Power System-I (EX-404) - Unit 5 Short Notes

1.0 ECONOMICS OF POWER GENERATION & LOAD ANALYSIS

1.1 Load Curves & Load Duration Curves

• Load Curve: Graph of load (kW/MW) versus time (hours/days). Shows variation.

  • Chronological Load Curve: Load vs. time of day (e.g., daily curve).

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

• Utility: Determines Maximum Demand, Load Factor, Plant Capacity Factor, and helps in economic dispatch and scheduling.

1.2 Key Performance Factors & Definitions

[!TIP] Proof: Diversity Improves System LF

System LF = (Total Energy) / (System MD × Time). Since System MD < Σ Individual MD (due to DFt > 1), for same total energy, a higher DFt reduces System MD, thus improving System LF.

1.3 Interconnection of Power Stations

• Advantages:

  • Reduces Reserve Capacity (standby plants).

  • Enables Economic Load Dispatch (cheapest stations loaded first).

  • Improves Reliability & Stability.

  • Facilitates sharing of Peak Loads.

• Disadvantages:

  • High Initial Cost (new lines, substations).

  • Complex Protection & Control.

  • Risk of Fault Propagation.

• Methods of Controlling Power Transfer (between Station A & B with slack bus):

  • Active Power (kW): Controlled by Scheduled Power (P_AB) setting. ΔP_A = -ΔP_B.

  • Reactive Power (kVAR): Controlled by Voltage Schedule at the interconnecting point. ΔQ_A = -ΔQ_B.

1.4 Effect of Load & Diversity Factors on Generation Cost

• Good Load Factor (LF → 1): Reduces Maximum Demand for same energy → Smaller plant capacity → Lower Fixed Costs (capital, interest, depreciation).

• Good Diversity Factor (DFt → large): Reduces System MD for same sum of individual MDs → Lower Fixed Costs for system.

• Overall Cost per kWh ↓ when LF & DFt ↑, as fixed cost component per unit decreases.

1.5 Energy Cost Analysis

Total Annual Cost = Fixed Cost (A) + Variable Cost (B × Energy).

Cost per kWh = $$\displaystyle \frac{A}{\text{Annual kWh}} + B $$.

• Impact of LF: If LF ↑, Annual kWh ↑ for same MD → $$\displaystyle \frac{A}{\text{kWh}} $$ ↓ → Cost/kWh ↓.

• Example (Jun 2023): Cost at 40% LF = 12 paise/kWh. If LF ↑ to 60%, fixed cost/kWh ↓. Given fuel cost ↑ 5%, net effect = New Cost = 12 × (40/60) × 1.05 = 8.4 paise/kWh.

1.6 Types of Loads

1.7 Power System Structure

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

• Non-Conventional (Renewable): Solar PV, Wind, Biomass, Geothermal, Tidal.

• Distributed Generation (DG): Small, modular sources (solar rooftops, small wind) located near load centers.

2.0 TRANSMISSION LINE PARAMETERS: INDUCTANCE & CAPACITANCE

2.1 Fundamental Concepts

• Flux Linkage (λ): Total flux linking a conductor.

• Inductance (L): $$\displaystyle L = \frac{\lambda}{I} $$ (H).

• Geometrical Mean Distance (GMD): For a set of distances D₁, D₂..., GMD = $$\displaystyle (D_1 \cdot D_2 \cdot ...)^{1/n} $$.

• Geometrical Mean Radius (GMR) / Self-GMD: For a solid round conductor of radius r, GMR = 0.7788r. For bundled conductors, GMR = $$\displaystyle (r' \cdot D)^{1/2} $$ for 2-conductor bundle, where $$\displaystyle r' = 0.7788r $$.

• Transposition: Periodic swapping of conductor positions to equalize inductance/capacitance of each phase over the line length.

