UNIT 1: ELECTRICAL POWER GENERATION & ECONOMY

Based on RGPV Past Papers (EX-503 A & C Series)

I. CONVENTIONAL / MAJOR POWER GENERATION TECHNOLOGIES

A. Hydroelectric Power Plants

1. Layout & Components

A hydroelectric plant converts potential energy of stored water into electrical energy.

Key Components & Functions:

• Dam/Reservoir: Stores water, creates head.

• Intake/Headrace: Admits water to penstock, screens debris.

• Penstock: Large pipe conducting water under pressure to turbine.

• Surge Tank: Relieves water hammer pressure in penstock during load changes.

• Turbine: Converts hydraulic energy to mechanical rotation.

*   **Pelton:** High head (300m+), impulse type, uses nozzles.

*   **Francis:** Medium head (30m-300m), reaction type.

*   **Kaplan:** Low head (<30m), reaction type with adjustable blades.

• Generator: Converts mechanical rotation to electrical energy.

• Tailrace: Discharges used water back to river.

2. Site Selection Factors

• Water Availability: Consistent, high annual flow; catchment area.

• Head: Height of water fall (higher head = smaller turbine, less civil work).

• Geology: Strong rock foundation for dam & powerhouse.

• Topography: Narrow gorge ideal for dam construction.

• Proximity to Load Centre: Minimizes transmission cost/losses.

3. Hydrographs & Duration Curves

• Hydrograph: Graph of discharge (flow) vs. time (daily, monthly, yearly). Shows river flow variability.

• Flow Duration Curve (FDC): Plot of flow magnitude vs. percentage of time it is exceeded. Critical for firm power estimation.

• Power Duration Curve (PDC): Derived from FDC using $$\displaystyle P = \rho g Q H \eta $$. Shows available power vs. time.

4. Pumped Storage Plants

• Working: During off-peak (low load), excess power pumps water from lower to upper reservoir. During peak load, water is released to generate power.

• Layout: Two reservoirs at different elevations, reversible pump-turbine unit.

• Merits: Excellent for peak load, quick start, improves system load factor.

• Demerits: High capital cost, 25-40% energy loss in cycle, requires suitable terrain.

5. Small Hydro Plants (SHP)

• Definition: Typically < 25 MW (India: < 10 MW for mini, < 5 MW for micro).

• Characteristics: Run-of-river (no large dam), minimal submergence, lower environmental impact, suitable for remote/hilly areas.

B. Thermal Power Plants (Steam)

1. Layout & Main Features

• Coal Handling: Unloading → Crushing → Storage → Pulverizing → Feeding to boiler.

• Boiler: Burns coal to produce high-pressure, high-temperature steam.

• Turbine: Steam expands through stages (HP, IP, LP), rotating shaft.

• Condenser: Condenses exhaust steam from turbine to water (creates vacuum, improves efficiency).

• Cooling Tower/Cooling Pond: Cools condenser cooling water (for recirculation).

• Path: Coal → Boiler → Turbine → Condenser → Cooling Tower → (Feedwater pumps) → Boiler.

2. Key Components & Functions

3. Site Selection Factors

• Fuel Availability: Proximity to coal mine/port (reduces transport cost).

• Water Source: Abundant water for boiler make-up & condenser cooling (river, sea).

• Land: Large, cheap, firm ground.

• Proximity to Load Centre: Reduces transmission losses/cost.

• Transport: Railways, roads, ports for coal & ash.

• Ash Disposal: Space for ash ponds, low water table area.

4. Water Treatment Plant

• Necessity: Prevents scale formation (Ca, Mg salts) in boiler tubes (reduces heat transfer, causes overheating) and corrosion (dissolved O₂, CO₂).

• Processes (Typical): Screening → Aeration (remove CO₂) → Chemical treatment (lime-soda softening, ion exchange) → Filtration → Demineralization (for high-pressure boilers) → De-aeration (remove O₂).

C. Nuclear Power Plants

1. Basic Principle & Reactor Components

• Nuclear Fission: Heavy nucleus (U-235, Pu-239) splits on neutron capture, releasing huge energy & neutrons (chain reaction).

• Fusion: Light nuclei combine (H to He) – experimental, not commercial.

