SCADA Systems & Applications (EX-802 (C)) - Unit 2 Short Notes

UNIT 2: SPECIAL ELECTRICAL MACHINES, POWER QUALITY & AI APPLICATIONS

I. SPECIAL ELECTRICAL MACHINES

A. Stepper Motors

Definition: Electromechanical devices converting electrical pulses into discrete angular movements (steps). Open-loop position control.

Types

Single Stack vs Multi Stack: Single stack = one stator/rotor stack. Multi stack = multiple identical stacks on same shaft, providing smaller step angles.

Hybrid Stepper Motor (Detailed)

• Construction: Stator has 2+ phases with toothed poles. Rotor is a permanent magnet (radially magnetized) with teeth (castellated poles). Teeth on rotor & stator are misaligned to create incremental attraction.

• Working: Sequential excitation of stator phases aligns rotor teeth with the nearest energized stator teeth, causing rotation. Each pulse = one step.

Static & Dynamic Characteristics

• Static: Torque vs. rotor position at fixed current. Shows detent torque (PM & Hybrid) and holding torque.

• Dynamic: Torque vs. speed curve. Pull-out torque (max torque at given speed without losing steps) decreases with speed. Pull-in torque (max torque to start/stop without missing steps) is lower.

Torque Equation & Characteristics

Instantaneous torque: $$\displaystyle T = \frac{1}{2} i^2 \frac{dL(\theta)}{d\theta} $$ (for VR/Hybrid, neglecting saturation).

• $i$ = phase current, $L(\theta)$ = phase inductance w.r.t. rotor position $\theta$.

• For Hybrid: $$\displaystyle T \propto i \cdot \sin(N_r \theta) $$ where $$\displaystyle N_r $$ = number of rotor teeth.

• Torque-Speed: Inversely related. At high speeds, inductance limits current rise, reducing torque.

Driver Circuits

• Dual Voltage Driver: Uses high voltage for fast current rise (overcome inductance) then switches to low voltage to maintain current. Reduces power loss.

  • Current Build-up: Fast initial rise (high V), then slower linear rise (low V) to set current level.

• Other: Unipolar, Bipolar (H-bridge), Chopper drive (constant current).

Microstepping

• Subdivision of a full step into smaller increments by proportionally controlling currents in two phases (e.g., sinusoidal currents).

• Result: Smooth motion, reduced vibration, increased resolution. Not perfect due to nonlinearities.

Speed Control

1. Pulse Rate Control: Vary frequency of input pulses.

2. Voltage Control: Vary phase voltage (affects torque capability).

3. Current Control (Chopper): Most effective, maintains constant torque at low speeds.

Load Angle Control

• Load Angle ($\delta$): Angular displacement between rotor's equilibrium position (under load) and the stator's magnetic axis.

• Control: By controlling phase current magnitude. Higher current increases magnetic force, reducing $\delta$ for a given load. Maximum stable $$\displaystyle \delta < 90° $$ (for Hybrid, typically $$\displaystyle < 45° $$).

Stepping Angle Calculation (Hybrid)

For a hybrid motor with:

• $$\displaystyle N_s $$ = number of stator teeth

• $$\displaystyle N_r $$ = number of rotor teeth

• $m$ = number of phases (stacks) Step Angle: $$\displaystyle \alpha = \frac{360°}{m \cdot N_r} $$ (if $$\displaystyle N_s $$ and $$\displaystyle N_r $$ differ by ±1 for each phase stack). Example (from paper): 8 main poles castellated to 5 teeth ($$\displaystyle N_s=5 $$ per phase?), rotor has 50 teeth ($$\displaystyle N_r=50 $$). For a 2-phase motor ($$\displaystyle m=2 $$): $$\displaystyle \alpha = \frac{360°}{2 \times 50} = 3.6° $$. (Verify exact configuration from question context).

Applications

• PV Water Pumping: Direct drive, precise control, good low-speed torque.

• Electric Vehicles: For actuators (mirrors, seats), pumps, fans. Not for main traction (low power density).

• General: Printers, plotters, CNC machines, robotics, valve control.

