Power Generation Concepts

Basic Concepts of Electrical Power Generation

Key Definitions for GATE

Load Factor (LF): Ratio of average load to maximum demand

\[\boxed{LF = \dfrac{\text{Average Load}}{\text{Maximum Demand}} = \dfrac{\text{Total Energy}}{\text{Max Demand} \times \text{Time Period}}}\]
Capacity Factor (CF): Ratio of actual energy produced to maximum possible

\[\boxed{CF = \dfrac{\text{Actual Energy Produced}}{\text{Plant Capacity} \times \text{Time Period}}}\]
Diversity Factor (DF):

\[\boxed{DF = \dfrac{\sum \text{Individual Max Demands}}{\text{System Max Demand}} \geq 1}\]
Plant Use Factor (PUF):

\[\boxed{PUF = \dfrac{\text{Energy Generated}}{\text{Plant Capacity} \times \text{Hours of Operation}}}\]
Base Load: Constant power demand met by reliable plants (e.g., nuclear, coal)
Peak Load: Maximum power demand met by flexible plants (e.g., hydro, gas)

Additional Important Factors

Demand Factor:

\[\boxed{\text{Demand Factor} = \dfrac{\text{Maximum Demand}}{\text{Connected Load}} \leq 1}\]
Utilization Factor:

\[\boxed{\text{Utilization Factor} = \dfrac{\text{Maximum Demand}}{\text{Rated Capacity}}}\]
Reserve Factor:

\[\boxed{\text{Reserve Factor} = \dfrac{\text{Plant Capacity} - \text{Max Demand}}{\text{Max Demand}}}\]
Loss Factor:

\[\boxed{\text{Loss Factor} = \dfrac{\text{Average Power Loss}}{\text{Peak Power Loss}}}\]
Plant Availability Factor:

\[\boxed{\text{PAF} = \dfrac{\text{Available Hours}}{\text{Total Hours}}}\]

Types of Power Plants - Thermal

Thermal Power Plants

Working: Rankine cycle
Efficiency: 30-40%
Fuel: Coal, Oil, Gas
Startup Time: 4-8 hours
Load Factor: 40-60%
Advantages: Reliable, flexible location
Disadvantages: Pollution, fuel cost
Typical Capacity: 100-1000 MW

Types of Power Plants - Hydro & Nuclear

Hydro Power Plants

Efficiency: 85-90%
Startup Time: Few minutes
Life: 50-100 years
Advantages: No fuel cost, clean, quick response
Disadvantages: Site dependent, seasonal variation
Typical Capacity: 10-1000 MW

Nuclear Power Plants

Process: Nuclear fission
Efficiency: 30-35%
Load Factor: 80-90%
Advantages: High energy density, no CO₂
Disadvantages: High capital cost, waste disposal
Typical Capacity: 500-2000 MW

Important for GATE

Hydro plants are best for peak load, thermal for base load, nuclear for base load operation.

Types of Power Plants - Renewable

Solar and Wind Power Plants

Solar PV:
- Efficiency: 15-22%
- Advantages: Clean, scalable
- Disadvantages: Intermittent, land use
Wind:
- Efficiency: \(30-50\%\)
- Advantages: Clean, low operational cost
- Disadvantages: Intermittent, noise
Typical Capacity: Solar: \(1-100 ~\text{MW}\), Wind:\(1-200~\text{ MW}\)

Load Curves and Their Significance

Daily Load Curve: Shows load variation over 24 hours
Monthly Load Curve: Shows peak demands over months
Annual Load Curve: Shows monthly peak demands
Load Duration Curve: Arranges loads in descending order
Chronological Load Curve: Represents load vs time in sequence

Key Point

Area under load curve = Energy consumed

AC and DC Transmission Concepts

AC vs DC Transmission - Technical Comparison

AC vs DC Transmission Comparison
Parameter	AC Transmission	DC Transmission
Initial Cost	Lower	Higher (converters)
Transmission Losses	Higher (skin effect, proximity)	Lower
Power Handling	Limited by stability	Only by thermal limits
Voltage Control	Reactive power needed	Only real power
Synchronization	Required	Not required
Breakeven Distance	\(< 500 ~\text{km}\) overhead	\(> 500~\text{km}\) overhead
	\(< 50 ~\text{km}\) underground	\(> 50 ~\text{km}\) underground
Fault Clearing	Circuit breakers available	Difficult
Reactive Power	Generates/consumes	No reactive power
Converter Efficiency	Not applicable	\(95-98\%\)