2. CANDU Reactor (CANada Deuterium Uranium)

• Features: Heavy water moderator & coolant; Natural U fuel (no enrichment); Pressure tubes (not vessel); On-power refuelling.

• Advantages: Uses natural U (cheaper), high neutron economy (can use thorium), flexible fuel cycle.

• Disadvantages: Heavy water expensive & can leak, large size, proliferation concerns (can breed Pu-239).

3. Nuclear Fuel Cycle & Availability in India

• Resources: Limited Uranium (Jaduguda, Singhbhum, Tummalapalle), abundant Thorium (Kerala, Odisha, Andhra Pradesh – world's largest).

• Strategy: Three-stage program: (1) PHWRs (U-238 → Pu-239), (2) Fast Breeder Reactors (Pu-239 + Th-232 → U-233), (3) Thorium-based reactors (U-233 + Th-232).

• Current: Primarily PHWRs (Uranium-based), moving towards FBRs.

4. Radioactive Pollution & Waste Disposal

• Sources: Fuel fabrication, reactor operation, spent fuel, decommissioning.

• Waste Classification: Low/Intermediate Level (LILW): Contaminated tools, filters. High Level (HLW): Spent fuel, reprocessing waste.

• Disposal Methods:

  • Storage: Wet (pool) → Dry (casks) for interim.

  • Reprocessing: Recover U, Pu (India: Trombay, Tarapur). Reduces HLW volume/radiotoxicity.

  • Final Disposal: Deep geological repository (stable rock formation, multiple barriers). Critical for public acceptance & safety.

• Importance: Long-term isolation (10⁴-10⁵ years) to protect biosphere from radiotoxicity.

5. Radiation Shielding

• Purpose: Attenuate α, β, γ, neutron radiation to safe levels for personnel & environment.

• Materials: Concrete (cheap, good for γ, neutrons with boron addition), Lead (excellent for γ), Water (neutrons), Steel (structural + shielding).

D. Gas Turbine Power Plants

1. Layout & Working Principle

• Working (Brayton/Joule Cycle):

  1. Compression: Air compressed in axial/centrifugal compressor (pressure ↑, temp ↑).

  2. Combustion: Fuel (gas, oil) injected & burned in combustion chamber (constant pressure, temp ↑ drastically).

  3. Expansion: Hot gases expand through turbine, producing work to drive compressor & load.

  4. Exhaust: Hot exhaust gases released (~450-650°C).

• Net Work: $$\displaystyle W_{net} = W_{turbine} - W_{compressor} $$. Low thermal efficiency (~30%) due to high exhaust loss.

2. Classification

• Based on Cycle: Open Cycle (air from atmosphere, exhaust to atmosphere – common). Closed Cycle (working fluid recirculated, heat added externally).

• Based on Application: Peak Load (quick start, low capital cost, high running cost). Base Load (combined cycle, high efficiency).

• Based on Fuel: Natural gas, liquid fuel.

3. Methods to Improve Thermal Efficiency

E. Diesel Power Plants

• Fuel System: Storage tanks → Filters → Injection pump → Injectors (atomize fuel into combustion chamber). High-pressure fuel injection (~150-200 bar).

• Exhaust System: Exhaust manifold → Silencer (muffler) → Stack. Contains pollutants (NOx, particulates, CO).

• Features: High efficiency (40-50%), quick start, good for standby/peak load, high maintenance, high running cost (diesel fuel).

II. RENEWABLE & NON-CONVENTIONAL POWER GENERATION

A. Solar Energy

1. Solar Radiation & Geometry

• Key Terms:

  • Latitude (φ): Angular distance from equator.

  • Longitude (L): Angular distance from Prime Meridian.

  • Declination (δ): Angle between sun-Earth line & equatorial plane. $$\displaystyle \delta = 23.45^\circ \sin\left(\frac{360}{365}(284 + n)\right) $$, where $n$ = day number.

  • Hour Angle (ω): Angular displacement of sun from local meridian. $$\displaystyle \omega = 15^\circ \times (t_s - 12) $$, $$\displaystyle t_s $$ = solar time (hrs).

