Power System-I (EX-404) - Unit 4 Short Notes

UNIT 4: Power Transmission and Distribution Systems

1. Power System Economics and Load Analysis

Load Curves and Load Duration Curves

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

• Load Duration Curve (LDC): Load values arranged in descending order vs. time (percentage of total period). Obtained from chronological curve.

• Utility: LDC directly gives:

  • Maximum Demand (peak of curve).

  • Load Factor = (Area under LDC) / (Area of rectangle with height = Max Demand).

  • Capacity Factor = (Area under LDC) / (Area of rectangle with height = Installed Capacity).

  • Economic Generation Planning: Helps in scheduling units (base, peak) and assessing reserve requirements.

[!TIP] Exam Tip: Be able to convert a given chronological table into an LDC and compute load factor/capacity factor.

Load Factors (Definitions & Interrelation)

Key Relationship: $$\displaystyle P_{max(station)} = \frac{\sum P_{max(i)}}{DF} $$. A high DF reduces the required station capacity.

Interconnection of Power Stations

• Advantages:

  1. Economy: Shared reserve, optimal dispatch (economic load sharing), reduced installed capacity.

  2. Reliability: Mutual aid during emergencies, improved load factor.

  3. Efficiency: Can use larger, more efficient units.

• Disadvantages:

  1. High Initial Cost: Transmission lines, substations.

  2. Complexity: Protection, control, stability issues.

  3. Fault Propagation: Trouble in one area can cascade.

• Control of Active & Reactive Power:

  • Active Power (kW): Controlled by governor settings (prime mover input). Increased mechanical power → increased electrical output. Tie-line power exchange is monitored and adjusted.

  • Reactive Power (kVAR): Controlled by excitation systems (field current). Increased excitation → increased voltage & reactive power output. Also via tap-changing transformers, shunt capacitors/reactors.

Economic Operation & Cost Analysis

• Effect of LF & DF on Cost:

  • High LF → More energy sold → Fixed cost per unit ↓.

  • High DF → Lower P_{max} → Smaller/cheaper plant & equipment → Lower capital cost.

• Annual Cost Components:

  • Fixed Cost: Interest, depreciation, taxes, insurance (independent of output).

  • Running Cost: Fuel, operation & maintenance (dependent on output).

• 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 $k$ = cost per kg of conductor, $r$ = resistivity, $l$ = length, $a$ = cross-sectional area.

    Capital Cost $$\displaystyle C_c = k \cdot a \cdot l $$.

    Annual Interest & Depreciation $$\displaystyle = C_c \cdot (i + d) = k a l (i+d) $$.

    Resistance $$\displaystyle R = \frac{\rho l}{a} $$.

    Annual Energy Loss $$\displaystyle E_{loss} = I^2 R \times \text{hours} \times LF_{loss} $$.

    Annual Cost of Loss $$\displaystyle C_l = \text{Cost per kWh} \times E_{loss} $$.

    Total Annual Cost $$\displaystyle C = C_c + C_l $$.

    For minimum $C$, $$\displaystyle \frac{dC}{da} = 0 \Rightarrow k l (i+d) - \frac{I^2 \rho l \cdot T \cdot LF_{loss}}{a^2} = 0 $$.

    \boxed{a = \sqrt{\frac{I^2 \rho T \cdot LF_{loss}}{k (i+d)}}}

  • Limitations:

    1. Ignores physical constraints (voltage drop, current density limits).

    2. Interest/depreciation rates & energy cost are estimates.

    3. Load factor for losses ($$\displaystyle LF_{loss} $$) is difficult to determine accurately.

    4. Does not account for future load growth.

• Effect of Transmission Voltage:

  • For same power $P$, $$\displaystyle I = P/(\sqrt{3} V \cos\phi) $$. Higher $V$ → Lower $I$.

  • Lower $I$ → Smaller conductor size (area $\propto I$) → Lower material cost.

  • Lower $$\displaystyle I^2R $$ loss → Higher efficiency.

