Power Generation Concepts
Basic Concepts of Electrical Power Generation
Key Definitions for GATE
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Load Factor (LF): Ratio of average load to maximum demand
\[\boxed{LF = \dfrac{\text{Average Load}}{\text{Maximum Demand}} = \dfrac{\text{Total Energy}}{\text{Max Demand} \times \text{Time Period}}}\]
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Capacity Factor (CF): Ratio of actual energy produced to maximum possible
\[\boxed{CF = \dfrac{\text{Actual Energy Produced}}{\text{Plant Capacity} \times \text{Time Period}}}\]
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Diversity Factor (DF):
\[\boxed{DF = \dfrac{\sum \text{Individual Max Demands}}{\text{System Max Demand}} \geq 1}\]
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Plant Use Factor (PUF):
\[\boxed{PUF = \dfrac{\text{Energy Generated}}{\text{Plant Capacity} \times \text{Hours of Operation}}}\]
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Base Load: Constant power demand met by reliable plants (e.g., nuclear, coal)
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Peak Load: Maximum power demand met by flexible plants (e.g., hydro, gas)
Additional Important Factors
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Demand Factor:
\[\boxed{\text{Demand Factor} = \dfrac{\text{Maximum Demand}}{\text{Connected Load}} \leq 1}\]
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Utilization Factor:
\[\boxed{\text{Utilization Factor} = \dfrac{\text{Maximum Demand}}{\text{Rated Capacity}}}\]
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Reserve Factor:
\[\boxed{\text{Reserve Factor} = \dfrac{\text{Plant Capacity} - \text{Max Demand}}{\text{Max Demand}}}\]
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Loss Factor:
\[\boxed{\text{Loss Factor} = \dfrac{\text{Average Power Loss}}{\text{Peak Power Loss}}}\]
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Plant Availability Factor:
\[\boxed{\text{PAF} = \dfrac{\text{Available Hours}}{\text{Total Hours}}}\]
Types of Power Plants - Thermal
Thermal Power Plants
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Working: Rankine cycle
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Efficiency: 30-40%
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Fuel: Coal, Oil, Gas
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Startup Time: 4-8 hours
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Load Factor: 40-60%
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Advantages: Reliable, flexible location
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Disadvantages: Pollution, fuel cost
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Typical Capacity: 100-1000 MW
Types of Power Plants - Hydro & Nuclear
Hydro Power Plants
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Efficiency: 85-90%
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Startup Time: Few minutes
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Life: 50-100 years
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Advantages: No fuel cost, clean, quick response
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Disadvantages: Site dependent, seasonal variation
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Typical Capacity: 10-1000 MW
Nuclear Power Plants
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Process: Nuclear fission
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Efficiency: 30-35%
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Load Factor: 80-90%
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Advantages: High energy density, no CO2
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Disadvantages: High capital cost, waste disposal
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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
Load Curves and Their Significance
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Daily Load Curve: Shows load variation over 24 hours
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Monthly Load Curve: Shows peak demands over months
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Annual Load Curve: Shows monthly peak demands
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Load Duration Curve: Arranges loads in descending order
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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
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
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Types:
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Applications:
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Advantages:
-
Disadvantages:
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High converter cost
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Complex control systems
Transmission Line Parameters
Transmission Line Parameters
Primary Parameters
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Resistance (R):
\[\boxed{R = \dfrac{\rho l}{A} \text{ } \Omega/\text{m}}\]
where \(\rho\) = resistivity, \(l\) = length, \(A\) = cross-sectional area
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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)
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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
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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
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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: \(Z_c \approx 50\Omega\)
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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
Capacitance
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Conductor spacing (inversely related)
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Conductor radius (directly related)
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Height above ground
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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
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!
