Jenkins N., Strbac G., Ekanayake J. Distributed Generation

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The relationship between the line and phase currents can be obtained by

assuming the three phase currents are balanced (i.e. the connected loads are equal).

From Figure I.15, the magnitude of the line current is equal to

ﬃﬃﬃ

times the mag-

nitude of the phase current.

I.9 Connection of loads

Loads can also be connected in star or delta. If the loading on each phase is equal,

then that load is called a balanced three-phase load. Figure I.16(a) and (b) shows

balanced star-connected and delta-connected loads.

(

)

Star connection

(

)

Delta connection

Z∠

Z∠f

Figure I.16 Star and delta connection of loads

I.10 Three-phase four-wire system

A three-phase star-connected generation system is connected to a three-phase star-

connected load as shown in Figure I.17 to form a three-phase four-wire system.

Three-phase balanced star-

connected supply

Z∠f

Three-phase balanced star-

connected load

Figure I.17 Configuration of the three-phase four-wire system

AC electrical systems 197

When the load is balanced, the current in the neutral wire is zero.

¼ I

þI

Zﬀf

½sin wt þ sinðwt  2p=3Þþsinðwt  4p=3Þ ¼ 0

ðI:12Þ

However, when the loads are not balanced (i.e. the three loads are not equal),

then there will be a current flowing through the neutral wire.

I.11 Three-phase delta-connected three-wire system

Three-phase balanced

delta-connected load

Three-phase balanced

delta-connected source

Z∠f

Figure I.18 Three-phase delta-connected source and a load

A three-phase delta-connected generator is connected to a three-phase delta-con-

nected load as shown in Figure I.18 to form the three-phase, three-wire system. In

the figure I

, I

and I

are line currents, and I

, I

and I

are phase currents.

¼ I

 I

; I

¼ I

 I

and I

¼ I

 I

ðI:13Þ

The corresponding phasor diagram of the system is shown in Figure I.19.

Defining the rms value of the magnitude of phase currents as I

and the rms value

of the magnitude of line currents as I

, and then from the phasor diagram, we get:

¼ 2  I

 cos 30



ﬃﬃﬃ

ðI:14Þ

198 Distributed generation

−I

30°

120°

−I

120°

Figure I.19 Phasor diagram of currents in a delta-connected load

I.12 Power in three-phase system

The power in a three-phase load is the sum of the power consumed in each phase.

Consider the load shown in Figure I.17.

For the load connected to the A phase:

¼ V

sin wt

Zﬀf

¼ I

sinðwt  fÞ

ðI:15Þ

where I

¼ V

=Z.

For the other phases:

¼ V

sinðwt  2p=3Þ v

¼ V

sinðwt  4p=3Þ

¼ I

sinðwt  2p=3  fÞ i

¼ I

sinðwt  4p=3  fÞ

ðI:16Þ

Instantaneous power in A phase ¼ v

 i

¼ V

 I

 sin wt  sinðwt  fÞ

Instantaneous power in B phase ¼ v

 i

¼ V

 I

 sinðwt  2p=3Þ

sinðwt  2p=3  fÞ

Instantaneous power in C phase ¼ v

 i

¼ V

 I

 sinðwt  4p=3Þ

sinðwt  4p=3  fÞ

By adding these equations, the total power consumed by the three-phase load is

obtained. It can be shown trigonometrically that

sin wt  sinðwt  fÞþsinðwt  2p=3Þsinðwt  2p=3  fÞ

þ sinðwt  4p=3Þsinðwt  4p=3  fÞ¼

cos f

Therefore, the total three-phase instantaneous power

cos f ¼ 3

ﬃﬃﬃ

cos f ¼ 3V

cos f ðI:17Þ

AC electrical systems 199

When (I.17) is compared with (I.9) for a single-phase case, it can be seen that

the total three-phase instantaneous power is equal to the addition of active power in

the three phases. Therefore, the active power, P, measured in watt for a three-phase

circuit is given by:

P ¼ 3V

cos f ðI:18Þ

where cos f is the power factor of the load.

Similarly as defined for a single-phase circuit, the apparent and reactive

powers are defined for three-phase circuits.

Apparent power S ¼ 3V

ðI:19Þ

Reactive power Q ¼ 3V

sin f ðI:20Þ

It is shown in Table I.2 that the power relationships for both star- and delta-

connected loads are equal.

