equations describing the system are linearized about a specificerating point and eigenvalues and eigenvectors of the linearized system found. The imaginary part of eachenvalue indicates the frequency of the oscillations associated with the eigenvalue; the real part indicatesmping of the oscillation. Usually, small signal stability analysis attempts to find disturbances and/or system

ditions that can lead to sustained oscillations (indicated by small damping factors) in the power system.all signal stability analysis does not provide oscillation magnitude information because the eigenvalues onlyicate oscillation frequency and damping. Additionally, the controllability matrices (based on the linearizedtem) and the eigenvectors can be used to identify candidate generating units for application of new orproved controls (i.e., power system stabilizers and new or improved excitation systems).

ansient Analysis

nsient analysis involves the analysis of the system (or at least several components of the system) whenjected to fast transients (i.e., lightning and switching transients). Transient analysis requires detailedponent information which is often not readily available. Typically only system components in the immediate

inity of the area of interest are modeled in transient analyses. Specialized software packages (most notablyTP) are used to perform transient analyses.

erational Analyses

eral additional analyses used in the day-to-day operation of power systems are based on the results of thealyses described above. Economic dispatch analyses determine the most economic real power output for eacherating unit based on cost of generation for each unit and the system losses. Security or contingency analyses

ess the systems ability to withstand the sudden loss of one or more major elements without overloading theaining system. State estimation determines the best estimate of the real-time system state based on a

undant set of system measurements.

e Power Flow Problem

wer flow analysis is fundamental to the study of power systems. In fact, power flow forms the core of powertem analysis. A power flow study is valuable for many reasons. For example, power flow analyses play a keye in the planning of additions or expansions to transmission and generation facilities. A power flow solution

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ften the starting point for many other types of power system analyses. In addition, power flow analysis andny of its extensions are an essential ingredient of the studies performed in power system operations. In this

ter case, it is at the heart of contingency analysis and the implementation of real-time monitoring systems.The power flow problem (popularly known as the load flow problem) can be stated as follows:

For a given power network, with known complex power loads and some set of specifications or restrictions onpower generations and voltages, solve for any unknown bus voltages and unspecified generation and finally forthe complex power flow in the network components.

ditionally, the losses in individual components and the total network as a whole are usually calculated.rthermore, the system is often checked for component overloads and voltages outside allowable tolerances.Balanced operation is assumed for most power flow studies and will be assumed in this chapter. Consequently, positive sequence network is used for the analysis. In the solution of the power flow problem, the networkment values are almost always taken to be in per unit. Likewise, the calculations within the power flowalysis are typically in per unit. However, the solution is usually expressed in a mixed format. Solution voltages usually expressed in per unit; powers are most often given in kVA or MVA.The given network may be in the form of a system map and accompanying data tables for the network

ponents. More often, however, the network structure is given in the form of a one-line diagram (such aswn in Fig. 63.1).

Regardless of the form of the given network and how the network data are given, the steps to be followeda power flow study can be summarized as follows:

1. Determine element values for passive network components.2. Determine locations and values of all complex power loads.3. Determine generation specifications and constraints.4. Develop a mathematical model describing power flow in the network.5. Solve for the voltage profile of the network.

FIGURE 63.1 The one line diagram of a power system.

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6. Solve for the power flows and losses in the network.7. Check for constraint violations.

rmulation of the Bus Admittance Matrix

e first step in developing the mathematical model describing the power flow in the network is the formulationthe bus admittance matrix. The bus admittance matrix is an nn matrix (where n is the number of busesthe system) constructed from the admittances of the equivalent circuit elements of the segments making up power system. Most system segments are represented by a combination of shunt elements (connectedween a bus and the reference node) and series elements (connected between two system buses). Formulationthe bus admittance matrix follows two simple rules:

1. The admittance of elements connected between node k and reference is added to the (k, k) entry of theadmittance matrix.

2. The admittance of elements connected between nodes j and k is added to the ( j, j) and (k, k) entries ofthe admittance matrix. The negative of the admittance is added to the ( j, k) and (k, j) entries of theadmittance matrix.

Off nominal transformers (transformers with transformation ratios different from the system voltage basesthe terminals) present some special difficulties. Figure 63.2 shows a representation of an off nominal turnsio transformer.The admittance matrix mathematical model of an isolated off nominal transformer is:

(63.1)

ereYe is the equivalent series admittance (referred to node j)c is the complex (off nominal) turns ratioIj is the current injected at node jVj is the voltage at node j (with respect to reference)

f nominal transformers are added to the bus admittance matrix by adding the corresponding entry of thelated off nominal transformer admittance matrix to the system bus admittance matrix.

ample Formulation of the Power Flow Equations

nsiderable insight into the power flow problem and its properties and characteristics can be obtained bysideration of a simple example before proceeding to a general formulation of the problem. This simple case

ll also serve to establish some notation.

FIGURE 63.2 Off nominal turns ratio transformer.

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A conceptual representation of a one-line diagram for a four-bus power system is shown in Fig. 63.3. Forerality, a generator and a load are shown connected to each bus. The following notation applies:

SG1 = Complex complex power flow into bus 1 from the generator

SD1 = Complex complex power flow into the load from bus 1

mparable quantities for the complex power generations and loads are obvious for each of the three otherses.The positive sequence network for the power system represented by the one-line diagram of Fig. 63.3 iswn in Fig. 63.4. The boxes symbolize the combination of generation and load. Network texts refer to this

twork as a five-node network. (The balanced nature of the system allows analysis using only the positiveuence network; reducing each three-phase bus to a single node. The reference or ground represents the fifthde.) However, in power systems literature it is usually referred to as a four-bus network or power system.For the network of Fig. 63.4, we define the following additional notation:

S1 =

SG1

SD1 Net complex power injected at bus 1

I1 = Net positive sequence phasor current injected at bus 1

V1 = Positive sequence phasor voltage at bus 1

The standard node voltage equations for the network can be written in terms of the quantities at bus 1fined above) and comparable quantities at the other buses.

FIGURE 63.3 Conceptual one-line diagram of a four-bus power system.

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(63.2)

(63.3)

(63.4)

(63.5)

e admittances in Eqs. (63.2) through (63.5),Yij, are the ijth entries of the bus admittance matrix for the

wer system. The unknown voltages could be found using linear algebra if the four currentsI1

I4 were

own. However, these currents are not known. Rather, something is known about the complex power andltage at each bus. The complex power injected into bus k of the power system is defined by the relationshipween complex power, voltage, and current given by Eq. (63.6).

(63.6)

Therefore,

(63.7)

substituting this result into the nodal equations and rearranging, the basic power flow equations for ther-bus system are given as Eqs. (63.8) through (63.11)

(63.8)

FIGURE 63.4 Positive sequence network for the system of Fig. 63.3.

I Y V Y V Y V Y V1 11 1 12 2 13 3 14 4= + + +

I Y V Y V Y V Y V2 21 1 22 2 23 3 24 4= + + +

I Y V Y V Y V Y V3 31 1 32 2 33 3 34 4= + + +

I Y V Y V Y V Y V4 41 1 42 2 43 3 44 4= + + +

S V Ik k k*

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(63.10)

(63.11)

Examination of Eqs. (63.8) through (63.11) reveals that, except for the trivial case where the generationals the load at every bus, the complex power outputs of the generators cannot be arbitrarily selected. In

t, the complex power output of at least one of the generators must be calculated last because it must take the unknown slack due to the, as yet, uncalculated network losses. Further, losses cannot be calculatedtil the voltages are known. These observations are a result of the principle of conservation of complex power., the sum of the injected complex powers at the four system buses is equal to the system complex powerses).Further examination of Eqs. (63.8) through (63.11) indicates that it is not possible to solve these equations the absolute phase angles of the phasor voltages. This simply means that the problem can only be solvedsome arbitrary phase angle reference.In order to alleviate the dilemma outlined above, suppose

SG4 is arbitrarily allowed to float or swing (in

er to take up the necessary slack caused by the losses) and thatSG1,

SG2, and

SG3 are specified (other cases

ll be considered shortly). Now, with the loads known, Eqs. (63.7) through (63.10) are seen as four simulta-ous nonlinear equations with complex coefficients in five unknowns

V1,

V2,

V3,

V4, and

SG4.

