Application of AI in Electrical/Electronics Engg. (EX-802 (D)) - Unit 3 Short Notes

UNIT 3: APPLICATION OF AI IN ELECTRICAL/ELECTRONICS ENGINEERING

I. NEURAL NETWORKS AND LEARNING SYSTEMS

A. Radial Basis Function (RBF) Networks

• Architecture: Three-layer feedforward network.

  • Input Layer: $n$ nodes, passes input vector $$\displaystyle \mathbf{x} \in \mathbb{R}^n $$ to hidden layer.

  • Hidden Layer: Contains $m$ RBF neurons. Each neuron computes a radial basis function (typically Gaussian) centered at a vector $$\displaystyle \mathbf{c}_j $$ with spread parameter $$\displaystyle \sigma_j $$.

$$\phi_j(\mathbf{x}) = \exp\left(-\frac{\|\mathbf{x} - \mathbf{c}_j\|^2}{2\sigma_j^2}\right)$$

*   **Output Layer:** Linear combination of hidden layer outputs. For regression: $$\displaystyle y_k(\mathbf{x}) = \sum_{j=1}^{m} w_{kj} \phi_j(\mathbf{x}) + b_k $$.

• Operational Principles:

  1. Centre Determination ($$\displaystyle \mathbf{c}_j $$): Often via k-means clustering on training data.

  2. Spread Parameter ($$\displaystyle \sigma_j $$): Controls the width of the Gaussian. Common heuristic: $$\displaystyle \sigma_j = \frac{d_{\text{max}}}{\sqrt{2m}} $$, where $$\displaystyle d_{\text{max}} $$ is max distance between chosen centres.

  3. Weight Adaptation ($$\displaystyle w_{kj} $$): Solved by linear least squares (pseudo-inverse) since output layer is linear. No iterative backpropagation needed for weights.

• Key Advantage: Faster training than MLP due to linear output layer adjustment.

• Application in Power Systems: Function approximation for load forecasting, security assessment.

[!TIP] Exam Focus: Be prepared to draw the RBF architecture and write the Gaussian RBF equation. Understand the role of $\sigma$ (generalization vs. overfitting).

B. Functional Link Networks (FLN)

• Architecture: Single-layer perceptron where the input pattern $$\displaystyle \mathbf{x} = [x_1, x_2, ..., x_n] $$ is non-linearly expanded to a higher-dimensional feature vector $\mathbf{F}(\mathbf{x})$ before linear combination.

$$y = \sum_{i=0}^{M} w_i \cdot F_i(\mathbf{x})$$

where $$\displaystyle F_0(\mathbf{x}) = 1 $$ (bias), and $M \gg n$.

• Types of Functional Expansions:

  • Polynomial: $$\displaystyle F(\mathbf{x}) = [1, x_1, x_2, x_1x_2, x_1^2, x_2^2, ...] $$

  • Trigonometric: $$\displaystyle F(\mathbf{x}) = [1, \sin(\pi x_1), \cos(\pi x_2), ...] $$

  • Product Unit: $$\displaystyle F(\mathbf{x}) = \prod_{i=1}^{n} x_i^{p_i} $$

• Advantages:

  • Can solve non-linear problems with a single linear layer.

  • Faster training than multi-layer networks (no backpropagation through multiple layers).

  • Avoids local minima problem of gradient descent.

• Application: Pattern recognition, classification tasks where input dimension is moderate.

C. Self-Organizing Maps (SOM) / Kohonen Networks

• Topology: 2D grid (usually rectangular or hexagonal) of neurons. Each neuron $i$ has a weight vector $$\displaystyle \mathbf{w}_i \in \mathbb{R}^n $$ of same dimension as input.

• Competitive Learning Algorithm:

  1. Initialization: Weights $$\displaystyle \mathbf{w}_i $$ initialized randomly or from data samples.

  2. Competition (Winner-Takes-All): For input $\mathbf{x}$, find Best Matching Unit (BMU) $c$ such that:

$$\|\mathbf{x} - \mathbf{w}_c\| = \min_i \|\mathbf{x} - \mathbf{w}_i\|$$

3.  **Cooperation:** BMU $c$ influences its **neighbourhood** $$\displaystyle N_c(t) $$ (defined by a neighbourhood function $$\displaystyle h_{ci}(t) $$, typically Gaussian decreasing with distance on grid and time $t$).

