AEE Electrical Engineering Core · Chapter 8

Per-Unit System, Load Flow, Voltage Control and Economic Operation

What to remember

Per-unit value = actual value ÷ base value. Base MVA is common to the whole system; base kV changes only across transformers. Transformer impedance in per-unit is the same on both sides.
Load flow solves for voltage magnitude and angle at every bus. Buses are slack, PV (generator) or PQ (load). Newton-Raphson converges fastest; Gauss-Seidel is simplest.
Economic operation = equal incremental cost. Without losses, all units run at the same incremental cost λ. With losses, each unit also carries a penalty factor.

Per-unit system

In the per-unit (pu) system every quantity is written as a fraction of a chosen base. Two bases are chosen freely (base MVA and base kV). The other two follow from them.

Base current (three-phase) Ibase = Sbase / (√3 × Vbase), where Sbase is three-phase MVA and Vbase is line-to-line kV.
Base impedance Zbase = (kVbase)² / MVAbase (ohm).
Z(pu) = Z(actual) / Zbase. Voltage (pu) = V / Vbase. Power (pu) = S / Sbase.

Change of base. A machine rating gives Z in pu on its own rating. To refer it to a new base:

Z(new) = Z(old) × (MVA new / MVA old) × (kV old / kV new)²

Why use per-unit?

Transformer impedance has the same pu value whether seen from the high-voltage or low-voltage side, so ideal transformers vanish from the diagram.
Pu values of machines of the same type lie in a narrow range, so errors are easy to spot.
Calculations become simple, and the √3 factors are avoided.

Worked example 1. Base 100 MVA, 11 kV. Zbase = 11² / 100 = 1.21 ohm. A reactance of 0.605 ohm is 0.5 pu.

Worked example 2. A generator is rated 50 MVA, 11 kV with X = 0.2 pu. On a 100 MVA, 11 kV base, X = 0.2 × (100/50) = 0.4 pu.

Worked example 3. A transformer of 0.1 pu on 50 MVA, 20 kV is moved to a base of 100 MVA, 10 kV. X = 0.1 × 2 × (20/10)² = 0.8 pu.

Choosing bases in a system with transformers. Pick one MVA base for the whole system. Pick the kV base in one zone, then find the other zones by the transformer turn ratio (line-to-line voltage ratio). Convert each equipment value to this base before drawing the pu impedance diagram.

Load flow (power flow) studies

A load flow study finds the steady-state voltages (magnitude and angle), line flows and losses for a given generation and load pattern. It is used for planning, operation and as the starting point for fault and stability studies.

Bus classification

Bus type	Specified	Unknown	Notes
Slack (swing, reference)		V	and angle δ	P and Q	One per system; takes up the losses
PV (generator, voltage-controlled)	P and	V		Q and δ	Q limits of the machine apply
PQ (load)	P and Q		V	and δ	Most buses are of this type

Bus admittance matrix (Ybus). Diagonal element Yii is the sum of all admittances connected to bus i. Off-diagonal element Yij is the negative of the admittance between buses i and j. Ybus is square, symmetric (no phase shifters) and sparse for large systems. The equation is I = Ybus × V.

Methods

Method	Key feature
Gauss-Seidel	Simple; uses Ybus; slow, needs many iterations; iterations grow with bus number
Newton-Raphson (NR)	Uses Jacobian matrix; quadratic convergence; few iterations; best for large systems
Fast decoupled (FDLF)	Decouples P-δ and Q-V; constant matrices; very fast; works when R/X is small

In NR the Jacobian is made of partial derivatives of P and Q with respect to δ and |V|. P is strongly coupled to δ and Q to |V|; the fast decoupled method exploits the weak P-|V| and Q-δ coupling, while full NR keeps all four Jacobian blocks. Real power mainly follows the angle difference; reactive power mainly follows the voltage magnitude difference.

Acceleration factor. In Gauss-Seidel, an acceleration factor (about 1.6) speeds convergence.

Outputs. Bus voltages, power flow on each line, losses, and the reactive power the generators must supply.

Voltage control

Voltage must stay within a small band of the rated value for consumers. Reactive power flow is the main reason for voltage change. A lagging load draws reactive power and lowers voltage.

Voltage regulation = (Vno-load − Vfull-load) / Vfull-load × 100%. If the no-load voltage is 10.5 kV and the full-load voltage is 10 kV, regulation is 5%.

