Woldia University

Woldia Institute of Technology

School of Electrical & Computer Engineering
Introduction to power systems
ECEG 4121

Prepared by: Tebeje T.

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 Chapter outlines
• One Line Diagram and impedance or reactance diagram,
• Per unit system
• Single phase and three phase transmission lines
• Single-phase solution of balanced three-phase networks

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 Methods of power system representation
In practice, electric power systems are very complex and their size is unwieldy. It is very
difficult to represent all the components of the system on a single frame. The complexities
could be in terms of various types of protective devices, machines (transformers,
generators, motors, etc.), their connections (star, delta, etc.), etc. Hence, for the purpose of
power system analysis, power system components are correctly represent. The most
common methods to represent the power system components are:

• Single Line Diagram/ One Line Diagram/

• Impendence diagram

• Reactance diagram

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 Single Line Diagram /SLD/
• Power system networks are represented by one-line diagrams using suitable
symbols for generators, motors, transformers and loads
• A one-line diagram of a power system shows the main connections and
arrangements of components
• Figure 1 shows the symbols used to represent the typical components of a power
system.
• Figure 2&3 is a one line diagram for a power system consisting of two generating
stations connected by a transmission line; note the use of the symbols of fig.1.
• The advantage of such a one-line representation is its simplicity: one phase
representation all three phases of the balanced system; the equivalent circuit of
the components are replaced by their standard symbols; and the completion of
the circuit through the neutral is omitted.

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 Cont’…

Figure 1: symbolic representation of power system components

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 Cont’…
• The one-line diagram is a symbolic representation of an electrical power system.
It simplifies a three-phase power system into a single-phase system. In a single-
phase system, there is a phase wire and a neutral wire. To create an equivalent
circuit, we use a symbolic representation as shown in the following figure.

Figure 2: single line diagram of small network

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 Cont’…
• N

Figure 3: single line diagram of large network

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 Network model

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 Impendence representation of the PS
• An impedance diagram represents electrical components in terms of impedances. We
can convert the one-line diagram into an impedance diagram, as shown in the following
figure.

In this diagram:
• Load impedances are represented by RL.
• The current limiting impedances between generators and grounds are omitted.
• The shunt impedances of transformers are omitted.
• The transmission line is represented by a π circuit.
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 Reactance Diagram
• A reactance diagram represents all components of a power system using
reactance, neglecting resistance. The reactance diagram is similar to the
impedance diagram, with the exception that loads of rotating machines are
usually neglected as they have no significant effect on line currents during a fault.
Synchronous motor loads are always considered during fault calculations, but the
generated EMF is considered in short circuit currents. Induction motors are
neglected when finding currents. The impedance diagram can be converted into
a reactance diagram, as shown in the following figure

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 Cont’ …

• The reactance diagram, in the figure above, is drawn by neglecting all

resistances, the static loads, and the capacitance of the transmission line.
• We use this method to analyze the effect of reactance on the given power system
network.

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 Required parameters in the power system model
• To draw the single line diagram and analysis a given power system network, the
following main parameters are required.
▪ Line data
• Resistance of the line
• Reactance of the line
• Capacitance of a line
▪ Bus data
• Voltage magnitude
• Source and load of active power
• Source and load of reactive power
▪ Protective devices: Circuit breaker, fuse, isolator…
• From the above parameters line and bus data are basic for power system
analysis such as stability, planning, power flow and so on, whereas protective
devices are mainly used for protection coordination.
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 PER-UNIT REPRESENTATION
▪ Computations for a power system having two or more voltage levels become very
cumbersome when it is necessary to convert currents to a different voltage level
wherever they flow through a transformer (the change in current being inversely
proportional to the transformer turns ratio).

▪ In an alternative and simpler system, a set of base values, or base quantities, is

assumed for each voltage class, and each parameter is expressed as a decimal fraction
of its respective base.

▪ For instance, suppose a rating voltage of 345 KV transmission line has been chosen,
and under certain operating conditions the actual system voltage is 334 kV: then the
ratio of actual to base voltage is 0.97.

