UNIT 2: Power System-I - Comprehensive Short Notes

I. Introduction to Power Systems & Generation Economics

A. Structure of Modern Power Systems

• Components: Generating stations → Transmission networks (EHV) → Sub-transmission → Distribution systems (MV/LV) → Substations (step-up/down, switching).

• Typical Voltage Levels:

  • Generation: 11 kV, 15 kV, 25 kV

  • Primary Transmission: 132 kV, 220 kV, 400 kV, 765 kV

  • Secondary Transmission/Sub-transmission: 33 kV, 66 kV

  • Primary Distribution: 11 kV, 33 kV

  • Secondary Distribution: 415 V (3-phase), 240 V (single-phase)

• Single Line Diagram (SLD): Represents 3-phase system with single lines and standard symbols for all major equipment.

B. Types of Power Generation

[!TIP] Wind Mills: Horizontal axis (higher efficiency, need yaw mechanism) vs. Vertical axis (omnidirectional, lower efficiency).

C. Interconnection of Power Stations

• Advantages:

  • Economy: Shared reserve capacity, optimal unit commitment.

  • Reliability: Mutual aid during outages.

  • Diversity: Smoothens total load curve (improves load & diversity factors).

• Disadvantages: Increased complexity, stability/control challenges, fault propagation risk.

• Power Transfer Control:

  • Active Power (kW): Controlled by power angle (δ) between terminal voltages. Increase δ → more power transfer.

  • Reactive Power (kVAR): Controlled by voltage magnitude. Increase sending-end voltage → more reactive power flow.

D. Load Analysis & Economic Principles

1. Load Curves & Duration Curves

• Chronological Load Curve: Load vs. time (24 hrs, annual). Shows variation.

• Load Duration Curve (LDC): Load magnitudes ranked in descending order vs. time. Area under LDC = Total annual energy (kWh).

• Utility: Determines capacity factor, load factor, and helps in economic scheduling.

2. Key Performance Factors

Proof: A high Diversity Factor (DF) reduces the system's Maximum Demand ($$\displaystyle P_{sys,max} = \frac{\sum P_{max(i)}}{DF} $$). Since Load Factor (LF) = $$\displaystyle \frac{\text{Energy}}{P_{sys,max} \times T} $$, a lower $$\displaystyle P_{sys,max} $$ for same Energy → Higher LF.

3. Economic Calculations

• Maximum Demand (MD): $$\displaystyle MD = \frac{\text{Total Energy (kWh)}}{LF \times \text{Total Hours}} $$

• Installed Capacity: $$\displaystyle P_{installed} \geq MD \times (1 + \text{Reserve \%}) $$. Choose standard unit sizes.

• Cost Impact: High LF & DF → Lower MD for same energy → Smaller plant size, lower fixed costs → Lower generation cost/kWh.

[!TIP] Common Pitfall: Confusing Demand Factor (≤1) with Diversity Factor (≥1). DF always ≥ 1, Demand Factor always ≤ 1.

II. Transmission Line Parameters

A. Resistance

• DC Resistance: $$\displaystyle R_{dc} = \frac{\rho l}{A} $$

• AC Resistance > DC Resistance due to:

  • Skin Effect: Non-uniform current density (higher at surface). Increases with frequency, conductor size, and material permeability.

  • Proximity Effect: Magnetic field from adjacent conductors causes further non-uniformity. Increases with close spacing.

• Conductor Comparison:

  • Copper: Higher conductivity (better), heavier, costlier.

  • Aluminium: Lower conductivity (larger size needed), lighter, cheaper, used with steel core (ACSR).

B. Inductance of Overhead Lines

Fundamental Concept: Flux linkage $$\displaystyle \lambda = L i $$. Inductance $$\displaystyle L = \frac{\lambda}{i} $$.

1. Single-Phase Two-Wire Line

• Flux Linkage per conductor: $$\displaystyle \lambda = \frac{\mu_0}{2\pi} i \left( \frac{1}{4} + \ln\frac{D}{r'} \right) $$

• Loop Inductance: $$\displaystyle L_{loop} = 2 \times L_{phase} = \frac{\mu_0}{\pi} \left( \ln\frac{D}{r'} + \frac{1}{4} \right) $$

  • $D$: Distance between conductors.

