PE Exam Exam > PE Exam Notes > Electrical & Computer Engineering for PE > Cheatsheet: Generation
Cheatsheet: Generation
1. Generator Fundamentals
1.1 Basic Operating Principles
| Principle | Description |
|---|---|
| Faraday's Law | e = -N(dφ/dt), induced voltage proportional to rate of flux change |
| Right-Hand Rule | Motion, field, and induced EMF directions are mutually perpendicular |
| Electromagnetic Induction | Relative motion between conductor and magnetic field induces voltage |
| Mechanical to Electrical | Prime mover provides torque to rotate conductors through magnetic field |
1.2 Generator Types Comparison
| Type | Key Characteristics |
|---|---|
| Synchronous | Rotor speed = synchronous speed (120f/p), DC excitation on rotor, AC output on stator |
| Induction | Rotor speed > synchronous speed (negative slip), no DC excitation required |
| DC Generator | Mechanical commutator converts AC to DC, field on stator or rotor |
2. Synchronous Generators
2.1 Construction Elements
- Stator: Stationary armature with three-phase windings, produces AC output voltage
- Rotor: Rotating field winding with DC excitation, creates magnetic field
- Salient Pole: Projecting poles for low-speed hydro turbines (4-50 poles)
- Cylindrical/Round Rotor: Smooth rotor for high-speed steam turbines (2-4 poles)
- Exciter: DC supply system for rotor field (brushless or static)
2.2 Key Equations
| Parameter | Formula |
|---|---|
| Synchronous Speed | ns = 120f/p (rpm), where f = frequency (Hz), p = poles |
| Generated Voltage | Ef = 4.44kNphfφ, k = winding factor, Nph = turns per phase |
| Frequency | f = (pn)/120, n = rotor speed (rpm) |
| Internal Voltage | EA = Vt + jIAXs + IARA |
| Synchronous Impedance | Zs = RA + jXs, Xs = Xd (reactance) |
| Power Output | P = 3VtIAcosθ (three-phase) |
| Power Angle Equation | P = (EfVt/Xs)sinδ, δ = power angle |
| Reactive Power | Q = (EfVt/Xs)cosδ - (Vt2/Xs) |
2.3 Equivalent Circuit Model
- Ef (EA): Internal generated voltage (excitation voltage)
- RA: Armature resistance (stator winding resistance)
- Xs (Xd): Synchronous reactance (sum of leakage and armature reaction reactances)
- Vt: Terminal voltage (output voltage)
- IA: Armature current (stator current)
2.4 Operating Characteristics
| Curve | Description |
|---|---|
| Open Circuit (OCC) | Terminal voltage vs. field current at no-load; shows saturation |
| Short Circuit (SCC) | Armature current vs. field current with shorted terminals; linear |
| External Characteristic | Terminal voltage vs. load current at constant field current and speed |
| Voltage Regulation | VR = [(Vnl - Vfl)/Vfl] × 100% |
2.5 Reactances
| Reactance Type | Symbol & Value |
|---|---|
| Direct-Axis Synchronous | Xd = 1.0 to 2.0 pu (salient pole), 1.0 to 1.5 pu (cylindrical) |
| Quadrature-Axis Synchronous | Xq = 0.6 to 0.8 pu (salient pole), Xq ≈ Xd (cylindrical) |
| Direct-Axis Transient | Xd' = 0.15 to 0.35 pu |
| Direct-Axis Subtransient | Xd" = 0.10 to 0.20 pu |
| Negative Sequence | X2 = 0.10 to 0.25 pu |
| Zero Sequence | X0 = 0.01 to 0.10 pu |
3. Parallel Operation
3.1 Synchronization Requirements
- Equal Voltage Magnitudes: Generator voltage = bus voltage
- Equal Frequency: Generator frequency = system frequency
- Same Phase Sequence: ABC or CBA rotation matches
- Phase Agreement: Voltages in phase (0° difference)
- Same Waveform: Voltage shapes match
3.2 Synchronizing Methods
| Method | Description |
