UNIT 5: POWER PLANT ENGINEERING - EXAM-FOCUSED SHORT NOTES

I. ENERGY SOURCES & DIRECT ENERGY CONVERSION METHODS

Primary vs. Secondary Energy Sources

• Primary Energy Sources: Naturally occurring sources found in raw form (e.g., coal, crude oil, natural gas, uranium, sunlight, wind, hydropotential). They require conversion for useful work.

• Secondary Energy Sources: Derived from primary sources after conversion (e.g., electricity, gasoline, refined fuels, hydrogen). They are carriers of energy.

• Inter-convertibility: Energy can be converted from one form to another (e.g., chemical → thermal → mechanical → electrical in a thermal plant; nuclear → thermal → mechanical → electrical in a nuclear plant). Direct conversion skips the mechanical stage (e.g., solar PV: radiant → electrical; fuel cell: chemical → electrical).

Classification of Direct Conversion Methods

Methods that convert non-mechanical energy (solar, chemical, nuclear, etc.) directly into electrical energy, bypassing the conventional Rankine or Brayton cycles (thermal → mechanical → electrical).

• Solar Photovoltaic (PV): Radiant (light) → Electrical.

• Fuel Cells: Chemical → Electrical.

• Magnetohydrodynamic (MHD): Thermal (kinetic energy of ionized gas) → Electrical.

• Thermoelectric: Thermal (temperature gradient) → Electrical (Seebeck effect).

• Direct Energy Conversion from Nuclear: Not yet commercial (e.g., betavoltaics).

Magnetohydrodynamic (MHD) Converter

• Working Principle: Based on Faraday's law of electromagnetic induction. A hot, ionized gas (plasma) from combustion is seeded with alkali metals (e.g., potassium) to increase conductivity. This conductive gas flows at high velocity through a magnetic field perpendicular to the flow direction. The motion of charged particles across the magnetic field induces an electromotive force (EMF) and thus direct current (DC) between electrodes placed on the channel walls.

  • Key Equation: Generated EMF, $$\displaystyle E = u B d $$, where $u$ = gas velocity, $B$ = magnetic field strength, $d$ = electrode spacing.

  • DiagramSEARCH: "MHD generator working principle schematic diagram"

• Factors Limiting Commercial Use:

  1. Material Challenges: Electrodes and channel walls must withstand extremely high temperatures (~2000°C) and corrosive, erosive plasma.

  2. Seed Recovery & Corrosion: Alkali metal seed (e.g., K₂CO₃) is expensive and must be recovered from the exhaust, adding complexity and cost.

  3. Low Efficiency & High Cost: Overall plant efficiency gains are marginal compared to modern combined cycles. High capital cost of superconducting magnets (for high B-field) and channel maintenance.

  4. Technical Complexity: Plasma stability, slag deposition, and integration with conventional steam bottoming cycle.

Fuel Cells

• Definition: An electrochemical device that converts the chemical energy of a fuel (typically hydrogen) and an oxidant (typically oxygen from air) directly into electricity and heat, with water as a by-product (if H₂/O₂ used). It is not a heat engine; efficiency is not limited by Carnot.

• Basic Principle: Similar to a battery but with continuous fuel supply. Involves two electrodes (anode, cathode) separated by an electrolyte.

  1. Anode Reaction: Fuel (H₂) is oxidized: $$\displaystyle H_2 \rightarrow 2H^+ + 2e^- $$.

  2. Cathode Reaction: Oxidant (O₂) is reduced: $$\displaystyle \frac{1}{2}O_2 + 2H^+ + 2e^- \rightarrow H_2O $$.

  3. Overall: $$\displaystyle H_2 + \frac{1}{2}O_2 \rightarrow H_2O + \text{Electricity} + \text{Heat} $$.

• Types & Working:

  • PEMFC (Polymer Electrolyte Membrane FC): Low temperature (~80°C), solid polymer electrolyte. Fast start-up, ideal for vehicles/backup power. Sensitive to CO poisoning.

  • SOFC (Solid Oxide FC): High temperature (~800-1000°C), ceramic electrolyte. High efficiency, fuel flexible (can use hydrocarbons internally reformed), but slow start-up. Used for stationary power.

  • AFC (Alkaline FC): Uses aqueous KOH electrolyte. High efficiency, used in space applications (Apollo). Sensitive to CO₂.