  • Trade-off: Higher voltage requires more expensive insulators, towers, and substation equipment. There is an optimum voltage for a given power and distance.

Renewable and Non-Conventional Energy Sources

• Classification:

  • Conventional: Thermal (coal, gas, nuclear), Hydro.

  • Non-Conventional / Renewable: Solar, Wind, Geothermal, Biomass, Tidal.

  • Distributed Generation (DG): Small-scale generation (solar panels, small wind) located near the load center.

• Wind Power:

  • Power Developed: $$\displaystyle P = \frac{1}{2} \rho A C_p V_w^3 $$

    • $\rho$: Air density (kg/m³)

    • $$\displaystyle A = \pi R^2 $$: Swept area (m²)

    • $$\displaystyle C_p $$: Power coefficient (Betz limit = 0.593)

    • $$\displaystyle V_w $$: Wind velocity (m/s)

  • Horizontal Axis Wind Mill (HAWM): Axis parallel to ground. Higher efficiency, needs yaw mechanism, more common.

  • Vertical Axis Wind Mill (VAWM): Axis perpendicular to ground. Omnidirectional, no yaw needed, lower efficiency, higher mechanical stress.

2. Transmission Line Parameters

Resistance and Skin Effect

• Skin Effect: Non-uniform current distribution at high AC frequencies. Current density is higher near the surface, reducing effective cross-section → AC resistance > DC resistance. Increases with frequency, conductor size, and material permeability.

• Proximity Effect: Distortion of current distribution due to magnetic fields from adjacent conductors. Increases AC resistance further. Significant in bundled conductors and closely spaced cables.

Inductance Calculations

• Geometrical Mean Distance (GMD): For a group of conductors, it is the $$\displaystyle n^{th} $$ root of the product of all mutual distances between conductors.

$$GMD = \left( \prod_{i=1}^{n} \prod_{j=1, j\neq i}^{n} D_{ij} \right)^{1/n}$$

• Geometrical Mean Radius (GMR) / Self-GMD: For a single conductor (stranded), it is the $$\displaystyle n^{th} $$ root of the product of all mutual distances between strands. For a round conductor of radius $r$: $GMR \approx 0.7788 r$. For a bundle of $n$ conductors with spacing $d$: $$\displaystyle GMR_{bundle} = (n \cdot d \cdot r)^{1/n} $$.

• Single-Phase Two-Wire Line:

$$L = \frac{\mu_0}{2\pi} \ln\left(\frac{D}{GMR}\right) \text{ H/m}$$

where $D$ = distance between conductors.

• Three-Phase Lines:

  • Symmetrically Spaced: $$\displaystyle L = \frac{\mu_0}{2\pi} \ln\left(\frac{GMD}{GMR}\right) $$.

  • Unsymmetrically Spaced: Transpose conductors to balance inductance. Equivalent $$\displaystyle GMD = \sqrt[3]{D_{12} D_{23} D_{31}} $$.

• Bundled Conductors: Reduces inductance & capacitance, increases critical corona voltage. $$\displaystyle GMR_{phase} = (n \cdot d \cdot r)^{1/n} $$, $$\displaystyle GMD_{phase} = \sqrt[3]{D_{12} D_{23} D_{31}} $$ (where $$\displaystyle D_{ij} $$ are distances between bundle centers).

• Transposition: Periodically exchanging conductor positions to make inductance & capacitance of all three phases equal. Necessary for unsymmetrical spacing.

Capacitance Calculations

• Single-Phase Transmission System:

$$C = \frac{\pi \epsilon_0}{\ln(D/r)} \text{ F/m}$$

(Assuming earth is infinitely far, effect negligible).

• Three-Phase Lines:

  • Equilateral Spacing: $$\displaystyle C = \frac{2\pi \epsilon_0}{\ln(GMD/GMR)} $$.

  • Unsymmetrical Spacing: Use method of images or transposition to find equivalent $C$.