|---|---|
| Lamp Method (Dark) | Three lamps across open breaker go dark simultaneously when in sync |
| Synchroscope | Rotating pointer indicates phase difference; close at 12 o'clock slow rotation |
| Automatic Synchronizer | Electronic control checks voltage, frequency, phase; closes breaker automatically |
3.3 Load Sharing
| Parameter | Control Method |
|---|---|
| Real Power (kW) | Adjust prime mover power (governor control), changes power angle δ |
| Reactive Power (kVAR) | Adjust field excitation current, changes Ef magnitude |
| Frequency (f) | System frequency determined by total generation vs. load balance |
| Voltage (Vt) | Maintained by coordinated excitation control of all generators |
3.4 Power Angle and Stability
- Power Angle (δ): Angle between Ef and Vt, increases with load
- Maximum Power: Pmax = EfVt/Xs at δ = 90°
- Steady-State Stability Limit: δ < 90°="" for="" stable="">
- Pull-Out Torque: Maximum torque before losing synchronism
- Synchronizing Power: dP/dδ = (EfVt/Xs)cosδ
4. Generator Ratings and Capability
4.1 Nameplate Ratings
| Rating | Description |
|---|---|
| kVA or MVA | Apparent power rating = √3 × VL-L × Irated |
| Power Factor | Rated pf (0.8 to 1.0 lagging), determines kW rating |
| Voltage | Rated line-to-line voltage (kV), at terminals |
| Current | Rated armature current (A), determines heating |
| Speed | Rated synchronous speed (rpm) |
| Frequency | 50 Hz or 60 Hz |
| Temperature Rise | Class B: 80°C, Class F: 105°C, Class H: 125°C |
4.2 Capability Curve Limits
- Armature Current Limit: Circle centered at origin, radius = Srated
- Field Heating Limit: Circle centered on reactive axis, limits underexcitation
- Prime Mover Limit: Vertical line at rated real power
- Stator End-Region Heating: Horizontal line limiting leading (capacitive) vars
- Steady-State Stability: Limit based on power angle approaching 90°
4.3 Efficiency Calculation
| Parameter | Formula |
|---|---|
| Efficiency | η = Pout/(Pout + Plosses) × 100% |
| Copper Losses | PCu = 3IA2RA |
| Core Losses | Pcore = hysteresis + eddy current losses (constant at rated speed) |
| Friction & Windage | Pf&w = mechanical losses (constant at rated speed) |
| Stray Load Losses | Pstray = additional losses under load (0.5-1% of rating) |
5. Excitation Systems
5.1 Excitation System Types
| Type | Description |
|---|---|
| DC Exciter | Separate DC generator on same shaft, requires slip rings and brushes |
| AC Exciter (Brushless) | AC generator with rotating rectifier, no brushes, lower maintenance |
| Static Exciter | Solid-state converter (thyristor or transistor), power from generator terminals or auxiliary source |
| Pilot Exciter | Small permanent magnet generator provides initial excitation |
5.2 Voltage Regulator Functions
- Automatic Voltage Regulation (AVR): Maintains constant terminal voltage under load changes
- Excitation Forcing: Rapidly increases field current during faults or disturbances
- Droop Compensation: Reactive power sharing among parallel generators
- Over/Under-Excitation Limiting: Protects field and armature from excessive current
- Power System Stabilizer (PSS): Damps power oscillations
5.3 Excitation Control Parameters
| Parameter | Range/Value |
|---|---|
| Exciter Voltage Response | 0.5 to 2.0 per unit ceiling voltage |
| Exciter Time Constant | 0.5 to 2.0 seconds |
| Voltage Regulator Gain | 20 to 400 pu |
| Regulator Time Constant | 0.01 to 0.1 seconds |
