Formula Sheet: Transmission

Transmission Line Parameters

Resistance (R)

DC Resistance: \[R_{dc} = \frac{\rho L}{A}\]

• R_dc = DC resistance (Ω)
• ρ = resistivity (Ω·m)
• L = conductor length (m)
• A = conductor cross-sectional area (m²)

AC Resistance: \[R_{ac} = R_{dc} \times (1 + y_s + y_p)\]

• y_s = skin effect factor
• y_p = proximity effect factor

Temperature Correction: \[R_2 = R_1 \left[\frac{T + t_2}{T + t_1}\right]\]

• R₁ = resistance at temperature t₁ (Ω)
• R₂ = resistance at temperature t₂ (Ω)
• T = temperature constant (234.5°C for copper, 228°C for aluminum at 20°C)
• t₁, t₂ = temperatures (°C)

Inductance (L)

Inductance of a Single Conductor: \[L = 2 \times 10^{-7} \ln\left(\frac{D}{r}\right) \text{ H/m}\]

• D = distance to return path (m)
• r = conductor radius (m)

Inductance per Phase (Three-Phase Line): \[L = 2 \times 10^{-7} \ln\left(\frac{D_{eq}}{r'}\right) \text{ H/m}\]

• D_eq = geometric mean distance (GMD) (m)
• r' = geometric mean radius (GMR) (m)

Geometric Mean Distance (Equilateral Spacing): \[D_{eq} = D\] Geometric Mean Distance (Unsymmetrical Spacing): \[D_{eq} = \sqrt[3]{D_{12} \times D_{23} \times D_{31}}\]

• D₁₂, D₂₃, D₃₁ = distances between phase conductors (m)

GMR for Bundled Conductors: \[GMR_b = \sqrt[n]{GMR \times d^{n-1}}\]

• n = number of conductors per bundle
• d = bundle spacing (m)
• GMR = geometric mean radius of individual conductor (m)

Inductive Reactance per Phase: \[X_L = \omega L = 2\pi f L\]

• X_L = inductive reactance (Ω/m or Ω/mile)
• f = frequency (Hz)
• L = inductance per unit length (H/m or H/mile)

Capacitance (C)

Capacitance to Neutral (Single Conductor): \[C = \frac{2\pi\epsilon_0\epsilon_r}{\ln(D/r)} \text{ F/m}\]

• ε₀ = permittivity of free space = 8.854 × 10^-12 F/m
• ε_r = relative permittivity (≈1 for air)
• D = distance to return path (m)
• r = conductor radius (m)

Capacitance per Phase (Three-Phase Line): \[C = \frac{2\pi\epsilon_0}{\ln(D_{eq}/r)} \text{ F/m}\] Simplified Form: \[C = \frac{0.0556}{\log_{10}(D_{eq}/r)} \text{ μF/mile}\] Capacitive Reactance per Phase: \[X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}\]

• X_C = capacitive reactance (Ω·m or Ω·mile)

Capacitive Susceptance: \[B = \omega C = 2\pi f C\]

• B = susceptance (S/m or S/mile)

Conductance (G)

Shunt Conductance: \[G = \omega C \tan\delta\]

• G = conductance per unit length (S/m)
• tan δ = loss tangent of insulation
• Note: Often neglected for overhead lines

Transmission Line Models

Short Line Model (length < 80="" km="" or="" 50="">

Assumptions:

• Capacitance is negligible
• Only series impedance is considered

Series Impedance: \[Z = R + jX_L = R + j\omega L\] ABCD Parameters: \[A = D = 1\] \[B = Z\] \[C = 0\] Voltage Relations: \[V_S = V_R + I_R Z\] \[I_S = I_R\]

• V_S = sending end voltage (V)
• V_R = receiving end voltage (V)
• I_S = sending end current (A)
• I_R = receiving end current (A)
• Z = total series impedance (Ω)

Medium Line Model (80 km < length="">< 250="">

Nominal π Model

Shunt Admittance (each end): \[Y' = \frac{Y}{2} = \frac{jB}{2}\] ABCD Parameters: \[A = D = 1 + \frac{YZ}{2}\] \[B = Z\] \[C = Y\left(1 + \frac{YZ}{4}\right)\] Voltage and Current Relations: \[V_S = \left(1 + \frac{YZ}{2}\right)V_R + Z I_R\] \[I_S = Y\left(1 + \frac{YZ}{4}\right)V_R + \left(1 + \frac{YZ}{2}\right)I_R\]