Advantages & Limitations

[!TIP] Exam: Be ready to calculate step angle for given pole/teeth configurations. Distinguish clearly between VR, PM, and Hybrid working principles.

B. Switched Reluctance Motors (SRM)

Definition: Doubly salient, singly excited motor. Torque produced by tendency of rotor to align with stator pole to minimize reluctance.

Construction

• Stator: Salient poles with concentrated windings.

• Rotor: Salient poles, no windings, no magnets (laminated steel). Can be solid (rugged, low cost) or laminated (reduces eddy currents).

• Pole Arcs: Stator pole arc ($$\displaystyle \beta_s $$) and rotor pole arc ($$\displaystyle \beta_r $$) are designed to ensure overlap during commutation for continuous torque.

Principle of Operation

• When a stator phase is excited, the rotor pole attracted to the nearest aligned position (minimum reluctance path).

• Torque Production: Due to variation of inductance with rotor position. $L(\theta)$ increases as rotor pole moves into alignment.

• Commutation: Phase excited only when rotor pole is approaching alignment. Turned off before pole passes alignment to avoid negative torque.

Torque Production

Instantaneous Torque Expression:

$$T = \frac{1}{2} i^2 \frac{dL(\theta)}{d\theta}$$

• Positive torque when $$\displaystyle \frac{dL}{d\theta} > 0 $$ (inductance increasing, rotor approaching alignment).

• Torque-Angle Characteristic: Parabolic for a given current. Maximum torque occurs at aligned position ($$\displaystyle \theta=0 $$) if current is maintained, but commutation turns off before that.

Energy Conversion per Stroke & Average Torque:

• Energy Input per Cycle (per phase): $$\displaystyle E_{in} = \int_0^{i_f} \psi(\theta, i) di $$ (area under $\psi-i$ curve).

• Energy Converted to Mechanical Work: $$\displaystyle W_m = \int_{\theta_1}^{\theta_2} T d\theta = \frac{1}{2} i^2 [L(\theta_2) - L(\theta_1)] $$.

• Average Torque: $$\displaystyle T_{avg} = \frac{W_m}{\text{stroke angle}} = \frac{1}{2} i^2 \frac{\Delta L}{\Delta \theta} $$, where $$\displaystyle \Delta L = L_{aligned} - L_{unaligned} $$.

Shaft Position Sensing

• Hall Effect Sensors: Common, digital output, mounted on stator.

• Resolvers / Encoders: Analog/digital, higher precision.

• Sensorless: Based on inductance measurement or back-EMF detection (more complex).

Advantages & Disadvantages

Torque Pulsation & Mitigation

• Causes: Fundamental torque production principle, pole geometry, switching.

• Mitigation:

  1. Optimal Pole Arc Design: Ensure smooth overlap.

  2. Multi-phase Operation (e.g., 8/6, 12/8).

  3. Advanced Control Strategies: Instantaneous torque control, fuzzy logic, neural networks.

  4. Skewing of stator/rotor poles.

  5. Current profiling (not rectangular).

Applications

• Appliances: Washing machines, vacuum cleaners.

• Industrial: Fans, pumps, compressors (where cost is critical).

• Automotive: Starter motors (already common), traction (research).

• Aerospace: Fuel pumps (safety-critical).

[!TIP] Exam: Be prepared to calculate instantaneous torque from given $L(\theta)$ profile and current. Know the torque expression derivation. Solid vs laminated rotor trade-offs are frequent.

C. Brushless DC Motors (BLDC)

Definition: DC motor with electronic commutation. Stator has 3-phase windings, rotor has permanent magnets.

Construction

• Stator: Similar to PMSM, 3-phase concentrated or distributed windings.

• Rotor: Surface-mounted or interior permanent magnets (NdFeB, SmCo).

• Electronic Commutator: Power electronic inverter + rotor position sensor.

Principle of Operation

• Trapezoidal Back-EMF: Due to concentrated windings and magnet placement.

• Six-Step Commutation: Each phase conducts for $120°$ electrical. Sequence: A+ B- → A+ C- → B+ C- → B+ A- → C+ A- → C+ B-.

