Power System-I (EX-404) - Unit 1 Short Notes

UNIT 1: POWER SYSTEM-I SHORT NOTES

1.0 POWER SYSTEM OVERVIEW AND STRUCTURE

1.1 Components of Modern Power System

A modern power system consists of three main components:

1. Generation: Production of electrical energy in power stations (thermal, hydro, nuclear, renewable). Typical voltage: 10-25 kV.

2. Transmission: Bulk transfer of power from generating stations to load centers via high-voltage lines. Typical voltage: 66 kV, 132 kV, 220 kV, 400 kV, 765 kV.

3. Distribution: Delivery of power to end consumers. Typical voltage: 33 kV/11 kV (primary), 415V/240V (secondary).

Diagram Suggestion: Search for "modern power system structure generation transmission distribution diagram".

1.2 Interconnection of Power Stations

Definition: Connecting two or more power stations in parallel to supply a common load.

Advantages:

• Increased reliability and security of supply.

• Economical operation (shared reserve capacity).

• Exchange of peak loads.

• Reduced plant capacity margin.

• Better utilization of generating units.

Disadvantages:

• Increased complexity in control and protection.

• Risk of fault propagation (cascading outages).

• Higher initial cost for interconnecting lines.

• Need for sophisticated coordination.

Methods of Power Transfer:

• Active Power (kW) Transfer: Controlled by adjusting the phase angle (δ) between the sending and receiving end voltages. Power transfer equation for a simple system: $$\displaystyle P = \frac{V_S V_R}{X} \sin \delta $$.

• Reactive Power (kVAR) Transfer: Controlled by adjusting the voltage magnitude difference. Reactive power flows from higher voltage to lower voltage.

1.3 Types of Power System Configurations

1. Radial System: Power flows from a single source along a primary feeder to distributors. Simple, low cost, but poor reliability (single point failure).

2. Ring Main System: The primary feeder forms a closed loop. Power can be supplied from both ends. Better reliability and voltage regulation than radial.

3. Interconnected System: Multiple sources supply a common load. Highest reliability, used for transmission and large distribution networks.

1.4 Problems with Large Interconnected Systems

• Complex Stability Issues: Maintaining synchronism (angle stability) under disturbances.

• Fault Propagation: A fault in one area can cascade through the network.

• Control Challenges: Coordinating frequency and voltage control across regions.

• Protection Complexity: Relaying must be selective under various fault types and system configurations.

• Economic Scheduling: Optimal power flow (OPF) becomes computationally intensive.

1.5 Comparison: Isolated vs. Interconnected Systems

2.0 ECONOMIC ASPECTS OF POWER GENERATION AND TRANSMISSION

2.1 Load Curves and Duration Curves

• Chronological Load Curve: Plot of load (kW or MW) vs. time (hours/days/years). Shows variation of load with time.

• Load Duration Curve (LDC): Load values arranged in descending order vs. time (percentage of period). Area under LDC = total energy consumed.

• Utility: LDC is crucial for economic operation, determining plant capacity factor, load factor, and for economic dispatch.

2.2 Key Performance Factors

Inter-relationships:

• $$\displaystyle LF \times DF = \frac{\text{Avg. Load}}{\text{System Max. Demand}} $$

• A high diversity factor reduces the system's maximum demand, improving the load factor.

2.3 Cost Implications

• Good Load Factor: Means the load is more uniform. Reduces need for large capacity units, lowers capital cost per unit energy, and improves plant utilization.

• Good Diversity Factor: Reduces the system's peak demand, allowing smaller total installed capacity, lowering fixed costs.

• Effect of Transmission Voltage: Higher transmission voltage reduces current for same power ($$\displaystyle P = \sqrt{3} V_L I_L \cos\phi $$), leading to:

  • Smaller conductor size (lower material cost).

  • Lower $$\displaystyle I^2R $$ losses (lower operating cost).

  • However, cost of insulators, towers, and substation equipment increases with voltage. An economic transmission voltage exists.

• Energy Cost with Load Factor: Annual cost = Fixed charges + Running charges (fuel). Fixed charges per kWh $$\displaystyle \propto \frac{1}{LF} $$. Improving LF reduces cost per kWh.

2.4 Kelvin’s Law for Economic Conductor Size

Statement: The most economical conductor size is that for which the annual cost of energy wasted equals the annual interest and depreciation on the capital cost of the conductor.

Derivation:

Let:

• $A$ = Cross-sectional area (m²)

• $$\displaystyle R = \frac{\rho l}{A} $$ (Resistance)

• Annual cost of energy wasted $$\displaystyle = K_1 \frac{I^2 R}{\text{LF}} \propto \frac{1}{A} $$

• Annual interest/depreciation on conductor cost $$\displaystyle = K_2 A $$

• Total annual cost $$\displaystyle C = K_2 A + \frac{K_1'}{A} $$

For minimum cost: $$\displaystyle \frac{dC}{dA} = 0 \Rightarrow K_2 = \frac{K_1'}{A^2} \Rightarrow A = \sqrt{\frac{K_1'}{K_2}} $$

\boxed{A \propto \sqrt{\frac{\text{Energy Cost per kWh} \times \text{Load Factor for Losses}}{\text{Rate of Interest & Depreciation}}}}

Limitations:

• Ignores mechanical stress, corona, skin effect.