  • Altitude (α): Angle of sun above horizon.

$$\sin \alpha = \sin \phi \sin \delta + \cos \phi \cos \delta \cos \omega$$

*   **Azimuth (γ):** Angle of sun's projection on horizontal plane from south (N. Hemisphere). 

$$\cos \gamma = \frac{\sin \delta \cos \phi - \cos \delta \sin \phi \cos \omega}{\cos \alpha}$$

2. Solar Thermal Power Generation

• Principle: Solar radiation → Heat (collector) → Steam (heat exchanger) → Turbine → Generator.

• Classification of Collectors:

  • Flat Plate: Absorber plate, glazing, insulation. Low temp (<100°C), low concentration.

  • Concentrating:

    • Parabolic Trough: Linear focus, tracks N-S axis. Fluid (oil) heated in tube.

    • Parabolic Dish: Point focus, high concentration, Stirling engine.

    • Solar Tower: Heliostats reflect to central receiver on tower.

• Flat Plate Collector Components:

  DiagramCANVAS: Cross-section: 1. Glazing (glass), 2. Absorber plate (black coating with tubes), 3. Insulation (back/sides), 4. Casing, 5. Fluid tubes

• Performance Factors: Insolation, collector orientation/tilt, ambient temp, wind speed, heat loss coefficient.

3. Solar Photovoltaic (PV) Systems

• Principle: Photoelectric effect. Photons excite electrons in p-n junction, creating voltage/current.

• Main Elements: Cell → Module (series/parallel cells) → Array → Inverter (DC→AC) → Balance of System (mounting, wiring, charge controller, batteries).

• PV Cell Diagram:

  DiagramCANVAS: Cross-section: 1. Anti-reflective coating, 2. n-type layer, 3. p-type layer, 4. p-n junction/depletion region, 5. Metal grid contacts (front), 6. Full back contact, 7. Substrate. Arrows show light photons generating e-h pairs.

• I-V Characteristics:

  • Open Circuit Voltage ($$\displaystyle V_{oc} $$): Voltage at $$\displaystyle I=0 $$.

  • Short Circuit Current ($$\displaystyle I_{sc} $$): Current at $$\displaystyle V=0 $$.

  • Maximum Power Point (MPP): $$\displaystyle (V_m, I_m) $$ where $$\displaystyle P_{max} = V_m I_m $$.

  • Fill Factor (FF):

$$\text{FF} = \frac{V_m I_m}{V_{oc} I_{sc}}$$

(Quality measure, typically 0.7-0.85).

*   **Efficiency (η):** 

$$\eta = \frac{P_{max}}{\text{Input Solar Power}} = \frac{V_m I_m}{A \cdot G}$$

, where $A$ = area, $G$ = irradiance (W/m²).

> **Example (from DEC 2024):** Given $$\displaystyle V_{oc}=0.24V $$, $$\displaystyle I_{sc}=10mA $$, $$\displaystyle V_m=0.14V $$, $$\displaystyle I_m=6.5mA $$, Intensity=24 W/m², Area=4 cm² = 0.0004 m².

> *   $$\displaystyle P_{max} = 0.14 \times 6.5 \times 10^{-3} = 0.91 \times 10^{-3} W = 0.91 mW $$

> *   Input Power = $$\displaystyle 24 \times 0.0004 = 0.0096 W = 9.6 mW $$

> *   $$\displaystyle \eta = \frac{0.91}{9.6} \times 100\% = 9.48\% $$

> *   FF = $$\displaystyle \frac{0.14 \times 6.5}{0.24 \times 10} = \frac{0.91}{2.4} = 0.379 $$

• Advantages: No moving parts, modular, low maintenance, silent.

• Limitations: Low efficiency (~15-20% commercial), intermittent (needs storage), high initial cost, area-intensive.

B. Wind Energy

1. Principle of Wind Power Generation

• Kinetic energy of wind → rotor blades → mechanical rotation → generator → electricity.

• Available Wind Power:

$$P_{wind} = \frac{1}{2} \rho A v^3$$

*   $\rho$ = air density (~1.225 kg/m³), $A$ = swept area ($$\displaystyle \pi R^2 $$), $v$ = wind speed.