HVDC Systems

High Voltage Direct Current

Types:
- Line Commutated Converter (LCC)
- Voltage Source Converter (VSC)
Applications:
- Long distance transmission
- Underwater cables
- Asynchronous interconnection
- Power flow control
Advantages:
- No reactive power transmission
- Better voltage regulation
- No stability issues
Disadvantages:
- High converter cost
- Complex control systems

Transmission Line Parameters

Primary Parameters

Resistance (R):

\[\boxed{R = \dfrac{\rho l}{A} \text{ } \Omega/\text{m}}\]
where \(\rho\) = resistivity, \(l\) = length, \(A\) = cross-sectional area
Inductance (L) (for single conductor):

\[\boxed{L = 2 \times 10^{-7} \ln\left(\dfrac{D}{r'}\right) \text{ H/m}}\]
where \(D\) = spacing between conductors, \(r'\) = \(0.7788r\) (r = radius)
Capacitance (C) (for single conductor):

\[\boxed{C = \dfrac{2\pi\epsilon_0}{\ln(D/r)} \text{ F/m}}\]
where \(\epsilon_0 = 8.854 \times 10^{-12}\) F/m
Conductance (G): Usually negligible for overhead lines

\[\boxed{G = \dfrac{2\pi\sigma}{\ln(D/r)} \text{ S/m}}\]
where \(\sigma\) = conductivity of medium

Secondary Parameters

Characteristic Impedance (Surge Impedance):

\[\boxed{Z_c = Z_0 = \sqrt{\dfrac{R + j\omega L}{G + j\omega C}} \approx \sqrt{\dfrac{L}{C}}}\]
For overhead lines: \(Z_c \approx 400\Omega\), For cables: \(Z_c \approx 50\Omega\)
Propagation Constant:

\[\boxed{\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}}\]
where \(\alpha\) = attenuation constant, \(\beta\) = phase constant
Surge Impedance Loading (SIL):

\[\boxed{P_{SIL} = \dfrac{V^2}{Z_c} \text{ MW}}\]
where \(V\) = line-to-line voltage in kV

Factors Affecting Line Parameters

Resistance

Material (Cu, Al, ACSR)
Temperature coefficient
Skin effect at high frequency
Proximity effect
Spiraling (for stranded conductors)

Inductance

Conductor spacing
Conductor radius
Bundling reduces inductance
Transposition equalizes inductance

Capacitance

Conductor spacing (inversely related)
Conductor radius (directly related)
Height above ground
Bundling increases capacitance

Key Formula

For bundled conductors:

\[L = 2 \times 10^{-7} \ln\left(\dfrac{D}{D_{sb}}\right)\]

where \(D_{sb}\) = bundle GMR

Transmission Line Models

Classification of Transmission Lines

Type	Length	Voltage	Model
Short	\(< 80 ~\text{km}\)	\(< 69~ \text{kV}\)	Series impedance only
Medium	\(80-240 ~\text{km}\)	\(69-138 ~ \text{kV}\)	Nominal \(\pi\) or T
Long	\(> 240 ~\text{km}\)	\(> 138 ~ \text{kV}\)	System: Distributed parameters

Important Note

Classification depends on electrical length, not just physical length!

Short Lines: \(\beta l < 15^{\circ}\)
Medium Lines: \(15° < \beta l < 45°\)
Long Lines: \(\beta l > 45°\)

where \(\beta = \omega\sqrt{LC}\) and \(l\) = line length

Short Transmission Lines

\(< 80~\text{km}\), Series Impedance Model

Assumptions:
- Capacitance neglected
- Uniform current throughout
- Only series impedance considered
Circuit Model: \(Z = R + jX = R + j\omega L\)
Phasor Relations:

\[\boxed{V_s = V_R + IZ}\]

\[\boxed{I_s = I_R = I}\]
ABCD Parameters:

\[A = D = 1, \quad B = Z, \quad C = 0\]

Medium Transmission Lines

\(80-240 ~\text{km}\), Nominal \(\pi\) Model

Assumptions:
- Capacitance lumped at both ends
- Half at sending end, half at receiving end
- Series impedance in middle
ABCD Parameters:

\[A = D = 1 + \dfrac{YZ}{2}\]

\[B = Z\]

\[C = Y\left(1 + \dfrac{YZ}{4}\right)\]
where \(Y = j\omega C\) and \(Z = R + j\omega L\)