• Effect of Earth: Increases capacitance. Using method of images, the presence of earth is equivalent to an image conductor below. The capacitance becomes:

$$C_{with earth} = \frac{2\pi \epsilon_0}{\ln\left(\frac{\sqrt{D_{12} D_{13} D_{23}}}{r \cdot \sqrt{D_{1'2'} D_{1'3'} D_{2'3'}}}\right)}$$

where primed distances are to image conductors.

• Single-Core Cable:

$$C = \frac{2\pi \epsilon}{\ln(r_i/r_c)} \text{ F/m}$$

where $$\displaystyle \epsilon = \epsilon_0 \epsilon_r $$, $$\displaystyle r_c $$ = conductor radius, $$\displaystyle r_i $$ = internal radius of sheath.

Conductor Selection and Economic Sizing

Conclusion: For same $P$, $V$, $\cos\phi$, and losses, 3-phase 3-wire requires least copper (only 3 conductors vs 4 in 4-wire, and lower current per conductor than single-phase).

3. Transmission Line Models and Performance

Short Transmission Lines (< 80 km)

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

• Phasor Diagram: $$\displaystyle \vec{V}_s = \vec{V}_r + I \vec{Z} $$.

• Voltage Regulation:

$$VR\% = \frac{|V_s|_{no-load} - |V_r|_{full-load}}{|V_r|_{full-load}} \times 100$$

Approximate formula (for lagging pf):

$$VR \approx \frac{I R \cos\phi_r + I X \sin\phi_r}{V_r} \times 100$$

• Transmission Efficiency:

$$\eta = \frac{P_r}{P_s} \times 100 = \frac{V_r I \cos\phi_r}{V_r I \cos\phi_r + I^2 R} \times 100$$

Medium Transmission Lines (80-250 km)

• Nominal-T Model: Shunt admittance ($Y/2$) placed in middle, series impedance ($Z$) split equally.

  • Sending end: $$\displaystyle I_s = I_r + \frac{Y}{2} V_r $$

  • $$\displaystyle \vec{V}_s = \vec{V}_r + I_r Z + \frac{Y}{2} V_r \cdot \frac{Z}{2} $$ (approx: $$\displaystyle \vec{V}_s = \vec{V}_r + I_r Z + \frac{Y}{2} V_r $$)

• Nominal-π Model: Shunt admittance split ($Y/2$ at each end), series impedance ($Z$) in middle.

  • $$\displaystyle I_s = I_r + \frac{Y}{2} V_s + \frac{Y}{2} V_r $$

  • $$\displaystyle \vec{V}_s = \vec{V}_r + I_r Z $$

• Generalized Circuit Constants (ABCD Parameters):

  For π-model:

$$A = D = 1 + \frac{YZ}{2}, \quad B = Z, \quad C = Y\left(1 + \frac{YZ}{4}\right) \approx Y$$

Relations: $$\displaystyle V_s = A V_r + B I_r $$, $$\displaystyle I_s = C V_r + D I_r $$.

• Sending End Calculations: Given $$\displaystyle V_r $$, $$\displaystyle I_r $$, $$\displaystyle \phi_r $$, and line parameters, find $$\displaystyle V_s $$, $$\displaystyle I_s $$, $$\displaystyle P_s $$, $$\displaystyle pf_s $$ using ABCD constants.

• Power Circle Diagram:

  • Construction: Plot $$\displaystyle P_r $$ vs $$\displaystyle Q_r $$ or $$\displaystyle P_s $$ vs $$\displaystyle Q_s $$ for constant $$\displaystyle |V_r| $$ or $$\displaystyle |V_s| $$. Circle equation derived from $$\displaystyle |V_s|^2 = |A V_r + B I_r|^2 $$.

  • Interpretation: Shows regions of power transfer for given voltages, limits for stability ($$\displaystyle dP_s/d\delta $$), and reactive power requirements.