• Assumes constant load factor and energy cost—often not true.

• Interest/depreciation rates are estimates.

• Practical conductor sizes are standardized; economic size may not be available.

2.5 System Comparison Economics

1. Copper (Conductor) Efficiency: 3-phase 3-wire vs. 3-phase 4-wire

• For same maximum voltage to earth ($$\displaystyle V_{ph} $$) and same power ($P$):

  • 3-phase 3-wire: $$\displaystyle P = \sqrt{3} V_L I_L \cos\phi $$, $$\displaystyle V_L = \sqrt{3} V_{ph} $$

  • 3-phase 4-wire: $$\displaystyle P = 3 V_{ph} I_{ph} \cos\phi $$

• Comparing conductor material (volume) for same losses leads to:

  \boxed{\frac{\text{Cu for 3-phase 4-wire}}{\text{Cu for 3-phase 3-wire}} = \frac{2}{3}}

  3-phase 3-wire saves 33.3% copper for same $$\displaystyle V_{ph} $$ and power.

2. Single-phase AC vs. DC Two-wire Systems (Equal Power, Equal Losses)

• For same power $P$, distance $l$, and same losses:

  • DC: $$\displaystyle P = V_{dc} I_{dc} $$, Loss $$\displaystyle = 2 I_{dc}^2 R_{dc} $$ (2 wires)

  • Single-phase AC: $$\displaystyle P = 2 V_{ac} I_{ac} \cos\phi $$ (2 wires, $$\displaystyle V_{ac} $$ is phase voltage)

• Derivation shows:

  \boxed{\frac{\text{AC Conductor Area}}{\text{DC Conductor Area}} = 2 \cos^2\phi}

  For $$\displaystyle \cos\phi = 0.8 $$, AC needs ~28% more conductor area than DC.

3. Additional Load with 3-phase Conversion (Same Voltage, Same Losses)

• Existing: Single-phase, 2-wire, line voltage $$\displaystyle V_L $$, power $$\displaystyle P_1 $$, current $$\displaystyle I_1 $$.

• Converted: 3-phase, 3-wire, same line voltage $$\displaystyle V_L $$, same losses.

• Power in 3-phase: $$\displaystyle P_2 = \sqrt{3} V_L I_2 \cos\phi $$

• Since losses same: $$\displaystyle I_2 = I_1 / \sqrt{3} $$ (for same conductor size and length).

• \boxed{\frac{P_2}{P_1} = \frac{\sqrt{3} V_L (I_1/\sqrt{3}) \cos\phi}{V_L I_1} = \cos\phi}

  3-phase transmits $\cos\phi$ times more power than single-phase. For $$\displaystyle \cos\phi=0.8 $$, 60% increase.

3.0 TRANSMISSION LINE PARAMETERS

3.1 Resistance & Skin/Proximity Effect

• DC Resistance: $$\displaystyle R_{dc} = \frac{\rho l}{A} $$ ($\rho$: resistivity, $l$: length, $A$: area).

• Skin Effect: Non-uniform current distribution in AC conductors due to self-induced EMF. Current density higher at surface. Increases effective AC resistance ($$\displaystyle R_{ac} > R_{dc} $$). More pronounced at higher frequency and larger conductor radius.

• Proximity Effect: Distortion of current distribution due to magnetic fields from currents in adjacent conductors. Also increases AC resistance. Significant in tightly packed conductors (e.g., cables, bundled lines).

3.2 Inductance of Overhead Lines

Fundamental Concept: Inductance depends on Geometrical Mean Distance (GMD) and Geometrical Mean Radius (GMR).

• GMD (or $$\displaystyle D_m $$): For a set of distances $$\displaystyle D_1, D_2, ..., D_n $$, $$\displaystyle GMD = \sqrt[n]{D_1 D_2 ... D_n} $$. Used for spacing between conductors.

• GMR (or $$\displaystyle D_s $$): For a composite conductor, $$\displaystyle GMR = \sqrt[n]{r_1 r_2 ... r_n} $$, where $$\displaystyle r_i $$ are distances from center to individual strands. For a round conductor of radius $r$, $$\displaystyle GMR = 0.7788r $$.

1. Single-Phase Two-Wire Line

Assumptions: parallel conductors, radius $r$, spacing $D$.

