The Electrical Engineering Handbook || Power System Analysis

Power System Analysis

Mani Venka tasubramanian 7.1 and Kevin Tomsovic 7.2 School of Electrical Engineering

and Computer Science, 7.3 Washington State University, Pullman, Washington, USA 7.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 7.2.1 Modeling �9 7.2.2 Power Flow Analysis Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 7.3.1 Modeling �9 7.3.2 Power System Stability Analysis Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The interconnected power system is often referred to as the largest and most complex machine ever built by humankind. This may be hyperbole, but it does emphasize an inherent truth: there is a complex interdependency between different parts of the system. That is, events in geographically distant parts of the system may interact strongly and in unexpected ways. Power system analysis is concerned with understanding the operation of the system as a whole. Generally, the system is analyzed either under steady-state operating conditions or under dynamic conditions during disturbances.

Electric power is primarily transmitted as a three-phase signal. Three ac current currents are sent that are out of phase by 120 '~ but of equal magnitude. Such balanced currents sum to zero and, thus, obviate the need for a return line. If the voltages are balanced as well, then the total power transmitted will be constant in time, which is a more efficient use of equipment capacity. For large scale systems analysis, the assumption is usually made that the system is balanced. Each phase can be then analyzed independently, greatly simplifying computations. In the following paragraphs, the implicit assumption is that three-phase systems are being used.

7.2 Steady-State Analysis

In steady-state analysis, any transients from disturbances are assumed to have settled down, and the system state is assumed as unchanging. Specifically, system loads including transmis-

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sion system losses, are precisely matched with power generation so that the system frequency is constant (e.g., 60 Hz in North America). Perhaps the foremost concern during steady- state analysis is economic operation of the system; reliability is also important as the system must be operated to avoid outages should disturbances occur. The primary analysis tool for steady-state operation is the so-called power flow analysis, where the voltages and power flow through the system are determined. This analysis is used for both operation and planning studies and throughout the system at both the high and low transmission voltages.

The power system can be roughly separated into three sub- components: generation, transmission and distribution, and load. The transmission and distribution network consists of power transformers, transmission lines, capacitors, reactors, and protection devices. The vast majority of generation is produced by synchronous generators. Loads consist of a large number and a diverse assortment of devices, from home ap- pliances and lighting to heavy industrial equipment and so- phisticated electronics. As such, modeling the aggregate effect is a challenging problem in power system analysis. In the following sections, the appropriate models for these components in the steady-state are introduced.

7.2.1 Modeling Transformers A transformer is a device used to convert voltage levels in an alternating current (ac) circuit. The device has numerous uses in power systems. To begin, it is more efficient to transmit

761

762 M a n i Venkatasubramanian and Kevin Tomsovic

power at high voltages and low current than low voltages and high current. Conversely, lower voltages are safer and more economic for end use. Thus, transformers are used to step-up voltages from generators and then used to stepdown the voltage for end use. Another wide use of transformers is instru- mentation: sensitive equipment can be isolated from the high voltages and currents of the transmission system. Transformers may also be used as means of controlling real power flow by phase shifting.

Transformers function by the linkage of magnetic flux through a core of ferromagnetic material. Figure 7.1(A) illus- trates a magnetic core with a single winding. When a current I is supplied to the first set of windings, called the pr imary windings, a magnetic field, H, will develop, and magnetic flux, 4', will flow in the core. Amp&e's law relates the enclosed current to the magnetic field encountered on a closed path. If H is constant throughout the path, then:

HI - NI , (7.1)

where l is the path length through the core; N is the number of turns of the winding on the core so that N I is the enclosed current by the path referred to as the magnetomotive force (mmf).

The magnetic field is related to the magnetic flux by the properties of the material, specifically the permeability. If a linear relationship is assumed (i.e., neglecting hysteresis and saturation effects), then the flux density, B, or the flux, +, is as follows:

NI NI B - I X H - tx i o r + - IxA I ' (7.2)

where A is the cross-sectional area of the core. This relationship between the flux flow in the core and the mmf is called the reluctance, R, of the core so that:

R + - NI . (7.3)

Now, if a second set of windings, the secondary windings, is wrapped around the core as shown in Figure 7.1(B), the two currents will be linked by magnetic induction. Assuming that no flux flows outside the core, then the two windings will see the exact same flux, 4'. Because the two windings also see the same core reluctance, the two mmfs are identical:

i . ~ 0 ~

(A) Flux Flowin9 Throu9 h

4 (B) Flux as kinked

FIGURE 7.1 (A) The Flux Flows Through Core from First Winding; (B) The Flux is Linked to a Second Set of Windings.

N111 -- N212. (7.4)

If the flux + or, equivalently, the current I are changing in time, then according to Faraday's law, a voltage will be induced. Assuming this ideal transformer has no losses, the power input will be the same as the power output, so:

V~ I~ - V212, (7.5)

where Vl and V2 are the primary and secondary voltages, respectively. Substituting (7.4) and rearranging shows:

V2 N2 = (7.6)

Vl N1

Thus, the voltage gain in an ideal transformer is simply the ratio of the primary and secondary windings' turns. A practical transformer experiences several non-ideal effects. Specifically, these include nonzero winding resistance, finite permeability of the core, eddy currents that flow in the core, hysteresis (the effect arising from the energy required to reorient the magnetic dipoles as the magnetic polarity changes), and magnetic saturation. For steady-state studies of the large system, linear circuit models are desired. These effects are typically modeled as a combination of series and parallel impedances in the following way:

�9 Series impedances: Because the transformer core has a finite permeability, some of the magnetic flux flows outside the core. This leakage flux will not link the primary and secondary windings. Thus, the voltage at the input sees not only the voltage that links the primary and secondary windings but also a voltage drop caused by this leakage inductance. Similarly, the finite winding resistance causes an additional voltage drop to be seen at the terminals.

�9 Shunt impedances: Finite permeability implies nonzero core reluctance and requires current to magnetize the core (i.e., a nonzero mmf). This difference between the primary and secondary mmfs can be modeled as a shunt inductance. Hysteresis and eddy currents lead to energy losses in the core that can be approximately modeled by a shunt resistor. Saturation is an important nonlinear effect that results in additional losses and the creation of odd order harmonics in the current and voltage signals. Because in steady-state system analysis only the 60-Hz component of the currents and voltages is considered, saturation effects are typically ignored.

An equivalent circuit for the transformer model just described is shown in Figure 7.2.

The main difficulty with the model in Figure 7.2 as it now stands concerns the ideal transformer component. Carrying this component around in the calculations creates unnecessary complexity. Further, from an engineering point of view, the

7 Power System Analysis 763

1 t II FIGURE 7.2 Transformer Circuit Model

voltages and currents in the system are most easily seen relative to their rated values. Thus, most system analysis is done on a normalization called the per unit system. In the per unit system, a system power base is established, and the rated voltages at each point in the network are determined. All system variables are then given relative to this value. These base quantities for the currents can be found as:

SB I i ~ - V R ' (7.7)

and for impedances:

V/~ Zsr = = (7.8)

Is3 S#r

This normalization has the great added advantage of redu- cing the need to represent the ideal transformer in the circuit. One must simply keep track of the nominal base voltage in each part of the network. In this way, the equivalent transformer model is as given in Figure 7.3. Note that phase shifting and off-nominal transformer ratios result in asymmetric circuits and require some additional manipulation in the per unit framework. Those details are omitted here for brevity.

Transmission Line Parameters

As mentioned previously, electric power is transmitted in three phases. This accounts for the common site of three lines or for dual circuits of six lines seen strung between transmission towers. Typically, a high-voltage transmission line has several feet of spacing between the three conductors. The conductors are stranded wire for improved mechanical as well as electrical properties. If the currents are expected to be large, several conductors may be strung per phase. This improves cooling

FIGURE 7.3 System

Simplified Transformer Circuit Model Under Per Unit

x

FIGURE 7.4 Infinite Transmission Line

compared to using one large conductor. This geometry is important as it impacts the electrical properties of the line.