The problem of too many unknowns (which would result in an infinite number of solutions) is solved bycifying another variable. Designating bus 4 as the slack bus and specifying the voltage

V4 reduces the problem

four equations in four unknowns. The slack bus is chosen as the phase reference for all phasor calculations,magnitude is constrained, and the complex power generation at this bus is free to take up the slack necessaryorder to account for the system real and reactive power losses.The specification of the voltage

V4 decouples Eq. (63.11) from Eqs. (63.8) through (63.10), allowing calcu-

ion of the slack bus complex power after solving the remaining equations. (This property carries over toger systems with any number of buses.) The example problem is reduced to solving only three equationsultaneously for the unknowns

V1,

V2, and

V3 . Similarly, for the case of n buses, it is necessary to solve n-1

ultaneous, complex coefficient, nonlinear equations.Systems of nonlinear equations, such as Eqs. (63.8) through (63.10), cannot (except in rare cases) be solved closed-form techniques. Direct simulation was used extensively for many years; however, essentially all powerw analyses today are performed using iterative techniques on digital computers.

V Buses

all realistic cases, the voltage magnitude is specified at generator buses to take advantage of the generatorsctive power capability. Specifying the voltage magnitude at a generator bus requires a variable specified in simple analysis discussed earlier to become an unknown (in order to bring the number of unknowns backo correspondence with the number of equations). Normally, the reactive power injected by the generatoromes a variable, leaving the real power and voltage magnitude as the specified quantities at the generator bus.

It was noted earlier that Eq. (63.11) is decoupled and only Eqs. (63.8) through (63.10) need be solvedultaneously. Although not immediately apparent, specifying the voltage magnitude at a bus and treating bus reactive power injection as a variable results in retention of, effectively, the same number of complexknowns. For example, if the voltage magnitude of bus 1 of the earlier four bus system is specified and thective power injection at bus 1 becomes a variable, Eqs. (63.8) through (63.10) again effectively have three

plex unknowns. (The phasor voltagesV2 and

V3 at buses 2 and 3 are two complex unknowns and the

gle 1 of the voltage at bus 1 plus the reactive power generation QG1 at bus 1 result in the equivalent of ard complex unknown.)


D3*

3*

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Bus 1 is called a

voltage controlled bus

because it is apparent that the reactive power generation at bus 1 isbeing used to control the voltage magnitude. This type of bus is also referred to as a

P-V

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cified quantities. Typically, all generator buses are treated as voltage controlled buses.

s Classifications

ere are four quantities of interest associated with each bus:

1. real power, P2. reactive power, Q3. voltage magnitude, V4. voltage angle,

At every bus of the system two of these four quantities will be specified and the remaining two will beknowns. Each of the system buses may be classified in accordance with the two quantities specified. Thelowing classifications are typical:

Slack busThe slack bus for the system is a single bus for which the voltage magnitude and angle arespecified. The real and reactive power are unknowns. The bus selected as the slack bus must have a sourceof both real and reactive power, because the injected power at this bus must swing to take up theslack in the solution. The best choice for the slack bus (since, in most power systems, many buses havereal and reactive power sources) requires experience with the particular system under study. The behaviorof the solution is often influenced by the bus chosen. (In the earlier discussion, the last bus was selectedas the slack bus for convenience.)

Load bus (P-Q bus)A load bus is defined as any bus of the system for which the real and reactive powerare specified. Load buses may contain generators with specified real and reactive power outputs; however,it is often convenient to designate any bus with specified injected complex power as a load bus.

Voltage controlled bus (P-V bus)Any bus for which the voltage magnitude and the injected real powerare specified is classified as a voltage controlled (or P-V) bus. The injected reactive power is a variable (withspecified upper and lower bounds) in the power flow analysis. (A P-V bus must have a variable sourceof reactive power such as a generator or a capacitor bank.)

eneralized Power Flow Development

e more general (n bus) case is developed by extending the results of the simple four-bus example. Consider case of an n-bus system and the corresponding n+1 node positive sequence network. Assume that the buses numbered such that the slack bus is numbered last. Direct extension of the earlier equations (writing thede voltage equations and making the same substitutions as in the four-bus case) yields the basic power flowations in the general form.

e Basic Power Flow Equations (PFE)

(63.12)

d

(63.13)

S P jQ V Y V

for k = 1, 2, 3, , n 1

k*

k k k*

ki i

n

i =1= =

P jQ V Y Vn n n*

ni i

n

i =1 =

2

Equation (63.13) is the equation for the slack bus. Equation (63.12) represents n-1 simultaneous equations inn-1 complex unknowns if all buses (other than the slack bus) are classified as load buses. Thus, given a set ofspeOn

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cified loads, the problem is to solve Eq. (63.12) for the n-1 complex phasor voltages at the remaining buses.ce the bus voltages are known, Eq. (63.13) can be used to calculate the slack bus power.

Bus j is normally treated as a P-V bus if it has a directly connected generator. The unknowns at bus j aren the reactive generation QGj and j because the voltage magnitude, Vj , and the real power generation, Pgj ,

ve been specified.The next step in the analysis is to solve Eq. (63.12) for the bus voltages using some iterative method. Once bus voltages have been found, the complex power flows and complex power losses in all of the network

ponents are calculated.

lution Methods

e solution of the simultaneous nonlinear power flow equations requires the use of iterative techniques forn the simplest power systems. Although there are many methods for solving nonlinear equations, only twothods are discussed here.

e Newton-Raphson Method

e Newton-Raphson algorithm has been applied in the solution of nonlinear equations in many fields. Theorithm will be developed using a general set of two equations (for simplicity). The results are easily extendedan arbitrary number of equations.A set of two nonlinear equations are shown in Eqs. (63.14) and (63.15).

f1(x1, x2) = k1 (63.14)

f2(x1, x2) = k2 (63.15)

Now, if x1(0) and x2(0) are inexact solution estimates and x1(0) and x2(0) are the corrections to the estimatesachieve an exact solution, Eqs. (63.14) and (63.15) can be rewritten as:

f1(x1+ x1(0), x2+ x2(0)) = k1 (63.16)

f2(x1+ x1(0), x2+ x2(0)) = k2 (63.17)

Expanding Eqs. (63.16) and (63.17) in a Taylor series about the estimate yields:

(63.18)

(63.19)

ere the superscript, (0), on the partial derivatives indicates evaluation of the partial derivatives at the initialimate and h.o.t. indicates the higher order terms.Neglecting the higher order terms (an acceptable approximation if x1(0) and x2(0) are small), Eqs. (63.18)d (63.19) can be rearranged and written in matrix form.

f x , xf

xx

f

xx h.o.t. k1 1

(0)2(0) 1

1

(0)

1(0) 1

2

(0)

2(0)

1( ) + + + =

f x , xf

xx

f

xx h.o.t. k1 1

(0)2(0) 2

1

(0)

1(0) 2

2

(0)

2(0)

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(63.20)

e matrix of partial derivatives in Eq. (63.20) is known as the Jacobian matrix and is evaluated at the initialimate. Multiplying each side of Eq. (63.20) by the inverse of the Jacobian yields an approximation of theuired correction to the estimated solution. Since the higher order terms were neglected, addition of therection terms to the original estimate will not yield an exact solution, but will provide an improved estimate.e procedure may be repeated, obtaining sucessively better estimates until the estimated solution reaches asired tolerance. Summarizing, correction terms for the th iterate are given in Eq. (63.21) and the solutionimate is updated according to Eq. (63.22).