2.2 Inductance Calculations

• Single-Phase Two-Wire Line:

$$L = 2 \times 10^{-7} \ln\left(\frac{D}{r'}\right) \text{ H/m}$$

where D = distance between conductors, r' = GMR.

• Three-Phase Lines:

  • Symmetrically Spaced (D₁₂=D₂₃=D₃₁=D):

    $$\displaystyle L = 2 \times 10^{-7} \ln\left(\frac{D}{r'}\right) \text{ H/m per phase} $$.

  • Unsymmetrically Spaced: Use GMD (D_s) for mutual distances and GMR (r') for self.

$$L = 2 \times 10^{-7} \ln\left(\frac{D_s}{r'}\right) \text{ H/m per phase}$$

    where $$\displaystyle D_s = (D_{12} D_{23} D_{31})^{1/3} $$.

*   **Double Circuit (Regular Hexagonal)**: Phase conductors on circle of radius R. GMD between phases of same circuit = D (side of hexagon). GMD between circuits = 2R. Equivalent GMD for inductance = $$\displaystyle (D \cdot D \cdot 2R \cdot 2R \cdot 2R \cdot 2R)^{1/6} = (4R^4 D^2)^{1/6} $$.

*   **Bundled Conductors (2 per phase)**: GMR<sub>bundle</sub> = $\sqrt{r' \cdot D}$ (D = spacing between conductors in bundle). Phase GMD (D<sub>s</sub>) as per configuration.

• Effect of Conductor Material: $$\displaystyle L \propto \mu_r $$ (relative permeability). Steel (μ_r≈50-100) has higher inductance than copper/aluminum (μ_r≈1).

• Skin Effect: Non-uniform current distribution at high freq. (AC) → current concentrates at surface → Effective AC resistance > DC resistance, Internal inductance decreases.

• Proximity Effect: Magnetic field of one conductor distorts current distribution in adjacent conductor → increases AC resistance.

2.3 Capacitance Calculations

• Single-Phase System (with Earth):

$$C = \frac{2\pi\epsilon_0}{\ln\left(\frac{2h}{r}\right)} \text{ F/m}$$

(h = height, r = radius). Earth effect increases capacitance.

• Three-Phase Overhead Line:

  • Symmetrical Spacing: $$\displaystyle C = \frac{2\pi\epsilon_0}{\ln(D/r')} $$ F/m per phase to neutral.

  • Unsymmetrical: Use GMD (D_s). $$\displaystyle C = \frac{2\pi\epsilon_0}{\ln(D_s/r')} $$.

• Single-Core Underground Cable:

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

where ε = ε<sub>0</sub>ε<sub>r</sub>, d = conductor diameter, D<sub>i</sub> = internal sheath diameter.

2.4 Transmission Line Configurations

3.0 TRANSMISSION LINE PERFORMANCE & MODELS

3.1 Classification of Lines

3.2 Short Transmission Line Model

• Phasor Diagram: V_R + I_R drop in series with V_S. For lagging PF, V_S > V_R.

• Voltage Regulation (VR):

$$\text{VR} = \frac{|V_S| - |V_R|}{|V_R|} \times 100\%$$

**Approximate VR** (for small R):

$$\text{VR} \approx \frac{I R \cos\phi_R + I X \sin\phi_R}{V_R} \times 100\%$$

• Transmission Efficiency (η):

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

3.3 Medium Transmission Line Models

• Nominal-T Model:

  • Total shunt admittance (Y = jωC) placed at midpoint.

  • Series impedance (Z = R + jX) split equally at ends.

  • ABCD Constants:

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

• Nominal-Π Model:

  • Total series impedance (Z) in middle.

  • Shunt admittance (Y/2) at both ends.

  • ABCD Constants:

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

*   **Phasor Diagram**: V<sub>R</sub> → drop across Z → V'<sub>S</sub> → add shunt current drop → V<sub>S</sub>.