Flux linkage per conductor $$\displaystyle \lambda = \frac{\mu_0}{2\pi} I \left( \frac{1}{4} + \ln\frac{D}{r} \right) $$

Inductance per conductor: $$\displaystyle L = \frac{\lambda}{I} = \frac{\mu_0}{2\pi} \left( \ln\frac{D}{r} - \frac{1}{4} \right) + \frac{\mu_0}{8\pi} $$

Using $$\displaystyle \mu_0 = 4\pi \times 10^{-7} $$ H/m:

\boxed{L = 0.2 \ln\frac{D}{r} \text{ (mH/km per conductor)}}

Loop inductance $$\displaystyle L_{loop} = 2L $$.

2. Three-Phase Lines

• Symmetrical Spacing (equilateral triangle, side $D$):

  \boxed{L = 0.2 \ln\frac{D}{GMR} + 0.2 \times 10^{-3} \text{ (mH/km/phase)}}

• Unsymmetrical Spacing: Use GMD of the three spacings. $$\displaystyle L = 0.2 \ln\frac{GMD}{GMR} + 0.2 \times 10^{-3} $$.

• Transposition: Rotating conductor positions along the line to balance inductance and capacitance. Makes the line "equivalent" to symmetrically spaced line. Necessary for long lines to avoid unbalanced voltages.

3. Bundled Conductors

• GMR of Bundled Phase Conductor (for $n$ conductors of radius $r$, spaced $d$):

  \boxed{GMR_{bundle} = \sqrt[n]{n r \cdot d^{n-1}}}

  For 2-conductor bundle: $$\displaystyle GMR = \sqrt{2 r d} $$.

• Effect: Increases GMR → decreases inductance → increases line capacity (surge impedance loading).

4. Double Circuit Three-Phase Line

Two 3-phase circuits on same tower. Inductance per phase calculated by:

1. Find GMR for each phase bundle (considering all sub-conductors in that phase).

2. Find GMD between equivalent conductors of the two circuits for the same phase (e.g., GMD between phase A of circuit 1 and phase A of circuit 2).

3. Use: $$\displaystyle L = 0.2 \ln\frac{GMD}{GMR} + 0.2 \times 10^{-3} $$.

3.3 Capacitance of Overhead Lines

Concept: Capacitance exists between conductors and between conductor and earth (via method of images).

1. Single-Phase Line

Two conductors radius $r$, spacing $D$.

Capacitance to neutral: $$\displaystyle C_n = \frac{2\pi\epsilon_0}{\ln\frac{D}{r}} $$ (F/km)

Line capacitance (between conductors): $$\displaystyle C = 2C_n = \frac{4\pi\epsilon_0}{\ln\frac{D}{r}} $$

Including earth (method of images) gives same expression if $$\displaystyle D >> r $$.

2. Three-Phase Lines

• Symmetrical Spacing: Capacitance to neutral per phase:

  \boxed{C_n = \frac{2\pi\epsilon_0}{\ln\frac{D}{GMR}} \text{ (F/km/phase)}}

• Unsymmetrical Spacing: Use GMD of the three spacings: $$\displaystyle C_n = \frac{2\pi\epsilon_0}{\ln\frac{GMD}{GMR}} $$.

3. Single-Core Cable

Conductor radius $r$, internal sheath radius $R$.

Electric stress at radius $x$: $$\displaystyle E_x = \frac{V}{x \ln(R/r)} $$

Maximum stress at conductor surface: $$\displaystyle E_{max} = \frac{V}{r \ln(R/r)} $$

Minimum at sheath: $$\displaystyle E_{min} = \frac{V}{R \ln(R/r)} $$

Capacitance per km: \boxed{C = \frac{2\pi\epsilon_0 \epsilon_r}{\ln(R/r)}}

3.4 Composite Conductors: Self-GMD & GMD

• Self-GMR (GMR): For a conductor made of $n$ strands of radius $r$, with $d$ spacing between strands:

  $$\displaystyle GMR = \sqrt[n]{r \cdot d^{n-1}} $$ (for a single layer).

  For a 7-strand conductor (1 central, 6 outer): $$\displaystyle GMR = 0.7688 \times \text{strand radius} $$.

• GMD: Used for spacing between different conductors or bundles. Always the geometric mean of distances between their centers.

4.0 TRANSMISSION LINE MODELING AND ANALYSIS

4.1 Classification of Lines

4.2 Short Transmission Line Model

• Equivalent Circuit: Series impedance $$\displaystyle Z = R + jX $$ only. Shunt capacitance neglected.

• Phasor Diagram: $$\displaystyle V_S = V_R + I_R Z $$ (for receiving end reference).

• Voltage Regulation:

  \boxed{\text{Reg. %} = \frac{|V_S| - |V_R|}{|V_R|} \times 100}

  Approximate: $$\displaystyle \text{Reg.} \approx \frac{I_R R \cos\phi_R + I_R X \sin\phi_R}{V_R} $$ (for lagging load).