(63.21)

x(+1) = x() + x() (63.22)

The solution of the original set of nonlinear equations has been converted to a repeated solution of a systemlinear equations. This solution requires evaluation of the Jacobian matrix (at the current solution estimate)each iteration.The power flow equations can be placed into the Newton-Raphson framework by separating the power flowations into their real and imaginary parts and taking the voltage magnitudes and phase angles as the

knowns. Writing Eq. (63.21) specifically for the power flow problem:

(63.23)

The underscored variables in Eq. (63.23) indicate vectors (extending the two equation Newton-Raphsonvelopment to the general power flow case). The (sched) notation indicates the scheduled real and reactivewers injected into the system. P() and Q() represent the calculated real and reactive power injections based the system model and the th voltage phase angle and voltage magnitude estimates. The bus voltage phasegle and bus voltage magnitude estimates are updated, the Jacobian re-evaluated and the mismatch between scheduled and calculated real and reactive powers evaluated in each iteration of the Newton-Raphsonorithm. Iterations are performed until the estimated solution reaches an acceptable tolerance or a maximummber of allowable iterations is exceeded. Once a solution (within an acceptble tolerance) is reached, P-V busctive power injections and the slack bus complex power injection may be evaluated.

st Decoupled Power Flow Solution

e fast decoupled power flow algorithm simplifies the procedure presented for the Newton-Raphson algorithmxploiting the strong coupling between real po wer and bus voltage phase angles and reactive power and bus

x x

f

x

f

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x

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2

(0)1(0)

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f

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f

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voltage magnitudes commonly seen in power systems. The Jacobian matrix is simplified by approximating thepartial derivatives of the real power equations with respect to the bus voltage magnitudes as zero. Similarly, thepaas the

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rtial derivatives of the reactive power equations with respect to the bus voltage phase angles are approximatedzero. Further, the remaining partial derivatives are often approximated using only the imaginary portion of bus admittance matrix. These approximations yield the following correction equations:

(63.24)

(63.25)

ere B is an approximation of the matrix of partial derviatives of the real power flow equations with respectthe bus voltage phase angles and B is an approximation of the matrix of partial derivatives of the reactivewer flow equations with respect to the bus voltage magnitudes. B and B are typically held constant duringrative process, eliminating the necessity of updating the Jacobian matrix (required in the Newton-Raphsonution) in each iteration.The fast decoupled algorithm has good convergence properties despite the many approximations used duringdevelopment. The fast decoupled power flow algorithm has found widespread use since it is less computa-nally intensive (requires fewer computational operations) than the Newton-Raphson method.

mponent Power Flows

e positive sequence network for components of interest (connected between buses i and j) will be of them shown in Fig. 63.5.An admittance description is usually available from earlier construction of the nodal admittance matrix. Thus,

(63.26)

erefore, the complex power flows and the component loss are:

(63.27)

(63.28)

(63.29)

FIGURE 63.5 Typical power system component.

( ) ( )l l= [ ] ( ) [ ]B schedP PV Q QB sched( ) ( )l l= [ ] ( ) [ ]

I

I

Y Y

Y Y

V

Vi

j

a b

c d

i

j

=

S V I V Y V Y Vij i i*

i a i b j= = +[ ]*S V I V Y V Y Vji j j

*i c i d j= = +[ ]*

S S Sloss ij ji= +

2

The calculated component flows combined with the bus voltage magnitudes and phase angles provide extensiveinformation about the operating point of the power systems. The bus voltage magnitudes may be checked toenrat

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sure operation within a prescribed range. The segment power flows can be examined to ensure no equipmentings are exceeded. Additionally, the power flow solution may used as the starting point for other analyses.An elementary discussion of the power flow problem and its solution is presented in this chapter. The powerw problem can be complicated by the addition of further constraints such as generator real and reactive powerits. However, discussion of such complications is beyond the scope of this chapter. The references provideailed development of power flow formulation and solution under additional constraints. The references alsovide some background in the other types of power system analysis discussed at the begining of the chapter.

efining Terms

s admittance matrix: The nodal admittance matrix for an electric network resulting from a power system.wton-Raphson algorithm: An iterative technique for solving a system of nonlinear algebraic equations.st-decoupled algorithm: An extension of the Newton-Raphson iterative technique for solving power flow

equations for a power system.e power flow problem: A model of a power system, the solution of which provides the system voltage

profile and power flows from the sources to the loads.

lated Topics

Node and Mesh Analysis 3.5 Three-Phase Circuits

ferences

R. Bergen, Power Systems Analysis, Englewood Cliffs, N.J.: Prentice-Hall.I. Elgerd, Electric Energy Systems Theory - An Introduction, 2nd ed., New York: McGraw-Hill.. Glover and M. Sarma, Power System Analysis and Design, 2nd ed., Boston, Mass.: PWS Publishing.

A. Gross, Power System Analysis, 2nd ed., New York: John Wiley & Sons. D. Stevenson, Elements of Power System Analysis, 4th ed., New York: McGraw-Hill.

rther Information

e references provide clear introductions to the analysis of power systems. An excellent review of many issuesolving the use of computers for power system analysis is provided in Proceedings of the IEEE (Special Issue Computers in the Power Industry), July 1974. The quarterly journal IEEE Transactions on Power Systemsvides excellent documentation of more recent research in power system analysis.

.2 Voltage Instability

bert A. Schlueter and Nassar Alemadi

ltage Stability Overview

tention of voltage stability and viability is the ability of a power system to preserve the voltage at an operatingilibrium under normal condition and to maintain an acceptable voltage at all buses after being subjected to a

turbance. A loss of voltage viability occurs when voltage declines below acceptable levels but does not decline togressively lower values. A system loses voltage stability when a disturbance, changes in system operating condi-

n, or increase in load demand causes a progressive and spreading drop in voltage [Kundur, 1994]. The incapabilitythe power system to meet the reactive power demand is the main cause of voltage instability. The drop in voltageults in increased network reactive losses due to (1) reduced shunt capacitive reactive supply and (2) increasedgnetic field due to increased current flow. The increased network losses results in (1) reduced reactive powerw to the region that needs the most reactive supply and (2) exhaustion of the reactive reserves on generators,

2

synchronous condensers, or SVCs causing loss of voltage control. This loss of voltage control can cause furthervoltage drop and further increase in network reactive losses that produce a voltage collapse [Schlueter, 1998d].

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Voltage instability has become the principal constraint on power system operation for many utilities. Thermalerheating) constraints or transient stability constraints were the principal limitations on power system

eration just 15 years ago. Low or high voltage limit violation (voltage viability) constraints and voltagetability constraints have become the major operational limitation on many utilities. Voltage instability is aique problem because it can produce an uncontrollable, cascading instability that results in blackout for age region or an entire country. Major blackouts have affected the Pennsylvania, New Jersey, Marylanderconnection, the Western System Coordinating Council (WSCC) system, Florida, France, Sweden, andan. A more complete list of voltage instabilities incidents is contained in [Taylor, 1994; NERC 1991]. Thermal,rload voltage limit violation transient angle instabilities do not have the potential to cause the uncontrollablecading instability that affects such large regions as does voltage instability.The modeling required and simulation tools used to accurately assess retention and loss of voltage stabilityreviewed in the next section. Voltage instability has been studied using both a loadflow (algebraic) and aferential algebraic model, and both are discussed. The kinds, classes, and agents of voltage instability that develop are discussed in in the third section. A bifurcation is a sudden change in response, usually stable

unstable, for a smooth, continuous, slow change in load or operating condition. Saddle node, Hopf, singu-ity induced, and algebraic bifurcation are the kinds of bifurcation that have been observed on a power systemferential algebraic model, and all these different bifurcation have been associated with voltage instability.gging and loss of control voltage instability are the two kinds of bifurcation in a loadflow (algebraic) model.