Nominal T Model

Series Impedance (each arm): \[Z' = \frac{Z}{2}\] ABCD Parameters: \[A = D = 1 + \frac{YZ}{2}\] \[B = Z\left(1 + \frac{YZ}{4}\right)\] \[C = Y\]

Long Line Model (length > 250 km or 150 miles)

Propagation Constant: \[\gamma = \alpha + j\beta = \sqrt{ZY}\]

• γ = propagation constant (1/km)
• α = attenuation constant (Np/km)
• β = phase constant (rad/km)
• Z = series impedance per unit length (Ω/km)
• Y = shunt admittance per unit length (S/km)

Characteristic Impedance: \[Z_c = \sqrt{\frac{Z}{Y}}\]

• Z_c = characteristic impedance (Ω)

For Lossless Lines: \[Z_c = \sqrt{\frac{L}{C}}\] ABCD Parameters: \[A = D = \cosh(\gamma l)\] \[B = Z_c \sinh(\gamma l)\] \[C = \frac{1}{Z_c}\sinh(\gamma l)\]

• l = line length (km)

Voltage and Current at any point x: \[V(x) = V_R \cosh(\gamma x) + I_R Z_c \sinh(\gamma x)\] \[I(x) = I_R \cosh(\gamma x) + \frac{V_R}{Z_c}\sinh(\gamma x)\] Sending End Relations: \[V_S = V_R \cosh(\gamma l) + I_R Z_c \sinh(\gamma l)\] \[I_S = I_R \cosh(\gamma l) + \frac{V_R}{Z_c}\sinh(\gamma l)\] Velocity of Propagation (Lossless Line): \[v = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}}\]

• v = velocity (m/s)

Wavelength: \[\lambda = \frac{v}{f} = \frac{2\pi}{\beta}\]

Transmission Line Performance

Voltage Regulation

Voltage Regulation: \[VR = \frac{|V_S/a| - |V_R|}{|V_R|} \times 100\%\]

• VR = voltage regulation (%)
• a = transformer turns ratio (if applicable, otherwise a = 1)
• Measured at no-load vs. full-load conditions

Approximate Formula: \[VR \approx \frac{IR\cos\theta_R + IX\sin\theta_R}{V_R} \times 100\%\]

• θ_R = receiving end power factor angle
• Positive for lagging power factor
• Negative for leading power factor

Transmission Efficiency

Efficiency: \[\eta = \frac{P_R}{P_S} \times 100\% = \frac{P_R}{P_R + P_{loss}} \times 100\%\]

• η = transmission efficiency (%)
• P_R = receiving end power (W)
• P_S = sending end power (W)
• P_loss = power loss in line (W)

Three-Phase Power Loss: \[P_{loss} = 3I^2 R\]

• I = line current per phase (A)
• R = resistance per phase (Ω)

Power Flow

Three-Phase Real Power (Receiving End): \[P_R = \sqrt{3}V_L I_L \cos\theta_R\]

• V_L = line-to-line voltage (V)
• I_L = line current (A)
• cos θ_R = power factor

Three-Phase Reactive Power: \[Q_R = \sqrt{3}V_L I_L \sin\theta_R\] Three-Phase Apparent Power: \[S_R = \sqrt{3}V_L I_L = \sqrt{P_R^2 + Q_R^2}\] Power at Sending End: \[P_S = \sqrt{3}V_{LS} I_{LS} \cos\theta_S\]

Charging Current

Charging Current (No-Load): \[I_C = V \times B = V \times \omega C\]

• I_C = charging current per phase (A)
• V = voltage to neutral (V)
• B = susceptance per phase (S)

Three-Phase Charging MVA: \[Q_C = \frac{3V^2}{\omega C} = 3V^2 B\]

Surge Impedance Loading (SIL)

Surge Impedance Loading: \[SIL = \frac{V_L^2}{Z_c}\]

• SIL = surge impedance loading (MW)
• V_L = rated line-to-line voltage (kV)
• Z_c = surge impedance (Ω)

Typical Values:

• Z_c ≈ 400 Ω for single conductor
• Z_c ≈ 300 Ω for bundled conductors

Condition:

• When line is loaded at SIL, reactive power generated by capacitance equals reactive power consumed by inductance
• Voltage profile is flat along the line

Symmetrical Components and Sequence Impedances

Symmetrical Components Transformation

Phase to Sequence Transformation: \[\begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} = \frac{1}{3}\begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix}\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix}\] Sequence to Phase Transformation: \[\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{bmatrix}\begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix}\]

• V₀ = zero-sequence component
• V₁ = positive-sequence component
• V₂ = negative-sequence component
• a = 1∠120° = e^j2π/3
• a² = 1∠240° = e^j4π/3
• Note: 1 + a + a² = 0