3.4 Long Transmission Line

• Rigorous Solution (Telegrapher's Equation):

$$\frac{d^2 V}{dx^2} = ZY V, \quad \frac{d^2 I}{dx^2} = ZY I$$

Solution: $$\displaystyle V = A_1 e^{\gamma x} + A_2 e^{-\gamma x} $$, $$\displaystyle I = \frac{1}{Z_0}(A_1 e^{\gamma x} - A_2 e^{-\gamma x}) $$

where $$\displaystyle \gamma = \sqrt{ZY} = \alpha + j\beta $$ (propagation constant), $$\displaystyle Z_0 = \sqrt{Z/Y} $$ (surge impedance).

• Interpretation: Traveling waves with attenuation (α) and phase shift (β).

• Equivalent Equation (for length l):

$$\begin{bmatrix} V_S \\ I_S \end{bmatrix} = \begin{bmatrix} \cosh\gamma l & Z_0 \sinh\gamma l \\ \frac{1}{Z_0}\sinh\gamma l & \cosh\gamma l \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix}$$

ABCD constants: $$\displaystyle A = D = \cosh\gamma l $$, $$\displaystyle B = Z_0\sinh\gamma l $$, $$\displaystyle C = \frac{1}{Z_0}\sinh\gamma l $$.

• Tuned Power Line: Line is "tuned" when $$\displaystyle \omega L = \frac{1}{\omega C} $$ (reactive power balance) → SIL (Surge Impedance Loading) = $$\displaystyle \frac{V^2}{Z_0} $$.

3.5 Performance Calculations

• Sending End Quantities: From ABCD: $$\displaystyle V_S = A V_R + B I_R $$, $$\displaystyle I_S = C V_R + D I_R $$.

• Voltage Regulation & Efficiency: As for short line, but using computed V_S, I_S.

• Power Circle Diagram:

  • Real Power Circle: $$\displaystyle P_S = \frac{|V_S|^2}{B} \cos\theta - \frac{|V_S||V_R|}{B} \cos(\delta + \theta) $$.

  • Reactive Power Circle: $$\displaystyle Q_S = \frac{|V_S|^2}{B} \sin\theta - \frac{|V_S||V_R|}{B} \sin(\delta + \theta) $$.

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

  • Interpretation: Intersection with axes gives max P_S (at unity PF) and max Q_S.

3.6 Special Phenomena

• Ferranti Effect: Voltage rise at receiving end (V_R < V_S) for lightly loaded or no-load long/medium lines due to dominant capacitive charging current.

  • Derivation (No-load): I_R = 0, I_C = jωCV_R/2 (Π model). V_S = V_R + I_C(Z/2) → V_S > V_R.

• Power Loss in Open-Circuited Line:

$$P_{\text{loss}} = \frac{|V_R|^2}{2} \left( \frac{R}{Z^2} \right) \text{ (per phase)}$$

(From I<sub>C</sub> = jωCV, P<sub>loss</sub> = I<sub>C</sub>²R).

4.0 MECHANICAL DESIGN OF OVERHEAD LINES

4.1 Supports & Sag

• Types of Supports:

  • Wooden: Low voltage, short spans, cheap.

  • Steel Tubular/Poles: Medium voltage, urban.

  • Concrete: Medium/long spans, rural.

  • Lattice Steel Towers: EHV, long spans, high wind/ice.

• Shape of Sag Curve: Parabola (for small sag/span ratio). Proof: Differential equation $$\displaystyle y'' = \frac{w}{H} $$, where w = load/unit length, H = horizontal tension ≈ constant.

• Sag for Level Supports (weight only):

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

where l = span, T<sub>m</sub> = tension at lowest point.

**Maximum Tension**: T<sub>max</sub> = T<sub>m</sub> + wS.

• Effect of Wind & Ice Loading:

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

  • Sag: $$\displaystyle S' = \frac{w' l^2}{8 T_m} $$ (assuming T_m unchanged).