Methods for assessing proximity to voltage instability in the loadflow model are reviewed. A Voltage Stabilityurity Assessment and Diagnosis Method [Schlueter, 1998d] is discussed that can answer the voltage instabilitygnostic questions of where, when, why, and proximity and cure for each equipment outage, transaction

bination, or both.Proximity to voltage instability has also been studied in a differential algebraic model. It has been shownt bifurcation sequences occur in a differential algebraic model that can include saddle node, Hopf, singularityuced, or algebraic bifurcation [Zaborsky, 1993; Guo et al., 1994]. Instability in the dynamics can occurore the bifurcation affects the algebraic model. It has been shown that saddle node bifurcation in a differentialebraic model at equilibrium is a bifurcation in the loadflow model that includes both the algebraic submodeld differential submodel at equilibrium. [Schlueter, 1998e; Liu, 1998]. In other cases, the bifurcation is solelythe algebraic model and has no effects on generator dynamics (algebraic bifurcation) or is in the algebraicdel that produces very rapid changes in generator dynamics (singularity-induced bifurcation). The bifur-ion in the algebraic equations is almost always associated with the ultimate blackout, even when saddle nodeHopf bifurcation initiates the instability that results in blackout.

ltage Stability Models and Simulation Tools

ifferential algebraic model for a power system can be written as [Schlueter et al., 1994].

(63.30)

(63.31)

(63.32)

ere:

x1 = state of generator, automatic voltage regulator (AVR), power system stabilizer, field currentlimiter, and armature current limiter on each generator in the system

p1 = overexcitation limiter relay limits that disable the AVR and trip the generator, armature currentrelay limits that trip generators

, ,x f x x p1 1 1 3 1= ( ) , ,x f x x p2 2 2 3 2= ( )

o f x x x p= ( )3 1 2 3 3, , ,

2

x2 = state of large induction motors, large thermostatic loads where temperature control is per-formed; generic load that represents action of under load tap changers, switchable shunt

or

wh

gento insrattrawitheall redcapbymeeff000 by CRC Press LLC

capacitors, aggregate of small induction motor load, and aggregate of small thermostatic loadmodels

p2 = parameters of the load modelx3 = voltage and phase of network power balance equationsp3 = parameters of network under load tap changers and switchable shunt capacitorsf1 ( ) = model of generators, synchronous condensers, and FACTS devices and their controlsf2 ( ) = model of large induction motors; large thermostatic loads where temperature is controlled;

and generic load models that represent aggregated action of under load tap changes, switchableshunt capacitors, smaller induction motors, and smaller thermostatic controlled load and theircontrols

f3 ( ) = network algebraic model that can include load where temperature, energy, or voltage controldynamics have no effect, under load tap changers, and switchable shunt capacitors

The loadflow model is:

(63.33)

(63.34)

ere

Voltage instability is most accurately assessed by a differential algebraic model (Eq. (63.30-63.32), but canerally be accurately assessed using the loadflow model (Eq. (63.34)). The loadflow model simulation is used

screen for equipment outages, and for accurate assessment of retention or loss of voltage stability for thosetability problems identified via loadflow. The size and complexity of the loadflow model required to accu-ely determine the steady-state equilibrium xo after one or more equipment outages, transfer and wheelingnsaction combinations, or both have grown from less than 1000 buses to above 15,000 buses for utilitiesth voltage instability problems. The model has grown to include distribution system buses in the study system, transmission and subtransmission network electrically distant from the study system, as well as virtuallyreactive generation devices in the interconnection. The exact sequence of actions on (1) field current limitersuction of field current, (2) tap position changes of under load tap changers, and (3) switchable shuntacitors insertions have been shown to be quite important in accurately obtaining the equilibrium produced

the simulation of the differential algebraic model. A quasi steady-state (QSS) approximation is an improve-nt on loadflow, that incorporates the effects of the control delays on devices (13) without incorporating

ects of faster generator and load dynamics. QSS is being implemented by several software vendors as a means

f x x p

f x x p

f x x x p

1 1 3 1

2 2 3 2

3 1 2 3 3

0

0

0

, ,

, ,

, , ,

( ) =( ) =( ) =

f x p,( ) = 0

x

x

x

x

p

p

p

p

f x p

f

f

f

=

=

( ) =

( )( )( )

1

2

3

1

2

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1

2

3

,

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of accurately obtaining an equilibrium without a full time simulation. The size and complexity of mid-termand long-term stability simulations that simulate the trajectory and determine the equilibrium also increasedin whterasyloacanfacreasup

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order to accurately simulate the response of a system to one or more equipment outages, transfer andeeling transaction combinations, or both. Timing of control actions must obviously be modeled in mid-m and long-term simulations in order to accurately simulate the trajectory and determine if it convergesmptotically or does not converge to an equilibrium or limit cycle. The actual tap position limits on under-d tap changers, field current limits on field current limiters, and switchable shunt capacitor capacity limits have dramatic effects on loadflow, QSS, or mid-term and long-term simulation results. This is due to the

t that loss of voltage control on these devices (1) eliminates actions that force reactive supply to areas withctive need or (2) choke off available reactive supply due to dramatic increases in I2x losses and shunt capacitiveply withdrawal that occur with voltage drop that results without proper voltage control.

nds, Classes, and Agents of Voltage Instability [Schlueter, 1994]

ltage instability can be classified in terms of the time frame required for voltage instability to develop. Long-m voltage instability is defined as developing over a time frame of 1 to 20 min [Taylor, 1994] and transientltage instability is defined as developing over a time frame of 1 to 10 s [Taylor, 1994]. Another method ofssifying voltage instability as short term or long term is to determine the dynamics that play the central rolethe development of voltage instability [Van Cutsem et al., 1998]. Short-term voltage instability [Van Cutsemal., 1998] occurs due to (1) motor stalling, (2) motor stalling after a short circuit occurs and is cleared, instability in generator flux decay dynamics when the AVR is disabled by an over-excitation limiter relayhlueter, 1998e], and (4) oscillations in generator dynamics, [Sauer, 1992] or between groups of generatorsbetween generators and induction motors [Van Cutsem et al., 1998]. All of these short-term stability problems be accurately captured in mid-term or long-term simulation but cannot be captured in a loadflow or QSSulation [Van Cutsem, 1998]. Action of tap changers, switchable shunt capacitors, generic load change,rmostatic load change, generator field current limiters and protection are long-term dynamics because theirponse can take 1 to 20 min or longer [Van Cutsem, 1998]. In many cases, these long-term dynamics resultinstability in short-term dynamics that ultimately produce lack of solution or instability in the algebraications. Loadflow and QSS models can capture most long-term voltage instability problems. The lack of a

dflow solution can indicate that there exists no steady-state equilibrium to the differential algebraic modeler some equipment outage, transaction combination, or both. Diagnostics can be applied to determine ifltage instability is the cause of the lack of an equilibrium solution [Schlueter, 1998d]. Voltage instability isown to have occurred if a cure can be found based on these diagnostics that corrects the lack of solution.e system matrix

(63.35)

be used to test for stability of the equilibrium if one can be computed by the loadflow. Thus, obtaining adflow solution does not guarantee stability of the dynamics at equilibrium. Bifurcation is a discontinuousnge in the qualitative behavior of the dynamics as some parameter changes slowly, continuously, andoothly and generally implies a change from a stable to an unstable response. There are many different kindsbifurcation that have been observed to occur in a power system model in both the long-term and short-m dynamics, as noted in the next paragraph. Different kinds of bifurcation can occur in the same dynamicsthe same subsystems or in different subsystems in a power system model. Bifurcation subsystem analysisen-Kilani, 1997] is being used to identify all of the different subsystems (classes) experiencing each kind ofurcation. Classes for each kind of bifurcation are the specific short-term (inertial dynamics of inductiontor or generators, flux decay and excitation control dynamics of generators or induction motors) or long-

m dynamics (tap changer, switchable shunt capacitor, thermostatic load control dynamics, field currentiter dynamics) that experience or produce the particular kind of bifurcation. The subsystem experiencing