• Torque Production: Lorentz force $$\displaystyle F = i \times B $$. Interaction between stator MMF (square-wave) and rotor PM field produces constant torque during each $120°$ conduction period.

Speed Control Methods

1. Voltage Control: Vary DC bus voltage (inefficient).

2. PWM Control: Vary duty cycle of inverter switches (most common).

3. Current Control: Inner current loop for torque control (vector control precursor).

Commutation & Armature Reaction

• Commutation: Switching of phases based on Hall sensor signals (60° apart) or back-EMF zero-crossing (sensorless).

• Armature Reaction: Distortion of main PM field by stator MMF. Can cause flux weakening at high currents, reducing back-EMF and affecting speed regulation. Mitigated by magnet design and control.

Winding Patterns

Position Sensing

• Hall Sensors: 3 sensors give 6 commutation states.

• Sensorless Control: Detect back-EMF zero-crossing in unenergized phase. Works only above ~10-20% rated speed (requires minimum speed for detectable back-EMF).

Applications

• Electric Vehicles: Power steering, coolant pumps, fans.

• PV Water Pumping: Direct drive, high efficiency over wide speed range.

• Industrial: Spindle drives, robotics, HVAC fans.

BLDC vs PMSM Comparison

Permeance Coefficient Derivation (PMBLDC)

• Permeance Coefficient ($$\displaystyle P_c $$): Relates air-gap flux density $$\displaystyle B_g $$ to magnet flux density $$\displaystyle B_r $$.

$$B_g = P_c \cdot B_r$$

• Derivation: Based on magnetic circuit. For surface-mounted magnet:

$$P_c = \frac{\mu_0 \cdot A_m}{l_g + \frac{A_m \cdot \mu_0}{\mu_r \cdot A_g}} \approx \frac{\mu_0 A_m}{l_g}$$

where $$\displaystyle A_m $$ = magnet area, $$\displaystyle A_g $$ = air-gap area, $$\displaystyle l_g $$ = air-gap length, $$\displaystyle \mu_r $$ = relative permeability of magnet (≈1 for NdFeB).

[!TIP] Exam: Clearly differentiate BLDC (trapezoidal, 6-step) from PMSM (sinusoidal, FOC). Know sensorless method (back-EMF zero-crossing). Permeance coefficient formula is key.

D. Permanent Magnet Synchronous Motors (PMSM)

Definition: AC synchronous motor with permanent magnet rotor. Requires variable frequency supply.

Construction

• Stator: 3-phase distributed windings (sinusoidal MMF).

• Rotor:

  • Surface-mounted (SPMSM): Magnets on surface. $$\displaystyle L_d = L_q $$ (non-salient).

  • Interior-mounted (IPMSM): Magnets buried. $$\displaystyle L_d < L_q $$ (salient, provides reluctance torque).

Principle of Operation

• Rotor locks to rotating magnetic field produced by stator. Speed $$\displaystyle n_s = \frac{120f}{P} $$ (synchronous).

• Sinusoidal Back-EMF: $$\displaystyle e_{ph} = 4.44 f N \phi k_w $$ (similar to synchronous generator).

EMF Equation Derivation (Per-Phase)

For a distributed winding:

$$E_{ph} = \sqrt{2} \pi N k_w f \phi$$

where:

• $N$ = turns per phase

• $$\displaystyle k_w $$ = winding factor (accounts for distribution & pitch)

• $f$ = frequency

• $\phi$ = flux per pole (from PM)

• $\sqrt{2}\pi \approx 4.44$ (rms form factor)

Torque-Speed Characteristics (d-q Model)

• Phasor Diagram (Steady-state): $$\displaystyle V = E + I_a(R_a + jX_s) $$. Torque $$\displaystyle T \propto |E||I_a|\sin\delta $$.

• Circle Diagram (d-q axes):

$$T = \frac{3}{2} \frac{P}{2} \left[ \phi i_q + (L_d - L_q) i_d i_q \right]$$

*   $\phi$ = PM flux linkage

*   $$\displaystyle L_d, L_q $$ = d,q axis inductances

*   $$\displaystyle i_d, i_q $$ = d,q axis currents.

