Power Generation and Transmission Concepts GATE EE Exam Quick Notes

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

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

  1. Load Factor: \(LF = \dfrac{\text{Average Load}}{\text{Peak Load}}\)

  2. Surge Impedance: \(Z_c = \sqrt{L/C}\) (For lossless line)

  3. Surge Impedance Loading: \(P_{SIL} = \dfrac{V^2}{Z_c}\)

  4. Voltage Regulation: \(\%VR = \dfrac{|V_{no-load}| - |V_{full-load}|}{|V_{full-load}|} \times 100\)

  5. Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} \times 100\)

  6. Skin Depth: \(\delta = \sqrt{\dfrac{2\rho}{\omega\mu}}\)

  7. Sag: \(s = \dfrac{wL^2}{8T}\) (for equal supports)

Key Formulas to Remember - Part 2

  1. ABCD Parameters: \(\begin{bmatrix} V_s \\ I_s \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix}\)

  2. Symmetrical Network: \(AD - BC = 1\), \(A = D\)

  3. Propagation Constant: \(\gamma = \sqrt{zy} = \alpha + j\beta\)

  4. Ferranti Effect: \(\dfrac{V_R}{V_S} = \dfrac{1}{\cos(\beta l)}\) (no load)

  5. Corona Critical Voltage: \(V_c = 21.1 \times m \times \delta \times r \ln(d/r)\)

  6. Bundle Conductor GMR: \(D_{sb} = \sqrt[n]{d_1 \times d_2 \times ... \times d_n \times r^{n-1}}\)

  7. 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

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

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

    \[\text{System Max Demand} = \dfrac{150}{1.5} = 100 \text{ MW}\]
  3. 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.

    \[I = \dfrac{50 \times 10^6}{\sqrt{3} \times 132 \times 10^3 \times 0.8} = 273 \text{ A}\]
    \[\%VR = \dfrac{273(20 \times 0.8 + 80 \times 0.6)}{132000/\sqrt{3}} \times 100 = 8.6\%\]