Table I.2 Power relationships for star- and delta-connected loads

Load Star connected Delta connected

Current and voltage

relationship

ﬃﬃﬃ

and I

¼ I

¼ V

and I

ﬃﬃﬃ

Apparent power (VA) S ¼ 3V

¼ 3

ﬃﬃﬃ

S ¼3V

¼3V

ﬃﬃﬃ

Active power (kW)

P ¼ 3V

cos f ¼

ﬃﬃﬃ

cos f

Reactive power

(kVAr)

Q ¼ 3V

sin f ¼

ﬃﬃﬃ

sin f

EXAMPLE I.5

A three-phase, wye-connected generator is driven from a water turbine that

produces 15 kW mechanical shaft power . It is connected to a 500 V (V

three-phase, 60 Hz supply. The generator has an efficiency of 95% and

operates at a power factor of 0.85 lagging (exporting VArs). Calculate

(a) The output apparent power and hence the current in the connecting

circuit

(b) The active and reactive components of the current

Answer:

(a) Output power = input power  efficiency = 15  0.95 = 14.25 kW

Apparent power, S ¼

ﬃﬃﬃ

Active power, P ¼

ﬃﬃﬃ

cos f ¼ S cos f

Therefore, output apparent power ¼

cos f

14:25

0:85

¼ 16 :76 kVA

200 Distributed generation

(b) The line current, I

ﬃﬃﬃ

cos f

14:25  10

ﬃﬃﬃ

 500  0:85

¼ 19:36 A

Active componen t of the current = I

cos f ¼ 19:36  0:85 ¼ 16:45 A

Reactive component of the current = I

sin f ¼ 19:36  sinðcos

1

0:85Þ¼

10:2A

ﬃﬃﬃ

sin f ¼

ﬃﬃﬃ

 500  10:2 ¼ 8:83 kVAr

EXAMPLE I.6

A 60 Hz, wye-connected synchronous generator has a 0.5 W/phase synchro-

nous reactance and a negligible armature resistance. It is connected to a three-

phase 690 V (V

) system and is exporting 300 kW at a power factor of 0.85

lagging. Calculate the line current, the internal voltage and the power angle

of the generator.

Answer:

690 V

Generator

300 kW

P ¼

ﬃﬃﬃ

cos f

300  10

ﬃﬃﬃ

 690  I

 0:85

Therefore, the magnitude of the line current = I

¼ 295: 32 A

The angle of the line current = cos

1

0:85 ¼ 31:79



As the power factor is lagging I

¼ 295:32ﬀ31:8



From the figure, assuming that 690 V bus voltage is the reference phasor:

E ¼ I

 jX

690

ﬃﬃﬃ

ﬀ0



¼ 295: 32ﬀ32:8



 j0:5 þ 398:37

¼ 147: 66ﬀ57:2



þ 398:37

¼ 80 þ j124 :12 þ 398:37

¼ 494: 2ﬀ14:55



Internal voltage = 494.2 V

Power angle = 14.55



AC electrical systems 201

I.13 Problems

1. If v ¼ 10 sinðwt þ p=3Þ and i ¼ 2 sinðwt  p=6Þ, find (a) the rms value of

voltage and current, (b) the phasor expression in polar form and (c) phase shift

between the current and voltage.

(Answers: (a) 7.07 V, 1.41 A; (b) 7:07ﬀ60



,1:41ﬀ30



and



)

2. A 60 mF loss-free capacitor is connected in parallel with a coil having an

inductance of 0.1 H and a resistance of 10 W. The network is connected across

an AC voltage V ¼ 10ﬀ0



V at a frequency of 100 Hz. Draw the circuit and

calculate the value of the current in each branch and its phase relative to the

supply voltage. Sketch the calculated currents on a phasor diagram.

(Answers: Current through: capacitor 0:377 ﬀ90



, coil 0:157 ﬀ81



)

3. A load having a series connected resistor of 5 W and a reactor of 2 W is

connected across a 240 V AC supply.

(a) Find the active and reactive power consumption of the load and its power

factor.

(b) What is the value of power factor correction capacitor in Farad required

to improve the power factor to 0.95?

(Answers: (a) P = 9.93 kW, Q = 3.97 kVAr and power factor = 0.93;

(b) 0.26 F)

4. A star-connected load having 10 W resistance per phase is connected to a

three-phase 400 V supply. Calculate the current and power drawn by the load.

(Answers: I = 23.09 A; P = 5.33 kW)

5. A balanced delta-connected load of pure resistance of 10 W per phase is

connected in parallel with a balanced star-connected load having impedance

5 þ j2 W per phase. Both loads are connected to a 400 V three-phase supply.

Calculate the current drawn by each load and hence calculate the total current

drawn by the loads.

(Answers: current drawn by: delta load = 69:28ﬀ0



star load = 42:88ﬀ21:8



, total load = 110:26ﬀ8:3



)

I.14 Further reading

1. Hughes E., Hiley J., Brown K., McKenzie-Smith I. Electrical and Electronic

Technology. Prentice Hall; 2008.

2. Grainger J.J., Stevenson W.D. Power System Analysis. McGraw-Hill; 1994.

3. Walls R., Johnston W. Introduction to Circuit Analysis. West Publishing

Company; 1992.

202 Distributed generation

Tutorial II

AC machines

North Hoyle Offshore Wind Farm (60 MW)

This was the UK’s first major offshore wind farm and is located 4–5 miles off the

North Wales coast. Each turbine is 2 MW [RWE npower renewables]

II.1 Introduction

AC electrical machines can act as generators to convert mechanical energy into

electrical energy or motors to convert electrical energy into mechanical energy. For

example, in a hydro power station, the kinetic and potential energy in the water are

converted into mechanical rotational energy by the turbine and then into electricity

by a generator. On the other hand, an AC motor in a factory provides mechanical

energy when supplied with AC. In principle, an electrical machine of the same

construction can be used as either a motor or a generator.