Long Transmission Lines (> 250 km)

• Rigorous Solution: Use telegrapher's equation. Voltage & current are traveling waves.

$$\frac{\partial^2 V}{\partial x^2} = Z Y V, \quad \frac{\partial^2 I}{\partial x^2} = Z Y I$$

Solution: $$\displaystyle V = V_1 e^{-\gamma x} + V_2 e^{\gamma x} $$, $$\displaystyle I = \frac{V_1}{Z_c} e^{-\gamma x} - \frac{V_2}{Z_c} e^{\gamma x} $$

where **Propagation constant** $$\displaystyle \gamma = \sqrt{ZY} = \alpha + j\beta $$, **Surge impedance** $$\displaystyle Z_c = \sqrt{Z/Y} $$.

• Interpretation:

  • $$\displaystyle e^{-\gamma x} $$: Surge traveling toward receiving end (attenuated & phase-shifted).

  • $$\displaystyle e^{\gamma x} $$: Surge traveling toward sending end.

  • For lossless line ($$\displaystyle \alpha=0 $$): Pure sinusoid propagation, $$\displaystyle Z_c $$ is real.

• Ferranti Effect: Voltage rise at the receiving end of a long, lightly-loaded line compared to sending end. Caused by the capacitive charging current flowing through the line inductance ($$\displaystyle I_c jX $$ voltage drop adds to $$\displaystyle V_r $$). More pronounced in cables and EHV/UHV lines.

• Tuned Power Line: A long line designed so that its reactance is tuned (resonates) with its shunt capacitance at the operating frequency. Condition: $$\displaystyle X = 1/(\omega C) $$ or $$\displaystyle X \cdot \omega C = 1 $$. Results in $$\displaystyle Z_c = \infty $$ and $$\displaystyle \gamma = \sqrt{ZY} \approx \sqrt{jX \cdot j\omega C} = \sqrt{-\omega^2 X C} $$ → purely real? Actually, for tuned line, $$\displaystyle B = \omega C $$, $$\displaystyle X = 1/(\omega C) $$ so $$\displaystyle XY = -1 $$, $$\displaystyle \gamma = j\alpha $$? Careful: Tuned line means $$\displaystyle X = 1/(\omega C) $$ → $$\displaystyle Z = jX $$, $$\displaystyle Y = j\omega C $$, so $$\displaystyle ZY = -1 $$, $$\displaystyle \gamma = \sqrt{-1} = j $$, $$\displaystyle \alpha=0 $$, $$\displaystyle \beta=1 $$. The surge impedance $$\displaystyle Z_c = \sqrt{jX / j\omega C} = \sqrt{X / \omega C} = \sqrt{1/(\omega^2 C^2)} = 1/(\omega C) $$, finite. The key is that the line behaves like a pure resistance? Actually, a tuned line has constant voltage magnitude along its length. It's used in HVDC converter stations and some AC applications for power flow control.

Voltage Control Methods

• Overview:

  1. Tap-Changing Transformers: On-load (OLTC) or off-load. Adjusts $V$ ratio.

  2. Shunt Compensation: Shunt capacitors (raise voltage), shunt reactors (lower voltage).

  3. Series Compensation: Series capacitors (reduce line reactance, increase power transfer, improve stability).

  4. Synchronous Condensers: Over-excited synchronous motor (acts as capacitor), under-excited (acts as inductor).

  5. Static VAR Compensators (SVC)/STATCOM: Fast-acting power electronics.

• Detailed Example: Shunt Capacitors:

  • Purpose: Compensate for lagging reactive power of inductive loads/lines.

  • Connection: Delta or wye at substation or along the line.

  • Effect: Supplies leading current, reducing net reactive power flow from source → voltage rise at point of connection.

  • Calculation: Required $$\displaystyle Q_c = P (\tan\phi_{initial} - \tan\phi_{desired}) $$.

4. Mechanical Design of Overhead Lines

Sag and Tension Calculations (Level Supports)

• Shape: Parabolic approximation (valid for sag << span). Exact shape is catenary.

