How to plot capability curve of generator

Synchronous Machine Ratings And Synchronous Machine Transients

Department of Electrical Engineering

United Arab Emirates University

Date: 05/November 2020 Due date: 5/December 2020

ELEC 411 Electric Energy Conversion

Project: Synchronous machine ratings and synchronous machine transients

See Chapter 4 of the book "Electric Machinery Fundamentals 5th Edition", by

Chapman S. J.

See the sections "Transient Stability of Synchronous Generators" and "Synchronous

Generator Capability Curves"

Objectives:

1. To gain experience in studying the ratings of the synchronous machine that are required for a proper and safe operation.

2. To study and analyze the transient behavior of synchronous machine in response to sudden disturbances including

a. Changes in the mechanical input power for synchronous generator

b. Changes in the field current for synchronous generator

There are certain basic limits to the speed and power that may be obtained from a synchronous

machine. These limits are expressed as ratings on the machine. The purpose of the ratings is to

protect the machine from damage due to improper operation. To this end, each machine has a

number of ratings listed on a nameplate attached to it. Typical ratings on a synchronous machine

are voltage, frequency, speed, apparent power, power factor, field current, and service factor.

For instance, the voltage and frequency of a synchronous generator are imposed by the power

system to which the synchronous generator is connected. However, for a given frequency, the

maximum allowable field current, i.e., maximum flux puts a limit on how large the generator's

voltage could be. On the other hands, the breakdown voltage of the winding insulation limits

the maximum allowable voltage.

The power limits of electric machines are determined by two factors, namely, the mechanical

torque on the shaft of the machine, and the heating of the machine's windings. In all practical

synchronous machines, the shaft is strong enough mechanically to handle a much larger steady-

state power than the rated one. Therefore, the heating of the machine's windings puts a practical

limit on how large the steady-state limits of power could be. The main cause of the heating of

the machine's windings is the current (field current and armature current). In particular, the heat

of the armature windings depends mainly on the stator copper losses 𝑅𝐴𝐼𝐴 2, while the heat of

the field windings is due to the field copper losses 𝑅𝐹𝐼𝐹 2. This means that the maximum

allowable heating determines the maximum acceptable armature current, which in turn, sets the

rated apparent power of the machine. In addition, the maximum allowable heating sets a

maximum field current for the machine, which in turn puts a limit on how large the internal

generated voltage 𝐸𝐴 could be.

During the lecture, we learned that the maximum power that the synchronous generator can

supply corresponds to 𝛿 = 90 degrees. This maximum power is referred to as the static stability limit or the steady-state stability, meaning that the system can remain stable for all 𝛿 less than 90 degrees. Form theoretical standpoint, a synchronous generator should be able to supply up

to its maximum power without loss of synchronism. In practice, however, the maximum power

that can be supplied by the generator is limited to a much lower level by its dynamic stability

limit. The reason is that the transient response of the power angle 𝛿 can go above the static stability limit (90 degrees) even if the desired steady-state power is much lower than the

maximum power (static stability limit). When it does so, the machine losses the synchronism

and becomes unstable. This explains why, in practice, the power angle is limited to 20 or 30

degrees to allow the transient response to take large values during transients without exceeding

the statistic stability limit. This project will give an opportunity to the students to analyze the

transient response of synchronous machine to sudden changes in the torque/power.

Problem: A 13.8-kV, 187-MVA, 0.8-power-factor-lagging, 60-Hz, 20-pole Y-connected

synchronous generator has a synchronous inductance of 0.8104 mH and an armature resistance

𝑅 of 0.0204 Ω. The generator is delivering power at the rated terminal voltage to an infinite bus bar. The core losses, stray losses, and friction and windage losses are neglected.

Tasks: a. Plot the synchronous generator capability curves b. Analyze the transient behavior of synchronous machine under sudden disturbances.

I. Theoretical part: 1. Consider the following expressions

, E , V V 0S Z A AZ R jX Z E          

where 𝑋𝑆 is the synchronous reactance of the generator. For synchronous generator operation, show that

   

2

*3 3 3 A

A Z Z

E V V S V I

Z Z

 

        

and

    2

*3 3 3 AA

conv A A Z Z

E VE S E I

Z Z

        

Here, 𝑆 is the complex power at the output of the machine, while 𝑆𝑐𝑜𝑛𝑣 is the complex power associated with the converted power 𝑃𝑐𝑜𝑛𝑣.