*   **For SPMSM:** $$\displaystyle L_d = L_q $$, $$\displaystyle T \propto i_q $$ (only magnet torque).

*   **For IPMSM:** $$\displaystyle L_d < L_q $$, second term gives **reluctance torque** (maximized by $$\displaystyle i_d < 0 $$).

Power Electronic Controllers

• Inverters: 2-level, 3-level (NPC), multi-level for high power.

• Drive Topologies: Voltage Source Inverter (VSI) most common. Current Source Inverter (CSI) for special apps.

Torque Pulsation Reduction

1. Skewing: Rotor magnets or stator slots skewed.

2. Fractional Slot Winding: Distributed windings.

3. Optimal Magnet Arc: Shaping magnets.

4. Advanced Control: FOC with current regulation.

Speed Control Methods

1. V/f Control: Open-loop, constant flux. Simple, used up to base speed.

2. Vector Control (FOC): Decouples torque & flux. $$\displaystyle i_d=0 $$ control (max torque/amp) or MTPA (max torque per amp) for IPMSM. High performance.

3. Direct Torque Control (DTC): Direct torque & flux control, fast response, torque ripple.

Sensorless Control

• High-Frequency Injection: Injects HF signal, detects rotor position from impedance variation (works at zero speed).

• Back-EMF Detection: Like BLDC, works above ~10% speed.

• Sliding Mode Observer: Model-based state estimation.

Applications

• High-performance drives: CNC, robotics, industrial servos.

• Electric Vehicles: Main traction motor (high efficiency, power density).

• Aerospace: Fuel pumps, actuators.

• Appliances: Inverters for compressors.

Advantages over Conventional Motors

• Higher efficiency (no rotor losses).

• Higher power density & torque density.

• Better dynamic response.

• Simpler construction (no brushes, slip rings).

• Better reliability & lower maintenance.

[!TIP] Exam: Derive EMF equation clearly. Know d-q torque equation and significance of $$\displaystyle L_d \neq L_q $$. Compare V/f vs FOC. Sensorless at zero speed (HF injection) is a key differentiator from BLDC.

E. Other Motors & Materials

Permanent Magnet DC (PMDC) Motors

• Construction: Stator = PM, Rotor = armature with commutator & brushes.

• Applications: Automotive (windshield wipers, power windows), toys, small appliances. Low cost, simple speed control (vary voltage).

Brushed DC Motors

• Construction: Stator = field winding (or PM), Rotor = armature, commutator & brushes.

• Commutation: Mechanical switching via brushes/commutator to keep torque unidirectional.

• Armature Reaction: Distortion of main field by armature MMF. Causes flux weakening and commutation problems (sparking). Mitigated by interpoles (commutating poles) and compensating windings.

• Applications: Traction (older), power tools, automotive starters.

Ferromagnetic Materials

Solid Rotors

• Construction: Rotor made of solid steel (no laminations).

• Advantages: Rugged, low cost, high mechanical strength, good for high-speed.

• Disadvantages: High eddy current losses (cannot laminate), poor efficiency, high heating. Used only in low-power or very high-speed applications where lamination impractical.

Magnetic Reluctance

• Definition: Opposition to magnetic flux creation in a material. $$\displaystyle R_m = \frac{l}{\mu A} $$.

• Role in SRM: Fundamental principle—torque produced by tendency to minimize reluctance.

• Role in Transformers: Core reluctance determines magnetizing current.

II. POWER QUALITY ISSUES & MITIGATION TECHNIQUES

A. Power Quality Fundamentals

Definition: Characteristics of electricity at a point of common coupling that do not disturb the load. Increased Concern Due To:

• Sensitive electronic loads (computers, PLCs).

• Proliferation of nonlinear loads (rectifiers, UPS, LED drivers).

• Economic impact: downtime, equipment damage, wasted energy. Major Issues:

1. Voltage Disturbances: Sags, swells, interruptions, transients, flicker.

2. Waveform Distortion: Harmonics, interharmonics.

3. Voltage Fluctuations: Flicker.

4. Frequency Variations. Causes: Faults, switching operations, nonlinear loads, renewable integration, lightning.

B. Voltage Disturbances

Transient Recovery Voltage (TRV):

• Voltage across circuit breaker poles after current interruption.