• Thermodynamic Equations:

  • Reversible Voltage (Nernst Equation): $$\displaystyle E = E^0 - \frac{RT}{nF} \ln Q $$, where $$\displaystyle E^0 $$ = standard potential, $n$ = electrons transferred, $Q$ = reaction quotient.

  • Actual Cell Voltage: $$\displaystyle V = E - \eta_{activation} - \eta_{ohmic} - \eta_{concentration} $$.

  • Efficiency: $$\displaystyle \eta = \frac{V}{E^0} \times 100\% $$ (not Carnot-limited). For H₂/O₂, $$\displaystyle E^0 = 1.23\,V $$ at 25°C.

• Applications & Advantages: Backup power, distributed generation, electric vehicles, space. Advantages: high efficiency (40-60%, up to 85% with CHP), low emissions (only H₂O), modular, quiet.

[!TIP] Exam Focus: Be ready to draw a labeled sketch of a PEMFC or SOFC and write the half-cell reactions. Know the key difference: temperature range and electrolyte type.

II. THERMAL POWER PLANTS (FOSSIL FUEL BASED)

Coal Handling System

A complete system for receiving, storing, processing, and delivering coal to the boiler bunkers.

1. Unloading: Coal received by rail/road/ship → wagon tippler or grab unloader.

2. Screening & Crushing: To remove foreign material and reduce size for pulverizers.

3. Conveying: Belt conveyors (main, secondary, tertiary) transport coal.

4. Storage: Outdoor stockpile (for buffer) and indoor bunkers (for continuous supply).

5. Bunkering: Final storage above pulverizers for gravity feeding.

   • DiagramSEARCH: "coal handling system thermal power plant schematic"

Fuel Burning Systems

• Overfeed Stoker: Coal fed above the grate. Grate moves slowly, carrying burning coal forward. Ash falls off the end. Simple, used for low-grade coals.

• Underfeed Stoker: Coal fed below the grate. Burning zone moves upward. Better control, less smoke, used for bituminous coals.

• Pulverized Fuel (PF) System (Most Common): Coal ground to fine powder (70-80% < 75µm) in pulverizers (ball, bowl, impact type). Mixed with preheated air and burned in furnace as a flame. Allows precise control, high combustion efficiency, and large boiler sizes.

Fluidized Bed Combustion (FBC) System

• Working Principle: Solid fuel (coal, biomass) is burned in a bed of inert material (sand, limestone) suspended (fluidized) by an upward flow of air at a velocity between minimum fluidization and transport velocity.

  • Bubbling FBC (BFBC): Air velocity creates bubbles. Bed depth ~0.5-1m. Good for small/medium plants (~50-250 MW).

  • Circulating FBC (CFBC): Higher air velocity entrains fine particles, which are separated by cyclone and returned to bed. Allows larger scale (~300 MW+) and better sulfur capture.

  • DiagramSEARCH: "bubbling fluidized bed boiler diagram" OR "circulating fluidized bed boiler schematic"

• Advantages over Grate Firing:

  1. Fuel Flexibility: Can burn low-grade coals, biomass, waste.

  2. In-Situ SO₂ Capture: Limestone (CaCO₃) added to bed decomposes to CaO, reacts with SO₂ to form CaSO₄.

  3. Lower NOx Formation: Lower combustion temperature (~850-900°C) reduces thermal NOx.

  4. Higher Heat Transfer Rate: Due to high bed surface area.

  5. Less Ash Fusion Problems.

Gas Turbine Plants

• Simple Open Cycle: Air compressed → mixed with fuel → burned → expands through turbine (producing work) → exhausts to atmosphere. Efficiency limited by low pressure ratio and high exhaust temperature loss.

• Combined Cycle (CCGT): Gas turbine exhaust (still ~500-600°C) used as heat source for a Heat Recovery Steam Generator (HRSG). Generated steam drives a steam turbine. Significantly higher overall efficiency (55-62%).

• Reheating in Gas Turbines:

  • Process: After partial expansion in high-pressure turbine, gas is returned to a reheater (another combustion chamber) and then expanded in a low-pressure turbine.

  • How it Improves Efficiency:

    1. Increases average temperature of heat addition ($$\displaystyle T_{m,add} $$), improving Brayton cycle efficiency ($$\displaystyle \eta \propto 1 - (P_1/P_2)^{(\gamma-1)/\gamma} $$).