• Transmission Efficiency:

  \boxed{\eta = \frac{P_R}{P_S} \times 100 = \frac{V_R I_R \cos\phi_R}{V_R I_R \cos\phi_R + I_R^2 R} \times 100}

4.3 Medium Transmission Line Models

Nominal-T Model:

• Shunt admittance $Y/2$ at each end, series impedance $Z$ in middle.

• ABCD Constants:

  \begin{aligned}

  A &= 1 + \frac{YZ}{2} \

  B &= Z \

  C &= Y \left(1 + \frac{YZ}{4}\right) \

  D &= A = 1 + \frac{YZ}{2}

  \end{aligned}

• Phasor Diagram: Shows $$\displaystyle V_S $$, $$\displaystyle V_R $$, $$\displaystyle I_S $$, $$\displaystyle I_R $$ with shunt currents.

Nominal-π Model:

• Series impedance $Z$, shunt admittance $Y$ split: $Y/2$ at each end.

• ABCD Constants:

  \begin{aligned}

  A &= 1 + \frac{YZ}{2} \

  B &= Z \left(1 + \frac{YZ}{2}\right) \

  C &= Y \

  D &= A = 1 + \frac{YZ}{2}

  \end{aligned}

• Solution: Given $$\displaystyle V_R $$, $$\displaystyle I_R $$, $$\displaystyle \phi_R $$, first find $$\displaystyle I_{sh} = j\omega C V_R /2 $$ (per phase), then $$\displaystyle I_S = I_R + I_{sh} $$, then $$\displaystyle V_S = V_R + I_R Z + I_{sh} V_R /2 $$.

4.4 Long Transmission Line

• Rigorous Solution: Solve differential equations $$\displaystyle \frac{dV}{dx} = ZI $$, $$\displaystyle \frac{dI}{dx} = YV $$.

• Propagation Constant: $$\displaystyle \gamma = \sqrt{ZY} = \alpha + j\beta $$

• Characteristic Impedance: $$\displaystyle Z_c = \sqrt{\frac{Z}{Y}} $$

• General Equation:

  \begin{aligned}

  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)

  \end{aligned}

• ABCD Constants:

  \boxed{A = D = \cosh(\gamma l), \quad B = Z_c \sinh(\gamma l), \quad C = \frac{1}{Z_c} \sinh(\gamma l)}

• Equivalent π-Circuit:

  • $$\displaystyle Z' = Z_c \sinh(\gamma l) $$ (series)

  • $$\displaystyle Y'/2 = \frac{1}{Z_c} \tanh(\gamma l / 2) $$ (shunt)

• Interpretation: For long lines, voltage and current are traveling waves. $$\displaystyle \cosh(\gamma l) \approx \frac{e^{\gamma l}}{2} $$ for very long lines.

4.5 Tuned Power Lines

• Concept: Adjusting line inductance or capacitance so that surge impedance $$\displaystyle Z_c $$ equals the load impedance. Condition: $$\displaystyle Z_c = \sqrt{L/C} = Z_{load} $$.

• Application: For a compensated line, when load equals $$\displaystyle Z_c $$, there is no reflection of traveling waves, maximum power transfer, and voltage profile is flat (no Ferranti effect). Used in HVDC cable terminations and some AC applications.

4.6 Power Circle Diagram

• Concept: Graphical representation of receiving end power $$\displaystyle P_R + jQ_R $$ for constant sending end voltage $$\displaystyle V_S $$ and impedance $Z$.

• Receiving End Power Circle:

  \begin{aligned}

  P_R &= \frac{|V_S||V_R|}{|Z|} \cos(\theta - \delta) - \frac{|V_R|^2}{|Z|} \cos\theta \

  Q_R &= \frac{|V_S||V_R|}{|Z|} \sin(\theta - \delta) - \frac{|V_R|^2}{|Z|} \sin\theta

  \end{aligned}

  where $$\displaystyle \theta = \arg(Z) $$, $$\displaystyle \delta = \text{angle between } V_S \text{ and } V_R $$.

• Construction: Plot center at $$\displaystyle (-\frac{V_R^2}{|Z|}\cos\theta, -\frac{V_R^2}{|Z|}\sin\theta) $$, radius $$\displaystyle \frac{|V_S||V_R|}{|Z|} $$.

• Sending End Power Circle: Similar, with $$\displaystyle V_R $$ constant.

• Useful Information:

  • Maximum power transfer (at $$\displaystyle \delta = \theta $$).

  • Limits of $$\displaystyle P_R $$ and $$\displaystyle Q_R $$ for stable operation.

  • Reactive power requirements for voltage control.

5.0 TRANSMISSION LINE PERFORMANCE

5.1 Voltage Regulation

Definition: The change in receiving end voltage from no-load to full-load, with sending end voltage constant, expressed as percentage of receiving end voltage.

\boxed{\text{Reg. %} = \frac{|V_{S, FL}| - |V_R|}{|V_R|} \times 100}

• Short Line: Use approximate formula from phasor diagram.