  • Where $$\displaystyle w_{ice} = \frac{\pi}{4} \rho_{ice} [(d+2t)^2 - d^2] $$, $$\displaystyle w_{wind} = \text{wind pressure} \times (d+2t) $$.

• Sag Template & String Chart:

  • Sag Template: Full-scale template of sag curve for a standard conductor & tension. Used to check tower height and ground clearance.

  • String Chart: Plot of sag vs. span for various temperatures (tension changes). Used for stringing conductors.

4.2 Conductor Material & Sizing

• Kelvin's Law for Economical Conductor Size:

  • Statement: The most economical conductor size is that for which annual interest & depreciation on capital cost equals annual energy loss cost.

  • Derivation:

    Let a = cross-sectional area (m²).

    Capital cost ∝ a. Annual cost = C₁a.

    Resistance R ∝ 1/a. Annual loss cost ∝ I²R ∝ I²/a.

    Total annual cost = C₁a + C₂/a.

    For minimum cost: $$\displaystyle \frac{d}{da}(C_1a + C_2/a) = 0 $$ → $$\displaystyle C_1 = C_2/a^2 $$ → a² = C₂/C₁.

  • Limitations:

    1. Assumes constant load factor & current.

    2. Interest/depreciation rates may vary.

    3. Does not consider mechanical strength (minimum area required for strength may be larger).

    4. Voltage drop not considered.

4.3 Tension & Safety

• Maximum Allowable Tension:

$$T_{\text{allow}} = \frac{\text{Ultimate Strength}}{\text{Factor of Safety}}$$

• Required Height/Height of Support:

$$H = S + \text{Ground Clearance}$$

(Ensure H ≥ minimum clearance).

5.0 INSULATION SYSTEMS FOR OVERHEAD LINES

5.1 Types of Insulators

5.2 Insulator String Analysis

• Capacitances:

  • Self-Capacitance (C): Capacitance between metal pin & cap of each unit.

  • Pin-to-Earth Capacitance (C₁): Typically 0.1C to 0.2C.

• Voltage Distribution (n identical units):

  Let V₁, V₂...V_n be voltages across units from top (line) to bottom (earth).

  Using KCL at each node: $$\displaystyle C(V_i - V_{i+1}) = C_1 V_i + C(V_{i-1} - V_i) $$.

  With $$\displaystyle k = C_1/C $$, solution: $$\displaystyle V_i = V_1 \frac{\sinh[(n-i+1)\theta]}{\sinh\theta} $$, where $$\displaystyle \cosh\theta = 1 + k $$.

  Top unit voltage is maximum.

• String Efficiency:

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

(V<sub>string</sub> = line-to-earth voltage). Always < 100% due to non-uniform distribution.

5.3 Methods to Improve String Efficiency

• Grading of Insulator Units: Use different capacitance units (longer discs at bottom, shorter at top) to equalize voltage drop.

• Guard Ring (Grading Ring): Metallic ring electrically connected to line side & earth side, shields pin-to-earth capacitance, making it more uniform → more uniform voltage distribution.

5.4 Testing: Flash-over Voltage

• Flash-over: Unintended arc over insulator surface due to excessive voltage.

• Test: Apply increasing voltage until flash-over occurs. Critical Flash-over Voltage (CFOV) is noted.

5.5 Insulator Pollution & Grading

• Pollution: Deposition of salt, dust, industrial fumes → conductive layer → leakage current → flash-over.

• Grading (in context of pollution): Using silicone or hydrophobic materials that repel water, preventing continuous conductive layer.

6.0 UNDERGROUND CABLES & COMPARISONS

6.1 Construction & Types

• Single-Core: One conductor + insulation + sheath + armouring.

• Three-Core: Three conductors + common insulation + sheath + armouring.

• Insulation: Paper (impregnated), Rubber, XLPE (cross-linked polyethylene), PILC (paper-insulated lead-covered).

• Sheathing: Lead or aluminum for moisture barrier.