Individual Component Formulas: \[V_0 = \frac{1}{3}(V_a + V_b + V_c)\] \[V_1 = \frac{1}{3}(V_a + aV_b + a^2V_c)\] \[V_2 = \frac{1}{3}(V_a + a^2V_b + aV_c)\]

Sequence Impedances

Positive-Sequence Impedance: \[Z_1 = R + j\omega L_1\]

• Same as normal operating impedance
• L₁ calculated with normal GMD

Negative-Sequence Impedance: \[Z_2 = Z_1\]

• For static equipment, Z₂ = Z₁
• For rotating machines, may differ

Zero-Sequence Impedance: \[Z_0 = R + j\omega L_0\]

• Generally Z₀ > Z₁
• Depends on return path (ground wires, earth)

Zero-Sequence Inductance: \[L_0 = 2 \times 10^{-7}\ln\left(\frac{D_{eq0}}{GMR}\right)\]

• D_eq0 = equivalent GMD for zero-sequence = \(\sqrt[3]{D_{s1} \times D_{s2} \times D_{s3}}\)
• D_s = distances from phase conductors to their images in ground

Carson's Correction for Ground Return: \[Z_0 = R + 3R_e + j\left(X_L + 3X_e\right)\]

• R_e = earth resistance ≈ 9.869 × 10^-4f (Ω/mile)
• X_e = earth reactance (Ω/mile)
• f = frequency (Hz)

Power Transfer and Stability

Power Transfer Equations

Power Transfer (Lossless Line): \[P = \frac{V_S V_R}{X}\sin\delta\]

• P = real power transferred (W)
• V_S = sending end voltage magnitude (V)
• V_R = receiving end voltage magnitude (V)
• X = line reactance (Ω)
• δ = power angle (angle between V_S and V_R)

Maximum Power Transfer: \[P_{max} = \frac{V_S V_R}{X}\]

• Occurs when δ = 90°

Reactive Power Flow: \[Q = \frac{V_S V_R}{X}\cos\delta - \frac{V_R^2}{X}\] General Power Transfer (with resistance): \[P = \frac{V_S V_R}{|Z|}\cos(\theta_Z - \delta) - \frac{V_R^2}{|Z|}\cos\theta_Z\]

• Z = line impedance magnitude (Ω)
• θ_Z = impedance angle

Steady-State Stability Limit

Steady-State Stability Criterion: \[\frac{dP}{d\delta} > 0\] Synchronizing Power Coefficient: \[P_{sync} = \frac{dP}{d\delta} = \frac{V_S V_R}{X}\cos\delta\]

• System is stable when P_sync > 0
• Maximum stable angle is 90° for lossless line

Corona and Electric Field Effects

Corona Discharge

Critical Disruptive Voltage (Line-to-Neutral): \[V_c = 21.1 m \delta r \ln\left(\frac{D_{eq}}{r}\right) \text{ kV (rms)}\]

• V_c = critical voltage for corona (kV)
• m = conductor surface factor (0.8-1.0, smooth to weathered)
• δ = air density factor = \(\frac{3.92b}{273 + t}\)
• b = barometric pressure (cm of Hg)
• t = temperature (°C)
• r = conductor radius (cm)
• D_eq = geometric mean distance (cm)

Visual Corona Voltage: \[V_v = 21.1 m_v \delta r \left(1 + \frac{0.3}{\sqrt{\delta r}}\right)\ln\left(\frac{D_{eq}}{r}\right) \text{ kV}\]

• m_v = surface irregularity factor for visual corona

Peek's Formula for Corona Power Loss: \[P_c = \frac{241}{{\delta}}(f + 25)\sqrt{\frac{r}{D_{eq}}}(V - V_c)^2 \times 10^{-5} \text{ kW/km/phase}\]

• P_c = corona power loss
• f = frequency (Hz)
• V = operating voltage (kV, line-to-neutral)
• V_c = critical disruptive voltage (kV)

Electric Field at Conductor Surface

Maximum Surface Gradient (Single Conductor): \[E_{max} = \frac{V}{r \ln(D/r)}\]

• E_max = maximum electric field (V/m)
• V = voltage to neutral (V)
• r = conductor radius (m)
• D = equivalent spacing (m)

Sag and Tension in Overhead Lines

Catenary Equations

Sag (Parabolic Approximation for flat terrain): \[D = \frac{wL^2}{8T}\]

• D = sag at midpoint (m or ft)
• w = weight per unit length (N/m or lb/ft)
• L = span length (m or ft)
• T = horizontal tension (N or lb)