As current flows in each conductor, a magnetic field develops. Adjacent lines then may induce voltages in nearby conductors through mutual induction (as seen for transformers, only now with coupling that is not as tight). This interaction largely determines the inductance seen by the respective phase currents. To understand this phenomenon, consider a single line of radius r and infinite length with some current flow, /, as sketched in Figure 7.4. Similar to what is done for the transformer development, apply Amp~re's law to characterize the magnetic field. The magnetic field at some distance x from the line can be found by assuming that the field is constant at all points equal distance from the line. The closed path is then a circle with circumference 2-rrx, which gives:

2"rrxH - I o r H . ( 7 . 9 ) 27rx

If x is less than the line radius, the closed path will not link all of the current. Assuming an equal distribution of current throughout the wire:

x 2 Ix 2 " r r x H - l--v or H - (7 10)

27rr 2 �9 1-

Once again, the flux density is determined by the permeability of the material, which in this case is either the conductor itself if x < r or that of free space for x > r. Then, the flux relationship is the following:

Ix d+ - ~ ~ dx. (7.11)

For x > r, the flux linked up to some radial distance R per unit of length is simply:

R

~kexternal -- ,1" r

I b~oI R P~o dx - In - . (7.12)

2-rrx 2-rr r

For x < r, only the enclosed current will be linked. Con- tinuing with the even distribution of current assumption, the flux linked is as written here:

764 M a n i V e n k a t a s u b r a m a n i a n a n d Kev in Tomsov ic

i Ix 3 I ~kinternal ~ ],l, c 2,rrr4 dx -- b~c 8'rr

0

(7.13)

For simplicity, assume the permeability of the conductor is that of free space, and then the total flux linkage is as follows:

= l % I + i a " 0 I l n R _ p , 0 I l n R . . . . . .

8"rr 2"rr r 2"rr re- l~ 4 " (7.14)

Typically, re -1/4 is written as r'. Consider a three-phase transmission line of phase currents Ia, Ib, and Ic, with each line spaced equally by the distance D. Flux from each of the currents will link with each of the other conductors. The flux linkage for phase a out to some point R a far distance away from the conductors is approximately:

X~ - ~ I~ In 77 + lb In ~ + Ic In . (7.15)

Assuming balanced currents (i.e., 1~ + I~, + Ic = 0) and recalling that inductance is simply the ratio of flux linkage to current, the series inductance in phase a per unit length of line will be the following:

i . , , - 1*{--2 In D-. (7.16) 2'n r'

In practice, the phase conductors may not be equally spaced as they are in Figure 7.5. This results in unbalanced conditions due to the imbalance in mutual inductance. High-voltage transmission lines with such a layout can be transposed so that, on average, the distance between phases is equal canceling out the imbalance. The equivalent distance of separation between phases can then be found as the geometric mean of this spacing. Similarly, if several conductors are used per phase, an equivalent conductor radius can be found as the geometric mean.

Transmission lines also exhibit capacitive effects. That is, whenever a voltage is applied to a pair of conductors separated by a nonconducting mcdium, charge accumulates, which leads to capacitance. Similar to the previous development for inductance, the capacitance can be determined based on Gauss's

FIGURE 7.5

D

End View of Equally Spaced Phase Conductors

law. For a point P at a distance x from a conductor with charge q, the electric flux density D is"

D - - q . (7.17) 2'n'x

Assuming a homogeneous medium, the electric field density E is related to D by the permitivity ~ of the dielectric, which, in this case, will be assumed to be that of free space:

E = q (7.18) 2"rrg0x

Integrating E over some path (a radial path is chosen for simplicity) yields the voltage difference between the two end points:

R2

I q V~2--, 2xrg0x Ri

d x - q In R1- (7.19) 2"rrc0 R2

Now consider a three-phase transmission line again with each line spaced equally by the distance D. Superposition holds so that the voltage arising from each of the charges can be added. To find the voltage from phase to ground arising from each of the conductors, assume a balanced system with q,, + q~, + qc = 0 and a neutral system located at some far distance R from phase a:

2'rrc0 q,, In - + r qt, ln ~ + q,.ln ---q,,ln--.2,rrc0 r (7.20)

Now since the capacitance is the ratio of charge to voltage, the capacitance from phase a to ground per unit length of line will be:

Ca,z - qa 2"rrCo -- V,,,, = In D/r" (7.21)

If the conductors are not evenly spaced, transposition results in an equivalent geometric mean distance, and using bundled conductors per phase can also be accommodated by using a geometric mean.

Finally, conductors have finite resistances that depend on the temperature, the frequency of the current, the conductor material, and other such factors. For most systems analysis problems, these can be based on values provided by manufac- turers or from tables for commonly used conductors and typical ambient conditions.

Transmiss ion Line Circuit Models

Transmission lines may be classified according to their total length. If the line is around 50 miles or less, a so-called short line, capacitance can be neglected, and the series inductance


R L

FIGURE 7.6 Short Line Model

and resistance can be modeled as lumped parameters. Figure 7.6 depicts the short line model per phase. The series resistance and inductance are simply found by calculating the per unit distance parameters times the line length; therefore, at 60 Hz for line length l, the line impedance is the following:

Z --/~l + jl20'n'Ll -- R + iX. (7.22)

For lines longer than 50 miles, up to around 150 miles, capacitance can no longer be neglected. A reasonable circuit model is to simply split the total capacitance evenly with each half represented as a shunt capacitor at each end of the line. This is depicted as the -rr-circuit model in Figure 7.7. Again, the total capacitance is simply the per unit distance capacitance times the line length:

Y - j l20'rr(),,,,l - jB. (7.23)

For line lengths longer than 150 miles, the lumped parameter model may not provide sufficient accuracy. To see this, note that at 60 Hz for a low-loss line, the wavelength is around 3000 miles. Thus, a 150-mile line begins to cover a significant portion of the wave, and the well-known wave equations must be used. The relationship between voltage and current at a point x (i.e., distance along the line) to the receiving end voltage, Vr, and current, lr, is seen through the following equations:

V(x) - Vrcosh "yx + IrZcsinh "yx.

v~ I(x') -- Lcosh yx +--~- sinh yx.

Z.c

(7.24)

(7.25)

The Zc - X/~/y is the characteristic impedance of the line, and y - v/2y is the propagation constant. It would be useful,

r/2 T

R X

L Y/2

FIGURE 7.7 Medium Line Model

and it turns out to be possible, to continue to use the w-model for the transmission line and simply modify the circuit parameters to represent the distributed parameter effects. The relationship between the sending end voltage, Vs, and current, Is, to the receiving end voltage and current in a w-model can be found as:

Vs - Vr(1 + Y~Z2 ) + IrZ. (7.26)

Is-- VrY(I+Y--~Z4 ) + I r ( l + - ~ ) . (7.27)

Now equating 7.24 and 7.25 for a line of length l to equations 7.26 and 7.27 and solving shows the equivalent shunt admittance and series impedance for a long line as:

Z ' - Zcsinh y l - Z ~sinh yl (7.28) yl

Y' 1 tanh yl Y tanh yl/2 2 =Zc 2 = 2 yl/2 (7.29)

In these equations, the prime indicates the modified circuit values arising from a long line.

Generators

Three-phase synchronous generators produce the overwhelm- ing majority of electricity in modern power systems. Syn- chronous machines operate by applying a dc excitation to a rotor that, when mechanically rotated, induces a voltage in the armature windings due to changing flux linkage. The per phase flux for a balanced connection can be written as:

X - Kf !t sin 0,,,, (7.30)

where If is the field current, 0,,, is the angle of the rotor relative to the armature, and Kf is a constant that depends on the number of windings and the physical properties of the machine. The machine may have several poles so that the armature will "see" multiple rotations for each turn of the rotor. So, for example, a four pole machine appears electrically to be rotating twice as fast as two pole machine. For a machine rotating at to,, radions per second with p poles, the electric frequency is as follows:

- P (7.31) COs -- CO,,, 2 '

with m, as the desired synchronous frequency. If the machine is rotated at a constant'speed Faraday's law indicates that the induced voltage can be written as:

d~ V = dt = Kflfco, sin(cost + 0o). (7.32)

766 Mani Venkatasubramanian and Kevin Tomsovic

Xs

FIGURE 7.8 Simple Synchronous Generator Model

If a load is applied to the armature windings, then current will flow and the armature flux will link with the field. This effec- tively puts a mechanical load on the rotor, and power input must be matched to this load to maintain the desired constant frequency. Some of the armature flux "leaks" and does not link with the field. In addition, there are winding resistive losses, but those are commonly neglected. The circuit model shown in Figure 7.8 is a good representation for the synchronous generator in the steady-state. Note that most generators are operated at some fixed terminal voltage with a constant power output. Thus, for steady-state studies, the generator is often referred to as a PV bus since the terminal node has fixed power P and voltage V.