• Medium Line: Use ABCD constants. $$\displaystyle V_S = A V_R + B I_R $$.

• Long Line: Use rigorous equation with $\cosh(\gamma l)$, $\sinh(\gamma l)$.

5.2 Transmission Efficiency

Definition: Ratio of receiving end power to sending end power.

\boxed{\eta = \frac{P_R}{P_S} \times 100 = \frac{V_R I_R \cos\phi_R}{V_R I_R \cos\phi_R + I^2 R} \times 100}

• For medium/long lines, use $$\displaystyle P_S = \text{Re}(V_S I_S^*) $$.

5.3 Ferranti Effect

Explanation: The phenomenon of voltage rise at the sending end compared to receiving end when a long transmission line is lightly loaded or open-circuited.

• Cause: Dominant capacitive charging current ($$\displaystyle I_C $$) flowing through the line inductance ($L$), causing a voltage drop $$\displaystyle I_C X $$ that adds to $$\displaystyle V_R $$.

• Significance: Critical for long and medium lines (> 150 km). Can cause overvoltages, requiring voltage control measures (shunt reactors).

• Voltage Rise for Open-Circuited Line ($$\displaystyle I_R=0 $$):

  For long line: $$\displaystyle V_S = V_R \cosh(\gamma l) \approx V_R \cosh(\alpha l) $$ (since $\gamma l \approx j\beta l$ for lossless, but for lossy, $$\displaystyle \cosh(\gamma l) > 1 $$). So $$\displaystyle |V_S| > |V_R| $$.

  For nominal-π: $$\displaystyle V_S = V_R (1 + YZ/2) \approx V_R $$ if losses small, but with capacitance, $$\displaystyle V_S > V_R $$.

• Power Loss in Open-Circuited Line: $$\displaystyle P_{loss} = \frac{|V_R|^2}{2} \text{Re}(Y) $$ (shunt admittance loss).

5.4 Sending End Quantities Calculation

Given: $$\displaystyle V_R $$, $$\displaystyle I_R $$, $$\displaystyle \phi_R $$ (or $$\displaystyle P_R $$, $$\displaystyle Q_R $$), line parameters. Steps:

1. Choose appropriate model (Short: $$\displaystyle V_S = V_R + I_R Z $$; Medium: use ABCD; Long: use rigorous).

2. Compute $$\displaystyle I_S $$ (for medium/long: $$\displaystyle I_S = C V_R + D I_R $$).

3. Compute $$\displaystyle V_S $$.

4. Sending end power factor: $$\displaystyle \phi_S = \angle(V_S) - \angle(I_S) $$.

5. Voltage regulation and efficiency as above.

6.0 TRANSMISSION LINE DESIGN AND MECHANICAL ASPECTS

6.1 Sag and Tension (Parabolic Approximation)

Assumptions: $$\displaystyle Sag << Span $$, parabolic shape, uniform weight $w$ (kg/m), equal level supports, span $l$, tension $$\displaystyle T_0 $$ at lowest point.

• Equation of Parabola: $$\displaystyle y = \frac{w}{2T_0} x^2 $$, where $x$ is horizontal distance from center.

• Sag (at support, $$\displaystyle x = l/2 $$):

  \boxed{S = \frac{w l^2}{8 T_0}}

• Maximum Tension (at support): $$\displaystyle T_{max} = T_0 + wS $$ (if $S$ small, $$\displaystyle T_{max} \approx T_0 $$).

• Effect of Ice and Wind:

  • Total effective weight: $$\displaystyle w_{eff} = \sqrt{(w + w_{ice})^2 + w_{wind}^2} $$

  • Where $$\displaystyle w_{ice} = \text{volume of ice} \times \text{density} $$, $$\displaystyle w_{wind} = \text{wind pressure} \times \text{diameter (including ice)} $$.

  • Use $$\displaystyle w_{eff} $$ in sag formula. Sag increases significantly.

6.2 Sag Templates and String Charts

• Sag Template: A graphical tool (transparent sheet) with curves of sag vs. span for different tensions and temperatures. Placed on profile drawing to check clearances and determine required support height or tension.

• String Chart: A plot on graph paper showing the relationship between sag, tension, and temperature for a given conductor and span. Used for setting tension during stringing.

• Difference: Sag template is used for design (checking clearances on profiles), string chart for construction (setting actual stringing tension).

6.3 Conductor Selection and Configuration

• Bundled Conductors:

  • Advantages: Reduced inductance & corona, increased current capacity, reduced reactance, lower RI & AN.

  • Application: EHV lines (> 220 kV), where bundle spacing ~ 0.2-0.3 m.