• Derivation:

  Let $L$ = span length, $w$ = weight per unit length, $$\displaystyle T_m $$ = tension at lowest point (horizontal tension), $x$ = horizontal distance from lowest point, $y$ = vertical sag at $x$.

  Equilibrium: $$\displaystyle T_m \frac{d^2 y}{dx^2} = w \sqrt{1 + (dy/dx)^2} \approx w $$ (for small sag).

  Integrating: $$\displaystyle \frac{dy}{dx} = \frac{w}{T_m} x $$, $$\displaystyle y = \frac{w}{2T_m} x^2 $$.

  At support ($$\displaystyle x = L/2 $$, $$\displaystyle y = S $$): $$\displaystyle S = \frac{w L^2}{8 T_m} $$.

  \boxed{S = \frac{w L^2}{8 T_m}} \quad \text{and} \quad \boxed{T_m = \frac{w L^2}{8 S}}

  Total tension at support: $$\displaystyle T_s = \sqrt{T_m^2 + (w L/2)^2} \approx T_m + \frac{w^2 L^2}{8 T_m} $$.

• Effect of Wind and Ice Loading:

  • Total weight per unit length becomes: $$\displaystyle w' = \sqrt{(w + w_{ice})^2 + (w_{wind})^2} $$

  • where $$\displaystyle w_{ice} = \text{volume of ice per unit length} \times \text{density of ice} $$.

  • Wind load $$\displaystyle w_{wind} = \text{pressure} \times \text{projected diameter} $$ (conductor dia + 2×ice thickness).

  • Sag $$\displaystyle S' = \frac{w' L^2}{8 T_m} $$ (use maximum tension $$\displaystyle T_m $$ from safety criteria).

  • Maximum Sag occurs at the point of maximum $w'$ and maximum temperature (minimum $$\displaystyle T_m $$).

Sag Templates and String Charts

• Sag Template: A full-scale drawing of the sag curve for a specific conductor, span, tension, and temperature. Used on profile drawings to:

  1. Check clearance at any point.

  2. Determine required support height for given terrain.

  3. Find stringing tension for a given clearance.

• String Chart: Graph of tension vs. temperature for a fixed span and initial condition. Shows how tension changes with temperature (thermal expansion). Used to find stringing tension for a given temperature to achieve desired sag at maximum temperature.

• Difference: Sag template is a spatial tool (sag vs. distance). String chart is a tension-temperature relationship tool.

Line Supports and Structures

5. Insulators and Insulation Coordination

Types of Insulators (with Construction)

• Advantage of Suspension for HV: Economical (cheaper discs), flexible (reduces mechanical stress), easy to replace single disc, can be used for any voltage by adding discs.

Voltage Distribution in Insulator Strings

• Capacitance Model:

  • $C$: Self-capacitance (capacitance to earth) of each unit.

  • $$\displaystyle C_1 $$: Pin-to-earth (or cap-to-earth) capacitance of each unit (due to metal fittings).

  • Usually $$\displaystyle C_1 = mC $$, where $m \approx 0.1 - 0.2$.

• Derivation (for $n$ units):

  Let $$\displaystyle V_1, V_2, ..., V_n $$ be voltages across units from top (line side) to bottom (earth side).

  By Kirchhoff's current law at each node (except ends):

$$C(V_1 - V_2) + mC V_1 = C(V_2 - V_3) + mC V_2 \quad \Rightarrow \quad V_2 = \frac{1}{1+m} V_1 + \frac{m}{1+m} V_3$$

Solving recursively gives: $$\displaystyle V_k = V_1 \left[ \frac{\sinh((n-k)\theta)}{\sinh(n\theta)} \right] $$ where $$\displaystyle \cosh\theta = 1 + 2m $$.

For $$\displaystyle m << 1 $$, $\theta \approx \sqrt{2m}$.