Loads Modeling power system loads remains a difficult problem. The large number of different devices that could be connected to the network at any given time renders precise modeling in- tractable. Broadly speaking, loads may vary with voltage and frequency. In the steady-state, frequency is constant, so the only concern is voltage. For most steady-state analysis, a fixed (i.e., constant over an allowable w)ltage range) power consumption model can be used. Still, some analysis requires consideration of voltage effects to be useful, and then the traditional exponential model can be used to represent real power consumption P and reactive power consumption Q as:

P = Po V". (7.33)

Q = Q,) v I'. (7.34)

In these equations, the voltage V is normalized to some rated voltage. The exponents a and b can be 0, 1, or 2 where they could represent constant power, current, or impedance loads, respectively. Alternatively, they can represent composite loads with a generally ranging between 0.5 and 1.8 and b ranging between 1.5 and 6.0.

7.2.2 Power Flow Analysis

Power flow equations represent the fundamental balancing of power as it flows from the generators to the loads through the

transmission network. Both real and reactive power flows play equally important roles in determining the power flow properties of the system. Power flow studies are among the most significant computational studies carried out in power system planning and operations in the industry. Power flow equations allow the computation of the bus voltage magnitudes and their phase angles as well as the transmission line current magnitudes. In actual system operation, both the voltage and current magnitudes need to be maintained within strict tolerances for meeting consumer power quality requirements and for preventing overheating of the transmission lines, respectively. The difficulty in computing the power flow solutions arises from the fact that the equations are inherently nonlinear because of the balancing of power quantities. Moreover, the large size of the power network implies that power flow studies involve solving a very large number of simultaneous nonlinear equations. Fortunately, the sparse interconnected nature of the power network reflects itself in the computational process, facilitating the computational algorithms.

In this section, we first study a simple power flow problem to gain insight into the nonlinear nature of the power flow equations. We then formulate the power flow problem for the large power system. A classical power flow solution method based oll the Gauss-Seidel algorithm is studied. The popular Newton-Raphson algorithm, which is the most commonly used power flow method in the industry today, is introduced. We then briefly consider the fast decoupled power flow algorithm, which is a heuristic method that has proved quite effective for quick power flow computations. Finally, we will discuss the dc power flow solution that is a highly simplified algorithm for computing approximate linear solutions of the power flow problem and is becoming widely used for electricity market calculations.

Simple Example of a Power Flow Problem Let us consider a single generator delivering the load P + jQ

through the transmission line with the reactance x. The generator bus voltage is assumed to be at the rated voltage, and it is at 1 per unit (pu). The generator bus angle is defined as the phasor reference, and hence, the generator bus voltage phase angle is set to be zero. The load bus voltage has magnitude V and phase angle 8. Because the line has been assumed to be lossless, note that the generator real power output must be equal to the real

1/o

j x

FIGURE 7.9 A Simple Power System

P+jQ


power load P. However, the reactive power output of the generator will be the sum of the reactive load Q and the reactive power "consumed" by the transmission line reactance x.

Let us write down the power flow equations for this problem. Given a loading condition P + jQ, we want to solve for the unknown variables, namely, the bus voltage magnitude V and the phase angle 8. For simplicity, we will assume that the load is at unity power factor of Q = 0. The line current phasor I from the generator bus to the load bus is easily calculated as:

1 ZO - V / 8 I -- . (7.35) j~

Next, the complex power S delivered to the load bus can be calculated as:

S - 111" = VZ(8 + "rr/2) _ V 2 / ( 7 r / 2 ) . (7.36) X X

Therefore, we get the real and reactive power balance equations:

- V sin 8 I V 2 _~_ V cos 8 P - and Q - (7.37)

X X

After setting Q - 0 in equation 7.37, we can simplify equation 7.37 into a quadratic equation in V 2 as follows:

V 4 I V 2 _~_ x 2 p2 _ 0 . (7.38)

Therefore, given any real power load P, the corresponding power flow solution for the bus voltage V can be solved from equation 7.38. We note that for nominal load values, there are two solutions for the bus voltage V,, and they are the positive roots of V 2 in the next equation:

1 -]- v/1 --4x 2 p2 V 2 - - (7.39)

2

Equation 7.39 implies that there exist two power flow solutions for load values P < Pmax where Pmax = 1/(2X), and there exist no power-flow solutions for P > Pmax. A qualitative plot of the power flow solutions for the bus voltage V in terms of different real power loads P is shown in Figure 7.10.

From the plot and from the analysis thus far, we can make the following observations:

1. The dependence of the bus voltage Von the load P is very much nonlinear. It has been possible for us to compute the power flow solutions analytically for this simple system. In the large power system with hun- dreds of generators delivering power to thousands of loads, we have to solve for thousands of bus voltages and their phase angles from large coupled sets of non-

m

F ~max

P

FIGURE 7.10 Qualitative Plot of the Power-Flow Solutions

linear power flow equations, and the computation is a nontrivial task.

2. Multiple power flow solutions can exist for a specified loading condition. In Figure 7.10, there exist two solutions for any load P < Pmax. Among the two solutions, the solution on the upper locus with voltage V near 1 pu is considered the nominal solution. For the solutions on the lower locus, the bus voltage V may be unacceptably low for normal operation. The lower voltage solution also requires higher line current to deliver the specified load P, and the line current values can become unacceptably high. In general, for any specified loading condition, we would like to locate the power flow solution that has the most acceptable values of voltages and currents among the multiple power flow solutions. In this example of a single generator delivering power to a single load, there exist two power flow solutions. In a large power system, there may exist a very large number of possible power flow solutions.

3. Once the bus voltage V has been computed from equation 7.39, the bus voltage phase angle 8 can be computed from equation 7.37. Then, the line current phasor I can be solved from equation 7.35. Specifically, we would like to ensure that the magnitude of the line current I stays below the thermal limit of the transmission line for preventing potential damage to the expen- sive transmission line.

4. Power flow solutions may fail to exist at high loading conditions, such as when P > Pmax in Figure 7.10. The loading value Pmax beyond which power flow solutions do not exist is called the static limit in the power literature. Because power flow solutions denote the steady-state operating conditions in our formulation, lack of power flow solutions implies that it is not possible to transfer power from the generator to the load in a steady-state fashion, and the dynamic interactions of the generators and the loads become significant. Operating the power system at loading conditions


SGi

sL, l

t, >

S~

FIGURE 7.11 Complex Power Balance at Bus i

beyond the static limit may lead to catastrophic failure of the system.

Power flow Problem Formulation

In this subsection, we will construct the power flow equations in a structured manner using the admittance matrix Ybus

representation of the transmission network. The admittance matrix Ybus is assumed to be known for the system under consideration. Let us first look at the complex power balance at any bus, say bus i, in the network.

The power balance equation is given by:

S i - ViI~ -- S ( ; , - Sl.,. (7.40)

Let us denote the vector of bus voltages as _Vbu ~ and the vector of bus injection currents as /bu~. By definition, the admittance matrix Ybu., provides the relationship {bL,~ = _Ybu~_Vbu.~. Suppose the ith or jth entry Y!i of the Ybu~ matrix has the magnitude Y!i and the phase Y!i" Then, we can simplify the current injection I i as :

Ii - ~ Y(iVi - Z YijVi/(SJ + "Vii). (7.41) j i

Then, combining equations 7.40 and 7.41, we get the complex power balance equations for the network as:

s~ - s , . - s~., - ~ Y, V~Vi/(a~ - a j - ~ j ) .

J (7.42)

Taking the real and imaginary parts of the complex equation 7.42 gives us the real and reactive power flow equations for the network:

Pi - - P(;, - PL, - - Z Yij Vi Vi cos (8i - 8.i - yij). J

Qi - Q(; , - QL, - Z Yij Vi V) sin (8i - - 8 j - - " y / j ) .

