UNIT 2: AI Techniques and Applications in Power Systems
I. Neural Networks and Learning Paradigms
A. Radial Basis Function Network (RBFN)
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Architecture: A three-layer feedforward network.
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Input Layer: Receives the input vector $$\displaystyle \mathbf{x} \in \mathbb{R}^n $$. No computation.
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Hidden Layer: Contains radial basis functions (typically Gaussian). Each neuron computes the Euclidean distance between $\mathbf{x}$ and its center $$\displaystyle \mathbf{c}_i $$, then applies a basis function:
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$$ \phi_i(\mathbf{x}) = \exp\left(-\frac{\|\mathbf{x} - \mathbf{c}_i\|^2}{2\sigma_i^2}\right) $$
where $$\displaystyle \sigma_i $$ is the spread (width) of the $$\displaystyle i^{th} $$ RBF.
3. **Output Layer:** A linear combiner that computes a weighted sum of the hidden layer outputs:
$$ y_k(\mathbf{x}) = \sum_{i=1}^{m} w_{ki} \phi_i(\mathbf{x}) + b_k $$
where $$\displaystyle w_{ki} $$ are weights, $$\displaystyle b_k $$ is bias, $m$ is number of RBFs, and $k$ is output index.
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Key Idea: RBFNs implement localized receptive fields. Hidden layer neurons respond strongly only to inputs near their center $$\displaystyle \mathbf{c}_i $$. This makes training often faster than MLPs.
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Training: Usually a two-stage process:
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Determine centers $$\displaystyle \mathbf{c}_i $$ and spreads $$\displaystyle \sigma_i $$ (e.g., using K-means clustering on input data).
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Solve a linear regression problem to find output weights $$\displaystyle w_{ki} $$ (using least squares or pseudo-inverse).
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[!TIP] Exam Focus: Be prepared to draw the 3-layer architecture and write the Gaussian RBF equation. Contrast with MLP: RBF hidden layer is non-linear (fixed after center selection), output layer is linear.
B. Functional Link Network (FLN)
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Architecture: A single-layer network where the input is expanded into a higher-dimensional functional space before being presented to a linear neuron.
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Input: $$\displaystyle \mathbf{x} = [x_1, x_2, ..., x_n]^T $$
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Functional Expansion: The input vector is non-linearly transformed using a set of basis functions $$\displaystyle F(\mathbf{x}) = [f_1(\mathbf{x}), f_2(\mathbf{x}), ..., f_M(\mathbf{x})]^T $$, where $M \gg n$ (e.g., polynomial, trigonometric, or exponential functions).
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Output: $$\displaystyle y = \mathbf{w}^T F(\mathbf{x}) + b $$
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Key Idea: It enhances the non-linear separability of the input data by mapping it to a higher dimension, similar to the kernel trick in SVMs, but with an explicit, fixed expansion. The learning is linear in the expanded space.
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Example: For a 2D input $$\displaystyle \mathbf{x} = [x_1, x_2]^T $$, a quadratic expansion could be: $$\displaystyle F(\mathbf{x}) = [1, x_1, x_2, x_1^2, x_2^2, x_1x_2]^T $$.
[!TIP] Common Pitfall: Do not confuse FLN with a multi-layer perceptron. FLN has no hidden layer of adaptive neurons; the expansion is a fixed, pre-defined transformation.
C. Self-Organizing Maps (SOM) - Kohonen Maps
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Concept: An unsupervised, competitive learning neural network that produces a low-dimensional (typically 2D), topology-preserving representation of the input space. It performs vector quantization and dimensionality reduction.
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Working Principle:
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Initialization: Neurons on a 2D grid have associated weight vectors $$\displaystyle \mathbf{w}_i \in \mathbb{R}^n $$, initialized randomly.
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Competition: For an input $\mathbf{x}$, find the Best Matching Unit (BMU) $c$ with weight vector closest to $\mathbf{x}$ (using Euclidean distance).
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Cooperation: Define a neighborhood function $$\displaystyle h_{ci}(t) $$ (usually a Gaussian) around the BMU on the grid. Neurons within this neighborhood "cooperate".
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Adaptation: Update the weights of the BMU and its neighbors:
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$$ \mathbf{w}_i(t+1) = \mathbf{w}_i(t) + \alpha(t) \cdot h_{ci}(t) \cdot (\mathbf{x} - \mathbf{w}_i(t)) $$
where $\alpha(t)$ is the learning rate, and $$\displaystyle h_{ci}(t) $$ decreases with distance from BMU and time.
5. Repeat steps 2-4 for all inputs, decreasing $\alpha(t)$ and neighborhood radius over time.
- Applications in Power Systems: Load profiling, contingency analysis, security assessment, visualization of high-dimensional system states.