**Top unit voltage** is highest.

• String Efficiency:

$$\eta_{string} = \frac{\text{Voltage across string}}{\sum \text{Voltage across each unit}} \times 100 = \frac{V_1 + V_2 + ... + V_n}{n \cdot V_n} \times 100$$

(Often defined as $$\displaystyle V_{line} / (n \cdot V_{bottom}) $$ where $$\displaystyle V_{bottom} $$ is voltage on lowest unit).

\boxed{\eta_{string} = \frac{n}{1 + (n-1)m} \times 100\%} \quad (\text{for } m<<1, \text{ approximate})

Methods to Improve String Efficiency

1. Grading of Insulator Units:

   • Use discs with different capacitances (e.g., larger diameter for lower units).

   • Aim: Make voltage distribution uniform. $$\displaystyle C_{top} < C_{bottom} $$.

   • Capacitance Grading: Vary disc size.

   • Intersheath Grading: Insert metallic sheaths between discs (rarely used).

2. Guard Rings (Corona Rings):

   • Large diameter ring connected to top of string and to tower arm.

   • Increases capacitance from top units to earth ($m$ increases for top units), equalizing voltage distribution.

   • Also reduces corona on hardware.

Insulator Testing

• Flash-Over Voltage Testing:

  1. Apply increasing AC voltage to insulator string until flash-over occurs.

  2. Record voltage at flash-over.

  3. Repeat to get average (dry, wet conditions specified).

  4. Critical Flash-Over Voltage (CFOV): Voltage at which flash-over probability is high.

  5. Withstand Voltage: Voltage the insulator can sustain for a specified time without flash-over.

6. Underground Cables

Construction and Types

• Single-Core Cable: One conductor + insulation + sheath. Used for high voltage.

• Multi-Core Cable: 3 or 4 cores in common sheath. Used for distribution (3-phase 4-wire).

• Insulation Materials:

  • Paper (impregnated with oil): Traditional for HV.

  • Rubber (natural/synthetic): Flexible, LT.

  • PVC: General purpose, good mechanical properties.

  • XLPE (Cross-Linked Polyethylene): Modern, high dielectric strength, high temperature rating.

• Sheath: Lead or aluminum for moisture/chemical protection.

Capacitance and Insulation Resistance

• Capacitance (single-core, cylindrical): $$\displaystyle C = \frac{2\pi \epsilon}{\ln(r_i/r_c)} $$ F/km.

  • High capacitance → high charging current → limitation for long cables.

• Insulation Resistance: $$\displaystyle R_{ins} = \frac{\rho_l \ln(r_i/r_c)}{2\pi l} $$ (for radial geometry). Decreases with length.

Dielectric Stress in Cables

• Expression (radial, cylindrical symmetry):

$$E(r) = \frac{V}{r \ln(r_i/r_c)} \quad \text{(for r_c ≤ r ≤ r_i)}$$

where $V$ = conductor voltage (to sheath), $$\displaystyle r_c $$ = conductor radius, $$\displaystyle r_i $$ = insulation outer radius.

• Maximum Stress: At conductor surface ($$\displaystyle r = r_c $$).

$$E_{max} = \frac{V}{r_c \ln(r_i/r_c)}$$

• Minimum Stress: At insulation outer surface ($$\displaystyle r = r_i $$).

$$E_{min} = \frac{V}{r_i \ln(r_i/r_c)}$$

• Calculation:

  Given $$\displaystyle E_{max} $$ (material limit), $$\displaystyle r_c $$, find $$\displaystyle r_i $$: $$\displaystyle r_i = r_c e^{V/(r_c E_{max})} $$.

  Operating voltage $V$ is phase-neutral for single-core.

Grading of Cables

1. Capacitance Grading (Intersheath Grading):

   • Use multiple insulation layers of different $$\displaystyle \epsilon_r $$.

   • Place high $$\displaystyle \epsilon_r $$ material near conductor (where $E$ is high), low $$\displaystyle \epsilon_r $$ outside.

   • Homogeneous Dielectric: $E \propto 1/r$.

   • Graded Dielectric: $E$ becomes more uniform.