J

(7.43)

(7.44)

Generally speaking, our objective in this section is to solve for the bus voltage magnitudes Vi and the phase angles 8 i

when the power generations and loads are specified. For a power system with N buses, there are 2N number of power flow equations. At each bus, there are six variables: PGi, QGi, PLi, QLi, Vi, and 8i. Depending on the nature of the bus, four of these variables will be specified at each bus, leaving two unknown variables at each bus. We will end up with 2N unknown variables related by 2N equations 7.43 and 7.44, and our aim in the rest of this section is to develop algorithms for solving this problem.

Let us consider a purely load bus first, that is, with PGi -- QGi - O. In this case, the loads PLi and QLi a r e assumed to be known either from measurements or from load estimates, and the bus voltage variables Vi and 8i are the unknown variables. Purely load busses with no generation support are called PQ busses in power flow studies because both real- power injection Pi and reactive power Qi have been specified at these busses.

Typically, every generator in the system consists of two types of internal controls, one for maintaining the real power output of the generator and the other for regulating the bus voltage magnitude. In power flow studies, we usually assume that both these control mechanisms are operating perfectly and so the real power output PGi and Vi are maintained at their specified values. Again, the load variables PI, i and Qi.i are also assumed to be known. This leaves the generator reactive output Q(;i and the voltage phase angle 8i as the two unknown variables for the bus. In terms of injections, the real power injection Pi and the bus voltage Vi are then the specified variables; thus, the generator busses are normally denoted PV busses in power flow studies.

In reality, the generator voltage control for keeping the bus voltage magnitude at a specified value becomes inactive when the control is pushed to the extremes, such as when the reactive output of the generator becomes either too high or too low. This voltage control limitation of the generator can be represented in power flow studies by keeping track of the reactive output Q(;i. When the reactive generation Q(;i becomes larger than a prespecified maximum value of Q(;i, max or goes lower than a prespecified minimum value Q(;i, rain, the reactive output is assumed to be fixed at the limiting value Q(;i, max o r Q(;i, rain, respectively, and the voltage control is disabled in the formulation; that is, the reactive power Q(;i becomes a known variable, either a t QGi, max o r QGi, rain, and the voltage Vi then becomes the unknown variable for bus i. In power flow terminology, we say that the generator at bus /ha s "reached its reactive limits" and, hence, bus i has changed from a PV bus to a PQ bus. Owing to space limitations, we will not discuss generator reactive limits in any more detail in this section.

In addition to PQ busses and PV busses, we also need to introduce the notion of a slack bus in the power flow formulation. Note that power conservation demands that the real power generated from all the generators in the network must equal the sum of the total real power loads and the line losses on the transmission network:


Z P G , - - Z P L i A V Z ~ _ a P l o s s e s i j . (7 .45) i i i j

The line losses associated with any transmission line in turn depend on the line resistance and the line current magnitude. As stated earlier, one of the main objectives of power flow studies is to compute the line currents, and as such, the line current values are not known at the beginning of a power flow computation. Therefore, we do not know the actual values for the line losses in the transmission network. Looking at equation 7.45, we need to assume that at least of one of the variables PGi or PLi should be a free variable for satisfying the real power conservation. Traditionally, we assume that one of the generations is a "slack" variable, and such a generator bus is denoted the slack bus. At the slack bus, we specify both the voltage Vi and the angle 8i. The power injections Pi and Qi are the unknown variables. Again, by tradition, we set the voltage at slack bus to be the rated voltage or at 1 pu and the phase angle to be at zero.

Like in standard textbooks, slack bus is defined in this section to be the first bus in the network with 1/1 = 1 and the angle 81 = 0. Assuming the number of generators to be N(;, the busses 2 through N(; + 1 are set to be the PV busses. The remaining buses N~; 4-2 through N are then the PQ busses.

Gauss-Seidel Algorithm Let us consider a set of simultaneous linear equations of the form Ax - b, where A is an n • n matrix and xand bare n • 1 vectors. Clearly, there exists a unique solution to the problem when the matrix A is invertible, and the solution is given by x - A_ lb. When the matrix size is very large, it may not be possible to compute the inverse of the matrix _A for finding the solution; there exist other numerical techniques. Gauss-Seidel algorithm is one such classical algorithm that tries to arrive at the solution x - A- lb iteratively by starting from an approximate initial condition of x ~ The iteration for the solution x k+l from the previous iterate __x k proceeds as follows.

/ \ xp -~l = l [ b i - Z a i j x f + l - ' ~ - ~ a i j x ~ } for i = 1,2 . . . . . n (7.46)

aii \ .i<i " j>i

Here, aq denotes the ith or jth entry of the matrix A as usual. It can be shown that the iterative solution _x k converges to the exact solution A-lb for any initial condition _x ~ provided the matrix A satisfies certain "diagonal dominance" properties. The details are limited here to save space, and they can be found in standard numerical analysis textbooks.

In the previous section, we formulated the power flow problem as a set of simultaneous nonlinear equations of 7.42 and as such, it is not obvious how the Gauss-Seidel algorithm can be applied for solving these equations. The trick here is to visualize the power balance equations to be arising from the

network admittance equations ----Ybus-Y-Vbus---/bus" The matrix Ybus takes over the role of the matrix _A in the linear equations. We will be solving for the bus voltage vector Vbu s. The current injections/bus are not known per se. The current injections are in fact dependent on the bus voltages. As we see next, they can also be computed iteratively from the power injections Si by using the relationship Ii - Si*/Vi*. For a PQ bus, the injection Si is a specified variable and, hence, is known. For PV busses, only the real power injection Pi is known, while the reactive injection Qi is evaluated first using the latest estimate of bus voltages __Vbu s.

An outline of the Gauss-Seidel algorithm for solving the power flow equations of 7.42 is presented next. Let us start with an initial condition for the bus voltages 0 ___Vbu s, and we would like to compute the iterate V k+l from the previous ~ b u s iterate Vkus . Recall that bus 1 is a slack bus and, hence, V ~ - 1/_0 for all iterations. Also, busses 2 through Nc + 1 are PV busses; therefore we need to keep V/k at specified values Vi, specified -- V~ for all iterations for the PV busses.

Gauss-Seidel Iterations

�9 Slack bus ( i - 1)" V~ +1 - 1/0 �9 PV busses (i = 2 . . . . . Nr + 1):

(1) Compute the reactive power generation at bus i.

Q +l _ Z si - - �9 j n Y ij/ j<i

+ Z Y i j V ) V ~ s i n ( 8 ) - 8 ) - Y O ) " .i > i

(2) Update the bus voltage phasor V) + ~.

V~ " 1 ( Pi - jO~+' - - - Z - Z

i" .t p .i

(3) Normalize the magnitude V~ +l to be V~ ).

V~ +1 = V~+I V'].

V~ +l

�9 PQ busses (i = N(; + 2 . . . . . N): (4) Update the bus voltage phasor V~ +l.

V!+ ' = l__~ (Vi - jQ~i _ ~-'~ YiiV,~ ' _ Z YiiV,). Y ;; v I:" i%-7 ~ ~;

(7.47)

. (7.48)

(7.49)

(7.50)

At the end of the (k + 1)th iteration, we have the updated vk+l values of the bus voltages --busVk+l" The values of--bus can be

compared with the previous set of values Vku.~ to check whether the solution has converged.

For the power flow problems, the Gauss-Seidel algorithm has been known to converge even from poor initial conditions, which is one of its main strengths. The algorithm is typically used for "fiat starts" when all the initial voltage magnitudes are set to be at their rated values (V ~ = 1 pu for all the PV busses), and the bus voltage angles are set to be zero (8 0 - 0 for all

770 M a n i V e n k a t a s u b r a m a n i a n a n d K e v i n T o m s o v i c

buses). The relative simplicity of the computations involved in each iteration step in equations 7.46 to 7.49 implies that the algorithm is very fast to implement. On the other hand, the error convergence rate is typically linear in the sense that the ratios of the error norm from one iteration to the previous iteration tend to be constant. Therefore, while the Gauss-Seidel algorithm can converge to the proximity of the actual solution in tens of iterations, it typically takes a large number of iterations to get to an accurate solution estimate.

Newton-Raphson Algorithm Unlike the Gauss-Seidel algorithm, which was originally developed for solving simultaneous linear equations, the Newton-Raphson (NR) algorithm is specifically designed for solving nonlinear equations. The algorithm proceeds iteratively by linearizing the nonlinear equations into linear equations at each step and by solving the linearized equations exactly.