4.  **Weight Update:** Adapt weights of BMU and its neighbours:

$$\mathbf{w}_i(t+1) = \mathbf{w}_i(t) + \alpha(t) \cdot h_{ci}(t) \cdot (\mathbf{x} - \mathbf{w}_i(t))$$

    where $\alpha(t)$ is the learning rate.

5.  **Decay:** Both $\alpha(t)$ and neighbourhood radius $r(t)$ decrease over time.

• Applications:

  • Data Clustering & Visualization: Projects high-D data onto 2D map preserving topology.

  • Feature Mapping: E.g., in power systems, for load pattern clustering (identifying typical daily profiles).

  • Vector Quantization.

[!TIP] Exam Focus: Know the 4-step SOM algorithm (Competition, Cooperation, Adaptation, Decay). Be able to explain "neighbourhood function" and its decay.

D. Learning Paradigms

[!TIP] Common Pitfall: Do not confuse competitive learning (unsupervised, no target) with supervised error correction. SOM uses competitive learning.

II. FUZZY LOGIC SYSTEMS

A. Fuzzy Relations and Operations

• Fuzzy Relation $R$: A fuzzy subset of the Cartesian product $X \times Y \times ...$. Represented by a membership matrix $$\displaystyle \mu_R(x, y, ...) \in [0,1] $$.

  • Example: "x is close to y" on real numbers.

• Operations on Fuzzy Relations (for relations $R, S$ on same universe):

  • Union: $$\displaystyle \mu_{R \cup S}(x,y) = \max\left(\mu_R(x,y), \mu_S(x,y)\right) $$

  • Intersection: $$\displaystyle \mu_{R \cap S}(x,y) = \min\left(\mu_R(x,y), \mu_S(x,y)\right) $$

  • Complement: $$\displaystyle \mu_{\neg R}(x,y) = 1 - \mu_R(x,y) $$

  • Composition: Given $R \subseteq X \times Y$ and $S \subseteq Y \times Z$, the composition $S \circ R \subseteq X \times Z$.

    • Max-Min Composition: $$\displaystyle \mu_{S \circ R}(x,z) = \max_{y \in Y} \min\left(\mu_R(x,y), \mu_S(y,z)\right) $$

    • Max-Product Composition: $$\displaystyle \mu_{S \circ R}(x,z) = \max_{y \in Y} \left(\mu_R(x,y) \cdot \mu_S(y,z)\right) $$

B. Fuzzy Logic Principles

• Core Steps:

  1. Fuzzification: Convert crisp input $x$ to fuzzy set(s) using membership functions (Triangular, Trapezoidal, Gaussian, etc.). E.g., Temperature $$\displaystyle 30^\circ C $$ → $$\displaystyle \mu_{\text{Warm}}(30) = 0.7 $$.

  2. Inference Engine: Apply IF-THEN rules (e.g., IF temperature is Hot AND humidity is High THEN fan_speed is VeryHigh). Uses Mamdani (max-min implication, outputs fuzzy sets) or Sugeno (linear/constant function, outputs crisp) method.

  3. Aggregation: Combine outputs from all rules into a single fuzzy set (using max or sum).

  4. Defuzzification: Convert aggregated fuzzy output to crisp value.

     • Centroid (Center of Gravity): $$\displaystyle z^* = \frac{\int \mu(z) \cdot z \,dz}{\int \mu(z) \,dz} $$ (most common).

     • Bisector, Mean of Maximum (MOM), Largest of Maximum (LOM).

• Example: Temperature Control System

  • Inputs: Temperature (Cold, Warm, Hot), Humidity (Dry, Normal, Humid).

  • Rules: IF Temp is Hot AND Humidity is High THEN Fan_Speed is Fast.