There are two major types of AC machine, synchronous and asynchronous

(sometimes known as induction machines). In a synchronous machine, the rotor is

supplied by DC or has permanent magnets mounted on it. The stator carries a three-

phase winding. In an induction machine, both rotor and stator carry three-phase

windings.

II.2 Synchronous machines

II.2.1 Construction and operation

1. Rotor construction and its magnetic field

Three forms or rotor constructions are commonly used in synchronous machines:

permanent magnet, cylindrical rotor or salient pole, as shown in Figure II.1. The

rotors of the cylindrical rotor and salient pole synchronous machine carry a field

winding, which is fed from a source of DC. Even though a two-pole rotor structure

is shown in this figure, two, four or even more poles are often used.

(

)

Permanent ma

net

(

)

lindrical rotor

(

)

Salient pole

Stator

′

Armature

windings

Air gap

Axis of

phase –

Axis of

phase – B

Axis of

phase – C

′

Rotor

Figure II.1 Rotor construction of synchronous machines

In cylindrical rotor and salient pole machines, the rotor is fed with DC

through slip rings, as shown in Figure II.2.

⫹V

⫺

Three-phase AC

Slip rings

Brushes

Figure II.2 Schematic representation of a synchronous machine

204 Distributed generation

The DC on the rotor circuit (or the permanent magnets) produces a mag-

netic field, f

Rotor

, that is fixed to the rotor. As the rotor rotates at synchronous

speed (w

), the rotor magnetic field also rotates at the same speed.

2. Stator construction and its magnetic field

Figure II.1 also shows the stator of an AC machine. The stator core consists of

slotted ring shape laminations that are stacked and bolted together to form a

cylindrical core. The slots on this cylindrical structure carry the armature

winding. Even though only three armature winding s (A–A

, B–B

and C–C

)

are shown in Figure II.1 for simplicity, in a real machine the stator has several

armature coils in each phase.

When the rotor rotates, the magnetic field produced by the field winding

sweeps the three-phase armature windings. This in turn induces three voltages

in three windings, A–A

, B–B

and C–C

(initially assumed to be open circuit),

which are displaced by 120



. At the position shown in Figure II.1(c), the

induced voltage of the A phase is at its positive maximum and as the rotor

rotates its magnitude reduces. Once the rotor rotates by 180



the induced

voltage comes to its negative maximum. Hence, the frequency of the induced

voltage is directly proportional to the speed of rotation of the ro tor and is given

by f = w

/2p.

If the armature windings are connected to a balanced three-phase load, the

resulting three-phase currents are also displaced by 120



as shown in Figure II.3.

t  0 t

I

Time

Current

Figure II.3 Three-phase currents (t

= p/6w and t

= p/3w)

When t = 0 (Figure II.3) the current in phase A is I

and the currents in phases

B and C are equal to (I

/2). The currents in phases A, B and C produce compo-

nents of the stator magnetic field, f

, f

and f

. The magnitudes of these flux

components are proportional to the number of ampere-turns, NI

,NI

/2 and NI

(where N is the effective number of turns of each phase of the stator winding) and act

along the axis of phases A, B and C (see Figures II.1(c) and II.4). These three

magnetic fields combine and produce a resultant stator magnetic field, f

Stator

,att =0

as shown in Figure II.4. The distribution of this flux in the air gap is also shown in

AC machines 205

Figure II.5(a). Similarly, the resultant magnetic fields produced by the stator

current at t = t

and t

are shown in Figure II.5(b) and (c).

Axis of A

phase

Axis of B

phase

Axis of C

phase

A phase flux (

fA)

Stato

Figure II.4 Resultant stator magnetic field in the air gap at t = 0 (The x axis

shows the air gap, around the circumference of the machine.)

′

(a) At t = 0

(0 rad rotation)

(b) At t = t

(π rad rotation)

′

Figure II.5 Illustration of the rotating stator magnetic field

During the time t

=(p/6w

), the flux waveform has moved by p/6. This is the

speed of the stator magnetic field and is equal to synchronous speed. In normal

operation the rotor, and hence the field winding, rotates synchronously with the

flux developed by the stator. The relative angle between the rotor flux axis and the

stator field, the rotor angle (sometimes known as the load angle), is determined by

the torque applied to the shaft.

II.2.2 Electrical and mechanical angle

In the case of a two-pole synchronous machine, for 180



of rotation of the rotor the

induced voltage changes by half a cycle or 180



electrical (P radians). Now con-

sider the four-pole machine shown in Figure II.6. At the rotor position shown, the

A phase induced voltage is at positive maximum. Once the rotor rotates by 90



south pole comes immediately below the A phase and the induced voltage becomes

negative maximum. Hence, a 90



mechanical rotation corresponds to 180



electrical

variation in the induced voltage.

206 Distributed generation