  • $$\displaystyle r' = 0.7788 \, r $$ (GMR for solid round conductor).

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

2. Three-Phase Lines

• Symmetrical Spacing: $$\displaystyle L_{phase} = \frac{\mu_0}{2\pi} \ln\frac{GMD}{GMR} $$

  • GMD (Geometrical Mean Distance): $$\displaystyle GMD = \sqrt[3]{D_{12} D_{23} D_{31}} $$ (for equilateral, GMD = D).

  • GMR (Geometrical Mean Radius / Self-GMD): $$\displaystyle GMR = 0.7788 \, r $$ for solid conductor.

• Unsymmetrical Spacing & Transposition:

  • Transposition: Rotate conductor positions over equal lengths to balance inductance.

  • Inductance (with transposition): $$\displaystyle L = \frac{\mu_0}{2\pi} \ln\frac{GMD}{GMR} $$ (same as symmetrical).

3. Special Configurations

• Bundled Conductors (2 conductors per phase):

  • GMR (Bundle): $$\displaystyle GMR_b = \sqrt[d]{d \cdot (GMR)} $$ where $d$ = bundle spacing.

  • Effect: Decreases inductance (increases GMR), reduces corona, increases current capacity.

• Double Circuit 3-Phase Line (Hexagonal):

  • Phase conductors on corners of two equilateral triangles.

  • GMD between phases: Calculate using distances between all conductors of same phase in both circuits.

  • GMR (Bundle): For two conductors per phase: $$\displaystyle GMR_b = \sqrt{d \cdot r'} $$.

• Self-GMD of 7-Strand Conductor: For 7 strands (1 central, 6 outer):

  • $$\displaystyle GMR_{7-strand} = 0.3456 \, d $$, where $d$ = diameter of individual strand.

[!TIP] GMD vs GMR: GMD is between different conductors (phase spacing). GMR is self-property of a composite conductor (strands/bundles).

C. Capacitance of Overhead Lines

Fundamental: $$\displaystyle C = \frac{Q}{V} $$. For a cylindrical conductor, $$\displaystyle C' = \frac{2\pi\epsilon_0}{\ln\frac{D}{r'}} $$ per unit length (to neutral).

1. Single-Phase System (with earth):

• Method of images: Earth replaced by image conductor at depth $D$ below.

• Capacitance to neutral: $$\displaystyle C_{ph-n} = \frac{2\pi\epsilon_0}{\ln\frac{D}{r'}} $$

2. Three-Phase System

• Symmetrical Spacing: $$\displaystyle C_{ph-n} = \frac{2\pi\epsilon_0}{\ln\frac{GMD}{GMR}} $$

• Unsymmetrical Spacing (with transposition): Same formula using GMD.

3. Bundled Conductors:

• GMR (Bundle) increases → Capacitance increases (since $C \propto 1/\ln(GMD/GMR)$).

D. Underground Cable Parameters

1. Single-Core Cable (Cylindrical Capacitor):

• Capacitance: $$\displaystyle C = \frac{2\pi\epsilon_0 \epsilon_r}{\ln\frac{r_2}{r_1}} $$ per unit length.

  • $$\displaystyle r_1 $$: Conductor radius, $$\displaystyle r_2 $$: Internal sheath radius.

  • $$\displaystyle \epsilon_r $$: Relative permittivity of insulation.

• Dielectric Stress (Voltage Gradient): $$\displaystyle E = \frac{V}{r \ln\frac{r_2}{r_1}} $$

  • Maximum at conductor surface ($$\displaystyle r=r_1 $$): $$\displaystyle E_{max} = \frac{V}{r_1 \ln\frac{r_2}{r_1}} $$

  • Minimum at sheath ($$\displaystyle r=r_2 $$): $$\displaystyle E_{min} = \frac{V}{r_2 \ln\frac{r_2}{r_1}} $$

  • Ratio: $$\displaystyle \frac{E_{max}}{E_{min}} = \frac{r_2}{r_1} $$

2. Grading of Cables (to achieve uniform stress):

• Capacitance Grading: Use multiple layers of dielectric with different $$\displaystyle \epsilon_r $$. Costly.