    2. Reduces compressor work by allowing higher pressure ratios without excessive turbine inlet temperature.

    3. Allows use of higher overall pressure ratios.

  • Regenerative Cycle: Uses a heat exchanger (regenerator) to preheat compressed air using turbine exhaust heat. Improves efficiency at lower pressure ratios.

  • Intercooled Cycle: Compressor stages intercooled to reduce compression work. Often combined with regeneration.

Auxiliary Systems & Plant Performance

Feed Water Treatment Plant

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

• Elements (for high-pressure boilers):

  1. Aeration/Deaeration: Removes dissolved gases (O₂, CO₂) by heating and spraying.

  2. Filtration: Removes suspended solids (mechanical filters, cartridge filters).

  3. Chemical Treatment: Addition of phosphates (for hardness precipitation), hydrazine (for O₂ scavenging), ammonia (for pH control).

  4. Ion Exchange (Demineralization): For ultra-pure water. Cation exchanger (H⁺ form) → Anion exchanger (OH⁻ form) → Mixed bed. Produces water with conductivity < 0.1 µS/cm.

  • DiagramSEARCH: "feed water treatment plant thermal power plant flow diagram"

Plant Heat Balance (Energy Audit)

• Definition: A quantitative accounting of all energy inputs and outputs in a power plant over a given period (usually annually). Shows where energy is lost.

• For Fossil Fuel Plant:

  • Energy Input: Chemical energy of fuel (HHV basis) = $$\displaystyle m_f \times HHV $$.

  • Energy Output: Net electrical energy delivered = $$\displaystyle W_{net} \times 3.6 \times 10^6 $$ (to convert kWh to kJ).

  • Major Losses:

    1. Stack Loss (Lₛ): Heat carried away by flue gases. Largest loss.

    2. Cooling Water Loss (L_cw): Heat rejected in condenser.

    3. Radiation & Unaccounted Loss (L_r): From boiler surfaces, piping, etc.

  • Efficiency: $$\displaystyle \eta_{overall} = \frac{\text{Net Electrical Output}}{\text{Heat Input}} \times 100\% $$.

  • Example: If HHV = 25,000 kJ/kg, fuel consumption = 1000 t/h, net output = 500 MW, then $$\displaystyle \eta = \frac{500 \times 10^3 \times 3.6}{1000 \times 25,000} \times 100\% = 72\% $$.

Cooling Towers

• Purpose: Reject waste heat from condenser cooling water to the atmosphere via evaporation and sensible heat transfer.

• Types:

  1. Natural Draft: Large hyperbolic concrete chimney. Creates draft by density difference (hot, moist air inside vs. cool air outside). No fans, low operating cost, high capital cost. Used in large plants.

  2. Mechanical Draft: Uses fans to force/induce air flow.

     • Forced Draft: Fan at air inlet (pushes air in). Positive pressure inside tower.

     • Induced Draft: Fan at air outlet (pulls air out). Negative pressure inside tower (more common).

  • Principle: Hot water from condenser sprayed at top → flows down over fill material → air flows up (natural/mechanical) → partial evaporation cools water → cold water collected at bottom for recirculation.

  • DiagramSEARCH: "induced draft cooling tower working diagram"

III. NUCLEAR POWER PLANTS

Nuclear Fission Phenomenon

• Basic Principle: A heavy nucleus (e.g., ²³⁵U) absorbs a thermal (slow) neutron and becomes unstable, splitting into two lighter nuclei (fission products), releasing ~200 MeV energy, 2-3 fast neutrons, and gamma rays.

• Chain Reaction: Neutrons released can cause further fissions. For sustained chain reaction:

  • Neutron Life Cycle: Fast neutrons → slowed by moderator → thermal neutrons → cause fission → release fast neutrons.

  • Multiplication Factor (k): $$\displaystyle k = \frac{\text{Number of neutrons in one generation causing fission}}{\text{Number of neutrons in previous generation}} $$.

    • $$\displaystyle k = 1 $$: Critical (steady state).

    • $$\displaystyle k > 1 $$: Supercritical (power increasing).

    • $$\displaystyle k < 1 $$: Subcritical (power decreasing).

• Context in Reactor Operation: Control systems (rods, chemical shim) adjust neutron absorption to maintain $k \approx 1$ for steady power.

Reactor Components & Materials: Moderators

• Function: Slow down fast fission neutrons to thermal energies where fission cross-section for ²³⁵U is maximum.