Ferranti effect. In a long lightly loaded line, the receiving-end voltage rises above the sending-end voltage, because of line charging current flowing through the line inductance.

Methods of voltage control

Device	Action
Shunt capacitor	Supplies lagging VAr; raises voltage; also improves power factor
Shunt reactor	Absorbs VAr; lowers voltage on lightly loaded long lines
Series capacitor	Cancels part of line reactance; reduces voltage drop
Synchronous condenser	Over-excited: supplies VAr; under-excited: absorbs VAr; smooth control
On-load tap changer (OLTC)	Changes turns ratio under load; used in transformers at substations
Booster transformer	Injects a voltage in series with the line
SVC (static VAr compensator)	Thyristor-controlled reactor plus capacitor; fast, stepless control
STATCOM	Voltage source converter based; faster than SVC; output current does not fall with voltage
Automatic voltage regulator (AVR)	Controls generator excitation

Key points

Reactive power from a capacitor is proportional to V². At 0.9 pu voltage a 100 kVAr capacitor gives 81 kVAr.
Tap changers work in steps. A ±10% range in 1.25% steps gives 8 steps on each side of the nominal tap.
Tap changing controls voltage but does not generate reactive power. It only shifts the problem to another part of the network.
Excitation control affects generator terminal voltage and the reactive power it delivers.
Voltage control and reactive power are linked; real power is controlled by the turbine governor and frequency.

Economic operation of power systems

Fuel cost curve. The cost of a thermal unit is usually written as C = a + bP + cP² (Rs/h), where P is the output in MW.

Incremental cost (IC). IC = dC/dP (Rs/MWh). For C = 0.1P² + 20P, IC = 0.2P + 20.

Economic dispatch without losses. For minimum total cost with a fixed demand, all units that are not at limits operate at equal incremental cost: dC1/dP1 = dC2/dP2 = … = λ. Also ΣP = demand.

Worked example. Two units with IC1 = 0.2P1 + 20 and IC2 = 0.2P2 + 30; demand 200 MW.

Set 0.2P1 + 20 = 0.2(200 − P1) + 30. So 0.4P1 = 50, P1 = 125 MW, P2 = 75 MW, λ = 45 Rs/MWh.

With losses (coordination equation). dC/dP × L = λ, where L = 1 / (1 − ∂PL/∂PG) is the penalty factor. If ∂PL/∂PG = 0.2, penalty factor = 1/0.8 = 1.25. A plant far from the load has a higher penalty factor and is loaded less.

Transmission loss formula (B-coefficients). PL = Σ Pi Bij Pj. The loss coefficients are found from load flow data.

Generator limits. Each unit has Pmin and Pmax. A unit that reaches a limit is held there and the remaining demand is shared by the others at equal λ.

Related concepts

Unit commitment: deciding which units are on or off for each hour, considering start-up cost, minimum up/down time and reserve. A simple method is the priority list (switch on the cheapest full-load cost unit first).
Spinning reserve: extra capacity of units already running and synchronised, ready to take sudden load.
Hydrothermal scheduling: water is limited, so hydro units are scheduled to save fuel in thermal units. Run-of-river plants run as base load.
Load duration curve: hours for which load equals or exceeds a value; used to plan base, intermediate and peak plants.
Merit order: low-cost plants (hydro, nuclear, efficient thermal) are loaded first.
AP-specific: the state's generation is shared among thermal stations, hydro projects such as Srisailam, and solar and wind parks. Dispatch must respect water release rules and grid code limits.

Exam traps

Base MVA is common to the whole system; base kV changes across a transformer. Do not mix them up.
Zbase = kV² / MVA, not kV / MVA.
Transformer pu impedance is the same on both sides; actual ohm values are not.
Slack bus has |V| and δ specified; PV bus has P and |V| specified. P and |V| at PV and P and Q at PQ are easy to swap.
Newton-Raphson has quadratic convergence; Gauss-Seidel has linear convergence.
Capacitor VAr varies as V², not as V.
An OLTC changes voltage, not the generated reactive power.
Equal incremental cost is correct only when losses are ignored and limits are not hit.
Penalty factor greater than 1 means the plant contributes more to losses.
Ferranti effect needs a light load or open-ended long line; it is not a heavy-load effect.
Shunt reactor lowers voltage; shunt capacitor raises it.
Unit commitment decides on/off status; economic dispatch decides the output of running units.