• Inter sheath Grading: Insert metallic sheaths between dielectric layers at graded potentials.

3. Insulation Thickness & Operating Voltage:

• Given $$\displaystyle E_{max} $$ limit and $$\displaystyle r_1 $$, find $$\displaystyle r_2 $$ from $$\displaystyle E_{max} = \frac{V_{op}}{r_1 \ln\frac{r_2}{r_1}} $$.

• Operating voltage $$\displaystyle V_{op} $$ is line-to-ground voltage for 3-core cable.

E. Overhead Lines vs. Underground Cables

III. Transmission Line Models and Performance

A. Classification of Lines

• Short Line (< 80 km, < 100 kV): Capacitance neglected. Model: Series $$\displaystyle Z = R + jX $$.

• Medium Line (80-250 km, < 100 kV): Capacitance distributed, lumped as shunt admittance. Models: Nominal T and Nominal π.

• Long Line (> 250 km, > 100 kV): All parameters distributed. Rigorous solution (telegraph equation).

B. Equivalent Circuits & ABCD Constants

1. Short Transmission Line (Phasor Diagram)

• Receiving-end voltage $$\displaystyle V_R $$ as reference.

• $$\displaystyle V_S = V_R + I_R Z $$

• Voltage Regulation: $$\displaystyle VR\% = \frac{|V_S|_{nl} - |V_S|_{fl}}{|V_S|_{fl}} \times 100 $$ (approx. for lagging pf: $$\displaystyle VR \approx \frac{I_R R \cos\phi_R + I_R X \sin\phi_R}{V_R} $$)

• Transmission Efficiency: $$\displaystyle \eta = \frac{P_R}{P_S} = \frac{V_R I_R \cos\phi_R}{V_R I_R \cos\phi_R + I_R^2 R} \times 100\% $$

2. Medium Transmission Line

• Nominal T Model:

  • Series impedance: $$\displaystyle Z = R + jX $$ (total line).

  • Shunt admittance: $jB/2$ at each end (total line charging $jB$).

  • DiagramCANVAS: Nominal T model: series Z in middle, shunt B/2 at sending and receiving ends

• Nominal π Model:

  • Shunt admittance: $jB/2$ at each end (total $jB$).

  • Series impedance: $Z$ in middle.

  • DiagramCANVAS: Nominal π model: shunt B/2 at both ends, series Z in middle

• ABCD Constants (Generalized Circuit Constants):

  • Defined by: $$\displaystyle \begin{bmatrix} V_S \\ I_S \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix} $$

  • For Nominal T:

$$A = D = 1 + \frac{ZY}{4}, \quad B = Z, \quad C = Y\left(1 + \frac{ZY}{8}\right) \approx Y \text{ (if } ZY \text{ small)}$$

*   **For Nominal π:**

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

*   **Reciprocity:** $$\displaystyle AD - BC = 1 $$ (for passive, symmetrical network).

3. Long Transmission Line (Rigorous Solution)

• Telegraph Equations: $$\displaystyle \frac{dV}{dx} = (R + j\omega L)I $$, $$\displaystyle \frac{dI}{dx} = (G + j\omega C)V $$

• Solution: Propagation constant $$\displaystyle \gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)} $$, Surge impedance $$\displaystyle Z_c = \sqrt{\frac{R+j\omega L}{G+j\omega C}} $$.

• Voltage & Current: $$\displaystyle V(x) = V_R \cosh(\gamma x) + I_R Z_c \sinh(\gamma x) $$, $$\displaystyle I(x) = I_R \cosh(\gamma x) + \frac{V_R}{Z_c} \sinh(\gamma x) $$.

• Equivalent π/T: Can be represented by lumped parameters: $$\displaystyle Z' = Z_c \sinh(\gamma l) $$, $$\displaystyle Y' = \frac{2}{Z_c} \tanh(\frac{\gamma l}{2}) $$.

