UNIT 4: ELECTRICAL POWER GENERATION & ECONOMY - SHORT NOTES

I. OVERVIEW & CLASSIFICATION OF POWER GENERATION SOURCES

A. Conventional vs. Non-conventional / Renewable Energy

B. Global & Indian Energy Scenario

• Global: Shift towards renewables due to climate change (Paris Agreement). Solar & Wind leading new capacity additions.

• India:

  • Target: 500 GW non-fossil capacity by 2030.

  • Leadership: 4th largest in wind, 5th in solar globally.

  • Prominent Sources: Solar PV (largest growth), Wind (Tamil Nadu, Gujarat), Biomass (agricultural residue), Small Hydro.

  • Challenges: Grid integration, storage, land acquisition.

C. Future Energy Strategies & Hybrid Systems

• Hybrid Systems: Integration of two or more renewable sources (e.g., Solar-Wind, Solar-Diesel) with/without storage.

  • Aim: Mitigate intermittency, improve reliability, optimize cost.

  • Example: Solar + Wind + Battery Storage smooths output.

• Hydrogen Economy: Green H₂ from renewables as long-term storage/energy carrier.

II. CONVENTIONAL (THERMAL & NUCLEAR) POWER GENERATION

A. Hydroelectric Power Plants

1. Site Selection Criteria

• Water Availability: High, consistent rainfall/snowmelt; large catchment area.

• Topography: Narrow gorge for dam; steep gradient for head.

• Geology: Sound rock foundation for dam & powerhouse.

• Accessibility: Proximity to load centers, transportation for equipment.

• Environmental & Social Impact: Minimal displacement, submergence area.

2. Key Components & Layout

DiagramSEARCH: "hydroelectric power plant layout diagram dam reservoir penstock surge tank turbine draft tube powerhouse"

• Dam & Reservoir: Stores water, creates head.

• Intake Structure & Penstocks: Controls flow, carries water under pressure to turbine.

• Surge Tank: Protects against water hammer in penstocks during load changes.

• Hydraulic Turbines:

  • Pelton (Impulse): High head (300m+), low flow. Uses nozzle & bucket.

    DiagramSEARCH: "pelton turbine diagram"

  • Francis (Reaction): Medium head (30-300m), medium flow. Uses spiral casing, stay vanes, runner.

    DiagramSEARCH: "francis turbine diagram"

  • Kaplan (Axial Flow Reaction): Low head (<30m), high flow. Adjustable blades.

    DiagramSEARCH: "kaplan turbine diagram"

• Draft Tube: Converts kinetic energy to pressure, recovers head, provides vacuum at turbine exit.

• Power House: Houses turbine, generator, transformer, control equipment.

3. Analysis Curves

• Hydrograph: Discharge (Q) vs. Time (t) for a river. Shows seasonal variation.

• Flow Duration Curve (FDC): % time flow ≥ a given value vs. Flow. Shows reliability.

• Power Duration Curve (PDC): Derived from FDC & head. Shows available power vs. time.

4. Pumped Storage Hydro Plants (PSHP)

• Layout: Two reservoirs (Upper & Lower). Reversible pump-turbine.

• Working: Off-peak → Pump water to upper reservoir. Peak → Release to generate.

• Merits: Excellent for peak load, rapid start/stop, energy storage.

• Demerits: High capital cost, 2 sites needed, energy loss in cycle (~25%).

5. Small-Scale Hydro Plants

• Capacity < 10 MW (definition varies).

• Run-of-River (ROR): Minimal reservoir, depends on river flow. Low environmental impact.

B. Steam (Thermal) Power Plants

1. Site Selection Criteria

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

• Water: Abundant, good quality water source (river, sea).

• Transportation: Rail, road, port facilities.

• Ash Disposal: Land availability for ash ponds.

• Land & Labor: Cheap, available land; skilled workforce.

• Load Center: Near major demand area to reduce T&D loss.

• Environmental: Away from populated areas, consider stack height.

2. Layout & Main Features

DiagramSEARCH: "modern steam power plant layout schematic"

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

• Boiler/Steam Generator: Produces high-pressure, high-temperature steam. Includes economizer, superheater, reheater, air preheater.

• Steam Turbine: High-pressure (HP), Intermediate-pressure (IP), Low-pressure (LP) stages. Impulse (nozzle & bucket) for first stage, Reaction (fixed & moving blades) for later stages.

