Power System Analysis K. Tomsovic V. Venkatasubramanian School of Electrical Engineering and Computer Science

Washington State University Pullman, WA

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: the complex interdependency of 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. That is, three AC current currents are sent that are out of phase by 120o 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, 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, the implicit assumption is that three phase systems are being used. 2. Steady-State Analysis In steady state analysis, any transients from disturbances are assumed to have settled down and the system state is unchanging. Specifically, system load, including transmission 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 is economic operation of the system but 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 is determined. This analysis is used for both operation and planning studies and throughout the system at both the high transmission voltages and the lower distribution system voltages. The power system can be roughly separated into three subcomponents: 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 of, and a diverse assortment, of devices, from home appliances and lighting to heavy industrial equipment to sophisticated electronics. As such, modeling the aggregate effect is a challenging problem in power system analysis. In the

following, the appropriate models for these components in the steady-state are introduced. 2.a Modeling 2.a.1 Transformers A transformer is a device used to convert voltage levels in an AC circuit. They have numerous uses in power systems. To begin, it is more efficient to transmit power at high voltages and low current than low voltage and high current. Conversely, lower voltages are safer and more economic for end use. Thus, transformers are used to step-up voltages from the generators and then used to step-down the voltage for end use. Another wide use of transformers is for instrumentation so that 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 2.1a illustrates a magnetic core with a single winding. When a current I is supplied to the first set of windings, called the primary windings, a magnetic field, H, will develop and magnetic flux, , will flow in the core. Ampres Law relates the enclosed current to the magnetic field encountered on a closed path. If we assume that H is constant throughout the path then

NIHl = (2.1) where l is the path length through the core and N is the number of windings on the core so that NI is the enclosed current by the path referred to as the magnetomotive force (mmf).

I I Figure 2.1 a) flux flows through core from first winding, b) flux is linked to a second set

of windings The magnetic field is related to the magnetic flux by the properties of the material, specifically, the permeability. If we assume a linear relationship, i.e., neglecting hysteresis and saturation effects, then the flux density B or the flux is

lNIHB == or

lNIA = (2.2)

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

NIR = (2.3) Now, if a second set of windings, the secondary windings, is wrapped around the core as shown in Figure 2.1b, the two currents will be linked by magnetic induction. Assuming that no flux flows outside the core, then the two windings will be see the exact same flux, . Since, the two windings also see the same core reluctance, the two mmfs are identical, i.e.,

2211 ININ = (2.4) If the flux is changing in time, or equivalently the current I, then according to Faradays 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

2211 IVIV = (2.5) where V1 and V2 are the primary and secondary voltages, respectively. Substituting (2.4) and rearranging shows

1

2

1

2

NN

VV

= (2.6)

Thus, the voltage gain in an ideal transformer is simply the ratio of the number of primary and secondary windings. A practical transformer experiences several non-ideal effects. Specifically, these include non-zero winding resistance, finite permeability of the core, eddy currents that flow within 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, we desire linear circuit models. These non-ideal effects are typically modeled as a combination of series and parallel impedances in the following way:

Series impedances Since 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.

Shunt impedances - Finite permeability implies non-zero core reluctance and so requires current to magnetize the core (i.e., a non-zero 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 non-linear effect that results in additional losses and the creation of odd order harmonics in the current and voltage signals. Since in the steady-state system analysis, only the 60 Hz component of the currents and voltages are considered, saturation effects are typically ignored.

An equivalent circuit for the transformer model described above is shown in Figure 2.2.

Figure 2.2 Transformer circuit model

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

B

BB V

SI = (2.7)

and for impedances

B

B

B

BB S

VIVZ

2

== (2.8)

This normalization has the great added advantage of reducing 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 2.3. Note, 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 for brevity.

Figure 2.3 Simplified transformer circuit model under per unit system 2.a.2 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 six lines, seen strung between transmission towers. Typically, a high voltage transmission line has several feet of spacing between the three conductors. The conductors themselves are stranded wire for improved mechanical properties, as well as electrical properties. If the currents will be large, several conductors may be strung per phase. This improves cooling compared to using one large conductor. This geometry is important as it impacts the electrical properties of the line. To begin, as current flows in each conductor, a magnetic field develops. Adjacent lines then may induce voltages in nearby conductors through mutual induction (as we saw for transformers only now the coupling 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, I, as sketched in Figure 2.4. Similar to the transformer development, we will apply Ampres 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. Then the closed path is a circle with circumference 2x, which gives

I

Fig

If x is less than the line radiusan equal distribution of curren

x

ure 2.4 Infinite transmission line

IxH =2 or x

IH2

= (2.9)

, the closed path will not link all of the current. Assuming t throughout the wire, then

2

2

2rxIxH = or 22 r

IxH

= (2.10)