Af f

f f

f

ff f fx x

x x

x

x

x x x=

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1

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1 2

3

3

3 1 2

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particular kind of bifurcation indicates whether it is short-term or long-term instability as noted in [Vantsem, et al. 1998]. It may be that the long-term dynamic changes bring about the particular bifurcation in art-term dynamics subsystem, as noted above. It is necessary to specify the time frame (long term or short

m) for instability to develop, the kind of bifurcation (saddle node, Hopf, singularity induced), the class ofurcation, and the agent of that particular kind and class to describe a particular voltage instability event.ents that experience bifurcation are the specific devices in a class that are in specific locations that experienceproduce the particular kind of bifurcation. Bifurcation subsystem analysis is identifying the agents for eachd and class of bifurcation. Only the generic kinds of bifurcations that have been studied extensively will nowdiscussed, although a more extensive list of bifurcation kinds observed in power systems is given in [Schlueter,94].Saddle node bifurcation occurs if both J and A have a zero eigenvalue as p changes toward bifurcation value p*d certain transversality and genericity conditions hold [Zaborsky, 1994]. Saddle node bifurcation occursen the region of attraction, where the trajectory converges to the stable equilibrium, becomes a null set and dynamics associated with the bifurcation eigenvalue of A evaluated at the equilibrium become infinitelyw and then unstable. Saddle node bifurcation occurs when a loadflow experiences bifurcation [Canizaresal., 1992]. Hopf bifurcation occurs if the real part of a complex pair of eigenvalues cross the j axis as pnges toward bifurcation values p* if certain transversality and genericity conditions hold [Zaborsky, 1994].

e Hopf bifurcation occurs when a stable or unstable limit cycle (oscillation) is formed. Singularity inducedurcation occurs when f3x3 has an eigenvalue that approaches zero and the eigenvalue of A approaches infinity,omes negative infinity, and then approaches zero [Zaborsky, 1994]. Algebraic bifurcation occurs when f3x3

s an eigenvalue approaching zero and A has no eigenvalue experiencing bifurcation.There is often a sequence of bifurcations associated with a particular voltage instability event. One suchuence is a Hopf followed by singularity-induced bifurcation [Sauer et al., 1992] as shown in Figs. 63.6 and

.7. Figure 63.6 shows the PV curve stress test and Fig. 63.7 shows the pair of bifurcating eigenvalues thaterience the sequence of bifurcations. Another bifurcation sequence is saddle node followed by singularity-uced bifurcation, as shown in Figs. 63.8 and 63.9. Figure 63.8 shows the Q-V curve stress test and Fig. 63.9ws the sequence of bifurcations. It is proven that after the saddle node bifurcation, the unstable internalerator voltage decline can produce a dynamic Q-V stress test that inevitably leads to the singularity inducedurcation and blackout [Schlueter, 1998e]. Motor stalling is a saddle node bifurcation that can also produce

FIGURE 63.6 PV curve on an example system.

2

a d19

FIGexa000 by CRC Press LLC

ynamically administered stress test that leads to singularity-induced bifurcation and blackout [Van Cutsem,98].

URE 63.7 Bifurcation sequence of Hopf (A), node-focus (B), singularity-induced (C) bifurcation produced on anmple system by the PV curve in Fig. 63.6.

FIGURE 63.8 Q-V curve produced on an example system.

2

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oximity to Voltage Instability

is proximity to voltage instability is initially assessed using a loadflow model (Eq. (63.34)). If a particulartingency, transfer, or wheeling combination, or both is found to experience voltage instability based on a

dflow model, it will be confirmed using a mid-term or long-term simulation. Using a loadflow has beennd to be quite a satisfactory model for accurate assessment of proximity to long-term voltage instability.e loadflow model is f(x,p) = 0, where x is the n dimension state of the model and is of the same dimensionf(x,p) and p is an m vector of parameters that can change and produce bifurcation or instability if p changesoothly and continuously. The implicit function theorem can indicate when a solution exists and the solutionnique.

eorem (Implicit function theorem) [Apostol, 1974] f = (f1,fn) be a vector valued defined on an open set S in Rn+m with values in Rn. Suppose fC or that

s continuously differentiable on S. Let (x0;p0) be a point in S for which f(x0;p0) = 0 and for which theterminant of the n n jacobian det[fx(x0;p0)] 0. Then there exists an m-dimensional open set P0 containingand one, and only one, vector-valued function g, defined on P0 and having values in Rn, such that

(1) gCon P0

(2) g(p0) = x0

(3) f(g(p);p) = 0 for every p in P0 .

When the jacobian is nonsingular at a point in S, the implicit function theorem indicates there exist solutionst are unique for all pP0. When a solution exists, the system is stable when all eigenvalues of fx(x0, p0) are

sitive or unstable, depending on whether there are non-positive eigenvalues of the jacobian fx(x0, p0). When solution exists at p0, the system is considered unstable. The vector p can be changed, usually reduced, fromuntil solutions exist. Singularity of the loadflow jacobian can be used to detect the point of voltage instability < p0 where instability is initiated. When the det[fx(x*, p*)] is zero (or the jacobian fx(x*, p*) is singular),

URE 63.9 Bifurcation sequence of saddle node (A), followed by singularity-induced bifurcation (B) produced by theV curve in Fig. 63.8.

2

the implicit function theorem does not provide any information but it may imply no solution x* = x(p*) existsat p* or there are multiple solutions x*i(p*).

Pr

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oximity Indices

number of indices have been developed to test for loadflow bifurcation when p is changed smoothly andtinuously in some direction n in Rm:

p = p0 + kn (63.36)

change in k until det[fx(x*,p*)] = 0. The indices come from tracking the miminum eigenvalue *i [IEEE,93] using

(63.37)

ere *i (k) = mini[i (k)] and ui(k) is a right eigenvector or alternatively the minimum singular value *i (k)tained [IEEE, 1993] using

(63.38)

ere (k) = diag[1(k), 2(k), L, n(k)] and *i (k) = mini [i(k)]. The singular values i(k) are the eigenvalues

(63.39)

d satisfy

(63.40)

(63.41)

ere vi(k) and wi(k) are the right and the left singular vectors of i(k) and are columns of matrices V(k) and(k) above.

The Q-V and P-V curves [IEEE, 1993] are particular scalar (m = 1) proximity measures where:

1. For a Q-V curve, the direction n is a unit vector in Eq. (63.36) where the voltage at a bus i is the onlynonzero element, k is the real valued negative number that starts at zero and decreases, and Qi(Vi) isthe reactive load that is added at bus i for each value of Vi. The curve Qi(Vi), shown in Fig. 63.8 is thereactive injection at bus i obtained by (1) changing bus type from a load (PQ) bus to a generator bus(PV) and (2) reducing the voltage Vi until Qi(Vi) reaches a minimum at Vimin with maximum added loadQi(Vimin) = Qimin 0. The value of (Vimin, Qimin) defines the minimum of the Q-V curve when Qi/Vi =0, that corresponds to the bifurcation point (x*, p*).

2. The P-V curve, shown in Fig. 63.6, can add active power load at a bus i or at several load busessimultaneously

p = p0 + knload (63.42)

f x k p k u k k kx i i i( ) ( )( ) ( ) = ( ) ( ), u

f x k p k f x k p k W k k V kxT

xT( ) ( )( ) ( ) ( )( ) = ( ) ( ) ( ), ,

f x k p k f x k p kx xT( ) ( )( ) ( ) ( )( ), ,

f x k p k v k k w kx i i i( ) ( )( ) ( ) = ( ) ( ), w k f x k p k k v ki

Tx i i

T( ) ( ) ( )( ) = ( ) ( ),

2

and pick up that power at several generators

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Allin forinsinsinsmefun000 by CRC Press LLC

g = g0 + kngen (63.43)

where n is made up of ngen and nload and both are participation vectors wherei ni = 1. A P-V curve can

also be computed for transfer power from one set of generators with generation g* to another set ofgenerators with generation g.