• Factors: System inductance/capacitance, fault location, breaker type. High TRV can cause re-ignition.

Mitigation Techniques:

• Sag/Swell: Dynamic Voltage Restorer (DVR) - series VSC injecting voltage.

• Surge: Surge Protective Devices (SPDs) - clamp voltage, divert energy to ground.

• Capacitor Switching Transients: Zero-Voltage Crossing (ZVC) switching. Switch when voltage is near zero to minimize inrush current and transient overvoltage.

• Flicker: Reduce load fluctuations, use STATCOM/DVR.

[!TIP] Exam: Sag vs Swell (magnitude), Surge vs Swell (duration/transient vs sustained). ZVC for capacitor switching is a key mitigation technique.

C. Harmonics and Filters

Fundamentals of Waveform Distortion

• Periodic non-sinusoidal waveform can be expressed by Fourier series as sum of fundamental + harmonics (integer multiples of fundamental frequency).

• Distortion Factor: Ratio of RMS harmonic content to total RMS.

Sources of Harmonics

• Industrial: Arc furnaces, electric arc welders, rectifiers (DC drives), saturated transformers.

• Commercial/Residential: Computers, UPS, LED drivers, electronic ballasts, variable speed drives (VSDs).

Harmonic Indices

• THD (Total Harmonic Distortion): For voltage or current.

$$THD_V = \frac{\sqrt{\sum_{h=2}^{\infty} V_h^2}}{V_1} \times 100\%$$

$$THD_I = \frac{\sqrt{\sum_{h=2}^{\infty} I_h^2}}{I_1} \times 100\%$$

• TDD (Total Demand Distortion): THD referred to fundamental of demand current ($$\displaystyle I_{1,demand} $$). Better for varying loads.

$$TDD = \frac{\sqrt{\sum_{h=2}^{\infty} I_h^2}}{I_{1,demand}} \times 100\%$$

• Crest Factor: $$\displaystyle CF = \frac{I_{peak}}{I_{rms}} $$. High CF indicates high harmonic content.

• Interharmonics: Frequencies not integer multiples of fundamental. Caused by cycloconverters, arcing loads.

Passive Filters

• Design: Tuned LC circuits to shunt specific harmonic frequencies to ground.

• Types:

  • Tuned Filters: Single or multiple (e.g., 5th, 7th, 11th, 13th).

  • High-Pass Filters: (e.g., C-type, damped) to damp higher harmonics.

• Series vs Shunt: Shunt filters are common (connected across load/bus). Series filters block harmonic currents.

• Disadvantages: Fixed compensation, can be detuned by system changes, may cause resonance.

Active Harmonic Filters (AHF)

• Principle (Shunt): Current-source VSC injects equal-but-opposite harmonic currents into the system. Reference currents generated by detecting load harmonics.

• Advantages: Dynamic compensation, can mitigate multiple harmonics, compensates for flicker & unbalance.

• Disadvantages: High cost, complex control, limited power rating compared to passive.

Harmonic Analyzer

• Operation: Samples voltage/current waveforms, uses FFT (Fast Fourier Transform) to decompose into frequency spectrum.

• Measurement: Provides magnitude & phase of each harmonic, THD, TDD, power parameters (P, Q, S, PF).

[!TIP] Exam: THD vs TDD difference is crucial (demand current vs fundamental). Know passive filter types and their limitations. AHF principle: "inject opposite currents".

D. Power Factor Correction

Benefits:

1. Reduced $$\displaystyle I^2R $$ losses in supply system.

2. Improved voltage regulation (less drop).

3. Increased system capacity (same kVA for more kW).

4. Avoid utility penalties for low PF (<0.9 lagging typically). Reactive Power Compensation Penalty: Utilities charge for excessive reactive power consumption (kVAr) as it increases their system losses and reduces capacity.