Now, then once again the flux density is determined by the permeability of the material, in this case, either the conductor itself if xr. Then the flux relationship is

dxr

Ixd 22 = (2.11)

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

rRIdx

xIR

r

ln22

00external

== (2.12) For x

D

D D

Figure 2.5 End view of equally spaced phase conductors In practice, the phase conductors may not be equally spaced. This results in unbalanced conditions due to the imbalance in mutual inductance. High voltages 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 non-conducting medium, charge accumulates leading to capacitance. Similar to the previous development for inductance, we can determine the capacitance based on Gausss Law. For a point P at a distance x from a conductor with charge q, the electric flux density D is

xqD2

= (2.17)

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.

xqE

02= (2.17)

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

2

1

0012 ln22

2

1RRqdx

xqV

R

R == (2.18)

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 first, a balanced system with 0=++ cba qqq , and two, a neutral located at some far distance R from phase a so

rDq

DRq

DRq

rRqV acbaan ln2

1lnlnln2

1

00 =

++= (2.19)

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

rDV

qC

an

aan

ln

2~ 0== (2.20)

Again 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 upon the temperature, the frequency of the current, the conductor material, and so on. For most systems analysis problems, these can be based on values provided by manufacturers or compiled into tables for commonly used conductors and typical ambient conditions. 2.a.3 Transmission line circuit models Transmission lines may be classified based on 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 and resistance can be modeled as lumped parameters. Figure 2.6 depicts the short line model per phase. The series resistance and inductance is simply the per unit distance parameters times the line length so at 60 Hz for line length l the line impedance is

jXRlLjlRZ +=+= ~120~ (2.22)

Figure 2.6 Short line model 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 -circuit model in Figure 2.7. Again, the total capacitance is simply the per unit distance capacitance times the line length so

jBlCjY an ==~120 (2.23)

Figure 2.7 Medium line model 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-know 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, Ir, is

xZIxVxV crr sinhcosh)( += (2.24)

xZVxIxI

c

rr sinhcosh)( += (2.25)

where yzZc = is the characteristic impedance of the line and zy= is the propagation constant. It would be useful, and it turns out to be possible, to continue to use the -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 -model can be found to be

ZIYZVV rrs +

+=

21 (2.26)

++

+=

21

41 YZIYZYVI rrs (2.27)

Now equating (2.24) - (2.25) for a line of length l to (2.26)-(2.27) and solving shows the equivalent shunt admittance and series impedance for a long line to be

llZlZZ c sinhsinh == (2.28)

22tanh

2

2tanh1

2 llYl

ZY

c

==

(2.29)

where the prime indicates the modified circuit values arising from a long line. 2.a.4 Generators Three phase synchronous generators produce the overwhelming majority of electricity in the modern power systems. Synchronous 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

mff IK sin= (2.30) where If is the field current, m 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 m rad/s with p poles, the electric frequency is

2p

ms = (2.31)

with s the desired synchronous frequency. If the machine is rotated at a constant speed Faradays Law tells us the induced voltage will be

( 0sin )

+== tIKdtdV ssff (2.32)

If a load is applied to the armature windings, then current will flow and the armature flux will link with the field. This effectively puts a mechanical load on the rotor and so power input must be matched to this load in order 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 2.8 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.

Figure 2.8 Simple synchronous generator model

2.a.5 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 intractable. Broadly speaking, loads may vary with voltage and frequency. In the steady-state frequency is constant so one only needs to be concerned with the voltage. For most steady-state analysis, a fixed, i.e., constant over an allowable voltage 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

aVPP 0= (2.33)

bVQQ 0= (2.34) where 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.

2.b 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 amongst the most significant computational studies carried out in power system planning and operations in the industry. Power flow equations allow us to compute 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 arising from the balancing of power

quantities. Moreover, the large size of the power network implies that the 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. We first study a simple power flow problem in Section 2.b.1 to gain insight into the nonlinear nature of the power flow equations. We then formulate the power flow problem for the large power system in Section 2.b.2. In Section 2.b.3, a classical power flow solution method based on the Gauss-Seidel algorithm is studied. In Section 2.b.4, the popular Newton-Raphson algorithm, which is the most commonly used power flow method in the industry today, is introduced. In Section 2.b.5, we will briefly consider the fast decoupled power flow algorithm, which is a heuristic method that has proved quite effective for quick power-flow computations. Finally, in Section 2.b.6, we will learn about the DC power-flow solution that is a highly simplified algorithm for computing approximate linear solutions of the power-flow problem, and becoming widely used for electricity market calculations. 2.b.1 Simple example of a power flow problem:

1 0 V

P + j Q

j x

Figure 2.9 A simple power system

Let us consider a single generator delivering the load P + j Q through the transmission line with the reactance x. The generator bus voltage is assumed to be at the rated voltage and thus it is at 1 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 . Since the line has been assumed to be lossless, note that the generator real power output must be equal to the real 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 + j Q, we want to solve for the unknown variables, namely, the bus voltage magnitude V and the phase angle . For simplicity, we will assume that the load is at unity power factor, that is, Q = 0. The line current phasor I from the generator bus to the load bus is easily calculated as

xjV

=01I (2.35)

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

( ) ( )x

Vx

V 22 2

+== *IVS (2.36)

Therefore, we get the real and reactive power balance equations to be

xVVQ

xVP cos,sin

2 +=

= (2.37)

After setting Q=0 in (2.37), we can simplify the two equations in (2.37) into a quadratic equation in V2 as follows

02224 =+ PxVV (2.38) Therefore, given any real power load P, the corresponding power-flow solution for the bus voltage V can be solved from (2.38). We note that for nominal load values, there are two solutions for the bus voltage V and they are the positive roots of V2 in the next equation

2411 222 PxV = (2.39)

Equation (2.39) 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 2.10. From the plot and from the analysis thus far, we can make the following observations:

1. The dependence of the bus voltage V on 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 hundreds 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 nonlinear power-flow equations, and the computation is a nontrivial task.

2. Multiple power-flow solutions can exist for a specified loading condition. In Figure 2.10, 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 in order 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 (2.39), the bus voltage phase angle can be computed from (2.37). Then, the line current phasor I can be solved from (2.35). 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 expensive transmission line.

4. Power-flow solutions may fail to exist at high loading conditions, for instance, when P > Pmax in Figure 2.10. The loading value Pmax beyond which power-flow solutions do not exist is called the static limit in the power literature. Since the 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 beyond the static limit may lead to catastrophic failure of the system.

V

P

Pmax

1

0

Figure 2.10 Qualitative plot of the power-flow solutions

2.b.2 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.

Vi

Ii

Si

SL i

SG i

Figure 2.11 Complex power-balance at Bus i

The power balance equation is given by

(2.40)

ii LG

*iii SSIVS ==

Let us denote the vector of bus voltages as Vbus and the vector of bus injection currents as Ibus . By definition, the admittance matrix Ybus provides the relationship Ibus = Ybus Vbus. Suppose the (i,j)-th entry Yij of the Ybus matrix has the magnitude Yij and the phase ij. Then, we can simplify the current injection Ii as

( ) +==j

ijjjijj

VY jiji VYI (2.41)

Then, combining (2.40) and (2.41), we get the complex power-balance equations for the network as

( )ijjij

jiij VVY == ii LGi SSS (2.42) Taking the real and imaginary parts of the complex equation (2.42) gives us the real and reactive power-flow equations for the network.

( )ijji

jjiijLGi VVYPPP ii == cos (2.43)

( )ijji

jjiijLGi VVYQQQ ii == sin (2.44)

Generally speaking, our objective in this section is to solve for the bus voltage magnitudes Vi and the phase angles 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 i. 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 (2.43) and (2.44), 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 = 0. In this case, the loads PLi and QLi are assumed to be known either from measurements or from load estimates, and the bus voltage variables Vi, and i are the unknown variables. Purely load buses with no generation support are called PQ buses in the power-flow studies since both real-power injection Pi and reactive power Qi have been specified at these buses. 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 PLi and QLi are also assumed to be known. This leaves the generator reactive output QGi and the voltage phase angle i 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, and thus the generator buses are normally denoted PV buses 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, say 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 the power-flow studies by keeping track of the reactive output QGi. When the reactive generation QGi becomes larger than a prespecified maximum value say QGi,max or goes lower than a prespecified minimum value QGi,min, the reactive output is assumed to be fixed at the limiting value QGi,max or QGi,min respectively, and the voltage control is disabled in the formulation. That is, the reactive power QGi becomes a known variable, either at QGi,max or QGi,min and the voltage Vi then becomes the unknown variable for bus i. In power-flow terminology, we say that the generator at bus i has reacted 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 buses and PV buses, 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.

+=i j

lossesi

Li

G ijiiPPP (2.45)

The line losses associated with any transmission line in turn depends on the line resistance and the line current magnitude. As stated earlier, one of the main objectives of the power-flow studies 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 (2.45), 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 i. 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 text books, slack bus is defined in this section to be the first bus in the network with V1=1 and the angle 1=0. Assuming the number of generators to be NG, the buses 2 through NG+1 are set to be the PV buses. The remaining buses NG+2 through N are then the PQ buses. 2.b.3. Gauss-Seidel algorithm: Let us consider a set of simultaneous linear equations of the form A x = b, where A is an n X n matrix, and, x and b are n X 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-1 b. When the matrix size is very large, it may not be possible to compute the inverse of the matrix A for finding the solution and there exist other numerical techniques. Gauss-Seidel algorithm is one such classical algorithm that tries to arrive at the solution x = A-1 b iteratively by starting from an approximate initial condition say x0. The iteration for the solution xk+1 from the previous iterate xk proceeds as follows.