• Betz's Limit: Maximum theoretical fraction of wind power extractable by an ideal turbine = 16/27 ≈ 59.3%. Actual: 35-45%.

2. Wind Energy Conversion Systems (WECS)

• Classification:

  • Horizontal Axis Wind Turbine (HAWT): Main axis parallel to wind. Most common. Needs yaw mechanism.

  • Vertical Axis Wind Turbine (VAWT): Main axis perpendicular to wind. Darrieus (lift-based, high speed), Savonius (drag-based, low speed, self-starting).

• HAWT Components:

  • Blades (aerofoil shape), Rotor, Nacelle (gearbox, generator), Tower, Yaw system, Brakes.

• Wind Characteristics: Speed follows Weibull distribution ($k$ shape, $c$ scale). Mean speed alone insufficient; need distribution for energy estimate.

• Performance & Limitations: Intermittent (capacity factor 20-40%), site-specific (need avg. wind > 5-6 m/s), noise, visual impact, avian mortality.

3. Control Schemes

• Pitch Control: Rotate blades to regulate power at high winds.

• Yaw Control: Rotate nacelle to face wind.

• Stall Control: Fixed blades; aerodynamics cause stall at high wind, limiting power.

• Generation Control: Adjust generator torque/slip.

4. Site Selection for Wind Plants

• High Wind Speed/Density: Avg. > 6 m/s at hub height, low turbulence.

• Favorable Terrain: Hilltops, coastal areas, mountain passes (wind acceleration).

• Accessibility: For transport/erection.

• Grid Proximity: Minimize evacuation cost.

• Land Use: Non-agricultural, low population density.

5. Safety & Environmental Aspects

• Noise: Aerodynamic (blade) & mechanical (gearbox).

• Visual Impact: "Skyline pollution".

• Avian/Bat Mortality: Collision/barotrauma.

• Shadow Flicker: Stroboscopic effect on nearby areas.

C. Biomass Energy

1. Biomass Resources & Applications

• Types: Agricultural residue (straw, bagasse), wood waste, animal dung, municipal solid waste (MSW), energy crops.

• Applications:

  • Direct Combustion: For heat/steam (cogeneration).

  • Biogas: Anaerobic digestion → CH₄ (cooking, electricity).

  • Biofuels: Ethanol (sugarcane, corn), Biodiesel (jatropha, algae).

  • Pyrolysis: Thermal decomposition without O₂ → bio-oil, char, syngas.

2. Biogas Generation

• Principle: Anaerobic digestion (4 stages: hydrolysis, acidogenesis, acetogenesis, methanogenesis) by bacteria in absence of O₂.

  • Input: Biomass + water (slurry) → Digester → Output: Biogas (CH₄ ~55-65%, CO₂) + Slurry (fertilizer).

• Deen Bandhu (Floating Drum): Fixed dome, steel drum floats on slurry as gas collects. Common, but drum maintenance.

• Pragati/KVIC Design: Fixed steel dome, no moving part in gas chamber. More durable.

• Community Biogas Plant Problems: Feedstock collection/transport, consistent supply, management, social issues.

• Materials Used: Cattle dung (most common), poultry litter, food waste, agricultural residue, human waste (with caution).

3. Electricity from Biomass

• Methods:

  1. Direct Combustion: Burn biomass in boiler → steam → turbine.

  2. Gasification: Partial combustion → producer gas (CO, H₂) → engine/gas turbine.

  3. Anaerobic Digestion: Biogas → engine/generator.

• Advantages: Renewable, carbon neutral (in cycle), waste management, rural employment.

4. Environmental Problems from Biomass

• Open Burning: Releases pollutants (PM2.5/10, CO, VOCs, PAHs), causes severe air pollution & health hazards (especially in Punjab/Haryana post-harvest).

5. Pyrolysis

• Definition: Thermal decomposition of biomass at high temp (400-800°C) in absence of oxygen.

• Small-Scale Unit: Feedstock → Heater (external) → Pyrolysis reactor (produces bio-oil vapor, char, syngas) → Condenser (bio-oil) → Collection. Bio-oil can be used in engines/boilers.

D. Other Renewable Sources

1. Geothermal Energy

• Sources:

  • Hydrothermal: Hot water/steam reservoirs (most used).