6. Generator Protections
6.1 Protection Schemes
| Protection | Function & Device Number |
|---|---|
| Overcurrent (51) | Detects stator winding phase faults, time-delayed |
| Differential (87G) | Compares current entering and leaving stator, detects internal faults |
| Loss of Excitation (40) | Detects field loss, generator operates as induction machine |
| Reverse Power (32) | Detects motoring condition when prime mover fails |
| Overexcitation (24) | V/Hz relay, protects from core overflux |
| Negative Sequence (46) | Detects unbalanced currents, protects rotor from overheating |
| Ground Fault (51G, 64G) | Stator ground fault protection, 95% stator winding coverage |
| Out-of-Step (78) | Detects loss of synchronism |
| Overvoltage (59) | Protects from excessive terminal voltage |
| Undervoltage (27) | Detects abnormally low voltage |
| Underfrequency (81U) | Detects frequency decay, load shedding initiation |
6.2 Protection Limits
- I2t Limit: Rotor heating during unbalanced faults, K = 10-40 for 10 seconds
- Negative Sequence Current: I2 < 0.08="" to="" 0.10="" pu="">
- V/Hz Limit: 1.05 pu continuous, 1.18 pu for 60 seconds
- Field Current: 1.05 to 1.10 × rated field current continuous
7. Short-Circuit Analysis
7.1 Three-Phase Short Circuit
| Period | Characteristics |
|---|---|
| Subtransient (0-0.05 s) | I" = Ef/Xd", highest current, all damper windings effective |
| Transient (0.05-0.5 s) | I' = Ef/Xd', damper windings decay, field winding effective |
| Steady-State (>3 s) | Iss = Ef/Xd, only synchronous reactance effective |
7.2 Short-Circuit Current Ratios
- Subtransient: I" = 4 to 10 times rated current
- Transient: I' = 2 to 5 times rated current
- Steady-State: Iss = 1 to 2 times rated current
- DC Component: Decays with armature time constant Ta = 0.1 to 0.2 seconds
7.3 Time Constants
| Time Constant | Symbol & Range |
|---|---|
| Direct-Axis Transient (OC) | Td0' = 4 to 10 seconds |
| Direct-Axis Subtransient (OC) | Td0" = 0.02 to 0.05 seconds |
| Direct-Axis Transient (SC) | Td' = 0.5 to 2.0 seconds |
| Direct-Axis Subtransient (SC) | Td" = 0.01 to 0.03 seconds |
| Armature Time Constant | Ta = 0.1 to 0.2 seconds |
8. Generator Grounding
8.1 Grounding Methods
| Method | Ground Fault Current |
|---|---|
| Solidly Grounded | High (hundreds to thousands of amps), phase fault magnitude |
| Low-Resistance Grounded | 200 to 400 A, limits core damage, allows continued short-term operation |
| High-Resistance Grounded | 1 to 25 A (primary), limits to charging current, alarm only |
| Ungrounded | Capacitive charging current only (few amps), can operate with one ground |
| Grounded Through Distribution Transformer | Limited by transformer impedance and secondary burden |
8.2 Grounding Selection Criteria
- Unit-Connected Generators: High-resistance grounding (1-25 A)
- System-Connected Generators: Low-resistance grounding (200-400 A) or solidly grounded
- Generator Voltage < 15=""> High-resistance grounding preferred
- Stator Core Protection: Low-resistance limits core burning
9. Prime Movers
9.1 Prime Mover Types
| Prime Mover | Speed & Application |
|---|---|
| Steam Turbine | 3000/3600 rpm (2-pole), 1500/1800 rpm (4-pole), fossil/nuclear plants |
| Hydro Turbine (Francis) | 75-600 rpm, medium head (30-600 m), 6-80 poles |
| Hydro Turbine (Kaplan) | 50-300 rpm, low head (2-40 m), adjustable blades |
| Hydro Turbine (Pelton) | 200-600 rpm, high head (>150 m), impulse type |