Capacitor Bank Installation Locations:

1. Individual Compensation: Across individual large motors (most effective).

2. Group Compensation: Across a group of similar loads (e.g., lighting).

3. Central/Bus Compensation: At substation or main distribution board (system-level). Switching Strategies: Fixed, switched (on load change), or thyristor-switched (for dynamic PF correction). Harmonics Consideration: Capacitors can amplify existing harmonics due to parallel resonance with system inductance. Detuning (adding series reactor) is essential.

E. FACTS Devices

FACTS Concept

• Flexible: Fast, continuous, dynamic control of AC transmission parameters (impedance, voltage, phase angle).

• Based on: Power electronics (VSCs, thyristors).

Shunt vs Series Compensation

• Shunt Capacitance: Connected line-to-ground. Increases power transfer capability by raising voltage magnitude at receiving end. Provides reactive power support.

• Series Capacitance: Connected in-line. Reduces line impedance, increasing power transfer and improving stability.

SVC vs STATCOM

DSTATCOM

• Operation: Shunt-connected VSC. Acts as a controlled current source.

• Working Principle: DC capacitor provides DC bus. VSC generates AC output voltage $$\displaystyle V_{out} $$ with controllable magnitude & phase. Injects current $$\displaystyle I_{inj} = (V_{out} - V_{sys})/Z $$ to regulate PCC voltage.

• Block Diagram: DC Source → VSC → Coupling Transformer → AC Bus.

• Applications: Voltage regulation, flicker mitigation, harmonic compensation (with appropriate control), unbalanced load compensation.

Unified Power Quality Conditioner (UPQC)

• Principle: Series APF + Shunt APF back-to-back, sharing a common DC capacitor.

• Working & Operation:

  • Shunt Part: Compensates load harmonics, compensates reactive power, balances load.

  • Series Part: Injects voltage to compensate for sags/swells, isolates load from source harmonics.

  • Power Flow: Series part handles voltage correction, shunt part handles current correction. DC link power exchange balances active power.

• Classification:

  • By Configuration: UPQC-Q (shunt injects only reactive), UPQC-P (shunt injects active power).

  • By Supply: 3-phase 3-wire or 3-phase 4-wire.

• Advantages: Comprehensive mitigation (sags, swells, harmonics, flicker, unbalance). Fast response.

• Differences:

  • vs DSTATCOM: DSTATCOM is only shunt. UPQC has both series & shunt.

  • vs Standalone APF: APF only compensates harmonics. UPQC also handles voltage disturbances.

Best Harmonic Elimination Methods

1. Active Filters (Shunt/UPQC): Best dynamic performance, comprehensive.

2. Hybrid Filters: Combination (e.g., passive + active) for cost/performance balance.

3. Passive Filters: Cheapest, but fixed & risk of resonance. Must be detuned.

[!TIP] Exam: SVC vs STATCOM comparison is very important. UPQC working: series for voltage, shunt for current. Know why detuning is needed with passive filters.

F. Protection Techniques for High Voltage Equipment

• Insulation Coordination: Select insulation levels based on expected overvoltages (protective level).

• Lightning Arresters / Surge Arresters: Limit overvoltages to safe levels by diverting lightning/energy to ground.

• Shielding: Use shield wires/ground grids to protect from direct lightning.

• Safe Working Practices:

  • De-energize & lockout/tagout.

  • Use insulated tools & PPE (gloves, mats).

  • Maintain safe approach distances.

  • Proper grounding (equipment & personnel).

  • Follow procedures (single-person rule, permits).

III. ARTIFICIAL INTELLIGENCE IN POWER SYSTEMS

A. Neural Networks

Radial Basis Function (RBF) Network

• Architecture:

  • Input Layer: Passes inputs to hidden layer.

  • Hidden Layer: Radial basis neurons (e.g., Gaussian function). Each neuron computes distance from input to its center $$\displaystyle \mathbf{c}_i $$.

$$\phi_i(\mathbf{x}) = \exp\left(-\frac{\|\mathbf{x} - \mathbf{c}_i\|^2}{2\sigma_i^2}\right)$$

    where $$\displaystyle \sigma_i $$ = spread (width).

*   **Output Layer:** Linear combination of hidden outputs: $$\displaystyle y_k = \sum_{i} w_{ki} \phi_i(\mathbf{x}) $$.