  • Geopressured: Hot brine under pressure (contains methane).

  • Hot Dry Rock (HDR): Hot impermeable rock (needs fracturing).

  • Magma: Molten rock (very high temp, exploratory).

• Potential in India: Low/Medium. Himalayas (tectonic), Western Ghats, Cambay basin, Son-Narmada-Tapti valleys. Not commercially exploited yet (except Parvati Valley, Jammu).

• Types of Power Plants:

  • Dry Steam: Direct use of geothermal steam (rare, e.g., Larderello, Italy).

  • Flash Steam: High-pressured hot water flashed to steam in separator.

  • Binary Cycle: Geothermal fluid heats secondary fluid (low boiling point, e.g., isobutane) in heat exchanger → vapor drives turbine. Most common for low-temp resources.

• Working of Binary Fluid: Geothermal fluid (120-180°C) → Heat Exchanger (vaporizes secondary fluid) → Turbine → Condenser → Pump → Heat Exchanger. Geothermal fluid reinjected.

• Why Flashing May Not Be Possible: If geothermal fluid temperature is below saturation temperature for the given pressure (i.e., subcooled liquid), or if it contains non-condensable gases/minerals that cause scaling/corrosion in flash tanks. Binary cycle avoids this.

• Hybrid Geothermal-Fossil: Use fossil fuel (gas/coal) to supplement geothermal heat, increasing output/efficiency, especially during low geothermal flow.

2. Ocean Energy

• Tidal Energy:

  • Principle: Potential energy of rising/falling tides. Requires high tidal range (>4m).

  • Site Selection: Large tidal range, narrow inlet/bay (for barrage), firm foundation, minimal shipping.

  • Schematic Layout (Barrage): Dam across estuary → Sluice gates → Turbines in caissons → Basin. Generate during ebb/flood tides.

  DiagramSEARCH: tidal barrage power plant diagram single effect

• Wave Energy: Kinetic energy of surface waves. Devices: Oscillating water column, point absorber, attenuator. Highly variable.

• Ocean Thermal Energy Conversion (OTEC):

  • Principle: Temperature gradient between warm surface water (~25-30°C) and cold deep water (~5-10°C). Uses low-boiling-point fluid (ammonia).

  • Closed OTEC System:

    DiagramSEARCH: closed cycle OTEC system diagram evaporator turbine condenser pump

    Warm surface water → Evaporator (vaporizes ammonia) → Turbine → Condenser (cold deep water condenses ammonia) → Pump → back to evaporator.

3. Hydrogen Energy

• Advantages: High energy density (by mass), clean burning (H₂O), can be stored/transported, versatile feedstock.

• Disadvantages: Low density (by volume), storage/transport challenges, production cost (if from electrolysis, needs cheap electricity), safety (flammable, wide explosive range).

• Storage Methods:

  | Method | Principle | Pros | Cons | | :--- | :--- | :--- | :--- | | Compressed Gas | High pressure (350-700 bar) cylinders | Simple, mature technology | Heavy, bulky, energy-intensive compression | | Liquid Hydrogen | Cryogenic storage (-253°C) | High density (by volume) | High boil-off loss, expensive insulation | | Metal Hydrides | H₂ absorbed in metal lattice (e.g., LaNi₅) | Safe, moderate pressure/temp | Heavy, slow kinetics, high cost |

4. Fuel Cells

• Principle: Electrochemical device converting chemical energy (fuel + oxidant) directly to electricity, without combustion. Anode (oxidation, e⁻ release), Cathode (reduction, e⁻ consumption), Electrolyte (ion conductor).

• Classification (by Electrolyte):

  • AFC (Alkaline): KOH electrolyte, space applications.

  • PEMFC (Polymer Electrolyte Membrane): Solid polymer, low temp (80°C), quick start, vehicles/backup power.

  • SOFC (Solid Oxide): Ceramic, high temp (600-1000°C), high efficiency, fuel flexible (CH₄, H₂, CO), stationary power.

  • MCFC (Molten Carbonate): Molten carbonate salt, high temp (650°C), fuel flexible, large stationary.

  • PAFC (Phosphoric Acid): Liquid phosphoric acid, medium temp (~200°C), commercial CHP.