2. Inter-sheath Grading: Insert thin metallic intersheaths at specific potentials between insulation layers. Each layer operates at lower voltage gradient.

Comparison: Overhead Lines vs. Underground Cables

7. Distribution Systems and Substations

Types of Distribution Systems

Three-Phase Systems: 3-Wire vs. 4-Wire

• Transmission (3-Wire): Only line conductors. No neutral needed as loads are balanced (or single-phase loads not connected). Saves one conductor (25% saving in copper for same voltage between phases).

• Distribution (4-Wire): Includes neutral conductor. Allows:

  1. Single-phase loads (phase-to-neutral, 230 V) for domestic/commercial.

  2. Three-phase loads (phase-to-phase, 415 V) for industrial.

  3. Neutral carries unbalanced current & provides path for zero-sequence currents.

• Copper Efficiency Comparison:

  • For same power $P$, voltage between phases $$\displaystyle V_L $$, power factor $\cos\phi$, and losses:

  • 3-Wire System: $$\displaystyle I = P/(\sqrt{3} V_L \cos\phi) $$. Conductor area $$\displaystyle a_{3w} \propto I $$.

  • 4-Wire System: For balanced 3-phase load, same $I$ per phase. But neutral may carry some current. Roughly, 4-wire requires about 33% more copper than 3-wire for same phase-to-phase voltage and balanced 3-phase load. The trade-off is the ability to serve single-phase loads.

Substation Equipment and Layout (Single Line Diagram)

• Main Equipment:

  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 load breaking).

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

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

  6. Lightning Arrester: Protects against overvoltages (lightning, switching).

  7. Capacitor Bank: For power factor correction.

  8. Bus Bars: Collect and distribute power.

  9. Relays & Control Panels: Protection and control logic.

• Typical SLD: Shows incoming lines → Isolators → CTs → CB → Transformer → Bus Bars → Outgoing CBs/Isolators → feeders. All major equipment represented by standard symbols.

Bus Bar Arrangements

8. Special Topics and Comparative Studies

Bundle Conductors vs Double Circuit Lines

Air-Insulated Substations (AIS) vs Gas-Insulated Substations (GIS)

Problems in Modern Large Interconnected Power Systems

1. Stability Issues: Angle stability (transient, steady-state), voltage stability. Complex dynamics.

2. Frequency Control: Need for automatic generation control (AGC) across regions.

3. Protection Challenges: Faults affect large areas, need coordinated relaying, distance protection with communication.

4. Power Flow Control: Congestion management, loop flows.

5. Blackouts: Cascading failures, need for wide-area monitoring (WAMS) and control.

6. Renewable Integration: Variability, uncertainty, inverter-based resources reducing inertia.

7. Market Operations: Transmission congestion management, locational marginal pricing (LMP).

Tuned Power Lines (Brief)

• A long transmission line where the inductive reactance ($X$) is tuned to the capacitive reactance ($1/\omega C$) such that $$\displaystyle X \cdot \omega C = 1 $$.

• Characteristics:

  • Surge impedance $$\displaystyle Z_c = \sqrt{X / \omega C} = 1/(\omega C) $$ (pure resistive).

  • Propagation constant $$\displaystyle \gamma = \sqrt{ZY} = \sqrt{jX \cdot j\omega C} = \sqrt{-\omega^2 X C} = j \omega \sqrt{X C} $$? Wait: $$\displaystyle ZY = (jX)(j\omega C) = -\omega^2 X C $$. If $$\displaystyle X = 1/(\omega C) $$, then $$\displaystyle ZY = -1 $$, $$\displaystyle \gamma = \sqrt{-1} = j $$. So $$\displaystyle \alpha=0 $$, $$\displaystyle \beta=1 $$ rad/km. No attenuation, only phase shift.

  • Voltage magnitude remains constant along the line.

• Application: Used in HVDC converter stations (as AC filter/tuned filter) and some special AC applications for power flow control.

Isolated vs Interconnected Power Systems