  • Output: Crisp fan speed command after defuzzification.

C. Fuzzy Rule-Based Systems

• Structure:

  • Rule Base (Knowledge Base): Collection of IF-THEN rules. Form: IF (x is A) AND (y is B) THEN (z is C).

  • Database: Defines membership functions for all linguistic variables (A, B, C).

  • Reasoning Mechanism: Implements fuzzification, inference (Mamdani/Sugeno), aggregation, defuzzification.

• Design of Rule Base:

  1. Identify input/output variables and their linguistic terms.

  2. Elicit rules from expert knowledge or data (e.g., using clustering).

  3. Example for Load Frequency Control (LFC):

     • ACE (Area Control Error) inputs: Negative_Large, Negative_Small, Zero, Positive_Small, Positive_Large.

     • Control_Signal output: Decrease_High, Decrease_Low, No_Change, Increase_Low, Increase_High.

     • Rule: IF ACE is Positive_Large THEN Control_Signal is Decrease_High (reduce generation).

• Inference Process (Mamdani):

  1. Fuzzify crisp inputs.

  2. For each rule, compute antecedent strength (using min for AND, max for OR).

  3. Implication: Clip/shrink consequent fuzzy set by antecedent strength (min operation).

  4. Aggregation: Combine all clipped consequent sets using max.

  5. Defuzzify aggregated set to get crisp output.

[!TIP] Exam Focus: Distinguish Mamdani (fuzzy output) vs. Sugeno (crisp output) inference. Be ready to design a simple 2-3 rule base for a control problem. Know the centroid defuzzification formula.

III. EVOLUTIONARY COMPUTATION

A. Genetic Algorithms (GA): Operators

GA evolves a population of candidate solutions (chromosomes) through iterative application of operators.

B. Mutation Operator: In-Depth

• Primary Purpose: Maintain population diversity within the search space. Acts as a "random walk" to explore unvisited regions.

• Impact of Mutation Rate ($$\displaystyle p_m $$):

  • Low $$\displaystyle p_m $$: Risk of premature convergence to suboptimal solution (population becomes too similar).

  • High $$\displaystyle p_m $$: Turns GA into a random search, destroying good building blocks (excessive disruption).

  • Typical Range: $$\displaystyle p_m \in [0.001, 0.1] $$ per gene. Often starts higher and decays.

• Exploration vs. Exploitation Balance:

  • Crossover is primary exploitation operator (recombines known good solutions).

  • Mutation is primary exploration operator (introduces novelty).

  • Effective GA tuning balances both.

• Applications in Power System Optimization:

  • Constraint Handling: Repair mutated solutions to satisfy constraints (e.g., generator limits in ELD).

  • Parameter Tuning: Mutate GA's own parameters (adaptive GA).

  • Escaping Local Optima: In multimodal cost functions (e.g., ELD with valve-point effects).

[!TIP] Exam Focus: Know the difference in purpose between Crossover (exploit) and Mutation (explore). Be able to explain premature convergence and how mutation helps prevent it. For ELD, mutation might randomly adjust one generator's output within limits.

IV. POWER SYSTEM OPTIMIZATION AND CONTROL USING AI

A. Economic Load Dispatch (ELD)

• Problem Formulation:

  • Objective: Minimize total fuel cost $F$ over $N$ generators.

$$\min F = \sum_{i=1}^{N} F_i(P_i) = \sum_{i=1}^{N} (a_i P_i^2 + b_i P_i + c_i)$$

    where $$\displaystyle P_i $$ = power from unit $i$, $$\displaystyle a_i, b_i, c_i $$ = cost coefficients.

*   **Constraints:**

    1.  **Power Balance:** $$\displaystyle \sum_{i=1}^{N} P_i = P_D + P_L $$ (Demand + Losses). Often losses approximated by $B$-coefficients: $$\displaystyle P_L = \sum_{i}\sum_{j} P_i B_{ij} P_j $$.

    2.  **Generator Limits:** $$\displaystyle P_i^{\min} \leq P_i \leq P_i^{\max} $$.