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

• Cooling Tower: Cools condenser cooling water.

  • Natural Draft: Hyperbolic shape, no fan.

  • Mechanical Draft: Uses fans (induced/forced draft).

• Feed Water Cycle: Condensate → Feed Water Heater (extraction steam) → Economiser (flue gas heat) → Boiler.

• Flue Gas Path: Boiler → Air Preheater (APH) (recovers heat for combustion air) → Dust Collector (ESP/Bag filter) → Chimney.

3. Water Treatment Plant

• Necessity: Prevent scaling, corrosion, fouling in boiler/turbine (impurities cause deposits, reduce efficiency, damage).

• Process: Clarification → Filtration → Softening (lime-soda, ion exchange) → Demineralization (mixed bed) → Oxygen removal (deaeration).

4. Diesel Power Plants

• Fuel System: Storage tank → Filters → Fuel injection pump → Injectors.

• Exhaust System: Silencer (muffler) → Stack.

• Use: Peak load, standby, remote areas (< 100 MW).

C. Nuclear Power Plants

1. Nuclear Fission vs. Fusion

2. Reactor Components

• Core: Fuel rods (UO₂ pellets) where fission occurs.

• Moderator: Slows neutrons (Light/H₂O, Heavy/D₂O, Graphite).

• Control Rods: Absorb neutrons (Boron, Cadmium). Insert to shut down.

• Coolant: Removes heat (Light/H₂O, Heavy/D₂O, CO₂, Liquid Na).

• Shield: Concrete/lead to absorb radiation.

• Pressure Vessel: Contains core, coolant under high pressure.

3. Reactor Types

• Pressurized Water Reactor (PWR):

  • Coolant/Moderator: Light water (H₂O).

  • Layout: Primary loop (pressurized) transfers heat to secondary loop (steam generator). Steam drives turbine. Most common worldwide.

    DiagramSEARCH: "PWR diagram"

• Boiling Water Reactor (BWR):

  • Coolant/Moderator: Light water (H₂O).

  • Layout: Water boils directly in reactor core; generated steam goes to turbine. Simpler, but turbine becomes radioactive.

    DiagramSEARCH: "BWR diagram"

• CANDU (CANada Deuterium Uranium):

  • Moderator/Coolant: Heavy water (D₂O).

  • Fuel: Natural uranium (no enrichment needed).

  • Layout: Pressure tubes (not vessel), on-power refueling.

  • Advantages: Fuel flexibility (thorium, spent fuel), high neutron economy.

  • Disadvantages: Heavy water expensive, large size, proliferation risk.

4. Radioactive Pollution & Waste Management

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

• Waste Classes:

  • Low & Intermediate Level: Shielded storage → Near-surface disposal.

  • High-Level (Spent Fuel): Initial cooling (pool) → Dry cask storage → Geological disposal (deep stable rock formations, e.g., KBS-3 method).

• Goal: Isolate waste from biosphere for >10,000 years.

5. Nuclear Fuel in India

• Uranium: Limited reserves (Jaduguda, Singhbhum, Tummalapalle). Import-dependent.

• Thorium: World's largest reserves (Kerala, Odisha, Andhra). Three-stage program: PHWR (U-233) → Fast Breeder (Pu-239 + Th-232 → U-233) → Thorium-based reactors.

• Plutonium: From reprocessing spent fuel (Tarapur, Kalpakkam).

D. Gas Turbine Power Plants

1. Classification & Brayton Cycle

• Open Cycle (Most common): Air → Compressor → Combustor (fuel added) → Turbine → Exhaust to atmosphere.

• Closed Cycle: Working fluid (He, CO₂) recirculated; heat added externally.

• Combined Cycle (CCGT): Gas turbine exhaust heat → Heat Recovery Steam Generator (HRSG) → Steam turbine. Efficiency > 60%.

• Thermodynamic Cycle: Brayton/Joule Cycle (Constant pressure heat addition/rejection).

2. Layout of Simple Gas Turbine Plant

DiagramSEARCH: "simple gas turbine plant layout"

1. Compressor (Axial/centrifugal): Compresses air.

2. Combustion Chamber: Fuel (NG, diesel) sprayed, ignited. High T, P gases.

3. Turbine: Expands gases to produce work (drives compressor & load).

4. Exhaust: High T gases wasted (in simple cycle).

3. Efficiency Improvement Methods

• Regeneration: Use exhaust heat to preheat compressed air before combustion. Increases efficiency, reduces fuel.

