EED 307 – POWER ENGINEERING

2011

Ybus matrix with

regulating transformers in

the power system network

S H I V N A D A R U N I V E R S I T Y

1. Kamalanchali A AAA0201

2. Sankaralingam M AAA0217

3. R K Varadaraj Reddy AAA0231

4. N Swetcha AAA0242

5. Varsha K AAA0244

INTRODUCTION:

Electric power transmission needs optimization in order to determine the necessary real

and reactive power flows in a system for a given set of loads, as well as the voltages and

currents in the system. Power flow studies are not only used to analyze current power

flow situations, but also to plan ahead for anticipated disturbances to the system, such as

the loss of a transmission line to maintenance and repairs. The power flow study would

determine whether or not the system could continue functioning properly without the

transmission line. Only computer simulation allows the complex handling required in

power flow analysis because in most realistic situations the system is very complex and

extensive and would be impractical to solve by hand. The admittance matrix or the Y

Matrix is a tool in that domain. It provides a method of systematically reducing a

complex system to a matrix than can be solved by a computer program. The equations

used to construct the Y matrix come from the application of Kirchhoff’s current law and

Kirchhoff’s voltage law to a circuit with steady-state sinusoidal operation. These laws

give us that the sum of currents entering a node in the circuit is zero, and the sum of

voltages around a closed loop starting and ending at a node is also zero. These principles

are applied to all the nodes in a power flow system and thereby determine the elements of

the admittance matrix, which represents the admittance relationships between nodes,

which then determine the voltages, currents and power flows in the system.

First step in solving the power flow is to create what is known as the bus

admittance matrix, often call the Ybus.

The Ybus gives the relationships between all the bus current injections, I, and all the

bus voltages, V,

I = Ybus V

The Ybus is developed by applying KCL at each bus in the system to relate the bus

current injections, the bus voltages, and the branch impedances and admittances.

Y bus Derivation:

To determine the voltage-current relationships of the network, it is to be noted that this

relation can be written in terms of the node (bus) voltages V1 to V4 and injected currents

I1 and I2 as follows:

. . . . . . (1)

Or

. . . . . . (2)

Consider node (bus) 1 that is connected to the nodes 2 and 3. Then applying KCL at this

node we get,

I1=Y11V1 + Y12(V1-V2) + Y13(V1-V3)

=(Y11+Y12+Y13)V1 - Y12V2 - Y13V3 . . . . . . (3)

In a similar way application of KCL at nodes 2, 3 and 4 results in the following equations

I2 = Y22V2 + Y12(V2-V1) + Y23(V2-V3) + Y24(V2-V4)

= -Y12V1 + (Y22+Y12+Y23+Y24)V2 – Y23V3 – Y24V4 . . . . . . (4)

0 = Y13(V3-V1) + Y23(V3-V2) + Y34(V3-V4)

= -Y13V1 –Y23V2 + (Y13+Y23+Y34 )V3 - Y34V4 . . . . . . (5)

0 = Y24(V4-V2) + Y34(V4-V3)

= -Y24V2 – Y34V3 + (Y24+Y34)V4 . . . . . . (6)

From (3) and (6)

. . . . . . (7)

From (1) and (7)

. . . . . . (8)

In general the format of the Ybus matrix for an n-bus power system is as follows

EXAMPLE:

Consider the same diagram as above, calculate the Ybus Matrix.

Sol:

Where,

Y11 =y 13 + y12

Y12 = -y12 = Y21

Y13= - y 13 = Y31

Y14= 0= Y41

Y22= y23+y24+y12

Y23= -y23= Y32

Y24= -y24= Y42

Y33= y31+y32+ y34

Y34 = -y34 = Y43

Y44= y43 + y42

Thus the Ybus Matrix is:

REGULATION OF A TRANSFORMER

#Voltage regulation in a transformer is defined as “The change in secondary voltage

when rated load at a specified power is removed

Let E2 = secondary terminal voltage at no load

V2 = secondary terminal voltage at full load

Then % regulation =

VOLTAGE REGULATION OF A TRANSFORMER

Recall

Secondary voltage on no-load

V2 is a secondary terminal voltage on full load

Substitute we have

• The purpose of voltage regulation is basically to determine the percentage of

voltage drop between no load and full load.

• Voltage Regulation can be determine based on 3 methods:

voltageload-no

voltageload-fullvoltageload-noregulationVoltage

p

s

p

s

N

N

V

V

1

212

N

NVV

1

21

2

1

21

regulationVoltage

N

NV

VN

NV

100V

VE

2

22

a) Basic Defination

b) Short – circuit Test

c) Equivalent Circuit

• In this method, all parameter are being referred to primary or secondary side.

Can be represented in either

Down – voltage Regulation

Up – Voltage Regulation

Tap Changer

A transformer tap is a connection point along a transformer winding that allows a

certain number of turns to be selected.

By this means, a transformer with a variable turns ratio is produced, enabling voltage

regulation of the output. The tap selection is made via a tap changer mechanism.

The output voltage of a transformer varies with the load even if the input voltage remains

constant. This is because a real transformer has series impedance within it. Full load

Voltage Regulation is a quantity that compares the output voltage at no load with the

output voltage at full load, defined by this equation:

%100.

NL

FLNL

V

VVRV

%100.

FL

FLNL

V

VVRV

%100down Regulation

%100up Regulation

,

,,

,

,,

nlS

flSnlS

flS

flSnlS

V

VV

V

VV

To determine the voltage regulation of a transformer, it is necessary understand the

voltage drops within it.