E. Integrated & Emerging Concepts

1. Hybrid Systems

• Concept: Combine two or more renewable sources (e.g., solar-wind, solar-biomass, wind-diesel) with/without storage to improve reliability and output stability.

• Examples: Solar + Wind (complementary – wind often at night), Solar + Biomass (biomass provides base, solar peak).

• Advantages: Reduced intermittency, better capacity factor, optimized sizing/cost, improved power quality.

2. Cogeneration (Combined Heat and Power - CHP)

• Definition: Simultaneous generation of electricity and useful thermal energy (heat/steam) from a single fuel source.

• Principle: Capture waste heat from power generation (e.g., exhaust from gas turbine, steam from turbine extraction) for industrial processes, district heating.

• Benefits: Dramatically increases overall fuel efficiency (from ~35-50% for condensing plants to 70-90% for CHP), reduces fuel cost & emissions per unit of useful energy.

III. ECONOMIC OPERATION & POWER SYSTEM PLANNING

A. Fundamentals of Power Plant Economics

1. Costs of Power Generation

$$C = F + V \cdot P$$

| $C$ = total cost/hr, $F$ = fixed cost/hr, $V$ = variable cost/Rupee per MWh, $P$ = power output (MW). |

2. Performance Factors

$$LF = \frac{\text{Avg Load}}{\text{Max Demand}} = \frac{\text{Energy (kWh)}}{\text{Max Demand (kW)} \times \text{Time (h)}}$$

| Load varies; max demand is peak. | | Capacity Factor (CF) | Measure of plant utilization over time. |

$$CF = \frac{\text{Actual Energy Output}}{\text{Rated Capacity} \times \text{Time}}$$

| Plant not always at full capacity (maintenance, low load). | | Utilisation Factor (UF) | Measure of how much of installed capacity is actually used. |

$$UF = \frac{\text{Max Demand}}{\text{Installed Capacity}}$$

| Installed capacity > max demand for reliability/spinning reserve. | | Demand Factor (DF) | Ratio of max demand to connected load. |

$$DF = \frac{\text{Max Demand}}{\text{Connected Load}}$$

| Not all connected load operates simultaneously. |

• Relationship: $$\displaystyle LF = CF \times UF $$. Energy cost $\propto 1/LF$ (higher LF spreads fixed cost over more units).

3. Load Curves & Forecasting

• Load Curve: Graph of load (kW) vs. time (typically 24 hrs). Shows variation.

• Load Duration Curve (LDC): Load values ranked in descending order vs. time percentage. Used for economic dispatch & capacity planning.

• Daily/Annual Load Curves: Show daily/seasonal patterns.

• Flow Duration Curve (FDC): For hydro – flow vs. % time exceeded.

• Load Forecasting: Crucial for generation scheduling & capacity expansion.

  • Short-term (1 day - 1 week): Unit commitment, economic dispatch.

  • Medium-term (1 month - 1 year): Maintenance scheduling, fuel procurement.

  • Long-term (>1 year): Capacity planning, transmission expansion.

  • Methods: Time series, regression, neural networks, expert systems.

B. Economic Dispatch & Scheduling

1. Economic Load Scheduling (ELS) / Economic Dispatch

• Objective: Minimize total system fuel cost for a given total load $$\displaystyle P_T $$ (and losses) while satisfying generator limits.

• Lossless System Criterion: For optimal dispatch, incremental fuel cost (λ) of all units must be equal.

$$\lambda = \frac{dC_i}{dP_i}$$

where $$\displaystyle C_i = a_i + b_i P_i + c_i P_i^2 $$ (common cost function).

• Procedure: Solve $$\displaystyle \frac{dC_1}{dP_1} = \frac{dC_2}{dP_2} = ... = \lambda $$ subject to $$\displaystyle \sum P_i = P_T $$ and $$\displaystyle P_{i min} \le P_i \le P_{i max} $$.

2. Problems & Calculations

Example 1 (DEC 2024, Lossless):

$$\displaystyle C_1=50+2P_1+0.005P_1^{2} $$, $$\displaystyle C_2=100+2P_2+0.01P_2^{2} $$, $$\displaystyle P_T=350 MW $$.