nixaxaba

xij ij

kjij

kjiji

ii

ki ,...,2,1for

1 11 =

=

< >

++ (2.46)

Here aij denotes the (i,j)-the entry of the matrix A as usual. It can be shown that the iterative solution xk converges to the exact solution A-1 b for any initial condition x0, provided the matrix A satisfies certain diagonal dominance properties. The details are limited here to save space and they can be seen in standard numerical analysis text-books. In the previous section, we have formulated the power-flow problem as a set of simultaneous nonlinear equations (2.42) 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 Vbus = Ibus. The matrix Ybus takes over the role of the matrix A in the linear equations. We will be solving for the bus voltage vector Vbus. The current injections Ibus 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 buses, only the real power injection Pi is known while the reactive injection Qi is evaluated first using the latest estimate of bus voltages Vbus. An outline of the Gauss-Seidel algorithm for solving the power-flow equations (2.42) is presented next. Let us start with an initial condition for the bus voltages Vbus0 and we would like to compute the iterate Vbusk+1 from the previous iterate Vbusk. Recall that bus 1

is a slack bus and hence V for all iterations. Also, buses 2 through N01=k1 G+1 are PV buses, and hence, we need to keep Vik at their specified values Vi, specified = Vi0 for all iterations for the PV buses. Gauss-Seidel iterations: a. Slack bus (i=1): 011 =+k1Vb. PV buses (i=2, , NG+1):

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

ij

kj

kiijij

kj

ki

ij

kj

kiij

ki VVYVVYQ +=

+

++

+

ij ij

kjij

kjijk

i

kii

ii

ki

jQP VYVYVY

V 1*1

1 1 (2.48)

(3) Normalize the magnitude Vik+1 to be Vi0. 0

1

11

iki

kik

i VV ++

+ =VV (2.49)

c. PQ buses (i=NG+2, , N): (4) Update the bus voltage phasor Vik+1.

=

< >

++

ij ij

kjij

kjijk

i

ii

ii

ki

jQP VYVYVY

V 1*1 1 (2.50)

At the end of the (k+1)-th iteration, we have the updated values of the bus voltages Vbusk+1. The values of Vbusk+1 can be compared with the previous set of values Vbusk to check whether the solution has converged or not. For the power-flow problems, 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 flat starts when all the initial voltage magnitudes are set to be at their rated values (say Vi0 = 1 pu for all the PV buses) and the bus voltage angles are set to be zero (that is, i0=0 for all buses). The relative simplicity of the computations involved in each iteration step in (2.47) to (2.50) 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 tends to be a constant. Therefore, while the Gauss-Seidel algorithm can converge to the proximity of the actual solution in tens of iterations, it typically takes large number of iterations to get to an accurate solution estimate. 2.b.4 Newton-Raphson algorithm: Unlike the Gauss-Seidel algorithm, which was originally developed for solving simultaneous linear equations, 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 F : is a smooth nonlinear function. We have been given an initial condition nn x0. Then, for computing the estimate xk+1 from xk, we first linearize the functions F(x) at xk as follows

( )kx

k xxxFxFxF

k

+= )()(0 (2.51)

The solution to the linearized equations (2.b.1.17) is defined as the iteration estimate xk+1.

( kx

kk xFxFxx

k

1

1

+

= ) (2.52) Note that the linearization (2.51) will be a good approximation if the estimate xk is close to the true solution say x* since F(x*)=0. The NR algorithm stated in (2.52) can be proved to converge to the true solution x*, when the initial condition x0 is sufficiently close to x*. On the other hand, for initial conditions away from x*, the approximation (2.51) 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 tends 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 (2.43) and (2.44). We will solve for the unknown variables 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.

( )TNNNNN VVx GGG ...,,,...,,,...,, 2212 +++= (2.53) The corresponding power-flow equations are as follows

( )

( )( )

( )( )

( )

=

=

+++++

+++++

+++++

++

++

++

jNjNj

jNjNN

jNjNj

jNjNN

jNjNj

jNjNN

jNjNj

jNjNN

jNjNj

jNjNN

jjj

jj

NN

NN

NN

NN

NN

VVYQ

VVYQ

VVYP

VVYP

VVYP

VVYP

xqQ

xqQxpP

xpPxpP

xpP

xF

GGGG

GGGG

GGGG

GG

GG

GG

,,

,222,22

,,

,222,22

,111,11

,222,22

22

22

11

22

sin

sin

cos

cos

cos.

cos

)(

)()(

)()(

.)(

)(

............