C. Performance Calculations & Analysis

1. Sending-end Quantities:

• $$\displaystyle V_S = A V_R + B I_R $$

• $$\displaystyle I_S = C V_R + D I_R $$

• $$\displaystyle P_S = \text{Re}[V_S I_S^*] $$, $$\displaystyle Q_S = \text{Im}[V_S I_S^*] $$

2. Ferranti Effect

• Phenomenon: Voltage at sending end ($$\displaystyle V_S $$) exceeds receiving-end voltage ($$\displaystyle V_R $$) for a lightly loaded or open-circuited long line.

• Cause: Dominant capacitive charging current flowing through line inductance.

• Expression (for lossless line, $$\displaystyle R=G=0 $$):

$$|V_S| = |V_R| \cosh(\gamma l) = |V_R| \cos(\beta l) \quad (\text{since } \gamma = j\beta)$$

For $l \to \lambda/4$, $\cos(\beta l) \to 0$, $$\displaystyle |V_S| $$ can be very high.

3. Power Circle Diagram (Lossless Line)

• Assumptions: $$\displaystyle R=0 $$, $$\displaystyle A=D=1 $$, $$\displaystyle B=jX $$, $$\displaystyle C=jB $$.

• Receiving-end Power Circle:

$$P_R = \frac{|V_S||V_R|}{X} \sin\delta - \frac{|V_R|^2 B}{2}$$

$$Q_R = \frac{|V_R|^2}{X}(\cos\delta - 1) - \frac{|V_R|^2 B}{2}$$

Center: $$\displaystyle (0, -\frac{|V_R|^2 B}{2}) $$, Radius: $$\displaystyle \frac{|V_S||V_R|}{X} $$.

• Interpretation: Shows limits of $$\displaystyle P_R $$, $$\displaystyle Q_R $$ for given $$\displaystyle V_S $$, $$\displaystyle V_R $$. Stability limit when $$\displaystyle \delta = 90^\circ $$ for lossless line ($$\displaystyle P_{R,max} = \frac{|V_S||V_R|}{X} $$).

4. Voltage Control Methods

• Tap-changing Transformers: Adjust voltage ratio.

• Shunt Capacitors/Reactors: Inject/absorb reactive power.

• Synchronous Condensers: Over-excited (capacitive), under-excited (inductive).

• Static VAR Compensators (SVC)/STATCOM: Fast reactive power control.

IV. Mechanical Design of Overhead Lines

A. Line Supports & Conductors

• Supports: Wooden (low voltage, rural), Steel (medium/high voltage), RCC (distribution), Lattice Steel Towers (EHV/UHV).

• Conductors: ACSR (Aluminium Conductor Steel Reinforced), AAAC, ACAR. Stranding improves flexibility, GMR.

B. Sag and Tension Calculations

1. Parabolic Approximation (for sag << span, supports at same level)

• Equation of Parabola: $$\displaystyle y = \frac{w}{2T_m} x^2 $$, where $w$ = weight per unit length, $$\displaystyle T_m $$ = tension at lowest point.

• Sag ($S$): At $$\displaystyle x = L/2 $$, $$\displaystyle S = \frac{wL^2}{8T_m} $$

• Tension at Support: $$\displaystyle T_s = T_m + \frac{wS}{2} \approx T_m $$ (if sag small).

• Key Assumption: $$\displaystyle T_m \approx T_s $$ (constant tension).

2. Effect of Wind and Ice

• Total Load per unit length:

$$w_{total} = \sqrt{(w + w_{ice})^2 + w_{wind}^2}$$

*   $$\displaystyle w_{ice} = \rho_{ice} \times \pi (r + t_{ice})^2 - \pi r^2 $$ (volume of ice per m).

*   $$\displaystyle w_{wind} = P_{wind} \times 2(r + t_{ice}) $$ (projected area per m).

• Maximum Sag: Use $$\displaystyle w_{total} $$ in parabolic formula. Tension at support $$\displaystyle T_s = \frac{w_{total} L^2}{8S} + \frac{w_{total} S}{2} $$.

• Factor of Safety (FoS): $$\displaystyle FoS = \frac{\text{Ultimate Strength}}{T_s} $$.