(63.44)

(63.45)

Note ngen, nload, n*, and n are unit vectors where one or more elements are nonzero and ini = 1. The

P-V curve plots voltage at some bus i for change in k = Psystem where it represents system power loadchange or k = Ptransfer represents the total power transfer change.

3. Optimization-based methods have been used to calculate Q-V and P-V curves [Reppen et al., 1991].These scalar optimization-based methods optimize performance index Qi to produce a Q-V curve withloadflow equality constraints

F(x, p, u) = 0 (63.46)

and inequality constraints on voltage controls u, states, x, and parameter p. These controls can includeunder load tap changer tap position, switchable shunt capacitor susceptance, and possibly generatorexcitation voltage control setpoints. The P-V curve computed by loadflow for varying k = Psystem or Ptransferhas all or most of these controls fixed rather than optimizing their values. The P-V curve computed viaan optimal power flow program would optimize k for a particular transfer or wheeling transaction withthe same loadflow model, same controls u, and inequality constraints on controls u, states, and param-eters used in computing the Q-V curve. The particular transfer or wheeling transaction is defined viaspecification of ngen, nload, n*, and n. In [Van Cutsem, 1991], a scalar optimization-based method wasused to maximize the reactive power stress when n in Eq. (63.36) was a unit vector with several nonzeroelements. This optimal power flow generalized Q-V curve allows added reactive load at several buses inthe participation factor normal direction, rather than just one as in a typical loadflow-based Q-V curvecalculation. The approach used in [Van Cutsem, 1991] eliminates the active power and phase anglerelationship, using active power generation as control, and imposes reactive power limits on the gener-ators. [Dobson et al., 1991] was first to develop a vector optimization-based method that optimizes thenormal direction vector n and the loading factor k in Eq. (63.36). The vector n is in the right eigenvectordirection wi of the bifurcating eigenvalue if the loadflow is continuously differentiable. This method andextensions [IEEE, 1993] computes the smallest change in the active or reactive load power and thus flowthat produces saddle node bifurcation. The proximity measure |p*p0| to saddle node bifurcation, wherep0 and p* represent the current load power and the bifurcation value of load power, respectively, wasfirst noted by Dobson [1991].

ltage Stability Security Assessment and Diagnosis

of these methods used [IEEE, 1993; Reppen, 1991; Dobson, 1991; and Van Cutsem, 1991] assess bifurcationa single mode due to continuous, smooth, scalar, or vector parameter variation. The methods are not practical assessing voltage instability or stability because the loadflow most often has no solution when voltagetability occurs, and all these methods require loadflow solutions to make any assessment of whether voltagetability occurs or has not occurred. This is true because these methods cannot determine whether voltagetability, algorithmic convergence difficulties, or round-off error is the reason for the lack of solution. Thesethods [IEEE, 1993; Reppen, 1991; Dobson, 1991; and Van Cutsem, 1991] can be viewed as based on implicitction theorem as long as the model is continuously differentiable. Implicit function theory and bifurcation

g g kn* * *= +0

g g kn= 0

2

theory assumptions are both violated for the case when a loadflow does solve after discontinuous parameterchange because the parameter variation is not continuous and smooth and the power system model may notbe coutinat sigfora fin mathaaba cbocomSec000 by CRC Press LLC

continuously differentiable at the point (x0, p0). The P-V curve, or Q-V curve, or eigenvalues and eigenvectorsld be computed and used to assess proximity to voltage instability after each equipment outage or discon-

uous parameter change when a loadflow solution exists to establish whether the solutions is stable or unstablevalues of p above p0. The computation of the P-V curve, Q-V curve, or eigenvalues and eigenvectors requiresnificant computation and is not practical for screening thousands of contingencies for voltage instability or assessing proximity to instability although they are used to assess stability and proximity to instability afterew selected contingencies. These methods also do not explicitly take into account the many discontinuitiesthe model and eigenvalues that occur for continuous parameter and discontinuous parameter changes. Inny cases, the eigenvalue changes due to discontinuities is virtually all the change that occurs in an eigenvaluet approaches instability [IEEE, 1993] and the above methods have particular difficulty in such cases. The

ove methods cannot assess the agents that lose voltage instability for a particular event and cannot diagnoseure when the loadflow has no solution for an equipment outage, wheeling or transaction combination, orth. These methods can provide a cure when a loadflow solution exists but its capabilities have not been

pared to the f Security Assessment and Diagnosis proposed cure. The Voltage Stabilityurity Assessment and Diagnosis (VSSAD) [Schlueter, 1998d] overcomes the above difficulties because:

1. It determines the number of discontinuities in any eigenvalue that have already occurred due to generatorPV to load PQ bus type changes that are associated with an eigenvalue compared to the total numberthat are needed to produce voltage instability when the eigenvalue becomes negative. The eigenvalue isassociated with a coherent bus group (voltage control area) [Schlueter, 1998a; f]. The subset of generatorsthat experience PV-PQ bus type changes (reactive reserve basin) for computing a Q-V curve at any busin that bus group are proven to capture the number of discontinuities in that eigenvalue [Schlueter,1998a; f]. An eigenvalue approximation for the agent, composed of the test voltage control area wherethe Q-V current is computed and its reactive reserve basin, is used to theoretically justify the definitionsof a voltage control area and the reactive reserve basin of an agent. The VSSAD agents are thus provento capture eigenvalue structure of the loadflow jacobian evaluated at any operating point (x0, p0). Thereactive reserve on generators in each voltage control area of a reactive reserve basin is proven to measureproximity to each of the remaining discontinuities in the eigenvalue required for bifurcation.

2. It can handle strictly discontinuous (equipment outage or large transfer or wheeling transaction changes)or continuous model or parameter change (load increase, transfer increases, and wheeling increases)whereas the above methods are restricted to continuous changes to assess stability or instability at apoint p0.

3. It can simultaneously and quickly assess proximity to voltage instability for all agents where each has abifurcating eigenvalue. Proximity to instability of any agent is measured by assessing (1) the percentageof voltage control areas containing generators in a reactive reserve basin with non-zero reserves, and(2) the percentage of base case reactive reserves remaining on reactive reserve basin voltage control areasthat have not yet exhausted reserves [Schlueter, 1998b; f].

4. It can assess the cure for instability for contingencies that do not have a solution. The cure can be either(1) adding needed reactive reserve on specific generators to obtain a solution that is voltage stable,(2) adding reactive supply resources needed in one or more agents, or (3) the reduction in generationand load in one or more agents or between one or more agents to obtain a solution and assure that itis a stable solution. These cures can be obtained in an automated fashion [Schlueter, 1998b; f]. Thediagnosis can also indicate if the lack of a solution is due to convergence difficulties or round-off errorif the diagnosis indicates the contingency combination does not produce sufficient network reactivelosses to cause instability or any agent.

5. It can provide operating constraints or security constraints on each agents reactive reserve basin reservesthat prevent voltage instability in an agent in a manner identical to how thermal constraints preventthermal overload on a branch and voltage constraints prevent bus voltage limit violation at a bus[Schlueter, 1998c; f].

2

6. The reactive reserve basin operating constraints allow optimization that assures that correcting onevoltage instability problem due to instability in one or more agents will not produce other voltage stability

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problems in the rest of the system [Schlueter, 1998c; f].7. The reactive reserve basin constraints after an equipment outage and operating change combination

allows optimization of transmission capacity that specifically corrects that particular equipment outageand transaction change induced voltage instability with minimum control change [Schlueter, 1998c; f].