............

............

............

............

............

(2.54) The entries of the F function in (2.54) 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 buses. In the power-flow problem, then we want to find a solution that makes the power mismatches in (2.54) to be zero. Suppose an initial condition x0 has been specified. Then, the NR algorithm for solving the power-flow equations (2.54) proceeds iteratively as follows: Newton-Raphson iterations:

(1) Compute the power mismatches F(xk) for step k from (2.54). If the mismatches are within desired tolerance values, the iterations stop.

(2) Compute the power-flow Jacobian kx

k

xFJ

= . Owing to the nice structure of the

equations in (2.54), 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 xk into these formulas.

(3) Compute the correction factors xk from (2.52) by solving a set of simultaneous linear equations

( )kkk xFxJ = (2.55)

The Jacobian matrix Jk is extremely sparse even for very large power systems, which facilitates the solution of xk in (2.55).

(4) Evaluate xk+1 from xk by adding the correction factors xk.

kkk xxx +=+1 (2.56)

As compared with the Gauss-Seidel algorithm, each iteration step in the NR algorithm is computationally much more intensive because of (i) evaluating the Jacobian and (ii) solving the linear equations in (2.55). 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, Gauss-Seidel algorithm is used only for flat starts with poorly known initial conditions. In most other situations, NR algorithm is the preferred choice. 2.b.5. 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 towards solving the power-flow problem. On the other hand, we study a power system specific method in this section called the fast decoupled power-flow algorithm, which is a heuristic method that is derived by exploiting power system specific properties. 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 Jk and solving the Jacobian equation (2.55). In this subsection, we will proceed to simplify this by replacing the iteration specific Jk 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 that all the Jacobian entries

of the form j

i

Vp

and j

iq

are all identically zero. Next, if 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, it can be shown that the NR algorithm greatly simplifies to the fast decoupled algorithm stated below. Let us split the power-flow state vector x in (2.53) into the voltage magnitude V and angle counterparts.

( ) ( )TNNTNNN VVV GGG ...,,,...,,,...,, 2212 +++ == (2.57) Similarly, we separate the real and reactive power mismatches in (2.54).

=

=++

++

++

)(

)(,

)(

)()(

.)(

22

22

11

22

xqQ

xqQQ

xpP

xpPxpP

xpP

P

NN

NN

NN

NN

NNGG

GG

GG ............

............

............

(2.58)

Suppose initial conditions for the voltage magnitude vector V0 and angle vector 0 have been specified. Then, the fast decoupled algorithm proceeds iteratively as follows. Fast decoupled iterations:

(1) Compute the real and reactive power mismatches P(xk) and 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.

=

=++

++

++

k

Nk

N

k

Nk

Nk

kN

kN

Nk

Nk

Nk

Nk

kk

k

VQ

VQQ

VP

VPVP

VP

PGG

GG

GG ............

............

............22

22

11

22

~,

.~ (2.59)

(3) Solve for the voltage magnitude and angle correction factors Vk and k by

using the constant matrices B and B which are extracted from the bus admittance matrix Ybus.

kkkk QVBPB ~,~ == (2.60)

Define Bij=imag(Yij). The matrices B and B are constructed as follows.

=

=

+

+++

NNNN

NNNN

NNN

N

BB

BBB

BB

BBB

G

GGG

,2,

,22,2

,2,

,22,2

,...

............

............

... (2.61)

(4) Update the voltage magnitude and angle vectors.

kkkkkk VVV +=+= ++ 11 , (2.62)

By using the constant matrices B and B in (2.60), the time taken for evaluating one iteration for the fast decoupled algorithm is considerably lesser 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. 2.b.6. DC power-flow algorithm: A further simplification of the fast decoupled algorithm is the highly approximate DC power-flow algorithm, which completely 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 Vk become irrelevant in (2.60). Moreover, the angle variables can be solved explicitly by the linear equations

PB = (2.63)

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. However, the limitations of the solution need to be kept in mind, in light of the drastic assumptions that were used to simplify the nonlinear power-flow equations into linear equations. 3. Dynamic Analysis The power system in practice is constantly undergoing changes either due to changing loads, planned outages of equipment for maintenance or other disturbances, such as, equipment failures, line faults, lightning strikes 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 and so the network models introduced in the previous are still valid. Still dynamic models for generator units models must be introduced in order 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. 3.a Modeling Electric Generators To understand modeling generators for dynamic analysis, some 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 not only to represent the mechanical speed of rotations but also since 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 so the effective coupling). During disturbances, these variations are most evident 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 frame of reference models are not developed here but instead a simple model approximating the variable inductances is presented. To begin, there are primarily two sets of windings of concern: 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 are distributed in slots around the stator so as 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 as they rotate induce a voltage in the armature windings. The current is controlled by the exciter in order to provide the desired voltage across the armature windings. This current must be constrained to avoid overheating of the windings and the modeling of these limits along with excitation control circuit is critical for analysis.

