Power Transmission & Distribut

Cheatsheet Content

### Transmission Lines #### 1. Line Parameters - **Resistance ($R$):** Opposition to current flow, causing power loss. $$ R = \frac{\rho L}{A} $$ where $\rho$ is resistivity of conductor material ($\Omega \cdot m$), $L$ is length of conductor (m), and $A$ is cross-sectional area of conductor ($m^2$). - **Inductance ($L$):** Property causing opposition to change in current, due to magnetic flux linkage. - Single-phase: $$ L = \frac{\mu_0}{\pi} \ln\left(\frac{D}{r'}\right) \quad \text{H/m per conductor} $$ where $\mu_0$ is permeability of free space ($4\pi \times 10^{-7}$ H/m), $D$ is spacing between conductors (m), and $r'$ is geometric mean radius (GMR) of conductor ($r' = 0.7788r$, where $r$ is actual radius). - Three-phase (symmetrical spacing): $$ L = \frac{\mu_0}{2\pi} \ln\left(\frac{D}{GMR}\right) \quad \text{H/m per phase} $$ where $D$ is the equilateral spacing between phases. - Three-phase (asymmetrical spacing, transposed): $$ L = \frac{\mu_0}{2\pi} \ln\left(\frac{D_{eq}}{GMR}\right) \quad \text{H/m per phase} $$ where $D_{eq} = \sqrt[3]{D_{ab}D_{bc}D_{ca}}$ is the equivalent equilateral spacing, and $D_{ab}, D_{bc}, D_{ca}$ are distances between phases. - **Capacitance ($C$):** Ability to store electric charge, due to electric field between conductors. - Single-phase: $$ C = \frac{\pi\epsilon_0}{\ln(D/r)} \quad \text{F/m per conductor} $$ where $\epsilon_0$ is permittivity of free space ($8.854 \times 10^{-12}$ F/m), $D$ is spacing between conductors, and $r$ is radius of conductor. - Three-phase (symmetrical spacing): $$ C = \frac{2\pi\epsilon_0}{\ln(D/r)} \quad \text{F/m per phase} $$ - Three-phase (asymmetrical spacing, transposed): $$ C = \frac{2\pi\epsilon_0}{\ln(D_{eq}/r)} \quad \text{F/m per phase} $$ - **Leakage Conductance ($G$):** Represents current leakage through insulation. Negligible for overhead lines. #### 2. Line Models - **Short Transmission Line (Length $L 240$ km):** - Distributed parameters are considered. - Propagation constant: $$ \gamma = \sqrt{ZY} $$ - Characteristic impedance: $$ Z_c = \sqrt{Z/Y} $$ - Sending end voltage and current: $$ V_S = V_R \cosh(\gamma l) + I_R Z_c \sinh(\gamma l) $$ $$ I_S = \frac{V_R}{Z_c} \sinh(\gamma l) + I_R \cosh(\gamma l) $$ where $l$ is the length of the line. - **ABCD Parameters:** $$ A = D = \cosh(\gamma l) $$ $$ B = Z_c \sinh(\gamma l) $$ $$ C = \frac{1}{Z_c} \sinh(\gamma l) $$ #### 3. Line Performance - **Voltage Regulation (VR):** A measure of voltage change from no-load to full-load. $$ VR = \frac{|V_{NL}| - |V_{FL}|}{|V_{FL}|} \times 100\% $$ where $|V_{NL}|$ is no-load receiving end voltage, and $|V_{FL}|$ is full-load receiving end voltage. - For short line (approximate): $$ VR \approx \frac{I_R(R \cos\phi_R + X \sin\phi_R)}{|V_R|} \times 100\% $$ where $I_R$ is full-load receiving end current, $R$ is total resistance, $X$ is total reactance, and $\phi_R$ is receiving end power factor angle. - **Efficiency ($\eta$):** Ratio of power output to power input. $$ \eta = \frac{P_{out}}{P_{in}} \times 100\% = \frac{P_{out}}{P_{out} + P_{losses}} \times 100\% $$ where $P_{out}$ is receiving end power, $P_{in}$ is sending end power, and $P_{losses}$ are total line losses. - **Surge Impedance Loading (SIL):** Power delivered when the line is terminated by its characteristic impedance, resulting in no reflections. $$ SIL = \frac{V_R^2}{Z_c} $$ where $V_R$ is receiving end voltage and $Z_c$ is characteristic impedance. - **Corona Effect:** Ionization of air around conductors at high electric field strengths, leading to power loss and audible noise. - Critical Disruptive Voltage ($V_c$): Minimum phase-to-neutral voltage at which corona starts. $$ V_c = g_0 \delta r \ln(D/r) \quad \text{kV (peak)} $$ where $g_0$ is dielectric strength of air, $\delta$ is air density factor, $r$ is conductor radius, and $D$ is spacing between conductors. - Power Loss due to Corona (Peek's formula): $$ P_c \propto (f+25) \sqrt{r/D} (V_{ph} - V_c)^2 \times 10^{-5} \quad \text{kW/km/phase} $$ where $f$ is frequency, $V_{ph}$ is phase voltage, and $V_c$ is critical disruptive voltage. ### Insulators and Cables #### 1. Insulators - **String Efficiency:** A measure of