One-liners

1. Per-unit value = actual value / base value; it has no unit.
2. Ibase = Sbase / (√3 Vbase) for a three-phase system.
3. Zbase = (kV)² / MVA in ohm.
4. Z(new pu) = Z(old pu) × (MVA new / MVA old) × (kV old / kV new)².
5. The slack bus supplies the system loss and provides the angle reference.
6. A load bus is a PQ bus; a generator bus is a PV bus.
7. Ybus diagonal element is the sum of connected admittances.
8. Newton-Raphson needs the Jacobian matrix and converges quadratically.
9. Fast decoupled load flow uses weak P-V and Q-δ coupling.
10. Shunt capacitors supply lagging reactive power and raise voltage.
11. Synchronous condenser is an over- or under-excited motor running with no mechanical load.
12. Economic dispatch: equal incremental cost; with losses, equal ICs multiplied by penalty factors.

Practice questions

What is the base impedance, in ohm, for a base of 33 kV and 100 MVA?
1. 3.3
2. 10.89
3. 108.9
4. 0.33
Answer
B. 10.89
Zbase = kV²/MVA = 1089/100 = 10.89 ohm.
What is the base current for a three-phase base of 100 MVA and 11 kV (line-to-line)?
1. 1.75 kA
2. 5.25 kA
3. 9.09 kA
4. 3.03 kA
Answer
B. 5.25 kA
Ibase = 100/(√3 × 11) = 5.25 kA.
A generator of 50 MVA, 11 kV has a reactance of 0.2 pu. What is its reactance on a base of 100 MVA, 11 kV?
1. 0.4 pu
2. 0.2 pu
3. 0.8 pu
4. 0.1 pu
Answer
A. 0.4 pu
X = 0.2 × (100/50) = 0.4 pu.
A transformer reactance is 0.1 pu on 50 MVA, 20 kV. What is it on a base of 100 MVA, 10 kV?
1. 0.2 pu
2. 0.4 pu
3. 0.8 pu
4. 0.05 pu
Answer
C. 0.8 pu
X = 0.1 × (100/50) × (20/10)² = 0.8 pu.
On a base of 100 MVA and 10 kV, an actual impedance of 5 ohm is equal to:
1. 0.05 pu
2. 50 pu
3. 0.5 pu
4. 5 pu
Answer
D. 5 pu
Zbase = 10²/100 = 1 ohm, so Z = 5/1 = 5 pu.
A bus voltage is 0.95 pu on a 220 kV base. The actual line-to-line voltage is:
1. 190 kV
2. 231 kV
3. 220.95 kV
4. 209 kV
Answer
D. 209 kV
V = 0.95 × 220 = 209 kV.
A system has 10 buses with one slack bus and three generator (PV) buses. How many load (PQ) buses are there?
1. 7
2. 5
3. 6
4. 3
Answer
C. 6
PQ buses = 10 − 1 − 3 = 6.
How many elements does the bus admittance matrix of a 4-bus system have?
1. 4
2. 16
3. 8
4. 12
Answer
B. 16
Ybus is a 4 × 4 matrix, so it has 16 elements.
A transformer has no-load secondary voltage of 10.5 kV and full-load secondary voltage of 10 kV. Its voltage regulation is:
1. 5%
2. 0.5%
3. 4.76%
4. 10%
Answer
A. 5%
Regulation = (10.5 − 10)/10 × 100 = 5%.
A transformer has taps of ±10% in steps of 1.25%. How many steps are there on each side of the nominal tap?
1. 4
2. 10
3. 16
4. 8
Answer
D. 8
10 ÷ 1.25 = 8 steps.
A shunt capacitor rated 100 kVAr at 1 pu voltage is operated at 0.9 pu voltage. The reactive power supplied is:
1. 100 kVAr
2. 110 kVAr
3. 81 kVAr
4. 90 kVAr
Answer
C. 81 kVAr
Q varies as V²: 100 × 0.81 = 81 kVAr.
Two units have IC1 = 0.2P1 + 20 and IC2 = 0.2P2 + 30 (Rs/MWh). For a total demand of 200 MW with no losses, the economic output of unit 1 is:
1. 100 MW
2. 125 MW
3. 75 MW
4. 150 MW
Answer
B. 125 MW
Equal IC: 0.2P1 + 20 = 0.2(200 − P1) + 30 gives P1 = 125 MW.
For a unit with C = 0.02P² + 10P + 100 (Rs/h), the incremental cost at P = 50 MW is:
1. 12 Rs/MWh
2. 10 Rs/MWh
3. 13 Rs/MWh
4. 11 Rs/MWh
Answer
A. 12 Rs/MWh
IC = 0.04P + 10 = 0.04 × 50 + 10 = 12.
If ∂PL/∂PG = 0.2 for a plant, its penalty factor is:
1. 0.8
2. 0.2
3. 1.25
4. 1.2
Answer
C. 1.25
PF = 1/(1 − 0.2) = 1.25.
In the per-unit system, a quantity is expressed as:
1. Actual value multiplied by base value
2. Actual value minus base value
3. Base value divided by actual value
4. Actual value divided by base value
Answer
D. Actual value divided by base value
Per-unit = actual/base; it is dimensionless.
The base impedance in ohm is given by:
1. (kV base)² / MVA base
2. kV base / MVA base
3. MVA base / (kV base)²
4. (MVA base)² / kV base
Answer
A. (kV base)² / MVA base
Zbase = kV²/MVA.
The per-unit impedance of a transformer, when the bases follow the voltage ratio, is:
1. Higher on the low-voltage side
2. The same on the primary and secondary sides
3. Higher on the high-voltage side
4. Zero on the secondary side
Answer
B. The same on the primary and secondary sides