• $$\displaystyle \frac{dC_1}{dP_1} = 2 + 0.01 P_1 $$

• $$\displaystyle \frac{dC_2}{dP_2} = 2 + 0.02 P_2 $$

• Set equal: $$\displaystyle 2 + 0.01 P_1 = 2 + 0.02 P_2 \Rightarrow P_1 = 2 P_2 $$

• $$\displaystyle P_1 + P_2 = 350 \Rightarrow 2P_2 + P_2 = 350 \Rightarrow P_2 = 116.67 MW $$, $$\displaystyle P_1 = 233.33 MW $$.

• $$\displaystyle \lambda = 2 + 0.01 \times 233.33 = 4.3333 $$ Rs/MWh.

Example 2 (JUN 2025, with Losses):

Given: $$\displaystyle \frac{dC_1}{dP_1}=0.15 P_1+150 $$, $$\displaystyle \frac{dC_2}{dP_2}=0.25 P_2+175 $$.

Operating at $$\displaystyle P_1=P_2=400 MW $$, $$\displaystyle \frac{\partial P_L}{\partial P_2}=0.2 $$.

• Penalty Factor (PF) for plant i: $$\displaystyle PF_i = \frac{\lambda}{(dC_i/dP_i)} $$ at optimum.

• For optimum with losses: $$\displaystyle \frac{dC_i}{dP_i} \cdot PF_i = \lambda $$ (same for all).

• Given $$\displaystyle \frac{\partial P_L}{\partial P_2} = 0.2 $$. For plant 2: $$\displaystyle PF_2 = \frac{1}{1 - \frac{\partial P_L}{\partial P_2}} = \frac{1}{1-0.2} = 1.25 $$.

• At given operating point (not necessarily optimum), we can find $\lambda$ from plant 2: $$\displaystyle \lambda_2 = (dC_2/dP_2) \times PF_2 $$ only if it's optimum. But we are asked PF of plant 1.

• Key: At optimum, $\lambda$ is same. We know $$\displaystyle PF_2 = 1.25 $$. We need $$\displaystyle PF_1 $$.

• From plant 2 at $$\displaystyle P_2=400 $$: $$\displaystyle dC_2/dP_2 = 0.25 \times 400 + 175 = 100 + 175 = 275 $$.

• If operating point is optimum, $$\displaystyle \lambda = 275 \times 1.25 = 343.75 $$.

• For plant 1 at $$\displaystyle P_1=400 $$: $$\displaystyle dC_1/dP_1 = 0.15 \times 400 + 150 = 60 + 150 = 210 $$.

• Then $$\displaystyle PF_1 = \lambda / (dC_1/dP_1) = 343.75 / 210 = 1.6375 $$.

• Answer: \boxed{1.6375} (or 1.64).

• Inclusion of Transmission Losses:

  • Loss Formula: $$\displaystyle P_L = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} P_i P_j $$ (where $$\displaystyle B_{ij} $$ are loss coefficients, symmetric, $$\displaystyle B_{ii}>0 $$, $$\displaystyle B_{ij}<0 $$).

  • Optimality Condition: $$\displaystyle \frac{dC_i}{dP_i} = \lambda \left(1 - \frac{\partial P_L}{\partial P_i}\right) $$.

  • Penalty Factor (PF): $$\displaystyle PF_i = \frac{1}{1 - \frac{\partial P_L}{\partial P_i}} $$. $$\displaystyle PF_i > 1 $$. Plant with higher PF gets less share.

C. Tariffs & Pricing

1. Tariff Definition & Types

• Tariff: Schedule of rates for supplying electrical energy to consumers.

• Types:

  | Tariff | Principle | Suitable For | Drawbacks | | :--- | :--- | :--- | :--- | | Simple Rate | Fixed charge per kWh | Residential, small consumers | No demand charge, not cost-reflective | | Flat Rate | Fixed charge per month/HP | Irrespective of consumption | Not equitable | | Block Rate | Slab system (rate ↓ as consumption ↑) | Domestic, commercial | Discourages high consumption? | | Two-Part Tariff | Fixed charge (demand) + Variable charge (energy) | Industrial, commercial (most common) | Fixed charge may be high even for low use | | Power Factor Tariff | Incentive/penalty based on PF (leading/lagging) | Industrial (inductive loads) | Requires PF measurement |

2. Peak Load Pricing

• Principle: Charge higher rates during system peak load hours (e.g., 6-10 PM) and lower rates during off-peak.