Suppose we want to solve the nonlinear equations _F(x) - 0, where x is a n x 1 vector and where F" :J~" ~ :J~" is a smooth nonlinear function. We have been given an initial condition x ~ Then, for computing the estimate _x k~-i from x k, we first lin- earize the functions F(__x) at x k as follows:

af 0 - E(x) ~ F(x k) + v -

r (x - xk). (7.51)

x 1,

The solution to the linearized equations 7.17 and 7.18 is defined as the iteration estimate xk+~:

1

_ - _ - ] E(x ) (7.52)

ables among the bus voltage magnitudes Vi and angles ~i first. That is, we define the vector x as consisting of all the PV and PQ bus angles and all the PQ bus voltages. PV bus voltages are known and, hence, they are not included in x:

X - ( ~ 2 . . . . . ~ N ( ; + I , ~)N(;+2 . . . . . ~)N, VN(;+2 . . . . . VN) T. (7.53)

The corresponding power flow equations are as follows:

f(_x) -

/ P2 -- p2(__x) '~

PN,;+I -- PN,;+I (X__) PN,;+2 -- pN,;+2(X_)

PN -- pN(X_.)

QN,;4 2 -- qf,, ~ 2 (x)

QN - q f ( x_) 7

P2 - Z Y2, j V2 Vj cos(~2 - ~j - ~2,1) j

. . . . . . . . .

] )N, , , l - - ~ - ~ Y N , , , I , IVN, IVICoS(~N, , , I - -~] - - "~N, , , I , j ) I

PN,,+2 - ~--~ Y N , , + 2 , 1 V N + 2 V I c ~ -- ~)I -- ~]N, ~ 2, i ) i

i

QN,,,2 ~_,Y,x',,,2,1V,x'.2V, s in (~ , \ ' , ,+2- ~ i - "Y,\', ,2,,) I

Q,v -- ~-~ Y:,,z, , V,v V, sin ( ~N -- ~, -- ~I;v, ) i ' j

(7.54)

Note that the linearization of equation 7.51 will be a good approximation if the estimate x k is close to the true solution of x* since _F(x*) - 0. The NR algorithm stated in equation 7.52 can be proved to converge to the true solution x* when the initial condition x ~ is sufficiently close to x*. On the other hand, for initial conditions away from x*, the approximation of equation 7.51 becomes poorly justified, and the iterations can quickly diverge away from x*. When the iterations converge, owing to the linearized nature of the algorithm, the norm of the error decreases to zero in a "quadratic" fashion. Roughly speaking, the ratios of the error norm from one iteration to the square of the error norm in the previous iteration tend to be a constant. An example would be that the error norms decrease from 0.1 in one iteration, to 0.01 in the next iteration, to 0.0001 in the following iteration. Therefore, given good initial conditions, the NR algorithm can typically get to an accurate solution estimate within a few iterations.

Let us apply the NR algorithm for solving the power flow equations 7.43 and 7.44. We will solve for the unknown vari-

The entries of the F function ill equation 7.54 are the differences between the specified power injections and the computed power injections from the current power flow solutions, and these are usually denoted as the real and reactive power mismatches at the different busses. In the power flow problem, we want to find a solution that makes the power mismatches in equation 7.54 equal zero.

Suppose an initial condition x ~ has been specified. Then, the NR algorithm for solving the power flow equations 7.54 proceeds iteratively as follows.

N e w t o n - R a p h s o n I t e ra t i ons

( 1 ) Compute the power mismatches F(X k) for step k from equation 7.54. If the mismatches are within desired tolerance values, the iterations stop.

(2) Compute the power flow Jacobian:

j k _SF (7.55) -- ~ X X~ _


Owing to the nice structure of the equations in 7.54, explicit formulas can be derived for the entries of the Jacobian matrix, and the Jacobian for step k can be evaluated by substituting the current values of_x k into these formulas.

(3) Compute the correction factors A_x k from equation 7.52 by solving a set of simultaneous linear equations"

]kAxk -- --F__(xk). (7.56)

The Jacobian matrix _]k is extremely sparse even for very large power systems, which facilitates the solution of A__x k in equation 7.55.

(4) Evaluate x k+~ from _x k by adding the correction factors Axk:

x k+l -- _x k + A_x k. (7.57)

As compared with the Gauss-Seidel algorithm, each iteration step in the NR algorithm is computationally much more intensive because of (a) evaluating the Jacobian and (b) solving the linear equations 7.55. On the other hand, the error convergence rate of the NR algorithm is spectacularly faster, and hence, the NR algorithm requires much fewer iterations to reach comparable solution accuracies. In usual practice, the Gauss-Seidel algorithm is used only for flat starts with poorly known initial conditions. In most other situations, the NR algorithm is the preferred choice.

Fast Decoupled Power Flow Algorithm Both the Gauss-Seidel algorithm and the Newton-Raphson algorithm are general methods for solving linear and nonlinear equations respectively, and they were tailored toward solving the power flow problem. On the other hand, we study a specific method for power systems in this section called the fast decoupled power flow algorithm, which is a heuristic method that is derived by exploiting specific properties for a power system.

The fast decoupled power flow algorithm is essentially a highly simplified and approximated version of the Newton- Raphson algorithm of the previous subsection. We recall that the NR iteration steps are computation intensive because of evaluating I k and solving the ]acobian equation 7.55. In this subsection, we will proceed to simplify this by replacing the iteration specific ]k with a constant matrix.

It is a well-known property of power systems that variations in the bus voltage magnitudes mostly affect the reactive power injections under nominal operating conditions. Similarly, the variations in the bus voltage phase angles mostly influence the real power injections. By idealizing this property, we assume

�9 ~p ~q that all the Jacobian entries of the form ~ and ~ are all

�9 I i

identically zero. Next, ff we assume that the bus voltage magnitudes are all close to one pu, and the voltage phase angle differences on the two ends of any transmission line are all close to zero, the NR algorithm greatly simplifies to the fast decoupled algorithm stated in equation 7.58.

Let us split the power flow state vector _x in equation 7.53 into the voltage magnitude _Vand angle 8 counterparts:

fi __ (8 2 . . . . . 8N(;+l ' 8N(;+2 . . . . . 8N ) r , .V_V--(VN~+2 . . . . . VN) T.

(7.58)

Similarly, we separate the real and reactive power mismatches in equation 7.54:

P2 -- p2(_x)

Ap = P,,;+~ - Pu~;+l (_x) and A Q = -- PN~;+2 -- pN~;+2(X)

PN -- pN(X)

/ QN~;+2 -- qNG+2(X) I

QN -- qN(X__)

(7.59)

Suppose initial conditions for the voltage magnitude vector V ~ and angle vector 8 ~ have been specified. The fast decoupled algorithm then proceeds iteratively as follows.

Fast Decoupled Iterations

(1) Compute the real and reactive power mismatches AP(_x k) and A Q(xk). If the mismatches are within desirable tolerance, the iterations end.

(2) Normalize the mismatches by dividing each entry by its respective bus voltage magnitude:

A i, ~ =

ae~/v~ . . . . . . . . .

aG,,,,/v ~ N ~ , + I

aG,,,,/v~,, ~, . . . . . . . . .

aG/v~

AQN.~2/VN~,~2

~nd aft ~ . . . . . . . . . . . (7.60) aQ~lV~

(3) Solve for the voltage magnitude and angle correction factors_ AV k and A_~ k by using the constant matrices and/3, which are extracted from the bus admittance matrix Ybus"

BA_~ k = A~ k and ~AV k -- A0 k. (7.61)

Dcfine B q - imag(Yij) . The matrices B and B are constructed as follows:

B2, 2 �9 �9 �9 B2, N

B - - . . . . . . . . . and

BN, 2 . . . BN, N

BN~;+2, N~;+2 . . . B ~

BN, N(;+2 . . .

BN,; + " "N N "1

(7.62)

(4) Update the voltage magnitude and angle vectors.

8_k+~ _ 8_ k + Aft k, gk+ ~ _ g k + ZXV k. (7.63)


By using the constant matrices B and B in equation 7.60, the time taken for evaluating one iteration for the fast decoupled algorithm is considerably less than that of the Newton- Raphson algorithm. In repeated power flow runs_ of the same power system, the inverses of the matrices B and B can directly be stored, which enables the implementation to be very fast. However, the convergence speed of the fast decoupled algorithm is not quadratic, and it takes considerably more iterations to converge to an accurate power flow solution. Fast decoupled algorithm is used in applications where quick approximate estimates of power flow solutions are required.