• Intercooling: Cool air between multi-stage compressors. Reduces compression work.

• Reheating: Expand gas in HP turbine, reheat, then expand in LP turbine. Increases output power.

• Combined Cycle: Best practical efficiency.

E. Magneto-Hydro Dynamic (MHD) Generation

1. Principle

• Faraday's Law: Conductor moving in magnetic field induces EMF.

• Working: Ionized hot combustion gases (seeded with alkali metal vapour, e.g., K₂CO₃) → conducting plasma → passed through magnetic field → Direct current generated across electrodes (no moving parts in generator).

2. Layout & Components

DiagramSEARCH: "MHD generator diagram channel electrodes magnet"

• Seed Recovery Plant: Recover seed material from exhaust.

• Combustor/Nozzle: Produces hot, seeded plasma.

• MHD Channel: High-temperature, non-magnetic material (ceramic). Electrodes on sides.

• Magnet: Superconducting (liquid He cooled) for strong field.

• Diffuser: Recovers pressure, reduces exhaust velocity.

• Bottoming Cycle: Exhaust still hot (~1500°C) → HRSG for steam turbine (combined cycle).

3. Advantages & Challenges

• Advantages: High theoretical efficiency (50-60% alone, >60% combined), no rotating parts in generator, fast start.

• Challenges: Material science (high T, corrosive plasma), seed recovery cost, electrode corrosion, low proven reliability. Not commercial yet.

III. RENEWABLE & NON-CONVENTIONAL POWER GENERATION

A. Solar Energy

1. Solar Radiation Basics

• Solar Constant (G_sc): ~1367 W/m² (outside atmosphere).

• Insolation: Solar radiation reaching earth's surface (varies with time, location, weather).

• Earth-Sun Angles:

  • Latitude (φ): Angular position N/S of equator.

  • Declination (δ): Angle between sun-earth line & equatorial plane. Varies ±23.45° annually.

  • Altitude (α): Angle of sun above horizon. sin α = sin φ sin δ + cos φ cos δ cos h

  • Azimuth (γ_s): Sun's projection on horizontal plane from South. cos γ_s = (sin α sin φ - sin δ) / (cos α cos φ)

  • Hour Angle (h): 15° per hour from solar noon.

2. Solar Thermal Power Generation

• Collectors Classification:

  • Flat Plate: Low T (<100°C). Absorber plate, glazing, insulation, casing.

    DiagramSEARCH: "flat plate solar collector diagram"

  • Concentrating:

    • Parabolic Trough: Linear focus, tracks N-S. Fluid in tube at focus.

    • Parabolic Dish: Point focus, tracks sun. Stirling engine at focus.

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

• Plant Layout: Collector field → Heat transfer fluid (HTF) → Steam Generator → Steam turbine → Condenser. May include thermal storage (molten salt).

3. Solar Photovoltaic (PV) Systems

• Principle: Photoelectric Effect in semiconductor (Si). Photon energy > bandgap → e-h pair generation → separation by p-n junction → DC current.

• Key Elements: Cell → Module (series/parallel cells) → Array → Inverter (DC→AC) → Transformer → Grid/Load. Balance of System (BoS): Mounting, wiring, controllers, batteries.

• I-V Characteristics:

  • Open Circuit Voltage (V_oc): I=0, max voltage.

  • Short Circuit Current (I_sc): V=0, max current.

  • Maximum Power Point (MPP): (V_m, I_m) where P_max = V_m * I_m.

  • Fill Factor (FF): FF = (V_m I_m) / (V_oc I_sc). Measures "squareness" of curve. Typical 0.7-0.8.

  • Efficiency (η): η = (P_max / (Intensity × Area)) × 100%.

  [!TIP] Numerical Example (From DEC 2024):

  Given: V_oc=0.24 V, I_sc=10 mA, V_m=0.14 V, I_m=6.5 mA, Intensity=24 W/m², Area=4 cm² = 0.0004 m²

  P_max = V_m × I_m = 0.14 × 0.0065 = 0.00091 W

  Input power = 24 × 0.0004 = 0.0096 W

  η = (0.00091 / 0.0096) × 100% = 9.48%

  FF = (0.14×0.0065) / (0.24×0.01) = 0.00091 / 0.0024 = 0.379

• Factors Affecting Performance: Insolation, temperature (↑T ↓V_oc), soiling, shading, inverter efficiency.