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 Cont’…
• A minimum of four base quantities is required to completely define a per- unit system: these are
voltage, current, power, and impedance (or admittance). If two of them are set arbitrarily, then the
other two become fixed.
• Base values can be determined by three ways
• Taking the largest value
• Taking the total sum
• Any arbitrary
• The per unit value of any quantity is the ratio of the actual value in any units to the chosen base
quantity of the same dimensions expressed as a decimal.
𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑎𝑛𝑦 𝑢𝑛𝑖𝑡𝑠
• 𝑃𝑒𝑟 𝑢𝑛𝑖𝑡 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 =
𝐵𝑎𝑠𝑒 𝑜𝑟 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑢𝑛𝑖𝑡
• In power systems the basic quantities of importance are voltage, current, impedance and power.
For all per unit calculations a base KVA or MVA and a base KV are to be chosen.
• Once the base values or reference values are chosen, the other quantities can be obtained as
follows.

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 Cont’…
▪ Where base KV and base MVA are the total or three phase values.
▪ If phase values are used
𝐵𝑎𝑠𝑒 𝐾𝑉𝐴
𝐵𝑎𝑠𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑖𝑛 𝑎𝑚𝑝𝑒𝑟𝑒𝑠 =
𝐵𝑎𝑠𝑒 𝐾𝑉

𝐵𝑎𝑠𝑒 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 Note, per unit

𝐵𝑎𝑠𝑒 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚 = conversion affects
𝐵𝑎𝑠𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
magnitudes, not
(𝑏𝑎𝑠𝑒 𝐾𝑉)2 the angles. Also,
1000
𝑏𝑎𝑠𝑒 𝐾𝑉𝐴 𝑝𝑒𝑟 𝑝ℎ𝑎𝑠𝑒 per unit quantities
no longer have
(𝑏𝑎𝑠𝑒 𝐾𝑉)2 units (i.e.,a voltage
𝐵𝑎𝑠𝑒 𝐼𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚 = is 1.0 p.u., not 1
𝑏𝑎𝑠𝑒 𝑀𝑉𝐴 𝑝𝑒𝑟 𝑝ℎ𝑎𝑠𝑒
▪ In all the above relations the power factor is assumed unity, so that p.u. volts)
▪ Base power KW = base K VA
(𝑎𝑐𝑡𝑢𝑎𝑙 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚)×𝑀𝑉𝐴
▪ 𝑁𝑜𝑤, 𝑃𝑒𝑟 𝑢𝑛𝑖𝑡 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 =
(𝐵𝑎𝑠𝑒 𝐾𝑉)2

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 Cont’…
• Selecting the total or 3-phase KVA as base KVA, for a 3-phase system

𝑏𝑎𝑠𝑒 𝐾𝑉𝐴
• 𝐵𝑎𝑠𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑖𝑛 𝑎𝑚𝑝𝑒𝑟𝑒𝑠 =
√3[𝑏𝑎𝑠𝑒 𝐾𝑉 𝑙𝑖𝑛𝑒 𝑡0 𝑙𝑖𝑛𝑒 ]

(𝑏𝑎𝑠𝑒 𝐾𝑉 𝑙𝑖𝑛𝑒−𝑡𝑜−𝑙𝑖𝑛𝑒 )2
• 𝐵𝑎𝑠𝑒 𝐼𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚𝑠 =
𝑏𝑎𝑠𝑒 𝑀𝑉𝐴

(𝑎𝑐𝑡𝑢𝑎𝑙 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚)×𝑀𝑉𝐴

• 𝑁𝑜𝑤, 𝑃𝑒𝑟 𝑢𝑛𝑖𝑡 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 =
𝑏𝑎𝑠𝑒 𝐾𝑉 2

• Some times, it may be required to use the relation

(𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑖𝑚𝑝𝑒𝑑𝑎𝑎𝑛𝑐𝑒)×(𝑏𝑎𝑠𝑒 𝐾𝑉)2

• 𝑎𝑐𝑡𝑢𝑎𝑙 𝐼𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑜ℎ𝑚 =
𝑏𝑎𝑠𝑒 𝑀𝑉𝐴

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 Cont’…
➢Very often the values are in different base values. In order to convert the per unit impedance from
given base to another base, the following relation can be derived easily.