Transformer Voltage Regulation

Fact: As the load current is increased, the voltage (usually) drops.

Transformer voltage regulation is defined as:

%100

/down Regulation

%100/

up Regulation

V

Vk noloadAt

,

,

,

,

p

s

xV

VkV

xV

VkV

nlS

flSP

flS

flSP

fls

flsnls

V

VVVR

,

,,

Ignoring the excitation of the branch (since the current flow through the branch is

considered to be small), more consideration is given to the series impedances (Req +jXeq).

Voltage Regulation depends on magnitude of the series impedance and the phase angle

of the current flowing through the transformer.

Phasor diagrams will determine the effects of these factors on the voltage regulation. A

phasor diagram consist of current and voltage vectors.

Assume that the reference phasor is the secondary voltage, VS. Therefore the reference

phasor will have 0 degrees in terms of angle.

Based upon the equivalent circuit, apply Kirchoff Voltage Law

SeqSeqSP IjXIRV

k

V

TRANSFORMER PHASOR DIAGRAMS

For lagging loads, VP / a > VS so the voltage regulation with lagging loads is > 0.

When the power factor is unity, VS is lower than VP so VR > 0.

With a leading power factor, VS is higher than the referred VP so VR < 0

Formation of a Y Bus of a system with regulating transformer between two

buses.

Firstly, we should consider per unit transformer admittances (the series and shunt using π

model).

Some Assumptions:

1. We assume the regulating transformer to be situated at the j-th bus of the line when the

pi model is included at the left side of the transformer.

2. We assume the regulating transformer to be situated at the i-th bus of the line when the

pi model is included at the right side of the transformer. (case A)

3. We assume the transformer to be placed at the ends of the line and nearest to the buses.

(case B)

Case A: When the regulating transformer is placed between two buses and the line model

is placed on the left hand side of the transformer (a:1).

Let a be the complex transformation ratio of the regulating transformer where it is given

by |a|<α.

Figure. 1 - Equivalent circuit of a line containing a regulating transformer between two

buses and placed at j-th bus.

Si and Sj are injected complex powers at the i-th and j-th bus receptively and Vi and Vj

are the respective bus voltages.

The complex transformation ratio a:1 corresponds to Vpri:Vsec.

Therefore,

Also, we take input power to be equal to output power.

Thus,

Applying KCL - Refer to Fig.1

[assuming yi0 = yj0 = y0]

and

Thus,

Rewriting the above equations,

In a matrix from they can be represented as:

It should be noted that a is complex and [Y] is not symmetric.

If a is a real quantity [(KV)base / (KV)tap] then the matrix [Y] becomes symmetric.

Case B: When the regulating transformer is placed between two buses and the line model

is placed on the right hand side of the transformer (1: a).

Figure. 2 - Equivalent circuit of a line containing a regulating transformer between two

buses and placed at i-th bus.

Like in case A, we follow the similar steps.

In this case also, we take input and output power to be equal while Ij’ as the secondary

current of the transformer.

Applying KCL - Refer to Fig.2

[Assuming yi0 = yj0 = y0]

At bus i:

At bus j:

We can represent this in a matrix form as:

In case a is a real quantity then the matrix [Y] becomes symmetric.

EXAMPLES:

Case A: Transformer at the receiving end

Example: A three-bus system is shown in the figure below. Assume an ideal transformer

to be connected between buses 2 and 3 in series with a line reactance of j0.5 p.u. If off-

nominal tap ratio of the transformer is 1:1.02, fine [Ybus].

Line no. From bus To bus R (in p.u.) X (in p.u.) Off-nominal tap ratio

of Xmer

1 1 2 0.05 0.15

2 1 3 0.05 0.15

3 2 3 0 0.5 1:1.02

Soln: The given system is a three bus system. Hence, the Ybus matrix obtained will be a

3 x 3 matrix. Ybus matrix for the given without considering the line with transformer is

calculated as explained earlier.

Figure. The system before considering the line with transformer

Impedance values:

Admittance values:

Ybus matrix values are obtained as,

When the bus is considered, the line admittance is modified according to the figure given.

Figure. model of the line with transformer

Now,

Ybus matrix values are,

The modified Ybus matrix due to the effect of transformers is given as,

p.u.

Case B: Consider for the same system, the transformer is connected between buses

1 and 2, at the sending end. Then for this case the Ybus matrix is calculated as

follows.

Impedance values:

Admittance values:

Ybus matrix values are obtained as,

When the bus is considered, the line admittance is modified according to the figure given.

Now,

Ybus matrix values are,

The modified Ybus matrix due to the effect of transformers is given as,

References:

1. Abhijit Chakrabarti, Sunita Halder, “Power System Analysis Operation and

Control”, Prentice Hall of India, New Delhi, pp. 57-64. September, 2006.

2. http://www.powershow.com/view1/2732be-

ZDc1Z/Transformer_Voltage_Regulation_powerpoint_ppt_presentation

3. http://www.docstoc.com/myoffice/recommendations?docId=34351482&downl

oad=1

4. http://en.wikipedia.org/wiki/Nodal_admittance_matrix

5. http://www.sari-

energy.org/PageFiles/What_We_Do/activities/CEB_Power_Systems_Simulatio

n_Training,_Colombo,_Sri_Lanka/Course_ppts/Y_Bus_chap3.pdf