Per-unit impedance is unchanged by the turns ratio.
In a load flow study, which bus has its voltage magnitude and angle specified?
1. Any load bus
2. Slack bus
3. PV bus
4. PQ bus
Answer
B. Slack bus
The slack bus is the reference; |V| and δ are fixed, and P and Q are found.
At a PV bus, which quantities are specified?
1. Real power and voltage magnitude
2. Real and reactive power
3. Reactive power and angle
4. Voltage magnitude and angle
Answer
A. Real power and voltage magnitude
A generator bus has P and |V| fixed; Q and δ are computed.
At a load (PQ) bus, the unknown quantities are:
1. Real power and angle
2. Reactive power and magnitude
3. Voltage magnitude and angle
4. Real and reactive power
Answer
C. Voltage magnitude and angle
P and Q are given; |V| and δ are found.
The diagonal element Yii of the bus admittance matrix equals:
1. The sum of all admittances connected to bus i
2. The negative of the admittance between buses i and j
3. The reciprocal of the bus impedance
4. The shunt admittance at bus i only
Answer
A. The sum of all admittances connected to bus i
Yii sums every admittance terminating at bus i; Yij is the negative of the admittance between i and j.
Which load flow method has quadratic convergence?
1. Gauss-Seidel with acceleration factor
2. Direct substitution
3. Gauss-Seidel
4. Newton-Raphson
Answer
D. Newton-Raphson
Newton-Raphson converges quadratically near the solution.
Fast decoupled load flow is based on the weak coupling between:
1. P and Q
2. P and angle, and Q and voltage magnitude
3. Voltage and current
4. P and voltage magnitude, and Q and angle
Answer
D. P and voltage magnitude, and Q and angle
P is mainly linked to angle and Q to magnitude; the cross couplings are weak and neglected.
The Ferranti effect refers to:
1. Skin effect in conductors
2. A rise in receiving-end voltage in a long, lightly loaded line
3. A fall in receiving-end voltage in a heavily loaded line
4. Loss of power by corona
Answer
B. A rise in receiving-end voltage in a long, lightly loaded line
Line charging current through the inductance raises the receiving-end voltage.
A shunt reactor is used mainly to:
1. Improve stability by adding real power
2. Reduce fault level to zero
3. Lower voltage on lightly loaded long lines
4. Raise voltage at heavy load
Answer
C. Lower voltage on lightly loaded long lines
A shunt reactor absorbs reactive power and so lowers voltage.
A synchronous condenser absorbs reactive power when it is:
1. Over-excited
2. Delivering real power
3. Under-excited
4. Running at no field
Answer
C. Under-excited
An under-excited machine takes lagging VAr (absorbs reactive power); over-excited supplies it.
Which of the following is a thyristor-based fast reactive power compensator?
1. Booster transformer
2. Static VAr compensator
3. Buchholz relay
4. Peterson coil
Answer
B. Static VAr compensator
An SVC uses a thyristor-controlled reactor with capacitors.
Voltage drop in a transmission line is mainly governed by:
1. The flow of reactive power
2. Frequency
3. Corona loss
4. The flow of real power
Answer
A. The flow of reactive power
Because X is much larger than R, the drop depends mostly on Q.
The condition for economic load dispatch without losses is that:
1. All units share equal load
2. All units run at full load
3. All units have equal efficiency
4. All units operate at equal incremental cost
Answer
D. All units operate at equal incremental cost
Total cost is minimum when dC/dP is the same for all units not at limits.
The penalty factor of a plant is given by:
1. 1 / (1 + ∂PL/∂PG)
2. ∂PL/∂PG
3. 1 − ∂PL/∂PG
4. 1 / (1 − ∂PL/∂PG)
Answer
D. 1 / (1 − ∂PL/∂PG)
Penalty factor L = 1/(1 − ∂PL/∂PG).
Unit commitment is the process of deciding:
1. Which generating units are on or off at each hour
2. The exact output of each running unit in real time
3. The tap setting of transformers
4. The bus voltages
Answer
A. Which generating units are on or off at each hour
Unit commitment is the on/off schedule; dispatch decides outputs.
Spinning reserve is:
1. Water stored in a dam
2. Spare capacity in running synchronised units
3. Capacity of cold-standby units
4. Reserve of coal at a station
Answer