Long Transmission Lines

\(> 240 ~\text{km}\), Distributed Parameter Model

Assumptions:
- Parameters uniformly distributed
- Hyperbolic functions used
- Exact solution of wave equations
ABCD Parameters:

\[\boxed{A = D = \cosh(\gamma l)}\]

\[\boxed{B = Z_c \sinh(\gamma l)}\]

\[\boxed{C = \dfrac{1}{Z_c} \sinh(\gamma l)}\]
Phasor Relations:

\[V_s = V_R \cosh(\gamma l) + I_R Z_c \sinh(\gamma l)\]

\[I_s = I_R \cosh(\gamma l) + \dfrac{V_R}{Z_c} \sinh(\gamma l)\]

Equivalent \(\pi\) Model for Long Lines

Equivalent Series Impedance:

\[Z' = Z_c \sinh(\gamma l)\]
Equivalent Shunt Admittance:

\[Y' = \dfrac{1}{Z_c} \tanh\left(\dfrac{\gamma l}{2}\right)\]
For lossless lines (\(R = G = 0\)):

\[\gamma = j\beta = j\omega\sqrt{LC}\]

\[Z' = jZ_c \sin(\beta l)\]

\[Y' = j\dfrac{1}{Z_c} \tan\left(\dfrac{\beta l}{2}\right)\]

Performance of Transmission Lines

Voltage Regulation

Key Performance Parameter

Definition: Change in receiving end voltage from no-load to full-load

\[\boxed{\%VR = \dfrac{|V_{R(no\ load)}| - |V_{R(full\ load)}|}{|V_{R(full\ load)}|} \times 100}\]
For short lines:

\[\%VR = \dfrac{IR\cos\phi + IX\sin\phi}{V_R} \times 100\]
Factors affecting VR:
- Load power factor (lagging increases VR)
- Line impedance (higher R, X increase VR)
- Load magnitude
- Line length
Good regulation: \(\mathrm{VR} < 10\%\)

Transmission Efficiency

Definition:

\[\boxed{\eta = \dfrac{P_{received}}{P_{sent}} \times 100 = \dfrac{P_{sent} - P_{loss}}{P_{sent}} \times 100}\]
For short lines:

\[P_{loss} = I^2R = \left(\dfrac{P}{\sqrt{3}V_L\cos\phi}\right)^2 \times 3R\]
Condition for maximum efficiency:

\[\dfrac{d\eta}{dP} = 0\]
This gives: Variable losses = Fixed losses
Ways to improve efficiency:
- Increase transmission voltage
- Reduce line resistance
- Improve power factor
- Reduce line length

Ferranti Effect

Voltage Rise in Long Lines

Definition: Receiving end voltage higher than sending end voltage at no load
Cause: Capacitive reactance dominates at light loads
Occurs in: Long lines and cables
Mathematical expression:

\[\dfrac{V_R}{V_S} = \dfrac{1}{\cos(\beta l)}\]
Mitigation:
- Shunt reactors
- Load compensation
- Voltage regulators

Corona Effect

Definition: Ionization of air around conductors at high voltage
Critical disruptive voltage:

\[V_c = 21.1 \times m \times \delta \times r \ln\left(\dfrac{d}{r}\right) \text{ kV}\]
where \(m\) = surface factor, \(\delta\) = air density factor, \(r\) = radius, \(d\) = spacing
Visual disruptive voltage:

\[V_v = 21.1 \times m \times \delta \times r \ln\left(\dfrac{d}{r}\right) \left[1 + \dfrac{0.3}{\sqrt{\delta r}}\right]\]
Corona power loss:

\[P_c = \dfrac{244 \times f \times (\delta/\delta_0)^2}{r} \left(\dfrac{V - V_c}{\log_{10}(d/r)}\right)^2 \times 10^{-6} \text{ MW/km}\]
Factors affecting corona: Surface condition, weather, conductor size, spacing

Cables vs Overhead Lines

Underground Cables vs Overhead Lines

Comparison of underground cables and overhead lines
Parameter	Overhead Lines	Underground Cables
Capital Cost	Lower	5-10 times higher
Maintenance Cost	Higher	Lower
Inductance	Higher	Lower
Capacitance	Lower (3-5 nF/km)	Higher (50-300 nF/km)
Surge Impedance	400-500 \(\Omega\)	20-80 \(\Omega\)
Thermal Rating	Higher	Lower
Fault Location	Easy	Difficult
Environmental Impact	Higher	Lower
Reliability	Lower	Higher
Life Span	25-40 years	30-50 years
Right of Way	More	Less