| Gas Turbine | 3000/3600 rpm, peaking/intermediate duty |
| Diesel Engine | 300-1800 rpm, backup/emergency generation |
| Wind Turbine | 12-30 rpm, gearbox to 1500/1800 rpm, or variable speed with converter |
9.2 Governor Control
- Speed Droop: R = (Δn/n)/(ΔP/P), 3-5% for stable load sharing
- Isochronous: Zero droop, maintains constant frequency (single generator)
- Speed-Load Characteristic: n = n0 - R(P/Prated)
- Governor Dead Band: 0.03-0.06 Hz, prevents hunting
10. Induction Generators
10.1 Operating Characteristics
| Parameter | Description |
|---|---|
| Slip | s = (ns - n)/ns, negative for generation (s <> |
| Speed Range | n > ns, driven above synchronous speed |
| Reactive Power | Must be supplied externally (capacitor banks or grid) |
| Voltage Regulation | Poor, depends on external reactive support |
| Frequency | Determined by grid, cannot control independently |
10.2 Key Equations
| Parameter | Formula |
|---|---|
| Power Output | P = (3I22R2/s) - (3I22R2) = 3I22R2[(1-s)/s] |
| Torque | T = (3V12R2/s)/(ωs[(R1+R2/s)2+(X1+X2)2]) |
| Reactive Power Demand | Q = Qmagnetizing + Qleakage |
10.3 Applications
- Wind Turbines: Fixed-speed induction generators with grid connection
- Small Hydro: Simple, robust construction, low maintenance
- Cogeneration: Grid-connected applications where voltage control not required
- Self-Excitation: Capacitor banks provide reactive power for isolated operation
11. Testing and Commissioning
11.1 Standard Tests
| Test | Purpose |
|---|---|
| Open-Circuit Test | Determine OCC, unsaturated and saturated synchronous reactance |
| Short-Circuit Test | Determine SCC, armature resistance, synchronous impedance |
| Insulation Resistance | Megohmmeter test, minimum 1 MΩ per kV + 1 MΩ |
| Polarization Index | PI = R10min/R1min, minimum 2.0 for good insulation |
| High-Potential (Hi-Pot) | AC or DC overvoltage test, 2 × rated voltage + 1000 V for 1 minute |
| Phase Rotation | Verify ABC or CBA sequence matches system |
| Heat Run Test | Full-load operation, verify temperature rise within class limits |
| Vibration Test | Measure bearing and frame vibration, limits per ISO 10816 |
11.2 Reactance Determination
- Unsaturated Xs: Xs(unsat) = VOC/ISC at same field current (air gap line)
- Saturated Xs: Xs(sat) = Vrated/ISC at field current for rated voltage
- Short-Circuit Ratio: SCR = If(rated V)/If(rated I SC) = 1/Xs(sat) pu
- Xd from Slip Test: Salient pole machines, rotor driven at non-synchronous speed
12. Power Factor and Reactive Power
12.1 Power Triangle Relationships
| Parameter | Formula |
|---|---|
| Apparent Power | S = √(P2 + Q2) = 3VLIL (VA) |
| Real Power | P = 3VLILcosθ = Scosθ (W) |
| Reactive Power | Q = 3VLILsinθ = Ssinθ (VAR) |
| Power Factor | pf = cosθ = P/S |
12.2 Excitation and Power Factor
- Underexcited: Low Ef, generator absorbs reactive power (leading pf)
- Normal Excitation: Ef adjusted for unity pf, Q = 0
- Overexcited: High Ef, generator supplies reactive power (lagging pf)
- V-Curves: Plot IA vs. If at constant power, minimum at unity pf
12.3 Reactive Power Limits
- Lagging pf (Overexcited): Limited by field heating and armature current
- Leading pf (Underexcited): Limited by stability, field heating, stator end heating
- Rated pf: 0.8 to 0.95 lagging for turbine generators
- Continuous Leading: 0.95 leading for many modern generators
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