• Types & Characteristics:

  | Moderator | Material | Neutron Slowing Power | Absorption Cross-Section | Advantages | Disadvantages | | :--- | :--- | :--- | :--- | :--- | :--- | | Light Water (H₂O) | Ordinary water | Good | High (absorbs neutrons) | Cheap, good coolant | Requires enriched uranium (3-5% ²³⁵U) | | Heavy Water (D₂O) | Deuterium oxide | Very Good | Very Low | Allows use of natural uranium (0.7% ²³⁵U) | Very expensive, leaks, radioactive (tritium) | | Graphite | Carbon | Moderate | Low | Cheap, stable, allows natural U | Low density, needs large core, fire risk (Windscale) | | Beryllium | Metal/oxide | Good | Low | Good neutron reflector | Toxic, expensive, limited use |

Reactor Types & Comparison

Pressurized Heavy Water Reactor (PHWR) - CANDU Type

• Working with Schematic:

  • Fuel: Natural uranium oxide (UO₂) in zircaloy cladding.

  • Moderator & Coolant: Heavy water (D₂O). Key Feature: Moderator and coolant are separate. Coolant is pressurized (high pressure) to prevent boiling. Moderator is at low pressure in a calandria (large tank).

  • Core Design: Pressure Tubes: Fuel channels (containing fuel bundles) run horizontally through the calandria. Each channel can be individually refueled on-power (using fueling machines). Allows high neutron flux and flexible operation.

  • Control: Adjustable absorber rods (in calandria) and liquid neutron absorber (D₂O with gadolinium) in moderator.

  • DiagramSEARCH: "CANDU reactor pressure tube calandria schematic"

• Advantages: Uses natural uranium (fuel cost low), on-power refueling (high capacity factor), good neutron economy (can breed Pu-239).

Advanced Gas-Cooled Reactor (AGR)

• Description: Second generation of UK's Magnox reactors.

  • Fuel: Enriched uranium (2.5-3.5% ²³⁵U) in stainless steel cladding.

  • Moderator: Graphite (in large blocks).

  • Coolant: Carbon dioxide (CO₂) at high pressure (~40 bar).

  • Core: Graphite moderator with fuel channels. CO₂ circulates through core, picks up heat, goes to steam generators (once-through).

• Fission Process Context: Thermal reactor using graphite moderator. Graphite slows neutrons effectively. Enriched uranium fuel compensates for higher neutron absorption in CO₂ coolant compared to helium. The large graphite mass provides good thermal inertia and stability.

Thermal vs. Fast Breeder Reactors (FBR) - Comparative Analysis

Reactor Operation & Safety

Principles of Reactor Control

1. Control Rods: Made of high neutron absorption materials (B₄C, Ag-In-Cd, Hf). Inserted/withdrawn from core to absorb excess neutrons. Primary shutdown mechanism.

2. Chemical Shim: Soluble neutron absorber (e.g., boric acid) dissolved in coolant/moderator (PWR). Allows fine, uniform power shaping.

3. Moderator Temperature Coefficient (MTC): Change in reactivity with moderator temperature. Negative MTC (most designs) is desirable: as moderator heats up, density decreases → fewer neutrons thermalized → power drops (inherent safety).

Reliability Features for Nuclear Plants

• Redundancy: Critical systems (pumps, diesels, control) have multiple, independent trains (e.g., 2x100%, 3x50%).

• Safety Systems: Diverse, fast-acting systems for shutdown (scram), emergency core cooling (ECCS), containment spray.

• Containment: Robust, leak-tight reinforced concrete dome surrounding reactor vessel. Designed to withstand pressure, earthquakes, aircraft impact. Multiple physical barriers: Fuel cladding → Reactor pressure vessel → Containment building.

• Defense-in-Depth: Multiple independent layers of safety (preventive, mitigative) to ensure core damage frequency is extremely low.

IV. HYDROELECTRIC POWER PLANTS

Site Selection Criteria

1. Hydrology: Head (available water fall, m) and Flow (discharge, m³/s). Determines potential capacity ($$\displaystyle P = \rho g Q H \eta $$). Must have reliable year-round flow.