B. Spare capacity in running synchronised units
Spinning reserve is available immediately from units already on the bus.
Statements on the per-unit system: 1. The base MVA is the same for the whole system. 2. The base kV is the same on both sides of a transformer.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Base kV changes across a transformer in the ratio of its voltages; base MVA is common.
Statements on bus types: 1. The slack bus has real and reactive power specified. 2. A PQ bus has voltage magnitude as an unknown.
1. 1 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 2 only
Answer
D. 2 only
The slack bus has |V| and δ specified, so statement 1 is wrong; a PQ bus has |V| unknown.
Statements on Newton-Raphson load flow: 1. It has quadratic convergence. 2. It uses a Jacobian matrix.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are true; it needs few iterations, each costing more computation.
Statements on voltage control: 1. A shunt capacitor raises voltage. 2. A shunt reactor raises voltage.
1. 2 only
2. 1 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 1 only
A shunt reactor absorbs reactive power and lowers voltage.
Statements on economic dispatch: 1. Without losses, units are loaded at equal incremental cost. 2. With losses, the penalty factor is included.
1. Both 1 and 2
2. 1 only
3. 2 only
4. Neither 1 nor 2
Answer
A. Both 1 and 2
Both are correct; the coordination equation is IC × penalty factor = λ.
Statements on the bus admittance matrix: 1. It is symmetric when there are no phase-shifting transformers. 2. It is dense for large practical systems.
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
D. 1 only
Ybus is sparse for large systems, since most buses connect to only a few others.
Statements on tap-changing transformers: 1. An OLTC changes the voltage ratio while on load. 2. An OLTC generates reactive power.
1. 2 only
2. Both 1 and 2
3. 1 only
4. Neither 1 nor 2
Answer
C. 1 only
An OLTC redistributes voltage but does not generate reactive power.
Statements on dispatch and commitment: 1. Incremental cost is dC/dP. 2. Unit commitment decides the output of each running unit minute by minute.
1. 2 only
2. 1 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 1 only
Unit commitment decides on/off status; economic dispatch sets the output.
Which pair is correctly matched?
1. Shunt reactor – raises voltage at heavy load
2. PV bus – P and Q specified
3. Ferranti effect – voltage rise in a lightly loaded long line
4. Slack bus – P and Q specified
Answer
C. Ferranti effect – voltage rise in a lightly loaded long line
The other pairs state wrong specified quantities or effects.
Statements on voltage regulation and reactive power: 1. Capacitor reactive power varies as the square of voltage. 2. A lagging load lowers the receiving-end voltage.
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
B. Both 1 and 2
Q = V²/Xc and a lagging load draws reactive power, increasing the voltage drop.
Statements on hydro and thermal scheduling: 1. Run-of-river plants are usually run as base load. 2. Water use in hydro plants has no limit, so fuel saving is not a consideration.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Water is limited, so hydrothermal scheduling aims to save fuel in thermal plants.

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1.Other than the nervous system, which other system helps in control and coordination in our body?
2.Which component is used to control the flow of electricity without changing the voltage source?
3.The central nervous system that receives information from all parts of the body and integrates it is constituted by:
4.The fundamental unit which is common in FPS and MKS systems is _________.
5.The figure given below shows the sectional view of the human female reproductive system. Identify the option stating correct functions of the labelled parts A and B.
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8.The system of scientific naming Organism was introduced by:
9.Through which part of the female reproductive system does the egg travel from the ovary to the uterus?
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