  • Comparison with Double Circuit:

    | Feature | Bundled Conductor (Single Circuit) | Double Circuit | | :--- | :--- | :--- | | Configuration | 2/3/4 conductors per phase on one tower | Two separate 3-phase circuits on one tower | | Inductance | Lower (due to larger GMR) | Higher (closer phase conductors) | | Reliability | Lower (single circuit failure) | Higher (one circuit can be备用) | | Cost | Lower tower cost, higher conductor cost | Higher tower cost, lower conductor cost per circuit | | Application | EHV AC transmission | High reliability needed, HVDC bipolar |

• Transposition of Conductors:

  • Necessity: To balance inductance and capacitance of unsymmetrically spaced lines, preventing unbalanced voltages and currents.

  • Methods: Complete transposition at regular intervals (e.g., 1/3 of line length for 3-phase), or using transposition towers.

6.4 Line Supports and Clearances

• Types:

  • Wooden: Low voltage, short spans, rural areas.

  • Steel Tubular: Medium voltage, urban distribution.

  • Reinforced Concrete (RCC): Medium voltage, longer spans, economical.

  • Lattice Steel: High/extra-high voltage, long spans, most common for transmission.

• Ground Clearance: Minimum height $h$ of conductor above ground at maximum sag.

  \boxed{h = S_{max} + \text{Minimum Ground Clearance (as per code)}}

  For example, for 400 kV, min. clearance ~ 8-10 m. Support height $$\displaystyle H = h + \text{pole/tower height above conductor attachment} $$.

7.0 OVERHEAD LINE INSULATORS

7.1 Types and Construction

Diagram Suggestion: Search for "pin suspension strain insulator construction diagram".

7.2 Voltage Distribution in Insulator String

Capacitance Model:

• $C$ = Self-capacitance (capacitance of each disc to earth/line).

• $mC$ = Capacitance from disc pin (metal fitting) to earth/tower (due to dirt, humidity).

• For $n$ discs, let $$\displaystyle V_1, V_2, ..., V_n $$ be voltages across discs from top (near line) to bottom (near tower).

Derivation (KVL on capacitive voltage divider):

For disc $k$: $$\displaystyle V_k = (V_{k-1} - V_k) mC + (V_k - V_{k+1}) mC + \text{current through } C $$

Solving yields: $$\displaystyle V_k = V_1 \frac{\sinh[(n-k+1)\phi]}{\sinh\phi} $$ where $$\displaystyle \cosh\phi = 1 + m $$.

Result: Voltage is non-uniform, highest across disc nearest line (top).

7.3 String Efficiency

Definition: Ratio of voltage across whole string to $n$ times voltage across the disc nearest to the conductor (or average disc voltage).

\boxed{\eta_{string} = \frac{V_{string}}{n \times V_{bottom}} \times 100%}

Where $$\displaystyle V_{bottom} $$ is voltage across disc nearest tower (or disc with max. voltage? Actually, typically $$\displaystyle V_{bottom} $$ is voltage across bottom disc, but string efficiency is often defined as $$\displaystyle \frac{V_{string}}{n \times V_{1}} $$ where $$\displaystyle V_1 $$ is voltage across top disc? Clarify: Standard definition: $$\displaystyle \eta = \frac{\text{Voltage across whole string}}{n \times \text{Voltage across disc nearest to conductor}} \times 100\% $$. Since top disc has highest voltage, this gives efficiency < 100%. Alternatively, some use bottom disc. Most common: $$\displaystyle \eta = \frac{V_{total}}{n \cdot V_1} \times 100\% $$, where $$\displaystyle V_1 $$ is voltage across top unit.

For $$\displaystyle m=0.1 $$ (10% pin-to-earth capacitance), 3-disc string: $$\displaystyle V_1:V_2:V_3 \approx 1:0.75:0.5 $$, $$\displaystyle V_{total}=2.25V_1 $$, $$\displaystyle \eta = \frac{2.25V_1}{3V_1} \times 100\% = 75\% $$.

7.4 Methods to Improve String Efficiency

1. Grading of Insulator Units:

   • Use discs with different capacitances (by varying diameter/thickness).

   • Top disc: smallest capacitance (to withstand highest voltage).

   • Bottom disc: largest capacitance.

   • Aim: Make voltage distribution uniform. Can achieve $\eta \approx 100\%$ but costly.

2. Guard Ring (Static Shielding):

   • A large metal ring (or additional discs) electrically connected to the bottom (tower) side of the bottom insulator.

   • Effect: It equalizes the potential of the metal fittings, reducing the pin-to-earth capacitance effect ($mC$).

   • Mechanism: The guard ring provides a capacitive path to earth that is similar for all pins, making the voltage drop across each disc more uniform.

   • Sketch: Show insulator string with a ring surrounding the bottom few discs, connected to the tower arm.

7.5 Insulator Testing

• Flash-over Voltage: Minimum voltage at which an arc forms over the insulator surface (between line and earth) without damaging the insulator.

• Testing Procedure:

  1. Apply voltage gradually (power frequency) until flash-over occurs. Record voltage.

  2. Repeat several times (usually 5) and take average.

  3. Also test with impulse voltages (standard lightning impulse: 1.2/50 μs) to simulate surges.

• Withstand Voltage: Voltage the insulator can tolerate for a specified time without flash-over.