➢Per unit impedance on new base

2
𝑏𝑎𝑠𝑒 𝑘𝑉𝑜𝑙𝑑 𝑏𝑎𝑠𝑒 𝑘𝑉𝐴𝑛𝑒𝑤
𝑍(𝑝𝑢)𝑛𝑒𝑤 = 𝑍(𝑝𝑢)𝑜𝑙𝑑 × ×
𝑏𝑎𝑠𝑒 𝑘𝑉𝑛𝑒𝑤 𝑏𝑎𝑠𝑒 𝑘𝑉𝐴𝑔𝑖𝑣𝑒𝑛
• In general
• Base current = base volt amperes/base voltage (in amperes)
• Base impedance = base voltage/ base current (in ohms)
• Per-unit voltage = actual voltage / base voltage (per unit, or pu)
• Per-unit current = actual current / base current (per unit, or pu)
• Per-unit impedance = actual impedance /base impedance (per unit. or pu)

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 Cont’…
• Advantage of Per unit system
▪ Avoids calculation complexity & errors due to referring quantities

▪ The p.u systems are ideal for the computerized analysis and simulation of complex power
system problems.

▪ Voltages, currents and impedances expressed in per unit do not change when they are
referred from one side of transformer to the other side.

▪ Per unit impedances of electrical equipment of similar type usually lie within a narrow range

▪ Transformer connections do not affect the per unit values.

▪ Manufacturers usually specify the impedances of machines and transformers in per unit or
percent of name plate ratings

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 Power in Balanced Three-phase System
• The power in a three-phase circuit is defined as the summation of the power
delivered to all the three phases. In a balanced three-phase circuit, though the
power in the individual phases varies sinusoidally at twice the supply frequency, it
will be seen that the total of the instantaneous power in the three-phase circuit is
a constant and independent of time.

• The concepts of instantaneous, active, reactive, apparent, and complex power

and power factor as applied to three-phase circuits will be presented in this
section

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 Cont’
1. Instantaneous Power in Three-phase Circuits
Power in a three-phase balanced network, having phase sequence A-B-C, can be
obtained by taking the sum of the products of instantaneous voltages and currents.
If p is the total instantaneous power, then

The instantaneous values of phase voltages VA,VB,VC and phase currents iA, iB, iC,
can be written as:

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 Cont’
• where Vp, and lp, are the rms values of phase voltage and phase current,
respectively, while 𝜑 is the phase angle between the phase voltage and
corresponding phase current. The instantaneous total power in the three phases
then becomes:

• After simplify the above equation, the instantaneous power becomes:

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 Cont’
• Hence, the average three-phase power is the integration of instantaneous power. Such that:

• The power in a star-connected system becomes:

• Similarly, for a delta-connected system the power becomes

In star connected system

In delta connected system
Therefore, it can be concluded that power in a balanced three-phase network, star or delta, is given
by:

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 Cont’
2. Reactive and Apparent Power

The reactive power in a three-phase system is equal to the sum of the reactive
powers of the individual phases. For a balanced three-phase system, the reactive
power Q is:

The apparent power or the volt-ampere for a three-phase circuit is the sum of the
volt-amperes of the three individual phases.

For a balanced three-phase system, the apparent power is given by

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 Cont’
3. Complex Power
For convenience in calculations, the concept of complex power is useful. It is a complex number the
real part of which is the total active power and the imaginary part is the total reactive power in the
circuit. Thus,

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 Power balance in three phase system
• Similarly, the balanced single phase system will be formulated as:

• After some simplification the active and reactive power of a single phase system is becoming:

• The product of the voltage and current of an ac circuit is termed the apparent power. It is usually
denoted by the symbol S. Thus,

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Summary

Quantity Terminology Descriptions

S Complex power Volteamperes, VA, KVA,
MVA
|S| Apparent power Volteamperes, VA, KVA,
MVA
P Active power Watt, W,KW,MW
Q Reactive power Volte ampere reactive,
Var, KVAr, MVAr

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