[!TIP] Exam Tip: Always explain the BMU concept and the neighborhood adaptation rule. The diagram is crucial—show the grid, BMU, and neighborhood.
D. Learning in Neural Networks
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1. Competitive Learning:
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Mechanism: Neurons compete to respond to an input. Only the "winner" (usually the one with smallest error/distance) updates its weights. Fundamental to SOMs and some clustering algorithms.
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Rule: $$\displaystyle \Delta \mathbf{w}_c = \alpha (\mathbf{x} - \mathbf{w}_c) $$ for winner $c$, others unchanged.
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Task: Clustering, vector quantization.
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2. Supervised Learning:
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Mechanism: Network learns a mapping from inputs to known target outputs using a error signal (difference between predicted and target). Uses gradient descent (backpropagation for MLPs).
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Rule: $$\displaystyle \Delta w_{ji} = -\eta \frac{\partial E}{\partial w_{ji}} $$, where $E$ is error function (e.g., MSE).
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Task: Function approximation, classification, regression (e.g., load forecasting).
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3. Learning Tasks:
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Pattern Association: Auto-associative (input = output) or hetero-associative (input ≠ output). Used in fault detection.
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Pattern Recognition: Classification of system states (e.g., stable/unstable).
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Function Approximation: Mapping input variables (load, generation) to output (cost, frequency deviation).
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Control: Adaptive control, system identification.
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II. Fuzzy Logic Systems
A. Operations on Fuzzy Relations
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Fuzzy Set Operations (on sets $A, B$ in universe $X$):
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Union: $$\displaystyle \mu_{A \cup B}(x) = \max[\mu_A(x), \mu_B(x)] $$
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Intersection: $$\displaystyle \mu_{A \cap B}(x) = \min[\mu_A(x), \mu_B(x)] $$
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Complement: $$\displaystyle \mu_{\bar{A}}(x) = 1 - \mu_A(x) $$
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Algebraic Product: $$\displaystyle \mu_{A \cdot B}(x) = \mu_A(x) \cdot \mu_B(x) $$
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Algebraic Sum: $$\displaystyle \mu_{A + B}(x) = \mu_A(x) + \mu_B(x) - \mu_A(x)\mu_B(x) $$
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Fuzzy Relation Composition: Given relations $R \subseteq X \times Y$ and $S \subseteq Y \times Z$.
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Max-Min Composition: $$\displaystyle T = R \circ S $$, where $$\displaystyle \mu_T(x,z) = \max_{y \in Y} \min[\mu_R(x,y), \mu_S(y,z)] $$
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Max-Product Composition: $$\displaystyle \mu_T(x,z) = \max_{y \in Y} [\mu_R(x,y) \cdot \mu_S(y,z)] $$
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[!TIP] Common Pitfall: Max-min is most common. Remember composition is like matrix multiplication but with min and max replacing multiplication and addition.
B. Fuzzy Logic System (FLS)
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Explanation with Practical Example (LFC):
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Inputs: Area Control Error (ACE) = $$\displaystyle \Delta P_{tie} + B \Delta f $$ (where $$\displaystyle \Delta P_{tie} $$ is tie-line power deviation, $\Delta f$ is frequency deviation, $B$ is frequency bias).
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Fuzzification: Convert crisp ACE into fuzzy sets (e.g., Negative Large, Negative Small, Zero, Positive Small, Positive Large).
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Rule Base: IF-THEN rules from operator experience.
Rule 1: IF ACE is Negative Large AND $\Delta$ACE is Negative THEN $$\displaystyle \Delta P_{ref} $$ is Decrease Fast.
Rule 2: IF ACE is Zero THEN $$\displaystyle \Delta P_{ref} $$ is No Change.
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Inference Engine: Applies rules (using Mamdani or Sugeno method). Mamdani uses min for AND, max for aggregation.
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Defuzzification: Converts fuzzy output to crisp control signal (e.g., using Centroid method: $$\displaystyle u_{crisp} = \frac{\int \mu_{agg}(u) \cdot u \, du}{\int \mu_{agg}(u) \, du} $$).
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Output: Change in reference power $$\displaystyle \Delta P_{ref} $$ for governors.
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Fuzzy Rule-Based System:
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Rule Formation: Derived from expert knowledge, operator experience, or data mining. Rules capture heuristic control strategies.
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Inference: For $n$ inputs and $m$ rules:
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Match: Compute firing strength $$\displaystyle \alpha_i $$ of each rule (e.g., $$\displaystyle \alpha_i = \min[\mu_{A_i}(x_1), \mu_{B_i}(x_2)] $$).