• Armouring: Steel tape/wires for mechanical protection.

6.2 Electric Stress in Cables

• Dielectric Stress (g): Electric field intensity (V/m).

$$g = \frac{V}{x \ln(D_i/d)} \text{ (radial, for coaxial cable)}$$

where x = radial distance from conductor center, d = conductor dia, D<sub>i</sub> = internal sheath dia.

• Maximum Stress (g_max): At conductor surface (x = d/2).

$$g_{\text{max}} = \frac{V}{d/2 \ln(D_i/d)} = \frac{2V}{d \ln(D_i/d)}$$

• Minimum Stress (g_min): At sheath inner surface (x = D_i/2).

$$g_{\text{min}} = \frac{V}{D_i/2 \ln(D_i/d)} = \frac{2V}{D_i \ln(D_i/d)}$$

• Operating Voltage: Must be ≤ g_max (breakdown strength of insulation).

• Insulation Thickness: From given g_max, g_min and d:

$$\frac{g_{\text{max}}}{g_{\text{min}}} = \frac{D_i}{d} \Rightarrow D_i = d \left(\frac{g_{\text{max}}}{g_{\text{min}}}\right)$$

Thickness t = (D<sub>i</sub> - d)/2.

6.3 Methods of Grading in Cables

To achieve uniform stress (g = constant):

1. Capacitance Grading: Use multiple layers of dielectric with different permittivities (ε_r decreasing radially).

2. Intersheath Grading: Use metallic intersheaths at specific voltages to control potential distribution.

6.4 Comparison: Overhead Lines (OHL) vs. Underground Cables (UGC)

6.5 Comparison: AIS vs. GIS

7.0 DISTRIBUTION SYSTEMS & SUBSTATIONS

7.1 Distribution System Configurations

• Radial: Single source → feeders → loads. Simple, cheap, but low reliability (single point failure).

• Ring Main: Sources at both ends → closed loop. Loads tapped along ring. Higher reliability (supply from either end).

• Interconnected: Multiple interconnections → Highest reliability (multiple supply paths), complex protection.

7.2 Distribution Practices

• Why 3-φ, 3-wire Transmission? For high voltage, no neutral needed → saves one conductor (cheaper for long distance). Only balanced 3-φ loads.

• Why 3-φ, 4-wire Distribution? To provide phase voltage (230V) for single-phase loads (lighting, domestic) along with line voltage (415V) for 3-φ loads. Neutral carries unbalanced current.

• Copper Efficiency Comparison (same max voltage to earth):

  • 3-φ, 3-wire: Power $$\displaystyle P = \sqrt{3} V_L I_L \cos\phi $$. Resistance per phase $$\displaystyle R = \frac{\rho l}{A} $$. Loss = 3I_L²R.

  • 3-φ, 4-wire: Same line voltage V_L, but neutral present. For same power & losses, required area A_4w > A_3w. 3-wire is more economical.

7.3 Bus Bar Systems in Substations

• Single Bus Bar with Sectionalization:

  • Arrangement: Single bus divided by a circuit breaker.

  • Merits: Can isolate a section without interrupting others. Some redundancy.

  • Demerits: Bus fault affects entire bus; breaker failure isolates section.

• Sectionalized Double Bus Bar:

  • Arrangement: Two parallel buses, each with its breaker, and coupling breaker between them.

  • Merits: High reliability (maintenance on one bus, transfer via coupling breaker). Flexible.

  • Demerits: Costly, complex protection.

• Ring Mains: Bus bars arranged in a ring. Similar reliability to double bus but with different switching.

7.4 Substation Equipment (Brief)

7.5 Single Line Diagram (Typical A.C. Distribution)

• Shows: Incoming transmission line → CTs/PTs → CB → Isolators → Main Bus → Power Transformer → Outgoing feeders with CBs/Isolators → Distribution lines.