8.0 UNDERGROUND CABLES

8.1 Comparison with Overhead Lines

8.2 Single-Core Cable Construction and Stress

• Construction: Conductor → Inner semi-conducting layer → Main insulation (XLPE, paper-oil) → Outer semi-conducting layer → Metallic sheath (lead, aluminum) → Armoring → Outer sheath.

• Dielectric Stress Distribution:

  • Radial stress at radius $x$: $$\displaystyle E_x = \frac{V}{x \ln(R/r)} $$ (for homogeneous dielectric).

  • Maximum Stress at conductor surface ($$\displaystyle x=r $$): $$\displaystyle E_{max} = \frac{V}{r \ln(R/r)} $$.

  • Minimum Stress at sheath inner surface ($$\displaystyle x=R $$): $$\displaystyle E_{min} = \frac{V}{R \ln(R/r)} $$.

  • Stress decreases hyperbolically from center to sheath.

• Insulation Thickness Calculation:

  Given $$\displaystyle E_{max} $$ (max. permissible stress), conductor radius $r$, operating voltage $V$ (line-to-neutral for 3-core, or phase voltage for single-core):

  \boxed{R = r \exp\left(\frac{V}{r E_{max}}\right)}

  Thickness $$\displaystyle t = R - r $$.

8.3 Cable Capacitance

For single-core cable (radius $r$, sheath radius $R$, permittivity $$\displaystyle \epsilon = \epsilon_0 \epsilon_r $$):

\boxed{C = \frac{2\pi\epsilon_0 \epsilon_r}{\ln(R/r)} \text{ (F/km)}}

• Effect of Earth: Method of images gives same formula if $R$ is sheath radius. Earth increases effective capacitance slightly if cable is close to surface, but usually neglected.

8.4 Insulation Grading for Cables

1. Homogeneous Dielectric: Single material with constant permittivity. Stress highest at conductor.

2. Graded Dielectric:

   • Capacitance Grading: Use multiple layers of different dielectrics with increasing permittivity towards conductor. Aims to make $$\displaystyle E_x $$ uniform. $$\displaystyle E_x = \frac{V}{\int_{r}^{R} \frac{dx}{\epsilon(x) x}} $$.

   • Resistance Grading: Use materials with decreasing resistivity towards conductor. Stress $$\displaystyle E_x \propto \frac{1}{x} $$ for uniform stress, requires $\rho(x) \propto x$. Difficult to implement.

9.0 SUBSTATIONS AND BUS BAR ARRANGEMENTS

9.1 Substation Equipment (List & Brief)

1. Power Transformer: Steps voltage up/down.

2. Circuit Breaker (CB): Makes/breaks normal & fault currents.

3. Isolator/Disconnecting Switch: Isolates equipment for maintenance (no current breaking).

4. Current Transformer (CT): Steps down current for metering/protection.

5. Potential Transformer (PT)/Capacitive Voltage Transformer (CVT): Steps down voltage for metering/protection.

6. Lightning Arrester/Surge Arrester: Protects equipment from overvoltages (lightning, switching).

7. Bus Bar: Conductors for distributing power within substation.

8. Isolating Switches: For maintenance isolation.

9. Relays & Control Panels: Protection and control.

10. Earthing Grid: For safety.

9.2 Single Line Diagram (SLD) of Typical A.C. Distribution Substation

Diagram Suggestion: Search for "typical 33/11 kV substation single line diagram". Typical Arrangement:

• Incoming 33 kV line → Isolator → Circuit Breaker → Main Bus Bar → Transformer (33/11 kV) → 11 kV Bus Bar → Outgoing feeders (each with CB, isolator, CT, PT, relay).

• Include lightning arresters on both sides, earthing points.

9.3 Bus Bar Systems

1. Single Bus Bar with Sectionalization:

   • Bus divided into sections by a sectionalizing CB or isolator.

   • Operation: Normally closed; can open to isolate a faulty section.

   • Merits: Partial continuity during fault, easier maintenance.

   • Demerits: Still single point of failure for CB, complex protection.

2. Sectionalized Double Bus Bar System:

   • Two parallel bus bars (Bus I, Bus II) with coupling CB.

   • Each circuit can be connected to either bus via bus coupler isolators.

   • Operation: One bus in service, other as spare; or both in service with appropriate coupling.

   • Merits: High reliability, flexible maintenance, no interruption during bus fault.

   • Demerits: High cost, more CBs and isolators.

3. Ring Main System (Distribution):

   • Distribution network forms a closed ring.

   • Power can be supplied from both ends.

   • Merits: Better reliability than radial, voltage regulation good.

   • Demerits: More complex protection, coordination needed.