• Training: Typically two-stage:

  1. Centers & Spreads: Determined by clustering (e.g., k-means) on input data.

  2. Weights: Solved by linear least squares (pseudo-inverse) since output layer is linear.

• Applications: Function approximation, pattern recognition, load forecasting, fault detection.

Functional Link Network (FLN)

• Architecture: Single-layer network. Input vector $\mathbf{x}$ expanded to a higher-dimensional space using functional link (e.g., trigonometric, polynomial basis functions) before linear combination.

  • $$\displaystyle \mathbf{z} = \mathbf{F}(\mathbf{x}) $$ (nonlinear expansion).

  • $$\displaystyle y = \mathbf{w}^T \mathbf{z} $$.

• Operation: Avoids hidden layer, reduces complexity. Learning is linear in expanded space.

• Use Cases: Simple nonlinear problems, where computational efficiency is key.

Self-Organizing Maps (SOM)

• Topology: Unsupervised, competitive learning. 2D grid of neurons, each with weight vector $$\displaystyle \mathbf{w}_i $$ (same dimension as input).

• Training Algorithm (Competitive Learning):

  1. Present input $\mathbf{x}$.

  2. Find Best Matching Unit (BMU): $$\displaystyle i^* = \arg\min_i \|\mathbf{x} - \mathbf{w}_i\| $$.

  3. Update BMU and its neighbors: $$\displaystyle \mathbf{w}_i(t+1) = \mathbf{w}_i(t) + \alpha(t) h_{i^*,i}(t) (\mathbf{x} - \mathbf{w}_i(t)) $$.

     • $\alpha(t)$ = learning rate, $h$ = neighborhood function (Gaussian).

  4. Decrease $\alpha$ & neighborhood size over time.

• Applications: Clustering, visualization (high-D to 2D), topology preservation. Used in power system state estimation, fault classification.

Learning Tasks

• Supervised Learning: Target known. Backpropagation for MLP. Error minimized.

• Unsupervised Learning: No target. Clustering (SOM, k-means). Finds structure.

• Competitive Learning: Subset of unsupervised. Neurons compete to respond to input (winner-takes-all).

B. Fuzzy Logic

Concept: Handles uncertainty via degrees of truth (0 to 1) instead of Boolean true/false.

Fuzzy Logic System Structure:

1. Fuzzification: Convert crisp input to fuzzy sets (membership functions $$\displaystyle \mu_A(x) $$).

2. Rule Base: IF-THEN rules (e.g., IF temperature is Hot AND pressure is High THEN valve is Slightly Open).

3. Inference Engine: Applies rules (Mamdani or Sugeno). Uses fuzzy operations (min/max for AND/OR).

4. Defuzzification: Convert fuzzy output to crisp value (e.g., Centroid method).

Example: Temperature Control

• Inputs: Error ($e$), Change in Error ($\Delta e$).

• Fuzzy sets: Negative Large (NL), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Large (PL).

• Rules: IF $e$ is NL AND $\Delta e$ is NL THEN output is Very High.

• Output: Crisp control signal (e.g., heater power).

Operations on Fuzzy Relations:

• Union: $$\displaystyle \mu_{A \cup B}(x) = \max[\mu_A(x), \mu_B(x)] $$

• Intersection: $$\displaystyle \mu_{A \cap B}(x) = \min[\mu_A(x), \mu_B(x)] $$

• Complement: $$\displaystyle \mu_{\bar{A}}(x) = 1 - \mu_A(x) $$

• Composition: $R \circ S$ (max-min or max-product composition for relational databases).

Fuzzy Rule-Based System:

• Structure: Input variables → Fuzzification → Rule Evaluation (inference) → Aggregation → Defuzzification → Output.

• Rule Formation: Based on expert knowledge or data.

• Membership Functions: Triangular, trapezoidal, Gaussian. Define linguistic terms.

C. Genetic Algorithms (GA)

Inspired by natural selection. Used for optimization.

Genetic Operators:

1. Selection: Choose parents for reproduction.

   • Roulette Wheel: Probability proportional to fitness.