2. Geology: Sound rock foundation for dam and powerhouse. Low seismic activity. No major faults/landslide zones.

3. Topography: Narrow gorge for dam, accessible area for powerhouse and reservoir. Steep valleys for high head.

4. Environmental & Social: Minimal displacement, impact on ecology, fisheries, forests. Reservoir submergence.

5. Proximity to Load Center: Reduces transmission cost and losses.

6. Other: Access for construction, availability of construction materials, sedimentation rate.

Hydraulic Turbines - Classification & Characteristics

• Selection Factors: Net Head (H) and Design Flow (Q). Specific Speed (Nₛ) is key dimensionless parameter: $$\displaystyle N_s = N \sqrt{P} / H^{5/4} $$ (metric). Higher Nₛ → suitable for higher flow, lower head.

  • High-Head (H > 300m): Pelton Wheel (Impulse type). High Nₛ? No, Pelton has LOW Nₛ (10-40). High efficiency (90-92%), single or multiple jets. Used in mountainous regions.

  • Medium-Head (H = 30-300m): Francis Turbine (Reaction type). Medium Nₛ (60-300). Spiral casing, adjustable wicket gates. Most common type.

  • Low-Head (H < 30m): Kaplan Turbine (Axial flow reaction, adjustable blades) or Bulb Turbine (Generator in water flow). High Nₛ (300-1000+). Used in river dams.

  • Comparison Table:

    | Turbine | Head Range | Specific Speed | Runner Type | Efficiency | Application | | :--- | :--- | :--- | :--- | :--- | :--- | | Pelton | > 300 m | 10 - 40 | Bucket (impulse) | 90-92% | High-head, mountain | | Francis | 30 - 300 m | 60 - 300 | Mixed flow (reaction) | 90-94% | Medium-head, most common | | Kaplan | 2 - 30 m | 300 - 1000 | Axial flow (reaction) | 88-92% | Low-head, high flow | | Bulb | 2 - 20 m | 500 - 1200 | Axial flow (reaction) | 88-91% | Very low-head, tidal/river |

Small Hydro Power (SHP)

• Micro Hydro (MH): Capacity ≤ 100 kW. Often run-of-river, minimal reservoir. Used for village electrification, isolated communities. Head: 2-40m, Flow: 0.1-5 m³/s. Technology: Pelton, Crossflow, Turgo.

• Pico Hydro (PH): Capacity ≤ 5 kW. Very small, often portable or micro-installations. Head: 1-20m, Flow: 0.01-0.5 m³/s. Technology: Simple impulse or reaction turbines, often with direct drive.

• Comparison: MH is for small village grids; PH is for single households or small institutions. MH has more formal engineering; PH is often simple, low-cost.

Spillways

• Purpose: Safely pass flood flows that exceed the reservoir's capacity, protecting the dam from overtopping and failure. Also used for sediment sluicing.

• Types:

  1. Overflow (Ogee) Spillway: Crest follows the shape of the lower nappe of a free-falling water jet. Most common for dams with adequate foundation.

  2. Chute (Open Channel) Spillway: Water flows over a crest into a steeply sloping open channel (chute) to a river bed. Used when foundation is weak.

  3. Shaft (Morning Glory) Spillway: Water enters through a horizontal or slightly inclined circular crest into a vertical shaft, then through a tunnel. Used in narrow gorges.

  4. Side Channel Spillway: Crest is parallel to dam axis; water flows into a side channel. Used when dam is too low for overflow.

• Role in Dam Safety: Primary safety valve. Must be designed for Probable Maximum Flood (PMF) with high freeboard.

V. RENEWABLE ENERGY SYSTEMS (SOLAR & WIND)

Solar Energy

• Factors Affecting Solar Intensity (Irradiance):

  • Latitude & Season: Higher latitudes receive less average insolation; summer vs. winter sun angle.

  • Time of Day: Maximum at solar noon.

  • Weather: Cloud cover, humidity, aerosols (pollution) scatter/absorb radiation.

  • Atmospheric Path Length: Air Mass (AM). AM1.5 is standard (sun at 48.2° from zenith).

• Advantages for Indian Conditions:

  1. High Insolation: 4-7 kWh/m²/day in most regions (Thar Desert, Gujarat, Rajasthan).

  2. Decentralized Potential: Can be installed on rooftops, wastelands.

  3. Low Operating Cost, No Fuel Cost.

  4. Abundant Resource in a densely populated, energy-deficient country.

• Limitations:

  1. Intermittency & Variability: No generation at night, reduced in monsoon/cloudy days. Needs storage or backup.

  2. High Initial Capital Cost (though decreasing).

  3. Large Land Requirement for utility-scale plants.

  4. Low Capacity Factor (15-25%).

  5. Grid Integration Challenges with high penetration.

Wind Energy

• Factors Affecting Wind Velocity:

  • Terrain & Roughness: Smooth surfaces (sea, desert) have higher wind speeds than urban/forest areas.