Conductor Length (Parabolic Approximation): \[l = L + \frac{8D^2}{3L}\]

• l = actual conductor length (m or ft)

Catenary Constant: \[c = \frac{T}{w}\]

• c = catenary constant (m or ft)

Exact Catenary Sag: \[D = c\left[\cosh\left(\frac{L}{2c}\right) - 1\right]\] Maximum Tension (at support points): \[T_{max} = T + wD = T\left(1 + \frac{wL^2}{8T^2}\right)\]

Effect of Temperature and Ice/Wind Loading

Change in Sag due to Temperature: \[\Delta D = \frac{\alpha \Delta t L^2 w}{8T}\]

• α = coefficient of linear expansion (1/°C)
• Δt = temperature change (°C)

Effective Weight with Ice and Wind: \[w_{eff} = \sqrt{w_c^2 + w_{wind}^2}\]

• w_c = weight of conductor plus ice (lb/ft or N/m)
• w_wind = horizontal wind load (lb/ft or N/m)

Wind Load: \[w_{wind} = P \times d_{ice}\]

• P = wind pressure (lb/ft² or N/m²)
• d_ice = conductor diameter plus ice thickness (ft or m)

Conductor Stress

Stress in Conductor: \[\sigma = \frac{T}{A}\]

• σ = stress (Pa or psi)
• T = tension (N or lb)
• A = conductor cross-sectional area (m² or in²)

Change in Length: \[\Delta l = \frac{TL}{AE} + \alpha L \Delta t\]

• E = modulus of elasticity (Pa or psi)
• α = coefficient of thermal expansion (1/°C)

Insulation Coordination and BIL

Basic Insulation Level (BIL)

BIL Definition:

• Standard impulse test voltage that equipment insulation can withstand
• Rated in kV peak
• Standard wave: 1.2/50 μs (1.2 μs rise, 50 μs to half-value)

Typical BIL Values for Distribution:

• 15 kV class: 95 kV or 110 kV BIL
• 25 kV class: 125 kV or 150 kV BIL
• 35 kV class: 200 kV BIL

Typical BIL Values for Transmission:

• 69 kV: 350 kV BIL
• 115 kV: 550 kV BIL
• 138 kV: 650 kV BIL
• 230 kV: 900 kV or 1050 kV BIL
• 345 kV: 1300 kV or 1550 kV BIL
• 500 kV: 1800 kV or 2050 kV BIL

Protective Margins

Protective Margin: \[PM = \frac{BIL - SPL}{SPL} \times 100\%\]

• PM = protective margin (%)
• BIL = basic insulation level (kV)
• SPL = surge protective level of arrester (kV)
• Typical margin: 20-30%

Ferranti Effect

Voltage Rise on Open-Ended Lines

Ferranti Effect:

• Receiving end voltage exceeds sending end voltage on lightly loaded or open lines
• Due to capacitive charging current and line inductance

Voltage Rise (Medium Line, No Load): \[V_R = V_S + I_C X_L\] Approximate Receiving End Voltage: \[V_R \approx V_S \left(1 + \frac{\omega^2 LC l^2}{2}\right)\]

• l = line length
• Valid for moderate lengths

Voltage Ratio (Long Line, No Load): \[\frac{V_R}{V_S} = \cosh(\gamma l) \approx \cosh(\beta l)\]

• For lossless line, γ = jβ

Reactive Compensation

Shunt Compensation

Shunt Reactor (for voltage control): \[Q_L = \frac{V^2}{X_L}\]

• Q_L = reactive power absorbed (VAR)
• X_L = reactor reactance (Ω)
• Used to absorb excess reactive power during light load

Shunt Capacitor (for voltage support): \[Q_C = \frac{V^2}{X_C} = V^2 \omega C\]

• Q_C = reactive power supplied (VAR)
• Used to supply reactive power during heavy load

Series Compensation

Series Capacitor Compensation: \[X_{comp} = X_L - X_C\]

• X_comp = compensated reactance (Ω)
• Reduces effective line reactance
• Increases power transfer capability

Degree of Compensation: \[k = \frac{X_C}{X_L}\]

• k = compensation factor (typically 0.25 to 0.75)

Power Transfer with Series Compensation: \[P = \frac{V_S V_R}{X_L(1-k)}\sin\delta\]

Per-Unit System for Transmission

Base Values

Base Power (Three-Phase): \[S_{base} = \sqrt{3} V_{base,LL} I_{base}\] Base Impedance (Line-to-Neutral): \[Z_{base} = \frac{V_{base,LN}^2}{S_{base,1\phi}} = \frac{V_{base,LL}^2}{S_{base,3\phi}}\]