• Key: All equipment represented by standard symbols; shows connectivity, not physical layout.

8.0 ECONOMIC DESIGN OF CONDUCTORS & SYSTEMS

8.1 Transmission Voltage Selection

• Higher Voltage → Lower Current for same power → Lower I²R losses and Smaller conductor size.

• Trade-off: Higher voltage requires more expensive insulation, larger tower size, higher clearance → Higher capital cost.

• Economic Voltage: Chosen where total cost (capital + losses) is minimized. Typically, long distances → higher voltage.

8.2 Conductor Material Comparison: DC 2-wire vs. Single-Phase AC

• Given: Same length (l), same power (P), same voltage (V_L), same % losses.

• DC 2-wire: P = V I_dc, Loss = 2 I_dc² R_dc.

• 1-φ AC: P = V I_ac cosφ, Loss = 2 I_ac² R_ac (R_ac > R_dc due to skin effect).

• For same losses & power:

  $$\displaystyle I_{ac} \cos\phi = I_{dc} $$ and $$\displaystyle I_{ac}^2 R_{ac} = I_{dc}^2 R_{dc} $$.

  → $$\displaystyle R_{ac} \cos^2\phi = R_{dc} $$.

  Since R_ac > R_dc, cosφ must be high (near 1) for AC to be competitive.

  Cross-sectional area A ∝ 1/R → A_ac / A_dc = R_dc/R_ac = cos²φ.

  → AC requires larger area than DC for same losses if cosφ < 1.

8.3 System Conversion Problem: 1-φ 2-wire to 3-φ 3-wire

• Given: Same voltage (V_L), same % losses, same length.

• 1-φ 2-wire: P₁ = V I₁ cosφ, Loss = 2 I₁² R₁.

• 3-φ 3-wire: P₃ = √3 V I₃ cosφ, Loss = 3 I₃² R₃.

• Same losses: 2 I₁² R₁ = 3 I₃² R₃.

  Same conductor material & length → R ∝ 1/A → R₃/R₁ = A₁/A₃.

• Additional load transmissible:

$$\frac{P_3}{P_1} = \frac{\sqrt{3} V I_3 \cos\phi}{V I_1 \cos\phi} = \sqrt{3} \frac{I_3}{I_1}$$

From loss equality: $$\displaystyle \frac{I_3}{I_1} = \sqrt{\frac{2 R_1}{3 R_3}} = \sqrt{\frac{2}{3} \frac{A_3}{A_1}} $$.

But for **same conductor size** (A<sub>3</sub> = A<sub>1</sub>), $$\displaystyle \frac{I_3}{I_1} = \sqrt{\frac{2}{3}} $$.

→ $$\displaystyle \frac{P_3}{P_1} = \sqrt{3} \times \sqrt{\frac{2}{3}} = \sqrt{2} \approx 1.414 $$.

→ **3-φ system transmits 41.4% more power** than 1-φ with same conductors, voltage, and losses.

8.4 Volume of Conductor Calculation

• Given: Load (P), Voltage (V), Efficiency (η), Length (l), System type, ρ.

• Step 1: Find P_loss = P(1/η - 1).

• Step 2: For system, find total resistance (R_total) from loss formula.

  • 1-φ 2-wire: Loss = 2 I² R → R = P_loss / (2 I²), I = P/(V cosφ).

  • 3-φ 3-wire: Loss = 3 I² R → R = P_loss / (3 I²), I = P/(√3 V cosφ).

• Step 3: R = ρ l / A → A = ρ l / R.

• Step 4: Volume = A × l × (number of conductors).

  • 1-φ 2-wire: Volume = 2 × A × l.

  • 3-φ 3-wire: Volume = 3 × A × l.

[!TIP] Common Pitfall: Forgetting number of conductors (2 for 1-φ, 3 for 3-φ) when calculating total volume/cost. Also, ensure consistent units (ρ in Ω·m, l in m, A in m²).