Power System Analysis - University of Tennessee

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, ¦Õ, will flow in the core. Amp¨¨re¡¯s 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

Hl = NI

(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

¦Õ

I

¦Õ

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 ¦Õ is

B = ?H = ?

NI

NI

or ¦Õ = ?A

l

l

(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

R¦Õ = NI

(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,

¦Õ. Since, the two windings also see the same core reluctance, the two mmf¡¯s are identical,

i.e.,

N1 I 1 = N 2 I 2

(2.4)

If the flux ¦Õ is changing in time, or equivalently the current I, 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

V1 I 1 = V2 I 2

(2.5)

where V1 and V2 are the primary and secondary voltages, respectively. Substituting (2.4)

and rearranging shows

V2 N 2

=

V1 N 1

(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 ¨C 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 mmf¡¯s 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

SB

VB

(2.7)

VB VB2

ZB =

=

I B SB

(2.8)

IB =

and for impedances

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 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.

Then the closed path is a circle with circumference 2¦Ðx, which gives

I

x

Figure 2.4 Infinite transmission line

2¦ÐxH = I or H =

I

2¦Ðx

(2.9)

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, then

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