In addition, in salient pole machines, there are solid conducting bars in the pole faces called damper windings, or amortisseurs windings, that influence the effective inductanc. 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 3.1. The induced voltage 'E , referred to as the voltage behind the reactance, is connected to the generator terminal through the transient reactance xd , where the subscript indicates a direct axis quantity and the superscript a transient quantity.

j xd

E

Figure 3.1 A simple model of a synchronous generator armature for dynamic studies

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 3.2. 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 supplementary stabilization control on large units referred to as a PSS (Power System Stabilizer). The interested reader is referred to the literature.

Figure 3.2 Basic components of simple voltage regulation and excitation system

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

)(1 emmm TTJdtd

== & (3.1)

where J is the rotational inertia, Tm is the mechanical torque input and Te the electromagnetic torque that arises from producing an electrical power output. Recall that power equals the torque times angular velocity so multiplying both sides of (3.1) by m yields

)(1 emmm PPJ=& (3.2)

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

B

m

SJH

2o )(21

= (3.3)

with the synchronous mechanical speed. If in addition, the speed and the real powers are expressed on a per unit basis, then substituting (3.3) into (3.2) gives

0m

)(2 em

s PPH

=& or )( ems PPH

f=

& (3.4)

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

(3.5) s =&

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. 3.b Power system stability analysis The power system is a nonlinear dynamical 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. Specifically, 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 fluctuations are denoted small-disturbances, and the ability of the power system to damp out such small disturbances is called small-signal stability. In Section 3.b.1, 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 Section 3.b.2, we will study techniques for verifying the transient stability of a power system following a specific large disturbance. 3.b.1 Small-signal stability: Consider a nonlinear system described by the following ordinary differential equation

smoothis,:,),( ffxxfdt

xd nnn = (3.6)

Suppose that x* is an equilibrium point for the system. That is, f(x*)=0. We define the

Jacobian matrix A for the equilibrium to be the matrix of partial derivatives xf

evaluated at x*. 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 i of the polynomial matrix characteristic equation det( In A ) = 0 where In denotes the n X n identity matrix. Also, the equilibrium will be locally unstable if any one of the eigenvalues has 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. A single generator that is connected to an ideal generator bus through a transmission line is shown in Figure 3.3. 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 connected to the generator terminal through the transient reactance xd. In the classical machine model, the internal induced voltage E is assumed to remain constant, and the rotor angle follows the second order swing equations as in (3.4) but here including a damping component Pd

dems

PPPH =

&&2 (3.7)

This can be rewritten into the standard form of

(

=

dems

s

PPPH2

&

&

) (3.8)

j xd

j x

E 1 0

Figure 3.3. A simple power system

For the power system in Figure 3.3, the electrical power output Pe is easily calculated as

sin'sin'

maxPxxEP

de =+

= (3.9)

where Pmax = E/(xd+x) is a constant. The remaining term, the damping power Pd, is usually defined as Pd = D ( s ) where D is known as the damping constant for the generator, which is a positive constant. Substituting the entries for Pe and Pd into (3.8), we get the dynamic equations for the generator as

((

=

sms

s

DPPH

sin2 max&

&

)) (3.10)

The equilibrium points for (3.10) can be easily solved by setting the derivatives to be zero. (*, s)T is an equilibrium point for this system if Pm = Pmax sin *. The equilibrium points can be identified visually by plotting Pe and Pm as shown in Figure 3.4. The intersection of the constant line Pm with the curve Pmax sin highlighted by the black marks depict the two possible equilibrium points when Pm < Pmax. Let us denote the equilibrium point between 0 and /2 by s and the other equilibrium between /2 and by u.

Pmu s

Pmax sin

Figure 3.4 Plot of Pe and Pm

We will show that the equilibrium (s, s)T is small-signal stable, while the other equilibrium (u, s)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 Jacobian is first derived as

( ) ( )

=

D

HP

Hxf

ss

2cos

2

10

max

(3.11)

The eigenvalues of the system matrix A can easily be computed at the equilibrium point (s, s)T as the solutions of the second order polynomial

0cos22 max

2 =++ sss P

HD

H

(3.12)

Since s by definition lies between 0 and /2, cos s is positive. Therefore, the two roots of the characteristic equation (3.12) have negative real parts and hence the equilibrium (s, s)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 (u, s)T is considered, note that cos u is negative since u lies between /2 and . Then, the characteristic equation (3.12) will have one positive real eigenvalue and one negative real eigenvalue, when s is replaced by u. Therefore, the equilibrium (u, s)T is small-signal unstable. Normally, the power system is operated only at the stable equilibrium point (s, s)T. The unstable equilibrium point (u, s)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. 3.b.2 Transient stability: Assume that the power system is operating at a small-signal stable equilibrium point. Suppose the system is suddenly subject 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 3.5.