2. Show that

    2

3 cos 3 cos AA

conv Z Z

E VE P

Z Z

     

3. Show that the generator phasor diagram for lagging power factor can be drawn as shown in Figure 1.

(a)

(b)

Figure 1. Generator phasor diagram in the PQ-plane (using power units)

II. Ratings of the synchronous generator: Use Matlab m-file to plot the required curves

1. Plot the synchronous generator capability curve in the PQ-plane, which is a plot of 𝑄 versus 𝑃. In the PQ-plane, the active power 𝑃 is on the horizontal axis and the reactive power 𝑄 is on the vertical axis. The rated power can be considered as the maximum acceptable apparent power. Note that the synchronous machine can only generate

active power, meaning that the active power should be positive, where 𝑃 varies from zero to the rated apparent power. Then, for each value of the active power 𝑃, the reactive power 𝑄 can be determined by considering the relationship between the rated power 𝑆, 𝑃, and 𝑄. Alternatively, for each value of 𝑸, varying from –𝑺 to 𝑺, the value of 𝑷 can be determined as

2 2P S Q 

Note that S is the rated power (apparent power).

2. Find the maximum acceptable armature current.

3. What is the maximum allowable internal generated voltage 𝐸𝐴? 4. Use the same PQ-plane of Question 1 to sketch the maximum field current circle.

Indeed, the field current limit appears as a circle corresponding to the rated (maximum

allowable) 𝐼𝐹 or 𝐸𝐴. In our case, the maximum allowable 𝐸𝐴 can be used to plot the maximum field current circle. From Figure 1, it follows that the center of such a circle,

in the PQ-plane is the point A. By considering the point O as the origin of the PQ-

plane, it can be concluded that the coordinates of the point A in the PQ-plane are given

by

       3 3

, cos , sinA A A A s s

V V x y V RI RI

X X

 

    

     

As shown from figure 1, the radius of such a circle is represented by the distance AM

which is given by

3 A E

s

E V D

X

 

The plot the maximum field current circle in the PQ-plane can be accomplished by

using the equation of a circle. By doing so, for each value of 𝑸, varying from –𝑺 to 𝑫𝑬 + 𝒚𝑨, the value of 𝑷 can be calculated as

    2 22 2

A E A E A AP x D Q y P D Q y x        

NB: Any point, defined with active and reactive power as coordinates, that lies within

both synchronous generator capability curve and the maximum field current circle is a

safe operating point for the generator.

III. Transient behavior of synchronous machine : Use Matlab Simulink to analyze the transient response of synchronous machine connected to an infinite bus

1. Initially, i.e., at 𝑡 = 0 𝑠, the generator is delivering 50 MVA to the bus bar at the rated voltage and at 0.95 lagging power factor. Find the generated internal voltage. What is the converted power?

.

2. At 𝑡 = 1𝑠, the power demand increases and the prime mover delivers 150 MW to the generator, while the field current is maintained constant. Find the power angle 𝛿, the power factor, the real power, and the reactive power of the generator. State whether the generator absorb or supply reactive power.

3. At 𝑡 = 4𝑠, the internal generated voltage 𝑬𝑨 is increased to make the generator operate at 0.8 lagging power factor, while the power of the prime mover is kept

constant. Find the armature current, the internal generated voltage, the power

angle, the active power, and the reactive power. Can this generator operates under such conditions, why and why not?

4. Simulate your system using Matlab/Simulink file provided with the project. Towards

this end, the simulation time can be set equal to 5 s to consider the three different

scenarios. 5. Plot the power angle, the rotor speed, the active power, the reactive power, the power

factor angle, and the current. Such waveforms should be plotted over the simulation

time of 5 s.

6. Compare the theoretical results with that obtained in simulation at steady-state conditions.

7. Comment about the transient behavior of the power angle and rotor speed in response to a sudden increase in the prime mover power, i.e., the transient response between 1 s

and 4 s

8. Comment about the transient behavior of the active power and the rotor speed in response to a sudden increase in the internal generated voltage, i.e., the transient