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Short Lines: \(\beta l < 15^{\circ}\)
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Medium Lines: \(15° < \beta l < 45°\)
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Long Lines: \(\beta l > 45°\)
where \(\beta = \omega\sqrt{LC}\) and \(l\) = line length
Short Transmission Lines
\(< 80~\text{km}\), Series Impedance Model
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Assumptions:
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Circuit Model: \(Z = R + jX = R + j\omega L\)
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Phasor Relations:
\[\boxed{V_s = V_R + IZ}\]
\[\boxed{I_s = I_R = I}\]
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ABCD Parameters:
\[A = D = 1, \quad B = Z, \quad C = 0\]
Medium Transmission Lines
\(80-240 ~\text{km}\), Nominal \(\pi\) Model
Long Transmission Lines
\(> 240 ~\text{km}\), Distributed Parameter Model
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Assumptions:
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Parameters uniformly distributed
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Hyperbolic functions used
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Exact solution of wave equations
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ABCD Parameters:
\[\boxed{A = D = \cosh(\gamma l)}\]
\[\boxed{B = Z_c \sinh(\gamma l)}\]
\[\boxed{C = \dfrac{1}{Z_c} \sinh(\gamma l)}\]
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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
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Equivalent Series Impedance:
\[Z' = Z_c \sinh(\gamma l)\]
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Equivalent Shunt Admittance:
\[Y' = \dfrac{1}{Z_c} \tanh\left(\dfrac{\gamma l}{2}\right)\]
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For lossless lines (\(R = G = 0\)):
\[\gamma = j\beta = j\omega\sqrt{LC}\]
\[Z' = jZ_c \sin(\beta l)\]
\[Y' = j\dfrac{1}{Z_c} \tan\left(\dfrac{\beta l}{2}\right)\]
Transmission Efficiency
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Definition:
\[\boxed{\eta = \dfrac{P_{received}}{P_{sent}} \times 100 = \dfrac{P_{sent} - P_{loss}}{P_{sent}} \times 100}\]
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For short lines:
\[P_{loss} = I^2R = \left(\dfrac{P}{\sqrt{3}V_L\cos\phi}\right)^2 \times 3R\]
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Condition for maximum efficiency:
\[\dfrac{d\eta}{dP} = 0\]
This gives: Variable losses = Fixed losses
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Ways to improve efficiency:
Ferranti Effect
Voltage Rise in Long Lines
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Definition: Receiving end voltage higher than sending end voltage at no load
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Cause: Capacitive reactance dominates at light loads
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Occurs in: Long lines and cables
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Mathematical expression:
\[\dfrac{V_R}{V_S} = \dfrac{1}{\cos(\beta l)}\]
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Mitigation:
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Shunt reactors
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Load compensation
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Voltage regulators
Corona Effect
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Definition: Ionization of air around conductors at high voltage
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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
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Visual disruptive voltage:
\[V_v = 21.1 \times m \times \delta \times r \ln\left(\dfrac{d}{r}\right) \left[1 + \dfrac{0.3}{\sqrt{\delta r}}\right]\]
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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
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
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Low Voltage Cables (up to 1.1 kV):
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PVC insulation
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Lead sheath
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Armoring
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High Voltage Cables (up to 33 kV):
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EHV Cables (above 33 kV):
Cable Parameters and Characteristics
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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:
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Heat generated by \(I^2R\) losses
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Limited by insulation temperature
-
Derating factors for grouped cables
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Dielectric losses:
\[P_d = V^2 \omega C \tan\delta \text{ W/m}\]
where \(\tan\delta\) = loss tangent of insulation
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Charging Current:
\[I_c = V \omega C \text{ A/m}\]
Key Formulas to Remember - Part 1
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Load Factor: \(LF = \dfrac{\text{Average Load}}{\text{Peak Load}}\)
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Surge Impedance: \(Z_c = \sqrt{L/C}\) (For lossless line)
-
Surge Impedance Loading: \(P_{SIL} = \dfrac{V^2}{Z_c}\)
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Voltage Regulation: \(\%VR = \dfrac{|V_{no-load}| - |V_{full-load}|}{|V_{full-load}|} \times 100\)
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Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} \times 100\)
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Skin Depth: \(\delta = \sqrt{\dfrac{2\rho}{\omega\mu}}\)
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Sag: \(s = \dfrac{wL^2}{8T}\) (for equal supports)
Key Formulas to Remember - Part 2
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ABCD Parameters: \(\begin{bmatrix} V_s \\ I_s \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix}\)
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Symmetrical Network: \(AD - BC = 1\), \(A = D\)
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Propagation Constant: \(\gamma = \sqrt{zy} = \alpha + j\beta\)
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Ferranti Effect: \(\dfrac{V_R}{V_S} = \dfrac{1}{\cos(\beta l)}\) (no load)
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Corona Critical Voltage: \(V_c = 21.1 \times m \times \delta \times r \ln(d/r)\)
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Bundle Conductor GMR: \(D_{sb} = \sqrt[n]{d_1 \times d_2 \times ... \times d_n \times r^{n-1}}\)
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Transposition: Equalizes impedance in 3-phase systems
Quick Tips for GATE Exam
Remember These Points
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Capacitance effect is negligible for short lines
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Ferranti effect occurs in long lines at light loads
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HVDC is better for long distances (\(> 500 ~\text{km}\) overhead)
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Bundle conductors reduce corona and improve power transfer
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Transposition equalizes impedances in 3-phase lines
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SIL represents natural loading of transmission line
Common Mistakes to Avoid
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Don’t forget to convert units (km to m, kV to V)
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Line-to-line voltage vs line-to-neutral voltage
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ABCD parameters: \(AD - BC = 1\) for passive networks
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Surge impedance is different for overhead lines vs cables
Practice Problems - Quick Check
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A 400 kV line has surge impedance of 400 \(\Omega\). Find SIL.
\[P_{SIL} = \dfrac{(400)^2}{400} = 400 \text{ MW}\]
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If diversity factor is 1.5 and sum of individual demands is 150 MW, find system maximum demand.
\[\text{System Max Demand} = \dfrac{150}{1.5} = 100 \text{ MW}\]
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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.
\[I = \dfrac{50 \times 10^6}{\sqrt{3} \times 132 \times 10^3 \times 0.8} = 273 \text{ A}\]
\[\%VR = \dfrac{273(20 \times 0.8 + 80 \times 0.6)}{132000/\sqrt{3}} \times 100 = 8.6\%\]