  • Pressure Gradients: Driven by temperature differences (e.g., sea breeze, mountain-valley winds).

  • Seasonal Winds: Monsoon (southwest), trade winds.

  • Height: Wind speed increases with hub height (logarithmic profile).

• Limitations for Indian Conditions:

  1. Monsoon Variability: Wind patterns are highly seasonal. Good wind sites (Tamil Nadu, Gujarat) have strong monsoon winds but off-season lull.

  2. Site Specificity: Requires high, consistent wind speeds (>6 m/s average). Good sites are often remote (coastal, off-shore potential).

  3. Grid Integration: Similar to solar, requires balancing for variability.

  4. Noise & Visual Impact, Avian/Bat Mortality.

Hybrid Energy Systems

• Definition: Integration of two or more renewable energy sources (with/without conventional backup like diesel) and often energy storage to provide a more reliable and consistent power output than any single source alone.

• Feasible Options in India:

  1. Solar-Wind-Diesel: Common for remote islands/villages. Solar peaks in day, wind often at night/monsoon, diesel backup.

  2. Solar-Wind-Hydro: Hydro (especially pumped storage) can act as flexible backup/ storage for solar-wind variability. Excellent synergy in Himalayan/peninsular regions.

  3. Solar-Wind-Biomass: Biomass provides stable base load.

  4. Solar-Wind with Battery Storage: For grid stabilization and firm power.

Comparative Analysis: Solar vs. Wind for India

VI. POWER PLANT ECONOMICS & LOAD ANALYSIS

Fundamental Definitions & Formulas

1. Maximum Demand (MD): The highest load (power, kW/MW) on the station during a given period (usually 1 hour). Determines installed capacity needed.

2. Load Factor (LF): Measure of utilization of installed capacity.

$$\boxed{\text{Load Factor} = \frac{\text{Average Load}}{\text{Maximum Demand}} = \frac{\text{Energy Produced in period}}{\text{Max Demand} \times \text{Hours in period}}}$$

*   Always ≤ 1. Higher LF → better utilization, lower cost per unit.

3. Diversity Factor (DF): Measures non-coincidence of peak demands of different consumers.

$$\boxed{\text{Diversity Factor} = \frac{\sum \text{Individual Max Demands}}{\text{Station Max Demand}}}$$

*   Always **≥ 1**. Higher DF → smaller station capacity needed.

4. Plant Factor (PF) / Capacity Factor: Measure of how much energy is produced vs. what could be produced at full load.

$$\boxed{\text{Plant Factor} = \frac{\text{Average Load}}{\text{Rated Capacity}} = \frac{\text{Energy Produced in period}}{\text{Rated Capacity} \times \text{Hours in period}}}$$

*   Similar to LF but denominator is **rated capacity**, not actual MD. PF ≤ LF usually.

*   **Note:** Some texts use "Plant Factor" and "Capacity Factor" interchangeably.

[!TIP] Common Pitfall: Do not confuse Load Factor (uses Maximum Demand) with Plant Factor (uses Rated Capacity). If MD < Rated Capacity (common), then LF > PF.

Load & Energy Calculations - Problem Solving

• Given: Individual peak loads ($$\displaystyle P_1, P_2, ... $$), Diversity Factor (DF), Annual Load Factor (LF).

• To Find:

  1. Station Maximum Demand (SMD): $$\displaystyle SMD = \frac{\sum P_i}{DF} $$.

  2. Average Load (AL): $$\displaystyle AL = SMD \times LF $$.

  3. Annual Energy Supplied (E): $$\displaystyle E = AL \times 8760\, \text{h} $$ (in kWh or GWh, where 1 GWh = 10⁶ kWh).

• Example (from Nov 2023 paper):

  • $$\displaystyle \sum P_i = 10+5+8+7 = 30\,MW $$

  • $$\displaystyle DF = 1.5 $$

  • $$\displaystyle SMD = 30 / 1.5 = \boxed{20\,MW} $$

  • $$\displaystyle LF = 0.6 $$

  • $$\displaystyle AL = 20 \times 0.6 = 12\,MW $$

  • $$\displaystyle E = 12 \times 8760 = 105,120\,MWh = \boxed{105.12\,GWh} $$

Depreciation & Costing

• Sinking Fund Method: Annual depreciation amount ($D$) is invested at compound interest ($i$) to accumulate the initial cost minus salvage value by end of life.

$$\boxed{D = \frac{C - S}{\left[ (1+i)^n - 1 \right] / i}}$$

where $C$ = Initial cost, $S$ = Salvage value, $n$ = life (years), $i$ = interest rate.