• System Types:

  • Standalone: With battery storage. For remote areas.

  • Grid-connected: No battery (net metering/feed-in). Most common.

B. Wind Energy

1. Principle

• Kinetic energy of wind → mechanical rotation → electrical energy.

• Power in Wind: P_wind = (1/2) ρ A V³

  • ρ = air density (~1.225 kg/m³ at STP)

  • A = swept area = πR² (R = blade length)

  • V = wind speed (m/s) → CUBIC dependency is critical.

• Betz Limit: Maximum theoretical power coefficient C_p,max = 16/27 ≈ 0.593. Real turbines: C_p ≈ 0.35-0.45.

2. Wind Energy Conversion Systems (WECS)

• Classification:

  • Horizontal Axis Wind Turbine (HAWT): Most common. Rotor shaft parallel to ground. Needs yaw mechanism to face wind.

    DiagramSEARCH: "horizontal axis wind turbine components nacelle gearbox generator tower"

  • Vertical Axis Wind Turbine (VAWT): Rotor shaft vertical. Omni-directional (no yaw). Lower efficiency (Darrieus, Savonius).

• Components (HAWT):

  • Rotor Blades: Capture wind energy (aerofoil shape).

  • Nacelle: Housing on top of tower. Contains gearbox (increases speed), generator.

  • Tower: Supports nacelle & rotor.

  • Yaw System: Motor & bearing to rotate nacelle into wind.

  • Control System: Pitch/blade angle control, brake.

3. Wind Characteristics & Performance

• Wind Speed Distribution: Follows Weibull distribution (k, c parameters).

• Capacity Factor (CF): CF = (Actual Energy Output) / (Rated Power × 8760 h). Typically 20-40%.

• Factors Affecting Performance: Wind speed (site), air density (T, P), turbine efficiency (C_p), availability, losses.

4. Site Selection for Wind Farms

• High average wind speed (> 6 m/s at hub height).

• Low turbulence intensity.

• Flat/gentle terrain, few obstacles.

• Proximity to grid (reduces evacuation cost).

• Environmental & social constraints (bird migration, noise, visual impact).

5. Control Schemes

• Start/Stop: Cut-in (~3-4 m/s), cut-out (~25 m/s) speeds.

• Power Regulation: Pitch control (blade angle), stall control (aerodynamic).

• Reactive Power Control: For grid support (voltage regulation).

C. Biomass Energy

1. Sources

Agricultural residue (straw, bagasse), animal waste (dung), municipal solid waste (MSW), energy crops (jatropha, switchgrass).

2. Biogas Generation (Anaerobic Digestion)

• Process: 4 stages in absence of oxygen:

  1. Hydrolysis: Complex organics → simple sugars, amino acids.

  2. Acidogenesis: Sugars → volatile fatty acids, alcohols.

  3. Acetogenesis: Acids → acetic acid, H₂, CO₂.

  4. Methanogenesis: Acetic acid/H₂+CO₂ → CH₄ (60-70%) + CO₂.

• Microbes: Consortia of bacteria (mesophilic ~35°C, thermophilic ~55°C).

3. Biogas Plant Types

DiagramSEARCH: "deen bandhu floating drum biogas plant diagram"

DiagramSEARCH: "KVIC pragati design biogas plant fixed dome diagram"

• Deen Bandhu (Floating Drum): Movable steel drum on slurry. Simple, visible gas production. Drum maintenance (corrosion).

• Pragati Design/KVIC (Fixed Dome): Brick masonry, fixed gas holder. No moving parts, cheaper. Gas pressure varies.

• Community Biogas Plants: Larger scale (cattle dung + night soil). Problems: Feedstock heterogeneity, management issues, maintenance, social acceptance.

4. Biomass Applications

• Direct Combustion: Boilers for steam/heat.

• Gasification: Partial combustion → producer gas (CO, H₂, CH₄) → engine/gas turbine.

• Pyrolysis: Thermal decomposition in absence of air → bio-oil, char, syngas.

• Biofuels: Bioethanol (sugarcane, corn), Biodiesel (jatropha, waste oil).

5. Landfill Gas Power Generation

• Organic MSW decomposes anaerobically in landfill → CH₄ (50%) + CO₂.

• Schematic: Wells drilled → Gas collection pipes → Flaring or Gas cleaning → IC engine driving generator.