• V_base,LN = base voltage line-to-neutral (V)
• V_base,LL = base voltage line-to-line (V)
• S_base,1φ = base power per phase (VA)
• S_base,3φ = base power three-phase (VA)

Per-Unit Impedance: \[Z_{pu} = \frac{Z_{actual}}{Z_{base}}\] Change of Base: \[Z_{pu,new} = Z_{pu,old} \times \frac{S_{base,new}}{S_{base,old}} \times \left(\frac{V_{base,old}}{V_{base,new}}\right)^2\]

Line Constants and Typical Values

Typical Impedance Values

Overhead Line Resistance:

• Varies with conductor size (AWG or kcmil)
• Aluminum: ρ ≈ 2.83 × 10^-8 Ω·m at 20°C
• Copper: ρ ≈ 1.72 × 10^-8 Ω·m at 20°C

Typical Inductive Reactance:

• X_L ≈ 0.4 to 0.8 Ω/mile for overhead lines at 60 Hz
• X_L ≈ 0.1 to 0.15 Ω/mile for bundled conductors

Typical Capacitive Susceptance:

• B ≈ 5 to 6 μS/mile for overhead lines

Typical Surge Impedance:

• Z_c ≈ 400 Ω for single conductor overhead line
• Z_c ≈ 300 Ω for double-bundled conductor
• Z_c ≈ 250 Ω for quad-bundled conductor
• Z_c ≈ 40-60 Ω for underground cable

Standard Voltage Classes

Transmission Voltage Levels (US):

• Sub-transmission: 69 kV, 115 kV, 138 kV
• High voltage: 230 kV, 345 kV, 500 kV
• Extra-high voltage: 765 kV

Temperature Effects and Conductor Rating

Ampacity

Heat Balance Equation: \[I^2 R = q_{solar} + q_{radiation} + q_{convection}\]

• Conductor temperature limited by sag and annealing considerations
• Typical maximum: 75-100°C for ACSR

IEEE Standard Ampacity Formula: \[I = \sqrt{\frac{q_{conv} + q_{rad} - q_{solar}}{R_{ac}}}\]

• q_conv = convective heat loss (W/ft)
• q_rad = radiative heat loss (W/ft)
• q_solar = solar heat gain (W/ft)
• R_ac = AC resistance at operating temperature (Ω/ft)

Lightning and Switching Surges

Traveling Waves

Voltage and Current Relation: \[V = Z_c I\]

• For wave traveling in positive direction

Reflection Coefficient (Voltage): \[\Gamma_V = \frac{Z_2 - Z_1}{Z_2 + Z_1}\]

• Z₁ = characteristic impedance of incident line (Ω)
• Z₂ = characteristic impedance of terminating line (Ω)

Refraction Coefficient (Voltage): \[\tau_V = 1 + \Gamma_V = \frac{2Z_2}{Z_2 + Z_1}\] Special Cases:

• Open circuit (Z₂ → ∞): Γ_V = +1, voltage doubles
• Short circuit (Z₂ = 0): Γ_V = -1, voltage cancels
• Matched impedance (Z₂ = Z₁): Γ_V = 0, no reflection

Lattice Diagram Analysis:

• Used to track multiple reflections at line discontinuities
• Each reflection multiplies by appropriate coefficient

Power Circle Diagrams

Receiving End Power Circle

Complex Power at Receiving End: \[S_R = P_R + jQ_R = V_R I_R^*\] Center of Receiving End Circle:

• Located on reactive power axis
• Locus of constant |V_R| and varying |I_R| forms circle

Radius: \[r = \frac{V_R^2}{|Z|}\]

Sending End Power Circle

Center and Radius:

• Complex center location depends on ABCD parameters
• Used for planning and operational studies

Line Loadability

Thermal Limit

Thermal Loading: \[P_{thermal} = \sqrt{3} V_L I_{max} \cos\theta\]

• I_max = maximum current limited by conductor temperature

Voltage Drop Limit

Voltage Drop Constraint: \[\Delta V \leq \Delta V_{max}\]

• Typically 5-10% for distribution, 5% for transmission

Stability Limit

Steady-State Stability Limit: \[P_{stability} = \frac{V_S V_R}{X}\]

• Practical limit usually 30-40% of theoretical maximum
• Includes stability margin

Line Loadability Comparison

• Short lines: Limited by thermal capacity
• Medium lines: Limited by voltage drop
• Long lines: Limited by steady-state stability