9.4 AIS vs. GIS Substations

10.0 DISTRIBUTION SYSTEMS

10.1 3-Phase 3-Wire vs. 3-Phase 4-Wire Systems

• Transmission (3-wire): Uses only the three phase conductors. No neutral. Voltage between phases is $$\displaystyle V_L $$, phase voltage $$\displaystyle V_{ph} = V_L/\sqrt{3} $$. Saves conductor material (as shown in 2.5).

• Distribution (4-wire): Includes a neutral conductor. Provides two voltage levels: $$\displaystyle V_L $$ (for 3-phase loads) and $$\displaystyle V_{ph} $$ (for single-phase loads, e.g., homes). Neutral carries unbalanced current.

• Reason for 4-wire in Distribution: To supply both 3-phase industrial loads and single-phase residential/commercial loads from the same network. The neutral is earthed at substation, providing a stable phase-to-neutral voltage.

• Conductor Material Comparison (for same max. voltage to earth $$\displaystyle V_{ph} $$):

  • 3-wire: $$\displaystyle P = \sqrt{3} V_L I_L \cos\phi = 3 V_{ph} I_{ph} \cos\phi $$

  • 4-wire: $$\displaystyle P = 3 V_{ph} I_{ph} \cos\phi $$ (same expression, but $$\displaystyle I_{ph} $$ is phase current in 4-wire).

  • For same power and same $$\displaystyle V_{ph} $$, the phase current is same. But 3-wire has 3 conductors, 4-wire has 4. However, in 3-wire, line current $$\displaystyle I_L = \sqrt{3} I_{ph} $$, so conductor size differs. Detailed calculation (see 2.5) shows 3-wire needs less copper for same $$\displaystyle V_{ph} $$ and power.

10.2 Distribution System Configurations

1. Radial System:

   • Features: Tree-like structure, single source per feeder.

   • Applications: Rural areas, long feeders, low reliability requirement.

   • Merits: Simple, low cost, easy protection.

   • Demerits: Poor reliability (single point failure), voltage drop at far end.

2. Ring Main System:

   • Features: Closed loop, normally open at one point (ring main unit).

   • Applications: Urban areas, high reliability needed.

   • Merits: Supply can be maintained during single fault, better voltage regulation.

   • Demerits: More complex protection, higher cost.

3. Interconnected System:

   • Features: Multiple interconnections, meshed network.

   • Applications: Dense urban centers, city centers.

   • Merits: Highest reliability, multiple supply paths.

   • Demerits: Very complex protection and operation, highest cost.

11.0 GENERATION: CONVENTIONAL AND NON-CONVENTIONAL

11.1 Conventional Generation

• Thermal: Coal, gas, nuclear. Uses steam/combustion turbines. Base load, high emissions (except nuclear).

• Hydro: Potential energy of water → turbines. Peak load, renewable, low operating cost, high capital cost, site-specific.

• Nuclear: Fission heat → steam. Base load, low fuel cost, high capital, waste disposal issues.

11.2 Non-Conventional and Renewable Sources

Classification: Solar (PV, thermal), Wind (HAWT, VAWT), Biomass, Geothermal, Tidal, Ocean Thermal, etc.

Wind Energy:

• Horizontal Axis Wind Turbine (HAWT):

  • Advantages: High efficiency, mature technology, high power output, yaw mechanism for wind direction.

  • Disadvantages: Need tall tower, noisy, requires yaw drive, difficult maintenance at height.

• Vertical Axis Wind Turbine (VAWT):

  • Advantages: No yaw mechanism (omnidirectional), generator at ground level, simpler.

  • Disadvantages: Lower efficiency, higher torque ripple, pulsating loads, lower rotational speed (needs gearbox).

• Power Developed by Wind Turbine (Betz's Limit):

  • Wind power per unit area: $$\displaystyle P_{wind} = \frac{1}{2} \rho A v^3 $$ ($\rho$: air density, $A$: swept area, $v$: wind speed).

  • Power extracted by rotor: $$\displaystyle P_{rotor} = \frac{1}{2} \rho A v^3 C_p $$, where $$\displaystyle C_p $$ = power coefficient.

  • Betz's Limit: Maximum $$\displaystyle C_p = 16/27 \approx 0.593 $$ (59.3%). No turbine can extract more.

  • Derivation from momentum theory: $$\displaystyle C_p = \frac{4a(1-a)^2}{1} $$, max at $$\displaystyle a=1/3 $$.

11.3 Distributed Generation (DG)

• Concept: Small-scale generation (kW to MW) located close to the load it serves, connected to the distribution network.

• Sources: Solar PV, small wind, microturbines, fuel cells, diesel gensets.

• Advantages:

  • Reduced transmission losses.

  • Improved reliability and power quality.

  • Deferred T&D upgrades.

  • Environmental benefits (renewables).

  • Peak shaving.

11.4 Comparison: Conventional vs. Non-Conventional vs. DG

END OF UNIT 1 NOTES