• Advantages: Reduces GHG emissions, utilizes waste, energy recovery.

D. Other Renewable Sources

1. Geothermal Energy

• Resources in India: Low-to-medium enthalpy (150-200°C). Puga Valley (Ladakh), Manikaran (Himachal), Tattapani (Chhattisgarh), Cambay Basin (Gujarat).

• Types of Power Plants:

  • Dry Steam: Direct use of natural steam (rare, The Geysers, USA).

  • Flash Steam: High-P hot water → flashed to steam in separator → turbine.

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

    DiagramSEARCH: "binary cycle geothermal power plant diagram"

• Hybrid Geothermal-Fossil: Geothermal preheats boiler feedwater or supplements steam cycle to improve output/efficiency.

2. Ocean Energy

• Tidal Power:

  • Principle: Potential energy of water at high tide → kinetic energy during ebb → turbine.

  • Site Selection: High tidal range (> 4-5 m), funnel-shaped bay (e.g., Gulf of Kutch, Cambay, Sundarbans).

  • Schematic Layout: Barrage/dam across estuary → sluice gates → Turbines (bulb, Kaplan, Francis) in tunnel.

    DiagramSEARCH: "tidal barrage power plant layout diagram"

• Wave Energy:

  • Concept: Convert kinetic & potential energy of waves (oscillating water column, point absorber, attenuator).

  • Challenges: Harsh marine environment, low efficiency, high cost.

• Ocean Thermal Energy Conversion (OTEC):

  • Principle: Temperature difference between warm surface water (~25-30°C) and cold deep water (~5-10°C) → heat engine.

  • Closed Cycle (Ammonia): Warm surface water evaporates ammonia in evaporator → ammonia vapor drives turbine → cold deep water condenses ammonia in condenser.

    DiagramSEARCH: "closed cycle OTEC diagram"

  • Challenges: Very low efficiency (3-4%), huge pipes, biofouling, site-specific.

3. Hydrogen & Fuel Cells

• Hydrogen as Energy Carrier:

  • Advantages: High energy density (mass), clean combustion (H₂O), versatile (fuel cells, combustion).

  • Disadvantages: Low energy density (volume), storage/transport challenges, production cost (if not from renewables), safety (flammable, embrittlement).

• Hydrogen Storage Methods:

  • Compressed Gas (350-700 bar): Common, but energy-intensive compression.

  • Liquefied (-253°C): Higher density, but high boil-off loss, cryogenic cost.

  • Chemical: Metal hydrides, chemical hydrides (NH₃, LOHCs). Safer, but heavy/slow kinetics.

• Fuel Cells:

  • Principle: Electrochemical conversion of fuel (H₂) & oxidant (O₂) directly to electricity, water, heat. Not combustion.

  • Classification by Electrolyte:

    • AFC (Alkaline): Spacecraft, high efficiency.

    • PEMFC (Polymer Electrolyte): Automotive, backup power. Low T (80°C), quick start.

    • SOFC (Solid Oxide): Stationary power. High T (700-1000°C), fuel flexible (can use hydrocarbons).

    • MCFC (Molten Carbonate): Stationary, medium T (650°C).

  • Basic Function: Anode: H₂ → 2H⁺ + 2e⁻; Cathode: ½O₂ + 2H⁺ + 2e⁻ → H₂O; Overall: H₂ + ½O₂ → H₂O + electricity + heat.

IV. POWER PLANT ECONOMICS & OPERATION

A. Cost Analysis

• Overall Cost of Electricity (COE): COE = (Annual Fixed Cost + Annual Operating Cost) / Annual Energy Generated (kWh). Units: ₹/kWh or ¢/kWh.

B. Tariffs & Pricing

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

• Types:

  • Flat Rate: Same rate per kWh for all consumers. Unfair (doesn't reflect cost of supply).

  • Block Rate: Different slabs with increasing/decreasing rates. Progressive for low consumption.

  • Two-Part Tariff (Most common for industrial/commercial): Total Bill = (Demand Charge ₹/kW or kVA) + (Energy Charge ₹/kWh). Recovers fixed cost via demand charge, variable via energy charge.

  • Power Factor Tariff: Incentive for high PF (≥0.9), penalty for low PF. Reduces system losses.

  • Seasonal Tariff: Different rates for summer/winter (reflects demand/cost variation).

  • Peak Load Pricing: Higher rates during system peak hours (e.g., 6-10 PM). Encourages load shifting, reduces need for peaking plants.