Direct Current Power Flow Algorithm

A further simplification of the fast decoupled algorithm is the highly approximate dc power flow algorithm, which com- pletely transforms the nonlinear power flow equations into linear equations by using drastic assumptions. In addition to the assumptions used in deriving the fast decoupled method, we also assume that all the voltage magnitudes are at 1 pu and all the transmission lines are lossless.

With these assumptions, the voltage correction factors A V k become irrelevant in equation 7.60. Moreover, the angle variables can be solved explicitly by the linear equations:

BS_- P, (7.64)

where P is the vector of bus power injections. The resulting solution for the bus voltage phase angles is called the dc power flow model. It gives approximate values for the phase angles across the power system. The phase angles can be used to approximate the real power flow on any transmission line by dividing the phase angle difference between the two ends of the transmission lines by the line reactance. The advantage of the dc power flow model is its extreme simplicity in finding a power flow solution. The limitations of the solution need to be kept in mind, however, in light of the drastic assumptions that were used to simplify the nonlinear power flow equations into linear equations.

7.3 Dynamic Analysis

The power system in practice is constantly undergoing changes because of such factors as changing loads, planned outages of equipment for maintenance or other disturbances (e.g., equipment failures, line faults, lightning strikes etc), or any number of other events that cause outages. During disturbances, the precise balance between generation and load is not maintained. These disturbances may lead to oscillations, and the system must be able to dampen these and reach a viable steady-state operating condition. Extremely fast electromagnetic transients, such as those that arise from lightning strikes or switching actions, are not considered from a system point of view, hence, the network models introduced in the previous

sections are still valid. Dynamic models for generator units models still must be introduced to understand system response. Load dynamics are also important, particularly from large induction motors, but such details are beyond the scope here. This section focuses on the transient response of the system over fractions of a second to oscillations over several seconds and then up to slow dynamics that occur with voltage problems over several minutes.

7.3.1 M o d e l i n g

Electric Generators

To understand modeling generators for dynamic analysis, more details on the physical construction are needed. Most generators are three-phase synchronous machines, which means they are designed to operate at a constant frequency. The machine rotor is driven mechanically by the prime mover governing system to control power output and speed. Synchronous machine rotors can be classified as either cylindrical or salient pole. Cylindrical rotors have an even air gap at all points between the stator, i.e., the stationary part of the machine, and the rotor. This construction is used for machines that rotate at high speed, typically steam-driven generators. Steam generators generally are two-pole or four-pole machines and so rotate, in North America, at 1800 RPM or 3600 RPM to produce the desired 60 Hz signal. Hydro generators, conversely, may have numerous pole pairs, as it is more efficient to drive them at lower speeds. These generators have a salient pole construction that leads to a variable air gap between the stator and the rotor.

For modeling purposes, the pole construction is important because it represents the mechanical speed of rotations and also because the transfer of power from the rotor to the stator through mutual inductance depends on the size of the air gap (i.e., the reluctance of the air gap and, thus, the effective coupling). During disturbances, these variations are most evi- dent, and the effective circuit inductances must be modeled accurately. In modern power system modeling for dynamic analysis, a rotating frame of reference is chosen to represent these effects. The circuit equations are then written in terms of direct and quadrature axes. For simplicity, the more involved rotating fl-ame of reference model.,, are llot developed here but instead a simple model approximating the variable inductances is presented. To begin, there are primarily two sets of windings of concern:

�9 Armature windings: The windings on the stator are referred to as the armature windings. The armature is the source from which the power generated will be drawn. A voltage is induced in the armature from the rotation of the field generated by currents on the rotor. For purposes of modeling, the self-inductance, including leakage, and mutual inductance between phases of the armature windings describe the circuit performance as current flows from the terminals. The armature windings


ix;

E'/_g_

FIGURE 7.12 A Simple Model of a Synchronous Generator Arma- ture for Dynamic Studies

are distributed in slots around the stator to produce a high-quality sinusoidal signal. Winding resistance may also be included but is small and often neglected in simple studies. Field windings: These windings reside on the rotor and provide the primary excitation for the machine. The windings are supplied with a dc current so that with rotation, they induce a voltage in the armature windings. The current is controlled by the exciter to provide the desired voltage across the armature windings. This current must be constrained to avoid overheating of the windings; the modeling of these limits and excitation of the control circuit are critical for analysis.

In addition, in salient pole machines, there are solid con- ducting bars in the pole faces called damper windings, or amortisseurs windings, that influence the effective inductance. These windings serve to damp out higher frequency oscillations. Similar effects are also seen in the cylindrical case through eddy current flows in the rotor even though damper windings may not be present. These details are not pursued further here.

The armature will be modeled as a voltage controlled by the rotor current behind a simple transient reactance as illustrated in Figure 7.12. The induced voltage E'/(3, referred to as the voltage behind the reactance, is connected to the generator terminal through the transient reactance if, t, where the sub- script indicates a direct axis quantity and the superscript a transient quantity.

To control the terminal voltage, the field current is controlled by the exciter, which in turn varies the voltage behind the reactance. A simple high gain exciter with limits can be

modeled as in Figure 7.13. For the field circuit, there is a time constant associated with the winding inductance and resistance. Together, these describe the basic electromagnetic time constants for a simple generator model. There is often a sup- plementary stabilization control on large units referred to as a power system stabilizer (PSS). The interested reader is referred to the literature.

Mechanically, the generator is similar to a mass spring system with some frictional damping. If there is a net imbalance of torque acting on the rotor, then, neglecting damping, the machine will accelerate according to the well-known swing equations:

d03m 1 = do,, = -;(Tin - Te), (7.65)

dt ]

where ] is the rotational inertia, T,, is the mechanical torque input, and Te the electromagnetic torque that arises from producing an electric power output. Recall that power equals the torque times angular velocity, so multiplying both sides of equation 7.65 by 03,,, yields:

1 &,, ,m, , , - - ; ( P , , , - P,.). (7.66)

1

Typically, engineers normalize the machine inertia based on the machine rating and use the per unit inertia constant H as:

1 1(03;;,) 2 H = - ~ , (7.67)

2 $I~

0 the synchronous mechanical speed. If in addition the with o3,,, speed and the real powers are expressed on a per unit basis, then substituting equation 7.67 into equation 7.66 gives:

do - 03, ~r~ ( P,, , _ P , . ) , (7.68) 2 H ( P , , , - P,.) or d ) - - -~ -

where & is now the acceleration relative to synchronous speed. The electrical power output is a function of the rotor angle t3, and so we need to write the simple differential equation that relates rotor angle to speed:

6 = co - 03s. (7.69)

Terminal voltage

Reference voltage

Exciter Field voltage limiter

FIGURE 7.13 Basic Components of Simple Voltage Regulation and Excitation System

774 M a n i V e n k a t a s u b r a m a n i a n a n d K e v i n T o m s o v i c

Thus, the mechanical equations can be expressed as a second- order dynamic system. In the next section, the analysis of this in a connected system is discussed.

7.3.2 Power System Stability Analysis

The power system is a nonlinear dynamic system consisting of generators, loads, and control devices, which are coupled together by the transmission network. The dynamic equations of the generator have been discussed in the previous section. The interactions of the generators with the loads and the control devices can result in diverse nonlinear phenomena. Specifi- cally, we are interested in understanding whether the system response can return to an acceptable operating condition following disturbances. Normally, we distinguish between two types of disturbances in the power system context:

Minor disturbances such as normal random load fluctu- ations, are denoted small-disturbances, and the ability of the power system to damp out such small disturbances is called small-signal stability. In the next section, we will learn that small-signal stability can be understood by computing the eigenvalues of the system Jacobian matrix that is evaluated at an equilibrium point. Major disturbances, such as transmission line trippings and generator outages, are denoted large disturbances, and the capability of the power system to return to acceptable operating conditions following a large disturbance is called the transient stability. In the following paragraphs, we will also study techniques for verifying the transient stability of a power system following a specific large disturbance.