1 0E

j x

j x

j xd

Figure 3.5 Pre-fault power system

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 3.5. We usually assume that the system is operating at a nominal equilibrium point before the fault occurs. For this pre-fault 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 pre-fault system are then given by

((

=

spre

ms

s

DPPH

sin2 max&

&

)) (3.13)

where 2'

'max xx

EPd

pre

+= . The equilibrium points can be solved by setting the derivatives to

zero in (3.13) and let us denote the stable equilibrium by (spre, s)T where spre is the equilibrium solution of the rotor angle between 0 and /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 Fig. 3.3 changes to the configuration shown in Fig. 3.6.

j x

j xd

E 1 0

j x/2

Figure 3.6 Fault-on power system

The computation of the generator electrical power output Pe for the fault-on system in Fig. 3.4 requires a little more work. Looking for the generator terminal bus, the effective

Thevenin voltage is 02/

2/

+ xxx which is 0

31 . 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

sin3'

3'sinmax xxEPP

d

faultfaulte +

== (3.14)

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

((

=

sfault

ms

s

DPPH

sin2 max&

&

)) (3.15)

Let us assume that the relays clear the fault at time t=tc by opening the lower transmission line at both the sending and receiving end. Then, the system configuration becomes as shown in Figure 3.3, and the dynamic equations for the post-fault system are given by

((

=

spost

ms

s

DPPH

sin2 max&

&

)) (3.16)

where xx

EPd

post

+=

''

max . If the system settles down to the stable equilibrium of the post-

fault system (3.16) after the fault is cleared, we say that the system is transient stable for the fault under study. In order to verify this numerically, we proceed as follows. Transient stability study by numerical integration:

(1) The system is at (spre, s)T before the fault occurs. Therefore, we start the simulation with (, )T = (spre, s)T at time t=0.

(2) During the fault-on period, the system dynamics is governed by (3.15) and the fault-on period is from time t=0 through time t=tc. Therefore, we integrate the equations (3.15) starting from the initial condition (spre, s)T at time t=0 for a time period from t=0 to t=tc. Let us denote the system state at the end of fault-on period numerical integration at time t=tc by the clearing state (c, c)T.

(3) At time t=tc, the system equations change to the post-fault equations (3.16). Therefore, we integrate the equations (3.16) starting from the clearing state (c, c)T at time t=tc for a period of several seconds to assess whether the system response converges to the stable equilibrium (spost, s)T or not. If the post-fault response settles down to the nominal equilibrium, we can determine the system to be transient stable for the disturbance. If the post-fault 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 formulation of the pre-fault, fault-on and post-fault models becomes a nontrivial task, and the numerical integration becomes highly time-consuming. Equal area criterion: In order to verify the transient stability, it is possible to derive analytical conditions for the power system in Figure 3.4. For simplicity, we assume that the damping constant D=0, and that the fault clearing is instantaneous (tc=0). We recall that the dynamics of the pre-fault system are described by

sin2 maxpre

ms

PPH =&& (3.17)

where 2'

'max xx

EPd

pre

+= . Similarly, the equations for the post-fault system are described by

sin2 maxpost

ms

PPH =&& (3.18)

where xx

EPd

post

+=

''

max . Clearly, it follows that > since x/2 < x. Intuitively, more

power can be transferred in the pre-fault configuration with two parallel lines as compared to one transmission line present in the post-fault configuration.

prePmaxpostPmax

As stated earlier, the system is operating at the pre-fault equilibrium (spre, s)T before the

fault occurs where prempre

s PP

max

=sin , and spre lies between 0 and /2. When the fault is

cleared instantaneously at time t=0, we would like to know whether the transient starting from (spre, s)T will settle to the post-fault system equilibrium (spost, s)T or whether it will diverge away. Convergence to (spost, s)T or divergence implies transient stability or instability, respectively. Let us start with the analysis. First, we note that spre < spost since > . For visualization, let us plot P

prePmaxpostPmax

e and Pm for the post-fault system as shown in Fig. 3.7.

spre

upost

Pmaxpost sin

Pm spost

Figure 3.7 Power-angle curve for the post-fault system

The dynamics of the transient starting from (spre, s)T is governed by the second order equation,

sin22 maxpost

mss

PPHH == &&& (3.19)