C. Load Management & Characteristics

1. Load Curves

• Load Curve: Power (kW/MW) vs. Time (hourly/daily/annual). Shows variation.

• Daily Load Curve: Typical shape: low night, morning peak, evening peak.

• Annual Load Curve: Derived from daily curves; shows seasonal variation.

2. Load Duration Curve (LDC)

• Construction: Rank load values from highest to lowest vs. cumulative time (hours).

• Significance: Directly gives number of hours a given load level is exceeded. Used for capacity planning, economic dispatch. Area under LDC = Total energy.

3. Key Performance Factors

• Maximum Demand (P_max): Peak load in a period (kW/MW).

• Load Factor (LF): LF = (Average Load) / (Maximum Demand) = (Total Energy / (P_max × Time)). Always < 1. Measures utilization. Higher LF → lower cost per unit.

• Demand Factor (DF): DF = (Maximum Demand) / (Connected Load). Always < 1. Measures how much of connected load is actually used.

• Capacity Factor (CF): CF = (Actual Energy Produced) / (Rated Capacity × Time). Always ≤ 1. Measures plant utilization over time.

• Utilisation Factor (UF): UF = (Actual Energy Produced) / (Maximum Possible Energy if run at full load continuously). Similar to CF but based on peak load period.

• Diversity Factor (DF): DF = (Sum of individual max demands) / (System max demand). Always > 1. Measures mutual diversity of loads.

[!TIP] Relationship: Higher Load Factor → Lower Cost of Energy because fixed costs are spread over more units.

4. Numerical Problem (From NOV 2022)

Given: P_max = 40 MW, Capacity Factor = 0.5, Utilisation Factor = 0.8

• i) Load Factor (LF): CF = (LF × P_max × T) / (Plant Capacity × T) → 0.5 = (LF × 40) / (Plant Capacity) ... need Plant Capacity first.

• ii) Plant Capacity (P_rated): UF = (Actual Energy) / (P_max × T). But CF = (Actual Energy) / (P_rated × T). So Actual Energy = CF × P_rated × T. Also UF = (CF × P_rated × T) / (P_max × T) = (CF × P_rated) / P_max. → 0.8 = (0.5 × P_rated) / 40 → P_rated = (0.8 × 40) / 0.5 = 64 MW.

• iii) Reserve Capacity: P_rated - P_max = 64 - 40 = 24 MW.

• iv) Annual Energy Production: E_annual = CF × P_rated × 8760 = 0.5 × 64 × 8760 = 280,320 MWh = 280.32 GWh.

D. Economic Operation & Scheduling

1. Economic Load Dispatch (ELD)

• Objective: Minimize total fuel cost while meeting load demand & generator limits.

• Equal Incremental Cost Criterion (Neglecting Losses): At optimum, incremental fuel cost (λ) of all units must be equal.

  dC_i/dP_i = λ for all i.

  Where C_i(P_i) = fuel cost function (₹/h), P_i = power output (MW).

2. Economic Scheduling (Two-Unit, No Losses)

• Given: P_D = P_1 + P_2, C_1(P_1), C_2(P_2).

• Condition: dC_1/dP_1 = dC_2/dP_2 = λ.

• Solution: Solve P_2 = P_D - P_1 and equate derivatives.

[!TIP] Numerical Example (From JUN 2025):

C_1 = 50 + 2P_1 + 0.005P_1², C_2 = 100 + 2P_2 + 0.01P_2², P_D = 350 MW.

dC_1/dP_1 = 2 + 0.01P_1, dC_2/dP_2 = 2 + 0.02P_2.

Set equal: 2 + 0.01P_1 = 2 + 0.02P_2 → 0.01P_1 = 0.02P_2 → P_1 = 2P_2.

Substitute: 2P_2 + P_2 = 350 → 3P_2 = 350 → P_2 = 116.67 MW, P_1 = 233.33 MW.

λ = 2 + 0.01×233.33 = 4.333 ₹/MWh.

3. Economic Dispatch with Transmission Losses

• Problem: Losses mean ΣP_i > P_D. Need Penalty Factors (F_i).

• Concept: F_i = λ / (dC_i/dP_i). At optimum, dC_i/dP_i = λ × F_i.

• Loss Coefficient Matrix (B-coefficients): For a system with n generators, transmission loss P_L = Σ_{i=1}^n Σ_{j=1}^n P_i B_{ij} P_j (MW). B_{ij} constants from network analysis.