3. Catenary vs. Parabola

• Catenary: Exact curve ($$\displaystyle y = \frac{T_m}{w}(\cosh\frac{wx}{T_m} - 1) $$). Use when sag is large relative to span.

• Parabola: Good approximation for sag/span < 0.1.

4. Sag Template & String Chart

• Sag Template: Full-scale template of conductor sag curve for given $$\displaystyle T_m $$, $w$, temperature. Used to check tower height and mid-span clearance.

• String Chart: Plot of tension vs. temperature for fixed span. Shows operating limits (max tension at min temp, max sag at max temp).

V. Insulators & Insulation Coordination

A. Types of Insulators

Advantages of Suspension (for HV):

• Cheaper for > 33 kV.

• Each disc operates at ~11 kV, flexible.

• Faulty disc can be replaced without de-energizing.

• Provides necessary flexibility.

B. Voltage Distribution in Suspension Insulator String

1. String Efficiency

• Definition: Ratio of voltage across whole string to the product of number of units and voltage across the unit farthest from the conductor (most stressed).

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

• Significance: Measures effectiveness of string. Lower efficiency → first unit (top) is most stressed → higher failure rate.

2. Derivation of Voltage Distribution

• Assumptions: Self-capacitance of each unit = $C$. Pin-to-ground capacitance = $mC$ (typically $m \approx 0.1 - 0.2$).

• Let $$\displaystyle V_1, V_2, ..., V_n $$ be voltages across units from top to bottom.

• Kirchhoff's Current Law at each node:

  • For top unit: $$\displaystyle C V_2 = C V_1 + mC V_1 \Rightarrow V_2 = (1+m)V_1 $$

  • General: $$\displaystyle V_{k+1} = (1+m) V_k - m V_{k-1} $$

• Solution: $$\displaystyle V_k = V_1 \left[ \cosh((k-1)\theta) + \frac{m}{\sinh\theta} \sinh((k-1)\theta) \right] $$, where $$\displaystyle \cosh\theta = 1 + m $$.

• Voltage across n-th unit (bottom, highest): $$\displaystyle V_n = V_1 \left[ \cosh((n-1)\theta) + \frac{m}{\sinh\theta} \sinh((n-1)\theta) \right] $$

• Total String Voltage: $$\displaystyle V_{string} = \sum_{k=1}^n V_k = V_1 \cdot \frac{\sinh(n\theta)}{\sinh\theta} $$

• String Efficiency:

$$\eta = \frac{V_{string}}{n V_n} = \frac{1}{n} \cdot \frac{\sinh(n\theta)}{\sinh((n-1)\theta) + \frac{m}{\sinh\theta} \sinh((n-1)\theta)}$$

For large $n$ and small $m$, $$\displaystyle \eta \approx \frac{1}{1 + (n-1)m} \times 100\% $$.

C. Methods to Improve String Efficiency

1. Grading of Insulator Units: Use discs with different capacitance values (top units higher capacitance) to equalize voltage distribution.

2. Guard Ring (Static Shielding): A large metal ring connected to the conductor and placed around the bottom insulator units. It introduces additional capacitance between units and guard ring, equalizing potential gradient.

VI. System Configuration, Distribution & Substations

A. Distribution System Configurations

B. 3-Phase Systems: Transmission vs. Distribution

• Why Transmission 3φ, 3-wire & Distribution 3φ, 4-wire?

  • Transmission: Only 3-phase power required. Neutral not needed → saves one conductor (cost). Balanced load, no neutral current.

  • Distribution: Single-phase loads (lighting, domestic) connected phase-to-neutral → requires neutral wire for 415V/240V supply.

Copper Efficiency Comparison (for same max potential difference V between phases & neutral):

• 3φ, 3-wire system: Power $$\displaystyle P_3 = 3 \frac{V^2}{R_{ph}} $$ (where $$\displaystyle R_{ph} $$ is resistance per phase conductor).

• 3φ, 4-wire system: Same phase voltage V, but neutral carries unbalanced current. For balanced load, neutral current = 0. Power same as 3-wire.