• Rationale: Reflects true cost of generation (peaking plants are expensive, fast-responding). Encourages load shifting (demand-side management), flattens load curve, defers capacity addition.

• Application: Time-of-Day (TOD) tariffs for industrial/commercial consumers.

IV. SPECIAL & ADVANCED TOPICS

A. Magneto-Hydro Dynamic (MHD) Generation

• Principle: Direct conversion of thermal energy (hot, ionized gas/plasma) to electrical energy without moving parts (bypasses Carnot limit).

  • Combustor burns fossil fuel with seed (e.g., potassium carbonate) → hot plasma (~2000-3000°C).

  • Plasma passed through magnetic field → charged particles deflected → electrodes collect DC current.

• System Components & Layout:

  DiagramSEARCH: MHD generator schematic diagram channel electrodes magnet

  1. Combustor/Ionizer: Produces conducting plasma.

  2. Nozzle: Accelerates plasma to high velocity (~1000 m/s).

  3. Channel (Electrodes): Insulated walls with electrode plates (cathode top, anode bottom). Plasma flows perpendicular to B-field.

  4. Magnet: Powerful electromagnet (superconducting) creates strong B-field (~5T).

  5. Seed Recovery: Cool exhaust, separate seed for reuse (critical for economics).

• Advantages: High efficiency (50-60% potential), fast start, high power density.

• Challenges: Material science (high temp, corrosive plasma), seed recovery cost, electrode life, magnet cost. Not yet commercial.

B. Energy Scenario & Strategies (India Focus)

1. Renewable Energy Scenario in India (as of ~2025)

• Installed Capacity: ~190-200 GW total renewable (solar, wind, biomass, small hydro) – ~45-50% of total installed capacity (~450 GW).

• Major Sources: Solar PV (dominant, >70 GW), Wind (~45 GW), Biomass (~10 GW), Small Hydro (~5 GW).

• Policies: National Solar Mission (100 GW target by 2022, now 280 GW by 2030), Wind Power Programme, Bioenergy Programme, International Solar Alliance (ISA) headquartered in India.

2. State-specific Scenario (e.g., Tamil Nadu)

• Leader in Wind: ~10 GW installed (Muppandal, etc.). High wind potential in southern districts.

• Growing Solar: Significant solar capacity.

• Challenges: Grid integration of variable renewables, curtailment during high wind/sun, land acquisition, transmission infrastructure.

3. Future Energy Strategies

• Diversification: Mix of large hydro, nuclear, thermal (with carbon capture), and distributed renewables.

• Grid Integration: Smart grids, forecasting, flexible generation (gas, hydro), energy storage (batteries, pumped hydro, green hydrogen).

• R&D Focus: Advanced solar (perovskites), offshore wind, green hydrogen, geothermal, tidal.

• Decarbonization: Shift from coal, promote electric vehicles, green hydrogen for industry.

4. Energy Resources Reserve

• Concept: Proven, probable, possible reserves of primary energy sources (coal, oil, gas, uranium, thorium, biomass potential). Not the same as installed capacity.

• Importance: Long-term energy security, planning for import dependence, R&D direction (e.g., India's thorium reserve → 3-stage program).

C. Miscellaneous Short Note Topics (from Past Papers)

• Cogeneration: See II.E.2 above.

• Magneto-Hydro dynamic systems: See IV.A above.

• Load Forecasting: See III.A.3 above.

• Peak Load Pricing: See III.C.2 above.

• Fuel Cells: See II.D.4 above.

• Biomass Energy: See II.C above.

• Wind Site Selection: See II.B.4 above.

• Energy Resources Reserve: See IV.B.4 above.

• Small scale hydro-electric plant: See I.A.5 above.

• Waste disposal in nuclear power plant: See I.C.4 above.

• Load curve: See III.A.3 above.

• Hybrid systems: See II.E.1 above.

END OF UNIT 1 NOTES