   • Tournament: Random k individuals, pick best.

2. Crossover (Recombination): Exchange genes between parents.

   • Single-point: Cut at one point, swap tails.

   • Multi-point: Multiple cuts.

   • Uniform: Each gene independently swapped.

3. Mutation: Randomly alter genes with low probability. Mechanism: Flip bit (binary) or add small random value (real-coded).

   • Application: Maintains diversity, prevents premature convergence, explores new regions.

D. Applications in Power Systems

Economic Load Dispatch (ELD)

• Problem: Minimize total fuel cost $$\displaystyle F = \sum_{i=1}^n F_i(P_i) $$ subject to:

  • Power balance: $$\displaystyle \sum P_i = P_D + P_L $$ (demand + loss).

  • Generator limits: $$\displaystyle P_{i,\min} \leq P_i \leq P_{i,\max} $$.

• Cost Function: Typically quadratic: $$\displaystyle F_i = a_i P_i^2 + b_i P_i + c_i $$.

• Solution via GA/Fuzzy:

  • GA: Chromosome = vector of $$\displaystyle P_i $$. Fitness = $1/F$ (or penalty for constraint violation). Evolves to optimal dispatch.

  • Fuzzy: Can handle uncertainty in load/cost, or for adaptive penalty functions.

• Example: 3 generators, given $a,b,c$, demand. Solve $\lambda$ (Lagrange multiplier) from $$\displaystyle \frac{dF_i}{dP_i} = \lambda $$ and power balance.

Load Frequency Control (LFC)

• Single Area System:

  • Block Diagram: Load change $$\displaystyle \Delta P_L $$ → Frequency deviation $\Delta f$ → Governor action (speed droop $R$) → Area Control Error (ACE) → Integrator → $$\displaystyle \Delta P_{ref} $$.

  • ACE: $$\displaystyle ACE = \Delta P_{tie} + B \Delta f $$ (for isolated area, $$\displaystyle \Delta P_{tie}=0 $$). $B$ = frequency bias constant.

• Two Area System:

  • Tie-line Power: $$\displaystyle \Delta P_{tie} = T(\Delta f_1 - \Delta f_2) $$ (synchronizing coefficient $T$).

  • Area Control Error (ACE) for Area 1: $$\displaystyle ACE_1 = \Delta P_{tie,1} + B_1 \Delta f_1 $$.

  • Interconnected Operation: Each area controls its own ACE to zero, maintaining scheduled tie-line flow & frequency.

• Parameters Affecting LFC:

  • Governor characteristics: Speed droop $R$ (regulation).

  • Load damping: $D$ (load changes with frequency).

  • Inertia constant: $H$ (affects initial frequency drop rate).

Small Signal Stability

• Concept: Ability to maintain synchronism after small disturbances (load fluctuations). Analyzed by eigenvalues of linearized system.

• Achievement via PSS: Power System Stabilizer adds damping to generator rotor oscillations (0.1-2 Hz). Provides supplementary signal to AVR (excitation system) based on speed/power deviation.

• AI-based Tuning: GA/Fuzzy used to optimize PSS parameters (gain, time constants) for robust damping across operating conditions.

Load Forecasting

• Types:

  • Short-term: Hourly/daily (unit commitment, dispatch).

  • Medium-term: Weekly/monthly (maintenance scheduling).

  • Long-term: Yearly (expansion planning).

• Methods:

  • Neural Networks: MLP, RBF, LSTM for time series.

  • Fuzzy Logic: Handle uncertainty in weather, economic factors.

  • Hybrid: NN + fuzzy, GA-optimized NN.

• Importance: Critical for SCADA/EMS functions (security analysis, economic dispatch).

[!TIP] Exam: ELD formulation (quadratic cost, lambda iteration). LFC single vs two-area block diagrams, ACE definition. PSS role in small signal stability. AI methods for load forecasting.

Final Note: This summary strictly follows the UNIT 2 blueprint and highlights topics from past RGPV papers. Focus on definitions, principles, key formulas (boxed), comparisons, and applications. Practice numerical problems for stepper angle, SRM torque, THD, ELD, and LFC.