Small-Signal Stability Consider a nonlinear system described by the following ordi- nary differential equation:

d x dt = f(x_),_ x_ E :J{", f:_ :~" --+ :J{", f_ is smooth. (7.70)

Suppose that _x* is an equilibrium point for the system. That is, f(x*) - 0. We define the lacobian matrix A for the equilibrium - b.l X * to be the matrix of partial derivatives ~ evaluated a t_ . Then,

classical analysis in nonlinear dynamical system theory tells us that the equilibrium _x* is locally stable if all the eigenvalues of the matrix A have negative real parts. Recall that the eigenvalues of a matrix A are defined as the solutions ki of the polynomial matrix characteristic equation det ( k / , , - A) - 0 , w h e r e / , denotes the n - X - n identity matrix. In addition, the equilibrium will be locally unstable if any one of the eigenvalues has a positive real part.

In the power system context, the concept of local stability is known as the small-signal stability or the small-disturbance stability. We will look at a simple power system example next for studying the concept in more detail.

jx' ,

E' A

jx

,/0 ()

FIGURE 7.14 A Simple Power System

A single generator that is connected to an ideal generator bus through a transmission line is shown in Figure 7.14. The ideal generator maintains its bus voltage at 1 pu irrespective of the external dynamics and is referred to as an infinite bus. It also defines the reference angle for this power system, and the angle is set at zero. The other generator is represented by the classical machine model with a voltage source of E'/O connected to the generator terminal through the transient reactance xit. In the classical machine model, the internal induced voltage E' is assumed to remain constant, and the rotor angle fl follows the second-order swing equations as in equation 7.68 but here includes a damping component P,i:

2H.. - - 0 - P , , , - P , . - P ,~ . ( 7 . 7 1 ) tO s

This can be rewritten into the standard form of:

-- .,, " (7.72) do ~ ( P,,, - P,, - P,~) "

For the power system in Figure 7.14, the electrical power output P,. is easily calculated as:

E' sin 0 P" - x' = Pm~,x sin 0, (7.73)

,t -+- x

where Pm,,x - E'/(x, ' t + x) is a constant. The remaining term, the damping power P,t. is usually defined as P,t - l ) ( m ' - m~),

where D is known as the damping constant for the generator, which is a positive constant. Substituting the entries for P,. and Pa into equation 7.72, we get the dynamic equations for the generator as:

-- ~o, " (7.74) & 2-~ (P,,, - Pm~,x sin O - D(co - cos)) "

The equilibrium points for equation 7.74 can be easily solved by setting the derivatives to be zero. The (0", cos) r is an equilibrium point for this system if P,,, = Pm.~x sin 0". The equilibrium points can be identified visually by plotting Pe

and Pm as shown in Figure 7.15. The intersection of the


FIGURE 7.15 Plot of Pe and P,,,

constant line P,,, with the c u r v e Pmax sin 0 highlighted by the black marks depict the two possible equilibrium points when P,, < Pm~x- Let us denote the equilibrium point between 0 and "rr/2 by 0s and the other equilibrium between "rr/2 and "rr by 0u.

We will show that the equilibrium (0s, COs)r is small-signal stable, while the other equilibrium (0,,, COs) T is small-signal unstable. For assessing the small-signal stability, we need to evaluate the Jacobian matrix A at the respective equilibrium point. The lacobian is first derived as:

oY-_ r 0 l L (I), ~_x ~ ( - P m . , x , cos0) ~ " ( - D )

(7.75)

O,)q

K2 + 2HmS DR + ~ Pm,,x cos 0, - 0.

The eigenvalues of the system matrix A can easily be computed at the equilibrium point (0s, co,) 1 as the solutions of the second-order polynomial:

(7.76)

t

jXd

Since 0, by definition lies between 0 and -rr/2, cos Os is positive. Therefore, the two roots of the characteristic equation 7.76 have negative real parts, and the equilibrium (0s, cos) T is small- signal stable. Therefore, the system will return to the equilibrium condition following any small perturbations away from this equilibrium point. On the other hand, when the equilibrium point (0,,, ms)T is considered, note that cos 0, is negative because 0,, lies between -rr/2 and "rr. Then, the characteristic equation 7.76 will have one positive real eigenvalue and one negative real eigenvalue when 0s is replaced by 0,,. Therefore, the equilibrium (0,, co,)r is small-signal unstable.

Normally, the power system is operated only at the stable equilibrium point (0s, cos) T. The unstable equilibrium point (Ou, COs) T also plays an important role in determining the large disturbance response of the system that we study in the next subsection. Unlike this simple system, assessing the small- signal stability properties of a realistic large power system is an extremely challenging task computationally.

Transient Stability Assume that the power system is operating at a small-signal stable equilibrium point. Suppose the system is suddenly sub- ject to a large disturbance such as a line fault. Then, the power system protective relays will detect the fault, and they will isolate the faulty portion of the network by possibly tripping some lines. The occurrence of fault and the subsequent line trippings will cause the system response to move away from the equilibrium condition. After the fault has been cleared, the ability of the system to return to nominal equilibrium condition is called the transient stability for that fault scenario. We will study a transient stability example for the simple system in Figure 7.16.

Suppose we want to study the occurrence of a solid three- phase-to-ground fault at the middle point of the lower transmission line in Figure 7.16. We usually assume that the system is operating at a nominal equilibrium point before the fault occurs. For this prefault system, the effective transmission line reactance is x/2, since there are two transmission lines in parallel each with reactance x. The dynamic equations for the prefault system are then given by:

to, ppre sin O - D(m - cos)) & ~-~ (P,,, - m~x (7.77)

where PI~(~;:• = E'/x~t + x/2. The equilibrium points can be solved by setting the derivatives to zero in equation 7.77. Let us denote the stable equilibrium by '"sr~l're' ms ) "1" where 0~pre is the equilibrium solution of the rotor angle between 0 and rr/2.

Next, let us say that the fault occurs at time t = 0. When the solid fault is present at the middle of the lower transmission line, the system in Figure 7.14 changes to the configuration shown in Figure 7.17.

The computation of the generator electrical power output P,, for the fault-on system in Figure 7.15 requires a little more work. Looking from the generator terminal bus, the effective

x _ l /0, which is 1/3/0. Thevenin voltage is 1/O[x/( ~ + x)] Next, the effective Thevenin reactance is the parallel equivalent of reactance x and the reactance x/2, which is x/3. Therefore, the electrical power Pe during the fault-on period is given by:

jx

1/o )

jx "'C6-"

FIGURE 7.16 Prefault Power System


1/0

jx

jX' d

FIGURE 7.17 Fault-On Power System

p~ault fault E'/3 - - P~ax sin 0 -- - 7 ~ sin 0. (7.78)

xj + x/3

The dynamic equations for the fault-on system are then given by:

to~ fault - Pickax sin 0 D(to - d) T~ (P,,, - to.<)) (7.79)

Let us assume that the relays clear the fault at time t = t, by opening the lower transmission line at both the sending and receiving end. Then, the system configuration becomes as shown in Figure 7.14, and the dynamic equations for the postfault system are given by:

m, post ,b ~7 ( P,,, - P.nax sin 0 - l)(to - to~ ) ) (7.80)

where P~'~I,~- E'/xd'+ x. If the system settles down to the stable equilibrium of the postfault system of equation 7.80 after the fault is cleared, we say that the system is transient stable for the fault under study. To verify this numerically, we proceed as follows.

Transient Stability Study by Numerical Integration Apre (1) The system is at (,,~ , to~) before the fault occurs.

Therefore, we start the simulation with (0, ( o ) 1 = (0~ 'r~', to~)s at time t - 0.

(2) During the fault-on period, the system dynamics is governed by equation 7.79, and the fault-on period is from time t - 0 through time t - re. Therefore, we integrate the equation 7.79 starting from the initial condition (A p~ to~)'r at time t - 0 for a time period ~'S ' -

from t -- 0 to t -- to. Let us denote the system state at the end of fault-on period numerical integration at t ime t - tr by the clearing state (0,., to,.)'r.

(3) At time t - t~, the system equations change to the postfault equation 7.80. Therefore, we integrate the equation 7.80 starting from the clearing state (0c, to,-)T at t ime t -- t~ for a period of several seconds to assess whether the system response converges to the

stable equil ibrium (f~p~ T ~ , COs) �9 If the postfault response settles down to the nominal equilibrium, we can determine the system to be transient stable for the disturbance. If the postfault system response diverges away, the system is not transient stable.