Therefore, the sign of the term determines whether the speed derivative sinmax

postm PP

& is positive or negative. Inspecting the plot Figure 3.7, we conclude that the rotor frequency increases whenever the rotor angle is below the Pm line since then

sinmaxpost

m PP

s =&

will be positive. Similarly, the rotor speed decreases in value when the angle is above the Pm line since then will be negative. sinmax

postm PP

s1

Let us recall that the post-fault system response starts at (spre, s)T at time t=0. As noted earlier, the initial rotor angle spre lies beneath the Pm line in Figure 3.7, and hence, the speed increases from the initial value =s at time t=0, as soon as the fault is cleared. When increases above s, the rotor angle starts to increase from spre since

for the machine dynamics. Therefore, the rotor angle moves up on the power-angle curve in Figure 3.7 and the rotor speed keeps increasing until the rotor angle reaches the value spost at the intersection point with the Pm line in Figure 3.7. Let us say that the angle takes time t1 seconds to increase from the initial value spre at time t=0, to the value spost. During this time period from t=0 to t=t1, the speed has increased from s to some higher value say 1. The dynamic state of the post-fault system at time t=t1 is then given by (spost, 1)T. Note that even though the rotor angle equals spost at time t=t1, the system is not in the equilibrium condition, since the speed equals 1 at time t1, and 1 is greater than the equilibrium speed value s. By construction, the speed value 1 is defined as follows by the dynamics (3.19).

( ) aspostmstt

t

AH

dPPH

dtpost

s

pres

====

=

2

sin2 max0

1

&

(3.20) where Aa is the shaded area shown in Figure 3.8. Since the rotor acceleration has been positive during this time period from t=0 to t=t

&&1, the angle has been accelerating in the

area shown, and hence, the area Aa is called the acceleration area.

Aa

spre

upost

Pmaxpost sin

Pm spost

Figure 3.8 Acceleration area for the post-fault system

When the transient reaches (spost, 1)T at time t=t1, the rotor angle keeps on increasing since at time t01 >= s& 1. However, for time t>t1, the rotor angle moves above the Pm line in Figure 3.7, and hence, the derivative of speed becomes negative. That is, the speed starts to decrease from the value 1 as the time increases from t=t1. Only after the speed has decreased below the synchronous speed s can the rotor angle start to decrease. Till then, the rotor angle will keep increasing and the rotor speed keeps decreasing as time increases from t=t1. Looking at the power-angle plot in Figure 3.8, the rotor angle stays above the Pm line only up to the unstable equilibrium value upost. If the rotor angle were to increase above spost, then the speed derivative & becomes positive again, and the speed will start to increase.In this case, there is no scope for speed to decrease to the synchronous speed s, and transient instability results. Therefore, for any chance of transient stability, and for response to settle down around (spost, s)T, we need the speed to decrease below s before the rotor angle reaches the critical value upost. Graphically, this implies that the maximum deceleration area Admax shown in Figure 3.9 needs to be larger than the acceleration area Aa shown in Figure 3.8.

Admax

spre

upost

Pmaxpost sin

Pm spost

Figure 3.9 Maximum deceleration area for the post-fault system

When Admax > Aa, as the rotor angle decelerates past spost from time t=t1, the deceleration area will become equal to the accelaration area at some intermediate value of between spost and upost at some time say t=t2. For time t>t2, the speed falls below s, and the rotor angle starts to decrease back towards spost. The alternating scenarios of rotor angle acceleration and deceleration will continue before the angle swings are damped out eventually by the rotor damping effects which have been ignored thus far. Therefore, we say that the system is transient stable whenever Admax > Aa.

On the contrary, when Admax < Aa, the rotor speed stays above s, when the rotor angle reaches upost. Then, the rotor speed starts to increase away from s monotonously. In this case, the rotor speed never recovers below s, 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. However, 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 section are commonly used by the power industry for studying the transient stability properties of large power systems. 4. Summary Remarks 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 security 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 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 A. Bergen and V. Vittal, Power Systems Analysis, Prentice Hall, 2nd Ed., New Jersey,

2000. S. Chapman, Electric Machinery and Power System Fundamentals, McGraw-Hill, New

York, 2002. J. Glover and M. Sarma, Power System Analysis and Design, 2nd Ed., PWS Publishing,

Boston, 1994. J.J. Grainger and W.D. Stephenson, Power System Analysis, McGraw-Hill, New York,

1994. P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994. H. Saadat, Power System Analysis, McGraw-Hill, 2nd Ed., New York, 2002.

K. Tomsovic V. Venkatasubramanian1. Introduction2.a Modeling2.a.1 TransformersFigure 2.1 a) flux flows through core from first winding, b) flux is linked to a second set of windingsFigure 2.2 Transformer circuit modelFigure 2.3 Simplified transformer circuit model under per unit system2.a.2 Transmission line parametersFigure 2.4 Infinite transmission line2.a.3 Transmission line circuit models2.a.4 Generators2.a.5 LoadsFigure 2.9 A simple power system3.a ModelingElectric Generators