voltage distribution uniformity across an insulator string. $$ \eta_{string} = \frac{\text{Voltage across string}}{n \times \text{Voltage across unit near conductor}} $$ where $n$ is the number of insulator units in the string. - **Methods to improve string efficiency:** Grading of insulators (using units with different capacitances), use of guard rings (to equalize capacitance). #### 2. Underground Cables - **Capacitance:** $$ C = \frac{2\pi\epsilon L}{\ln(R/r)} \quad \text{F/m} $$ where $\epsilon$ is permittivity of dielectric, $L$ is cable length, $R$ is radius of outer sheath, and $r$ is radius of conductor. - **Insulation Resistance:** $$ R_{ins} = \frac{\rho}{2\pi L} \ln(R/r) \quad \Omega $$ where $\rho$ is resistivity of insulation material. - **Dielectric Stress ($E$):** Electric field intensity within the insulation. - Maximum at conductor surface: $$ E_{max} = \frac{V}{r \ln(R/r)} $$ - Minimum at sheath surface: $$ E_{min} = \frac{V}{R \ln(R/r)} $$ where $V$ is the voltage across the insulation. - **Grading of Cables:** Technique to achieve uniform dielectric stress distribution, preventing premature breakdown. - **Capacitance Grading:** Uses layers of different dielectric materials with varying permittivities. - **Inter-sheath Grading:** Employs metallic sheaths between dielectric layers, maintained at specific potentials. ### Distribution Systems #### 1. Types of Distribution Systems - **Radial System:** Power flows in one direction from source to consumer. Simplest in design but least reliable. - **Ring Main System:** Feeds load from two directions, improving reliability. More complex protection. - **Interconnected System:** Multiple sources and interconnections, offering highest reliability and operational flexibility, but most complex. #### 2. Voltage Drop Calculations - **DC Distributor:** - Uniformly loaded: $$ V_D = I R_d L / 2 $$ where $I$ is total current, $R_d$ is resistance per unit length, and $L$ is total length. - Concentrated loads: Sum of voltage drops in each section, $V_D = \sum I_k R_k$. - **AC Distributor:** - Voltage Drop ($V_{drop}$) per phase: $$ V_{drop} = I_L (R_L \cos\phi + X_L \sin\phi) $$ where $I_L$ is load current, $R_L$ is resistance of line, $X_L$ is reactance of line, and $\phi$ is power factor angle. - For a feeder with uniformly distributed load of 'i' A/m: $$ V_{drop} = \frac{iL}{2} (R_L \cos\phi + X_L \sin\phi) $$ - Total voltage drop for a 3-phase system (line-to-line): $\sqrt{3} \times V_{drop}$. #### 3. Substations - **Functions:** Voltage transformation (step-up/step-down), switching operations, protection of equipment, power factor correction. - **Types:** - **Step-up substation:** Increases voltage for efficient long-distance transmission. - **Step-down substation:** Decreases voltage for distribution or industrial use. - **Distribution substation:** Further reduces voltage for local consumption. - **Switching substation:** Connects/disconnects transmission lines, without voltage transformation. ### Fault Analysis #### 1. Symmetrical Faults (3-Phase Fault) - **Per Unit System:** A system where quantities are expressed as a fraction of a base value, simplifying calculations. - Base Power ($S_{base}$), Base Voltage ($V_{base}$). - Base Current ($I_{base}$): $$ I_{base} = S_{base}/(\sqrt{3}V_{base}) \quad \text{for 3-phase} $$ - Base Impedance ($Z_{base}$): $$ Z_{base} = V_{base}^2/S_{base} \quad \text{for 3-phase} $$ - Per Unit Value = Actual Value / Base Value. - **Fault Current ($I_f$):** Current flowing during a fault condition. $$ I_f = \frac{V_{pre-fault}}{Z_{th}} $$ where $V_{pre-fault}$ is the pre-fault voltage at the fault point, and $Z_{th}$ is the Thevenin equivalent impedance seen from the fault point. #### 2. Unsymmetrical Faults - **Sequence Components:** Any unbalanced three-phase system can be resolved into three balanced systems: - **Positive Sequence ($V_1, I_1$):** Normal phase rotation (ABC). - **Negative Sequence ($V_2, I_2$):** Reverse phase rotation (ACB). - **Zero Sequence ($V_0, I_0$):** All phases in unison (in-phase). - Transformation from phase quantities to sequence quantities: $$ \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{bmatrix} \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} $$ where $a = e^{j2\pi/3} = -0.5 + j0.866$. - **Fault Types:** - **Single Line-to-Ground (SLG) Fault:** One phase conductor touches the ground. $$ I_f = \frac{3V_a}{Z_0 + Z_1 + Z_2 + 3Z_f} $$ where $V_a$ is pre-fault phase 'a' voltage, $Z_0, Z_1, Z_2$ are zero, positive, and negative sequence impedances, and $Z_f$ is the fault impedance. - **Line-to-Line (LL) Fault:** Two phase conductors come into contact. $$ I_f = \frac{\sqrt{3}V_a}{Z_1 + Z_2 + Z_f} $$ - **Double Line-to-Ground (LLG) Fault:** Two phase conductors touch the ground. $$ I_f = \frac{3V_a Z_1}{Z_1 Z_2 + Z_2 Z_0 + Z_0 Z_1 + Z_f(Z_1+Z_2+Z_0)} $$ ### Protection #### 1. Relays - **Inverse Definite Minimum Time (IDMT) Relay:** Operating time is inversely proportional to fault current, with a definite minimum time. $$ T = \frac{TMS \times A}{(PSM)^B - 1} $$ where $T$ is operating time, $TMS$ is Time Multiplier Setting, $A$ and $B$ are constants for the relay curve, and $PSM$ is Plug Setting Multiplier (fault current / pick-up current). - **Differential Relay:** Operates based on the current difference between two points in a protected zone. Used for transformers, generators, busbars. - **Distance Relay:** Measures the impedance of the line to the fault point to determine if the fault is within its protected zone. #### 2. Circuit Breakers - **Types:** Devices to interrupt fault currents and isolate faulty sections. - **Air Blast CB:** Uses high-pressure air to extinguish arc. - **Oil CB:** Uses insulating oil to quench arc. - **SF6 CB:** Uses Sulfur Hexafluoride gas for arc extinction. - **Vacuum CB:** Arc extinction occurs in a vacuum. - **Rating:** - **Breaking Capacity (MVA):** Maximum MVA that can be interrupted. - **Making Current (peak kA):** Maximum peak current it can withstand at the instant of closing. - **Rated Voltage (kV):** Nominal voltage of the system. #### 3. Earthing/Grounding - **Purpose:** Ensures safety by providing a low-resistance path for fault currents, protects equipment, stabilizes system voltage. - **Types:** - **Solid Grounding:** Direct connection of neutral to earth. - **Resistance Grounding:** Neutral connected to earth via a resistor. - **Reactance Grounding:** Neutral connected to earth via a reactor. - **Resonant Grounding (Petersen Coil):** Neutral connected to earth via an inductor tuned to compensate for line capacitance. ### Power Factor Correction - **Power Factor (PF):** The cosine of the angle ($\phi$) between voltage and current in an AC circuit. $$ \cos\phi = P/S $$ where $P$ is active power (W) and $S$ is apparent power (VA). - **Methods:** Improving power factor reduces losses and improves voltage regulation. - **Capacitors:** Most common method, connected in parallel with inductive loads. - **Synchronous Condensers:** Over-excited synchronous motors operating at no-load, drawing leading current. - **Capacitor bank rating ($Q_c$ in kVAR):** Required reactive power from capacitors to improve power factor. $$ Q_c = P (\tan\phi_1 - \tan\phi_2) $$ where $P$ is active power of load, $\phi_1$ is initial power factor angle, and $\phi_2$ is desired power factor angle. ### Economic Aspects - **Load Factor:** $$ \text{Load Factor} = \frac{\text{Average Load}}{\text{Maximum Demand}} $$ Indicates how efficiently the capacity is utilized over a period. - **Diversity Factor:** $$ \text{Diversity Factor} = \frac{\text{Sum of individual maximum demands}}{\text{Coincident maximum demand}} $$ Greater than 1, accounts for non-simultaneous maximum demands of consumers. - **Demand Factor:** $$ \text{Demand Factor} = \frac{\text{Maximum Demand}}{\text{Connected Load}} $$ Ratio of maximum demand on a system to total connected load. - **Utilization Factor:** $$ \text{Utilization Factor} = \frac{\text{Maximum Demand}}{\text{Rated Capacity}} $$ Indicates the proportion of plant capacity actually being used. - **Tariffs:** Pricing structures for electricity. - **Flat rate tariff:** Fixed price per unit consumed. - **Two-part tariff:** Fixed charge + charge per unit consumed. - **Three-part tariff:** Fixed charge + maximum demand charge + energy charge. - **TOD (Time-of-Day) tariff:** Different rates for different times of the day, encouraging off-peak consumption.