8. It requires very little computation per contingency and can find multiple contingencies that cause voltageinstability by simulating only a small percentage of the possible multiple contingencies [Schlueter, 1998d].

nds of Loadflow Instability

o kinds of voltage instability have been associated with a loadflow model: loss of control voltage instabilityd clogging voltage instability [Schlueter, 1998d]. Loss of control voltage instability is caused by exhaustionreactive power supply that produces loss of voltage control on some of the generators or synchronousdensers. Loss of voltage control on these reactive supply devices implies both lack of any further reactiveply from these devices and loss of control of voltage that will increase network reactive losses that absorbortion of the flow of reactive power supply and prevent it from reaching the subregion needing that reactiveply. Loss of voltage control develops because of equipment outages (generator, transmission line, and

nsformer), operating condition changes (wheeling, interchange, and transfer transactions), and load/gener-on pattern changes. Loss of control voltage instability occurs in the subtransmission and transmission systemhlueter, 1998d]. It produces either saddle node or singularity-induced bifurcation in a differential algebraicdel. On the other hand, clogging develops because of increasing reactive power losses, and switching shuntacitors and tap changers reaching their limits. These network reactive losses, due to increasing magnetic

ld and shunt capacitive supply withdrawal, can completely block reactive power supply from reaching theregion with need [Schlueter, 1998d]. Clogging voltage instability can produce algebraic bifurcation in a

ferential algebraic model. The VSSAD method can diagnose whether the voltage instability occurs due togging or loss of control voltage instability for each equipment outage, transaction combination, or both thatve no solution.

eoretical Justification of the Diagnosis in VSSAD

bifurcation subsystem analysis has been developed that theoretically justifies the diagnosis performed byhlueter, 1997; 1998a; b; d; f]. This bifurcation subsystem analysis for a loadflow model attempts to break loadflow model into a subsystem model and external model

(63.47)

d to break the state x into two components where xs is the dimension of fs(xs, xe, p) = Oni.

The bifurcation occurs at p* = po + *o n when

(63.48)

f x x pf x x p

f x x p

O

Os es s e

e s e

n

n

, ,, ,

, ,( ) = ( )( )

=

1

2

xx

x

s

e=

( ) ( ) ( ) ( )

( )( )

= ( )

((

f

xx*,p *

f

xx*,p *

f

xx*,p *

f

xx*,p *

u p *

w p *p *

u p

w p

s

s

s

e

e

s

e

e

i

ii

i

i

2

The vector is the right eigenvector of eigenvalue i(p*) = 0 at bifurcation point p*. A bifurcation

sub

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system exists if two conditions hold:

(63.49)

(63.50)

The first condition is called the bifurcation subsystem condition and the second is called the geometricoupling condition. Finding a bifurcation subsystem for any bifurcation of the full system model requires

ding the combination of correct dimension, correct subset of equations, and correct subset of variables sucht the subsystem experiences the bifurcation (Eq. (63.49)) of the full system model (Eq. (63.48)) but alsoduces that bifurcation since the external model is completely uncoupled from the bifurcation subsystem in direction of the right eigenvector (Eq. (63.50)). The right eigenvector is an approximation of the centernifold at bifurcation, and the center manifold is the subsystem that actually experiences the bifurcation andbtained via a nonlinear transformation of the model. The expectation of finding a bifurcation subsystem

any loadflow bifurcation, noting the above requirements for identifying such a bifurcation subsystem, ist the difficulty in finding a bifurcation subsystem would be great even though one may exist for someurcations. The results in [Schlueter, 1998b; f] prove that one cannot only describe the bifurcation subsystemhere) for every clogging voltage instability and for every loss of control voltage instability, but also canoretically establish diagnostic information on when, proximity, and cure for a specific bifurcation in a specificurcation subsystem for clogging or for loss of control voltage instability [Schlueter, 1998b; f].The analysis establishes that:

1. The real power balance equations are a bifurcation subsystem for angle instability when the loadflow

model is decoupled ( and are assumed null) [Schlueter, 1998b; f].

2. The reactive power balance equations are a bifurcation subsystem for voltage instability when theloadflow model is assumed decoupled [Schlueter, 1998b; f].

3. A voltage control area is the bifurcation subsystem (agent) for clogging voltage instability. The agent isvulnerable to voltage instability for loss of generation in the agent, line outage in the agent boundary,or increased real and reactive flow across the agent boundary based on analysis of the lower boundapproximation of the eigenvalue associated with that agent. The cure for clogging voltage instability inthis agent is to reduce the real and reactive flow across the boundary of the agent [Schlueter, 1998b; f].

4. A voltage control area and its associated reactive reserve basin are the bifurcation subsystem (agent) forloss of control voltage instability. The agent is vulnerable to voltage instability for loss of generation inthe agent, line outages, transfer or wheeling transactions that reduce reactive reserve basin reserves basedon analysis of the lower bound approximation of the eigenvalue associated with that agent. The cure forvoltage instability in the agent is to add reactive reserves on the reactive reserve basin via capacitorinsertion, generator voltage setpoint changes on reactive reserve basin generators, or reverse tap positionchanges on underload tap changers [Schlueter, 1998b; f].

5. The percentage of reserves unexhausted in the reactive reserve basin is theoretically justified as a proximitymeasure for clogging instability in any clogging voltage instability agent. The percentage of voltage controlareas in a reactive reserve basin with unexhausted reactive reserve is theoretically justified as a proximitymeasure for each loss of control voltage instability agent [Schlueter, 1998b; f].

6. Exhaustion of reactive reserves in a particular locally most vulnerable agents reactive reserve basin causescascading exhaustion of reactive reserves and loss of control voltage instability in agents with successively

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ture Research

search is needed to:

1. Develop improved nonlinear dynamic load models that are valid at any particular instant and that arevalid when voltage decline is severe. The lack of accurate load models makes it difficult to accuratelysimulate the time behavior and/or assess the cause of the voltage instability. The lack of knowledge ofwhat constitutes an accurate load model makes accurate postmortem simulation of a particular blackouta process of making trial and error assumptions on the load model structure to obtain as accurate asimulation as possible that conforms with time records of the event. Accurate predictive simulation ofevents that have not occurred is very difficult [Taylor, et al. 1998].

2. Explain (a) why each specific cascading sequence of bifurcations inevitably occurs in a differentialalgebraic model, and (b) the dynamic signature associated with each bifurcation sequence. Work isunderway to explain why instability in generator and load dynamics can inevitably cause a singularity-induced bifurcation to occur. The time signature for singularity-induced bifurcation changes dependenceon why it occurs is discussed in [Schlueter, 1998e; Liu, 1998].

3. Extend bifurcation subsystem analysis to the differential algebraic model and link the bifurcation sub-system in a differential algebraic model, to those obtained in the loadflow model. The bifurcationsubsystems for different Hopf and saddle node bifurcations can explain why the subsystem experiencesinstability, as well as how to prevent instability as has been possible for bifurcation subsystems in thealgebraic model. Knowledge of bifurcation subsystems in the algebraic model may assist in identifyingbifurcation subsystems in the differential algebraic model.

4. Develop a protective or corrective control for voltage instability. A protective control would use con-straints on the current operating condition for contingencies predicted to cause voltage instability if theyoccurred. These constraints on the current operation would prevent voltage instability if and when thecontingency occurred. A corrective control would develop a control that correct the instability in thebifurcation subsystems experiencing instability only after the equipment outages or operating changespredicted to produce voltage instability have occurred. The implementation of the corrective controlrequires a regional 5-s updated data acquisition system and control implementation similar to that usedin Electricit de France and elsewhere in Europe.

efining Terms

wer system stability: The property of a power system that enables it to remain in a state of operatingequilibrium under normal operating conditions and to converge to another acceptable state of equilib-rium after being subjected to a disturbance. Instability occurs when the above is not true or when thesystem loses synchronism between generators and between generators and loads.

all signal stability: The ability of the power system to maintain synchronism under small disturbances[Kundur, 1994].

nsient stability: The ability of a power system to maintain synchronism for a severe transient disturbance[Kundur, 1994].

tor angle stability: The ability of the generators in a power system to remain in synchronism after a severetransient disturbance [Kundur, 1994].

ltage viability: The ability of a power system to maintain acceptable voltages at all buses in the systemafter being subjected to a disturbance. Loss of viability can occur if voltage at some bus or buses arebelow acceptable levels [Kundur, 1994]. Loss of viability is not voltage instability.