• AI-Based Solution (e.g., GA/PSO):

  1. Chromosome/Position Representation: Vector $$\displaystyle \mathbf{P} = [P_1, P_2, ..., P_N] $$.

  2. Fitness Function: Inverse of cost $F$ (or $-F$ for minimization). Penalty functions for constraint violation.

  3. Algorithm: Initialize population → Evaluate fitness → Apply selection, crossover, mutation → Repair for constraints → Iterate.

• Worked Example (Simplified, Neglecting Losses):

  • Data: 2 generators.

    • Gen 1: $$\displaystyle F_1 = 0.001P_1^2 + 10P_1 + 200 $$, $$\displaystyle P_1^{\min}=50 $$, $$\displaystyle P_1^{\max}=200 $$ MW.

    • Gen 2: $$\displaystyle F_2 = 0.002P_2^2 + 8P_2 + 180 $$, $$\displaystyle P_2^{\min}=50 $$, $$\displaystyle P_2^{\max}=150 $$ MW.

    • Total Demand $$\displaystyle P_D = 250 $$ MW.

  • Solution (Lambda-Iteration / Equal Incremental Cost):

$$\frac{dF_1}{dP_1} = \lambda \Rightarrow 0.002P_1 + 10 = \lambda$$

$$\frac{dF_2}{dP_2} = \lambda \Rightarrow 0.004P_2 + 8 = \lambda$$

    With $$\displaystyle P_1 + P_2 = 250 $$.

    Solving: $$\displaystyle P_1 = 100 $$ MW, $$\displaystyle P_2 = 150 $$ MW, $$\displaystyle \lambda = 10.2 $$ $/MWh$.

    **Cost:** $$\displaystyle F = 0.001(100)^2 + 10(100) + 200 + 0.002(150)^2 + 8(150) + 180 = 10000 + 2000 + 180 + 450 + 1200 = \boxed{11830 \text{ units}} $$.

[!TIP] Exam Focus: Write the ELD objective function and constraints clearly. For numerical, show lambda-iteration steps. Understand how GA encodes solution ($\mathbf{P}$ vector) and handles constraints (penalty/repair).

B. Load Frequency Control (LFC)

• 1. Fundamental Concept: Maintain system frequency and tie-line power at scheduled values despite load fluctuations.

  • Area Control Error (ACE): $$\displaystyle \text{ACE} = \Delta P_{tie} + B \Delta f $$

    where $$\displaystyle \Delta P_{tie} $$ = deviation in tie-line power, $\Delta f$ = frequency deviation, $B$ = frequency bias factor (MW/Hz).

  • Goal: Integrate ACE to zero via Integral Controller.

• 2. Single-Area System Block Diagram:

  DiagramCANVAS: Show block diagram with: Load Change (ΔPL) -> (+) -> Frequency Deviation (Δf) <- (Turbine-Governor) <- (Integral Controller with gain Ki) <- (ACE). Governor has time constant Tg, turbine gain Kt.

  • Control Strategy: $$\displaystyle \Delta P_c = -K_i \int \text{ACE} \, dt $$ (ACE = BΔf for isolated area).

  • Key Parameters: Governor time constant $$\displaystyle T_g $$, turbine gain $$\displaystyle K_t $$, bias $B$.

• 3. Two-Area System:

  • Model: Two interconnected control areas (Area 1 & 2) via tie-line.

  • Tie-Line Power Dynamics: $$\displaystyle \Delta P_{tie1} = T_{12} (\theta_1 - \theta_2) \approx \frac{2\pi}{s} T_{12} (\Delta f_1 - \Delta f_2) $$, where $$\displaystyle T_{12} $$ = tie-line synchronizing coefficient (pu MW/rad).

  • Coordinated Control: Each area has its own ACE: $$\displaystyle \text{ACE}_1 = \Delta P_{tie1} + B_1 \Delta f_1 $$, $$\displaystyle \text{ACE}_2 = \Delta P_{tie2} + B_2 \Delta f_2 $$. Controllers act on local ACEs.

• 4. Key Parameters Summary:

  • $$\displaystyle T_g $$: Governor time constant (speed governor response delay).