Cable Construction and Types

Low Voltage Cables (up to 1.1 kV):
- PVC insulation
- Lead sheath
- Armoring
High Voltage Cables (up to 33 kV):
- XLPE insulation
- Metallic sheath
- Better thermal properties
EHV Cables (above 33 kV):
- Oil-filled or gas-filled
- Special cooling arrangements
- Expensive

Cable Parameters and Characteristics

Capacitance of single-core cable:

\[C = \dfrac{2\pi\epsilon}{\ln(R/r)} \text{ F/m}\]
where \(R\) = inner radius of sheath, \(r\) = conductor radius
Capacitance of 3-core cable:

\[C = \dfrac{2\pi\epsilon}{\ln(s/r)} \text{ F/m}\]
where \(s\) = center-to-center distance between cores
Thermal considerations:
- Heat generated by \(I^2R\) losses
- Limited by insulation temperature
- Derating factors for grouped cables
Dielectric losses:

\[P_d = V^2 \omega C \tan\delta \text{ W/m}\]
where \(\tan\delta\) = loss tangent of insulation
Charging Current:

\[I_c = V \omega C \text{ A/m}\]

Important Formulas Summary

Key Formulas to Remember - Part 1

Load Factor: \(LF = \dfrac{\text{Average Load}}{\text{Peak Load}}\)
Surge Impedance: \(Z_c = \sqrt{L/C}\) (For lossless line)
Surge Impedance Loading: \(P_{SIL} = \dfrac{V^2}{Z_c}\)
Voltage Regulation: \(\%VR = \dfrac{|V_{no-load}| - |V_{full-load}|}{|V_{full-load}|} \times 100\)
Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} \times 100\)
Skin Depth: \(\delta = \sqrt{\dfrac{2\rho}{\omega\mu}}\)
Sag: \(s = \dfrac{wL^2}{8T}\) (for equal supports)

Key Formulas to Remember - Part 2

ABCD Parameters: \(\begin{bmatrix} V_s \\ I_s \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix}\)
Symmetrical Network: \(AD - BC = 1\), \(A = D\)
Propagation Constant: \(\gamma = \sqrt{zy} = \alpha + j\beta\)
Ferranti Effect: \(\dfrac{V_R}{V_S} = \dfrac{1}{\cos(\beta l)}\) (no load)
Corona Critical Voltage: \(V_c = 21.1 \times m \times \delta \times r \ln(d/r)\)
Bundle Conductor GMR: \(D_{sb} = \sqrt[n]{d_1 \times d_2 \times ... \times d_n \times r^{n-1}}\)
Transposition: Equalizes impedance in 3-phase systems

Quick Tips for GATE Exam

Remember These Points

Capacitance effect is negligible for short lines
Ferranti effect occurs in long lines at light loads
HVDC is better for long distances (\(> 500 ~\text{km}\) overhead)
Bundle conductors reduce corona and improve power transfer
Transposition equalizes impedances in 3-phase lines
SIL represents natural loading of transmission line

Common Mistakes to Avoid

Don’t forget to convert units (km to m, kV to V)
Line-to-line voltage vs line-to-neutral voltage
ABCD parameters: \(AD - BC = 1\) for passive networks
Surge impedance is different for overhead lines vs cables

Practice Problems - Quick Check

A 400 kV line has surge impedance of 400 \(\Omega\). Find SIL.

\[P_{SIL} = \dfrac{(400)^2}{400} = 400 \text{ MW}\]
If diversity factor is 1.5 and sum of individual demands is 150 MW, find system maximum demand.

\[\text{System Max Demand} = \dfrac{150}{1.5} = 100 \text{ MW}\]
A short line has R = 20 \(\Omega\), X = 80 \(\Omega\). For a load of 50 MW at 0.8 pf lagging at 132 kV, find voltage regulation.

\[I = \dfrac{50 \times 10^6}{\sqrt{3} \times 132 \times 10^3 \times 0.8} = 273 \text{ A}\]

\[\%VR = \dfrac{273(20 \times 0.8 + 80 \times 0.6)}{132000/\sqrt{3}} \times 100 = 8.6\%\]