*   **Annual Capital Recovery (CR):** $$\displaystyle CR = D + i(C - S_{accumulated}) $$? Actually, simpler: $$\displaystyle CR = D + i \times (\text{avg. fund}) $$? For exam, use: **Total annual cost = Depreciation (sinking fund) + Interest on initial cost**? No, standard formula for **annual equivalent cost**:

$$\boxed{A = P \left[ \frac{i(1+i)^n}{(1+i)^n - 1} \right]}$$

where $P$ = present worth (initial cost - salvage). This is the capital recovery factor.

• Straight Line Method (SLM): Equal amount of depreciation each year.

$$\boxed{D = \frac{C - S}{n}}$$

*   Simple, but does not consider time value of money.

• Numerical Example (from Nov 2023 paper):

  • $$\displaystyle C = 90,000\,Rs. $$, $$\displaystyle S = 5,000\,Rs. $$, $$\displaystyle n=15\,y $$, $$\displaystyle i=6\% = 0.06 $$

  • SLM: $$\displaystyle D = (90,000 - 5,000)/15 = \boxed{5,666.67\,Rs./year} $$

  • Sinking Fund:

    $$\displaystyle D = \frac{90,000 - 5,000}{[(1+0.06)^{15} - 1] / 0.06} = \frac{85,000}{(2.3966 - 1)/0.06} = \frac{85,000}{23.275} \approx \boxed{3,650\,Rs./year} $$ (approx).

    • Note: The sinking fund deposit is lower, but you recover the initial cost via the fund at end. For annual capital recovery comparison, use the capital recovery factor formula above.

Load Duration & Curves

• Load Duration Curve (LDC): Loads arranged in descending order vs. time (percentage or hours). X-axis: time (hours/year), Y-axis: power (MW).

  • Significance:

    1. Capacity Planning: Area under curve = total energy. Shows how much time each load level occurs.

    2. Firm Power: The load level that can be supplied 100% of the time (intersection with 8760 hours).

    3. Used for Hydro-Thermal Scheduling: Base load (flat part) by thermal, peak load (steep part) by hydro.

  • Construction: Sort hourly load data for a year (8760 points) from highest to lowest. Plot.

  • DiagramSEARCH: "load duration curve power plant economics"

• Power Duration Curve: Similar to LDC but for hydro plants, it plots available power (from inflow and storage) vs. time. Used to determine firm power and secondary energy.

Tariffs & Pricing

• Objectives: Recover costs (fixed + variable), promote economic use, encourage load factor improvement.

• Types:

  1. Flat Rate: Fixed charge per unit of energy consumed ($/kWh). Simple, but no demand charge.

  2. Block Rate: Different rates for different consumption blocks (slab system). Progressive for domestic, regressive for industry.

  3. Two-Part Tariff: Most common for industrial/commercial.

     • Fixed Charge (Demand Charge): $$\displaystyle K_1 \times \text{MD} $$ (or connected load). Recovers fixed costs (capital, O&M).

     • Running Charge (Energy Charge): $$\displaystyle K_2 \times \text{units consumed} $$. Recovers variable costs (fuel).

     • Total Bill = Fixed Charge + Energy Charge.

  4. Power Factor Tariff: Incentive/penalty based on power factor (cos φ). Low PF increases current losses, so penalty applied. Encourages PF improvement (capacitor installation).

  5. Time-of-Day (TOD) Tariff: Different rates for peak, normal, off-peak hours. Shifts load, improves system LF.

[!TIP] Exam Focus: Be able to define all four economic terms (MD, LF, DF, PF) with formulas and explain their significance. Know Two-Part Tariff structure. Practice numericals for DF/LF and Depreciation (SLM & Sinking Fund). Sketch Load Duration Curve and state its use in hydro-thermal coordination.