• For given total copper volume: 4-wire uses more copper (extra neutral) but provides flexibility for single-phase loads.

% Additional Load Transmissible (Single-phase 2-wire → 3φ, 3-wire, same V, losses):

• For same voltage $V$ and same losses (hence same $$\displaystyle I^2R $$):

  • Single-phase 2-wire: Total conductor length = 2L, current = $$\displaystyle I_s $$.

  • 3φ, 3-wire: Total conductor length = 3L, phase current $$\displaystyle I_{ph} $$.

  • Losses equal: $$\displaystyle 2L I_s^2 R = 3L I_{ph}^2 R \Rightarrow I_{ph} = \sqrt{\frac{2}{3}} I_s $$

  • Power transmitted: $$\displaystyle P_{3\phi} = \sqrt{3} V I_{ph} \cos\phi $$, $$\displaystyle P_{1\phi} = V I_s \cos\phi $$ (assume same pf).

  • $$\displaystyle \frac{P_{3\phi}}{P_{1\phi}} = \sqrt{3} \times \sqrt{\frac{2}{3}} = \sqrt{2} \approx 1.414 $$

  • % Additional Load = $(\sqrt{2} - 1) \times 100\% \approx 41.4\%$.

C. Substation Equipment & Layout

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

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

7. Bus Bars: Collect and distribute power.

8. Isolating Switches, Earthing Switch, etc.

Single Line Diagram (SLD):

D. Bus Bar Arrangements

1. Single Bus Bar System:

   • Without Sectionalization: Simple, low cost. Entire bus de-energized for any fault/maintenance.

   • With Sectionalization: Bus divided by CBs. Allows partial operation during fault. More reliable, slightly complex.

2. Ring Main System: Bus arranged in ring. Supply from multiple sources → high reliability.

3. Sectionalized Double Bus Bar: Two parallel bus bars with coupler CB. Allows maintenance on one bus without interruption. High reliability, high cost.

VII. Special Topics & Comparative Studies

A. Bundle Conductors vs. Double Circuit 3-Phase Line

B. Tuned Power Lines

• Concept: Adjust line length so that it is an integer multiple of half wavelength at the operating frequency ($$\displaystyle l = n \frac{\lambda}{2} $$).

• Significance: At tuning frequency, line behaves as pure resistance ($$\displaystyle Z_{in} = Z_c $$), no reflection, maximum power transfer. Used in power line carrier communication (PLCC) and as quarter-wave transformers.

C. AIS vs. GIS Substation

D. Kelvin's Law for Economic Conductor Size

• Statement: Most economical cross-section is where annual cost of conductor = annual cost of energy wasted in conductor.

• Derivation:

  • Annual conductor cost: $$\displaystyle C_c = P_c \cdot A $$ (where $$\displaystyle P_c $$ = cost per unit area).

  • Annual energy loss cost: $$\displaystyle C_e = \frac{I^2 R \cdot \text{LF} \cdot T \cdot C_e}{\text{Voltage}^2} = \frac{J^2 A \cdot l \cdot \text{LF} \cdot T \cdot C_e}{\rho \cdot \text{Voltage}^2} $$ (where $J$ = current density).

  • Total annual cost: $$\displaystyle C_{total} = P_c A + K J^2 A $$ (K constant).

  • For minimum cost: $$\displaystyle \frac{dC_{total}}{dA} = 0 \Rightarrow P_c = 2K J^2 \Rightarrow J_{economical} = \sqrt{\frac{P_c}{2K}} $$.

• Limitations:

  • Ignores interest on capital for wasted energy (should include).

  • Assumes constant load factor & current density.

  • Does not consider voltage drop limits.

  • Practical conductor sizes are standard; economic size may not match.

[!TIP] Exam Focus: Be prepared to derive inductance (GMD/GMR), capacitance, voltage regulation (short/medium), ABCD constants, Ferranti effect, insulator string efficiency, sag (parabolic), dielectric stress in cable, and Kelvin's law. Numericals on load factors, inductance/capacitance calculation, line performance (nominal T/π), sag with wind/ice, insulator voltage distribution, and economic conductor size are very frequent.