The steps involved in the numerical integration procedure for a realistic large system are similar to the outline above for the simple system. The models are of very large dimensions for real- size power systems. Hence, the formulat ion of the prefault, fault-on, and postfault models becomes a nontrivial task, and the numerical integration becomes highly t ime-consuming.

Equal Area Criterion

To verify the transient stability, it is possible to derive analytical conditions for the power system in Figure 7.15. For simplicity, we assume that the damping constant D = 0 and that the fault clearing is instantaneous (t~ = 0). We recall that the dynamics of the prefault system are described by:

2 H . . pro - - 0 -- P,,, - Pm~,x sin 0, (7.81 ) r s

where P~,'(;,~- E' /xd '+ x/2. Similarly, the equations for the postfault system are described by:

2 H . . post - - 0 - P,,, - Pm,x sin 0, (7.82) tos

111ilX '.)pro IO,M where pt .... t _ E'/xd' + x. Clearly, it follows that t,,1,,x > P~il,,x because x/2 < x. Intuitively, more power can be transferred in the prefault configuration with two parallel lines compared to one transmission line present in the postfault configuration.

As stated earlier, the system is operating at the prefault equilibrium (0P r<', to~) "1 before the fault occurs where s in0prc_ p,,,/ p,c 0~,rc s Pl~,~,x, and lies between 0 and 'n'/2. When the fault is cleared instantaneously at t ime t = O, we would like to

pre O.l~)'l' will settle know whether the transient starting from (6< , ~posl T to the postfault system equilibrium (-,s , to,~) or whether it

will diverge away. Convergence to (0~ '''~t, to<)s or divergence implies transient stability or instability, respectively.

Let us start with the analysis. First, we note that 0 p'-< < 0 t .... 5 5 because pro ~(,st P~ax > P~,a,," For visualization, let us plot P,. and P,,, for the postfault system as shown in Figure 7.18.

[4prc T The dynamics of the transient starting from ("s , tos) are governed by the second-order equation:

2H .. 2H ~ - 0 - - - t b - P,,, - P,P~';'x t sin 0. (7.83) tos tos

Therefore, the sign of the term P,,,- PP~ii'tx sin 0 determines whether the speed derivative d) is positive or negative. Inspect- ing the plot Figure 7.18, we conclude that the rotor frequency co increases whenever the rotor angle 0 is below the P,,, line because then P , , , - PPm~ t sin 0 will be positive. Similarly, the


mmax p~ sin 0

OsP~ ~ OuP~ P., /

Os pre

FIGURE 7.18 Power-Angle Curve for the Postfault System

mmax p~ sin 0

OsP~ ~ OuP~

Aa

s pre

FIGURE 7.19 Acceleration Area for the Postfault System

rotor speed decreases in value when the angle 0 is above the P,,, post line because then P , , - Pm.,x sin 0 will be negative.

Let us recall that the postfault system response starts at (0~ 're, tos) "z at time t - 0. As noted earlier, the initial rotor angle 0~ 're lies beneath the P,,, line in Figure 7.18, hence, the speed to increases from the initial value to = m.~ at time t = 0 as soon as the fault is cleared. When to increases above to~, the rotor angle starts to increase from 0 p'~ because 0 - to - to, for 5 the machine dynamics. Therefore, the rotor angle moves up on the power-angle curve in Figure 7.18, and the rotor speed keeps increasing until the rotor angle reaches the value 0~ '''~t at the intersection point with the P,,, line in Figure 7.18. Let us say that the angle 0 takes time tl seconds to increase from the initial value 0!~ 're at time t - 0 to the value 0~ '''~t. During this time period from t = 0 to t - - t ~ , the speed to has increased from tos to some higher value, such as tol. The dynamic state of the postfault system at time t - t~ is then given by .(0 p~ , tol) T. Note that even though the rotor angle 0 equals 0~s '''~t at time t = tl, the system is not in the equilibrium condition because the speed to equals to, at time t~, and to~ is greater than the equilibrium speed value to.,.

By construction, the speed value col is defined as follows by the dynamics of equatioll 7.83:

t- tl ()!,,,~t

- t sin O] dO col - t o ~ = ( o d t _--3:_ ( P , , , - ) t . . . . t A,, �9 . . 2H

t -- 0 0{""

(7.84)

where A,, is the shaded area shown in Figure 7.19. Because the rotor acceleration iJ has been positive during this time period from t = 0 to t = tl, the angle has been accelerating in the area shown, and hence, the area A, is called the acceleration area.

When the transient reaches ({.)post T .-s ,col) at time t - t~,the rotor angle keeps on increasing because 0 = ml -cos > 0 at time t~. However, for time t > t~, the rotor angle moves above the P,,, line in Figure 7.18, and hence, the derivative of speed

becomes negative. That is, the speed m starts to decrease from the value col as the time increases from t = tl. Only after the speed to has decreased below the synchronous speed cos can the rotor angle start to decrease. Until then, the rotor angle will keep increasing, and the rotor speed keeps decreasing as time increases from t = t~.

Looking at the power-angle plot in Figure 7.19, the rotor angle stays above the P,,, line only up to the unstable equilib-

post rium value 0,, . If the rotor angle were to increase above 0~ '''st then the speed derivative 4~ becomes positive again, and the speed will start to increase. In this case, there is no scope for speed co to decrease to the synchronous speed cos, and transient instability results. Therefore, for any chance of transient stability, and for response to settle down around (0p~ , ms ) T, we need the speed to to decrease below m~ before the rotor angle

post. reaches the critical value 0,, Graphically, this implies that the maximum deceleration area ,l A 'l~ax shown in Figure 7.20 needs to be larger than the acceleration area A,, shown in Figure 7.19.

Pmax p~ sin e

Pm s 1 7 6 up~

Adma x

Os pre

,,,.. r 0

FIGURE 7.20 Maximum Deceleration Area for the Postfault System

778 M a n i Venkatasubramanian and Kevin Tomsovic

When A~ ax > Aa, as the rotor angle decelerates past 0sP~ from time t = h, the deceleration area will become equal to the accelaration area at some intermediate value of 0 between 0 p~ and 0 p~ at some time of t - t2. For time t > t2, the speed to falls below to,, and the rotor angle 0 starts to decrease back toward 0 p~ The alternating scenarios of rotor angle acceleration and deceleration will continue before the angle swings are damped out eventually by the rotor damping effects that have been ignored thus far. Therefore, we say that the system is transient stable whenever amax "'~d > Aa.

On the contrary, when A~ax< Aa, the rotor speed stays above to,, the rotor angle reaches 0 p~ Then, the rotor speed starts to increase away from to, monotonously. In this case, the rotor speed never recovers below to,, and the rotor angle continuously keeps increasing. The transient diverges away, thus resulting in transient instability.

The analytical criterion presented in this section for the simple system can be extended to multimachine models using Lyapunov theory and based on the concepts of energy functions. Development of analytical criteria for checking the transient stability of large representative dynamic models remains a research area. Numerical integration procedures outlined in the previous sections are commonly used by the power industry for studying the transient stability properties of large power systems.

7.4 Conclusion

This chapter has introduced the readers to the basic concepts in power system analysis, namely modeling issues, power flow studies, and dynamic stability analysis. The concepts have been illustrated on simple power system representations. In real power systems, power-flow studies and system stability studies are routinely carried out for enduring the reliability and secur- ity of the electric grid separation. While the basic concepts here have been summarized in this chapter on simple examples, the real power systems are large-scale, nonlinear systems. The large interconnected nature of electric networks makes the computation aspects highly challenging. We have highlighted some of these issues in this section, and the readers are encouraged to refer to advanced power system analysis textbooks for additional details.

References Bergen, A., and Vittal, V. (2000). Power systems analysis, 2 ''d Ed. New

Jersey: Prentice Hall. Chapman, S. (2002). Electric machinery and power system fundamen-

tals. New York: McGraw Hill. Glover, 1., and Sarma, M. (1994). Power system analysis and design, 2 '''t

Ed. Boston: PWD Publishing. Grainger, 1.1., and Stephenson, W.D. (1994). I)ower system analysis.

New York: McGraw Hill. Kundur, P. (1994). Power system stability and control. New York:

McGraw Hill. Saadat, H. (2002). Power system analysis, 2 '''t Ed. New York: McGraw

Hill.

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