• Penalty Factor: F_i = 1 / (1 - (∂P_L/∂P_i)) where ∂P_L/∂P_i = 2 Σ_{j=1}^n B_{ij} P_j.

[!TIP] Numerical Example (From JUN 2025):

Given: dC_1/dP_1 = 0.15P_1 + 150, dC_2/dP_2 = 0.25P_2 + 175. P_1 = P_2 = 400 MW (initial). ∂P_L/∂P_2 = 0.2.

At optimum, (dC_1/dP_1) × F_1 = (dC_2/dP_2) × F_2 = λ.

F_2 = 1 / (1 - ∂P_L/∂P_2) = 1 / (1 - 0.2) = 1.25.

dC_2/dP_2 at 400 MW = 0.25×400 + 175 = 100 + 175 = 275 ₹/MWh.

So λ = 275 × 1.25 = 343.75 ₹/MWh.

dC_1/dP_1 at 400 MW = 0.15×400 + 150 = 60 + 150 = 210 ₹/MWh.

F_1 = λ / (dC_1/dP_1) = 343.75 / 210 = 1.6375.

4. Economic Load Scheduling (ELS) vs. Dispatch

• Economic Dispatch (ED): Real-time allocation of load among online units to minimize instantaneous fuel cost. (λ constant over short period).

• Economic Load Scheduling (ELS): Commitment (which units to ON/OFF) over hours/days considering startup costs, minimum up/down times, ramp rates. A unit commitment problem.

E. Load Forecasting

• Need: For generation scheduling, fuel procurement, maintenance planning, reliability, economic operation.

• Types:

  • Short-term (1-7 days): Hourly/daily. Weather-sensitive. Used for ED/ELS.

  • Medium-term (1-12 months): Weekly/monthly. For fuel, maintenance.

  • Long-term (1-20 years): Annual peak & energy. For capacity expansion planning.

• Basic Methods:

  • Trend Analysis: Extrapolate past growth (linear, exponential).

  • Cyclical Analysis: Account for business cycles (5-10 yr).

  • Seasonal Analysis: Adjust for daily/weekly/monthly patterns.

  • Multiple Regression: Load = f(Temperature, Humidity, Time, Economic Index, ...).

  • Modern: AI/ML (Neural Networks, SVM).

V. SPECIAL TOPICS & INTEGRATED SYSTEMS

A. Cogeneration (Combined Heat & Power - CHP)

• Concept: Simultaneous generation of electricity and useful heat (steam, hot water) from same fuel source.

• Advantages: Overall efficiency 70-90% (vs. 30-40% for condensing steam plants), reduces fuel cost & emissions, improves economics.

• Applications: Process industries (sugar, paper, chemical, refineries), district heating.

B. Hybrid Power Systems

• Definition: Integration of two or more different generation/storage technologies to overcome limitations of individual sources.

• Common Combinations:

  • Solar-Wind-Battery: Smoothens variability.

  • Solar-Diesel: Diesel backup for reliability.

  • Wind-Hydro: Hydro provides storage/regulation for wind.

  • Renewable-Fossil (e.g., Solar-Thermal, Geothermal-Fossil): Ensure base load.

• Benefits: Improved reliability, optimized cost, reduced storage size, better resource utilization.

C. Energy Resources Reserve

• Concept: Strategic reserves of fuel (coal, gas, uranium, oil) to ensure security of supply during disruptions (geopolitical, natural disasters).

• Importance: Energy security, price stability, national security. Mandated by many countries (e.g., USA Strategic Petroleum Reserve).

D. Safety & Environmental Aspects (General)

• Thermal (Coal): Air pollution (SOx, NOx, PM, Hg), Ash disposal, Water consumption, GHG.

• Nuclear: Radioactive releases (normal/accident), thermal pollution, Nuclear Waste (long-term hazard).

• Hydro: Dam safety, submergence, displacement, ecological flow, siltation.

• Solar PV: Land use, material toxicity (CdTe), end-of-life recycling.

• Wind: Noise, bird/bat mortality, visual impact, shadow flicker.

• Biomass: Air pollution (if combustion incomplete), land use competition.

• Mitigation: ESP/Bag filters, FGD, SCR, cooling towers, ash utilization, stringent regulations, EIA, monitoring.

END OF UNIT 4 NOTES