  • $$\displaystyle K_t $$: Turbine gain (steam/power conversion).

  • $B$: Frequency bias factor (how much frequency deviates per MW imbalance). $$\displaystyle B = \frac{1}{R} + \frac{1}{K_p} $$, where $R$ = speed regulation, $$\displaystyle K_p $$ = load-frequency constant.

  • $$\displaystyle T_{12} $$: Tie-line synchronizing coefficient (electrical coupling strength).

[!TIP] Exam Focus: Draw single-area LFC block diagram and label all blocks/parameters. Write ACE formula for both single and two-area. Know that integral control eliminates steady-state error.

C. Small Signal Stability Analysis

• Definition: Ability of power system to maintain synchronism when subjected to small disturbances (e.g., small load changes, minor faults). Characterized by damped oscillations in rotor angles and power flows.

• AI/Computational Methods:

  • Eigenvalue Analysis: Linearize system dynamics around an operating point to get state matrix $A$. Compute eigenvalues $$\displaystyle \lambda_i = \sigma_i \pm j\omega_i $$.

    • Stability Criterion: All $$\displaystyle \sigma_i < 0 $$ (real parts negative).

    • Damping Ratio: $$\displaystyle \zeta_i = -\sigma_i / \sqrt{\sigma_i^2 + \omega_i^2} $$. $$\displaystyle \zeta > 0.05 $$ typically acceptable.

  • Role of AI:

    • Mode Identification: SOM or clustering to group similar oscillation modes from simulation/measurement data.

    • Parameter Estimation: Neural networks or GA to tune PSS parameters from system response data.

    • Online Stability Assessment: Real-time neural network approximators for critical eigenvalues.

• Power System Stabilizer (PSS): Supplementary controller in AVR excitation system. Provides damping torque to oscillatory modes by modulating generator excitation based on speed/power deviation signals.

D. Load Forecasting

• Types by Horizon:

  | Type | Horizon | Primary Use | Key AI Techniques | | :--- | :--- | :--- | :--- | | Very Short-Term (VST) | Seconds-minutes | Real-time control, security | RBF, SOM, fuzzy logic | | Short-Term (ST) | Hours-days | Unit commitment, dispatch | RBF, SOM, ANNs, fuzzy, hybrid | | Medium-Term (MT) | Weeks-months | Maintenance scheduling | ARIMA, ANNs | | Long-Term (LT) | Years | Planning, expansion | Regression, econometric models |

• AI Techniques & Applications:

  • RBF Networks: Good for STLF due to fast training and ability to approximate nonlinear relationships (load vs. temp, humidity, time).

  • SOM: Used for day-type clustering (group similar load patterns) before applying specific forecaster for each cluster.

  • Fuzzy Systems: Handle uncertainty in input variables (e.g., "very hot", "moderately humid"). Often hybridized with ANNs (neuro-fuzzy).

  • Hybrid Models: e.g., SOM + RBF (cluster first, then forecast per cluster), GA-ANN (GA optimizes ANN weights).

• Input Variables: Historical load, temperature, humidity, wind speed, day-of-week, holiday indicators, time-of-day.

• Model Evaluation Metrics:

  • Mean Absolute Percentage Error (MAPE): $$\displaystyle \text{MAPE} = \frac{100\%}{n} \sum_{t=1}^{n} \left| \frac{L_t - F_t}{L_t} \right| $$

  • Root Mean Square Error (RMSE): $$\displaystyle \text{RMSE} = \sqrt{\frac{1}{n} \sum_{t=1}^{n} (L_t - F_t)^2} $$

  • Where $$\displaystyle L_t $$ = actual load, $$\displaystyle F_t $$ = forecasted load at time $t$.

[!TIP] Exam Focus: Differentiate forecasting types by horizon. For STLF, RBF and SOM are key. Know MAPE and RMSE formulas. Explain how SOM aids in load pattern clustering before forecasting.

END OF UNIT 3 NOTES
Aligned with RGPV EX-802(D) May 2023 paper pattern.