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Voltage stability: The ability of the combined generation and transmission system to supply load after adisturbance, increased load, or change in system conditions without an uncontrollable and progressive

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decrease in voltage [Kundur, 1994]. Loss of voltage instability may stem from the attempt of load dynamicsto restore power consumption beyond the capability of the combined transmission and generation system.Both small signal and transient voltage instability can occur.

ltage collapse: An instability that produces a cascading (1) loss of stability in subsystems, and/or (2) outageof equipment due to relaying actions.

urcation: A sudden change in system response from a smooth, continuous, slow change in parameters p.

ferences

. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley Publishing, 1974.Ben-Kilani, Bifurcation Subsystem Method and its Application to Diagnosis of Power System Bifurcations

Produced by Discontinuities, Ph.D. Dissertation, Michigan State University, August 1997.. Canizares, F.L. Alvarado, C.L. DeMarco, I. Dobson, and W.F. Long, Point of collapse methods applied to

AC/DC power system, IEEE Trans. on Power System, 7, 673683, 1992.obson and Liming Lu, Using an iterative method to compute a closest saddle node bifurcation in the load

power parameter space of an electric power system, in Proceedings of the Bulk Power System VoltagePhenomena. II. Voltage Stability and Security, Deep Creek Lake, MD, 1991.

. Guo and R.A. Schlueter, Identification of generic bifurcation and stability problems in a power systemdifferential algebraic model, IEEE Trans. on Power Systems, 9, 10321044, 1994.

E Working Group on Voltage Stability, Suggested Techniques for Voltage Stability Analysis, IEEE PowerEngineering Society Report, 93TH0620-5PWR, 1993.

undur, Power System Stability and Control, Power System Engineering Series, McGraw-Hill, 1994.Liu, Bifureation Dynamics as a Cause of Recent Voltage Collapse Problems on the WSCC System, Ph.D.

Dissertation, Michigan State University, East Lansing, MI, 1998.D. Reppen and R.R. Austria, Application of the optimal power flow to analysis of voltage collapse limited

power transfer, in Bulb Power System Voltage Phenomena. II. Voltage Stability and Security, August 1991,Deep Creek Lake, MD.

rvey of Voltage Collapse Phenomena: Summary of Interconnection Dynamics Task Forces Survey on VoltageCollapse Phenomena, Section III Incidents, North American Reliability Council Report, August, 1991.

. Sauer, C. Rajagopalan, B. Lesieutre, and M.A. Pai, Dynamic Aspects of voltage/power characteristics, IEEETrans. on Power Systems, 7, 9901000, 1992.

. Schlueter, K. Ben-Kilani, and U. Ahn, Impact of modeling accuracy on type, kind, and class of stabilityproblems in a power system model, Proceedings of the ECC & NSF International Workshop on Bulk PowerSystem Voltage Stability, Security and Control Phenomena-III, pp. 117156. August 1994.

. Schlueter, A structure based hierarchy for intelligent voltage stability control in planning, scheduling, andstabilizing power systems, Proceedings of the EPRI Conference on Future of Power Delivery in the 21stCentury, La Jolla, CA, November 1997.

. Schlueter and S. Liu, Justification of the voltage stability security assessment as an improved modal analysisprocedure, Proceedings of the Large Engineering System Conference on Power System Engineering,pp. 273279, June 1998.

. Schlueter, K. Ben-Kilani, and S. Liu, Justification of the voltage security assessment method using thebifurcation subsystem method, Proceedings of the Large Engineering System Conference on Power Systems,pp. 266-272, June 1998.

. Schlueter and S. Liu, A structure based hierarchy for intelligent voltage stability control in operationplanning, scheduling, and dispatching power systems, Proceedings of the Large Engineering System Con-ference on Power System Engineering, pp. 280285, June 1998.

. Schlueter, A voltage stability security assessment method, IEEE Trans. on Power Systems, 13, 1423-1438,1998.

2

R.A. Schlueter, S. Liu, K. Ben-Kilani, and I.-P. Hu, Static voltage instability in generator flux decay dynamicsas a cause of voltage collapse, accepted for publication in the Journal on Electric Power System Research,

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July 1998.Schlueter, S. Liu, and N. Alemadi, Intelligent Voltage Stability Assessment Diagnosis, and Control of Power

Systems Using a Modal Structure, Division of Engineering Research Technical Report, December 1998and distributed to attendees of Bulk Power System Dynamics and Control IV; Restructuring, August 2428,1998, Santorini, Greece.

Taylor, Power System Voltage Stability, Power System Engineering Series, McGraw-Hill, New York, 1994.Taylor, D. Kostorev, and W. Mittlestadt, Model validation for the August 10, 1996 WSCC outage, IEEE Winter

Meeting, paper PE-226-PWRS-0-12-1997.Van Cutsem, A method to compute reactive power margins with respect to voltage collapse, in IEEE Trans.

on Power Systems, 6, 145156, 1991.Van Cutsem and C. Vournas, Voltage stability of electric power systems, Power Electronic and Power System

Series, Kluwer Academic Publisher, Boston, MA, 1998.Venkatasubramanian, X. Jiang, H. Schattler, and J. Zaborszky, Current status of the taxonomy theory of large

power system dynamics, DAE systems with hard limits, Proceedings of the Bulk Power System Voltage;Phenomena-III Stability, Security and Control, pp. 15103, August 1994.

rther Reading

ere are several good books that discuss voltage stability. Kundur [1994] is the most complete in describing modeling required to perform voltage stability as well as some of the algebraic model-based methods foressing proximity to voltage instability. Van Cutsem and Vournas book [1998] provides the only dynamicaltems discussion of voltage instability and provides a picture of the various dynamics that play a role inducing voltage instability. Methods for analysis and simulation of the voltage instability dynamics aresented. This analysis and simulation is motivated by a thorough discussion of the network, generator, andd dynamics models and their impacts on voltage instability. Taylor [1994] provides a tutorial review of

ltage stability, the modeling needed, and simulation tools required and how they can be used to perform anning study on a particular utility or system.The IEEE Transactions on Power Systems is a reference for the most recent papers on voltage viability andltage instability problems. The Journal of Electric Power Systems Research and Journal on Electric Machinesd Power Systems also contain excellent papers on voltage instability.

ContentsPower Systems63.1Power System AnalysisIntroductionTypes of Power System AnalysesPower Flow AnalysisFault AnalysisTransient Stability AnalysisExtended Stability AnalysisSmall Signal Stability AnalysisTransient AnalysisOperational Analyses

The Power Flow ProblemFormulation of the Bus Admittance MatrixExample Formulation of the Power Flow EquationsP-V BusesBus ClassificationsGeneralized Power Flow DevelopmentThe Basic Power Flow Equations (PFE)

Solution MethodsThe Newton-Raphson MethodFast Decoupled Power Flow Solution

Component Power Flows

63.2Voltage InstabilityVoltage Stability OverviewVoltage Stability Models and Simulation ToolsKinds, Classes, and Agents of Voltage Instability ...Proximity to Voltage InstabilityProximity IndicesVoltage Stability Security Assessment and Diagnosi...Kinds of Loadflow InstabilityTheoretical Justification of the Diagnosis in VSSA...

Future Research