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Activate: Apply $$\displaystyle \alpha_i $$ to the consequent fuzzy set $$\displaystyle C_i $$ (implication, often using min: $$\displaystyle \mu_{C_i}'(u) = \min(\alpha_i, \mu_{C_i}(u)) $$).
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Aggregate: Combine all activated consequents: $$\displaystyle \mu_{agg}(u) = \max_i (\mu_{C_i}'(u)) $$.
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Defuzzify: Get crisp output from $$\displaystyle \mu_{agg}(u) $$.
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III. Evolutionary Computation
A. Genetic Algorithms (GA)
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Overview: Search heuristic inspired by natural selection, operating on a population of candidate solutions (chromosomes).
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Genetic Operators:
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Selection: Choose parents for reproduction based on fitness (objective function value).
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Roulette Wheel Selection: Probability proportional to fitness.
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Tournament Selection: Randomly pick $k$ individuals, select best. Robust, common.
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Rank Selection: Select based on rank, not absolute fitness.
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Crossover (Recombination): Exchange genetic material between two parents to create offspring.
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Single-Point: Swap segments after a random cut point.
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Two-Point: Swap segments between two cut points.
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Uniform: Each gene independently chosen from either parent.
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Arithmetic/Blend: For real-valued GAs: $$\displaystyle child = \alpha \cdot parent1 + (1-\alpha) \cdot parent2 $$.
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Other Operators:
- Reproduction: Copying elite individuals directly to next generation (elitism) to preserve best solutions.
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Mutation Operator:
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Explanation: Random, small perturbation applied to a single gene with low probability $$\displaystyle p_m $$. Introduces genetic diversity, helps escape local optima.
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Mechanism: For binary: flip a bit. For real-valued: add small random noise (e.g., $$\displaystyle x_{new} = x_{old} + \mathcal{N}(0, \sigma) $$).
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Applications in Power System Optimization: ELD (minimize cost), Optimal Power Flow (OPF), capacitor placement, network reconfiguration. Balances exploration (mutation) and exploitation (crossover).
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[!TIP] Exam Focus: Know the difference between selection methods (roulette vs. tournament) and crossover types. Relate mutation to avoiding premature convergence in ELD problems.
IV. Applications in Power System Optimization and Control
A. Economic Load Dispatch (ELD)
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Problem Formulation: Minimize total fuel cost $$\displaystyle F_{total} = \sum_{i=1}^{N} F_i(P_i) $$ subject to:
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Power Balance: $$\displaystyle \sum_{i=1}^{N} P_i = P_D + P_{loss} $$ (where $$\displaystyle P_D $$ = demand, $$\displaystyle P_{loss} $$ = transmission loss).
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Generator Limits: $$\displaystyle P_i^{min} \leq P_i \leq P_i^{max} $$.
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Ramp Rate Limits: $$\displaystyle |P_i(t) - P_i(t-1)| \leq UR_i $$ (Up Ramp), $$\displaystyle |P_i(t) - P_i(t-1)| \leq DR_i $$ (Down Ramp).
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Cost Function: Typically quadratic: $$\displaystyle F_i(P_i) = a_i P_i^2 + b_i P_i + c_i $$ ($/hr$).
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AI-Based Solution Approaches:
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GA/PSO/DE: Treat $$\displaystyle [P_1, P_2, ..., P_N] $$ as a chromosome/particle. Fitness = $$\displaystyle -F_{total} $$ (or penalized for constraint violation). Handle power balance by slack generator or penalty function.
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Neural Networks: RBFN or FLN can approximate the incremental cost curve or solve the dispatch equations.
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Worked Example Concept:
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Given 3 generators with quadratic costs, demand $$\displaystyle P_D $$, and neglecting losses.
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Equal Incremental Cost Criterion: $$\displaystyle \lambda = \frac{dF_1}{dP_1} = \frac{dF_2}{dP_2} = \frac{dF_3}{dP_3} $$.
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Solve: $$\displaystyle 2a_1P_1 + b_1 = 2a_2P_2 + b_2 = 2a_3P_3 + b_3 $$ and $$\displaystyle P_1+P_2+P_3 = P_D $$.
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Check limits, adjust if needed (lambda iteration or gradient method).
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[!TIP] Critical: Always include power balance and generator limits. For AI, explain how constraints are handled (penalty function, repair operators).
B. Load Frequency Control (LFC)
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Concept & Significance: Maintains system frequency $\Delta f$ near nominal (50/60 Hz) and tie-line power $$\displaystyle \Delta P_{tie} $$ between control areas at scheduled values. Ensures stability in interconnected grids.
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Control Mechanism & Block Diagram:
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Signal: Area Control Error (ACE) = $$\displaystyle \Delta P_{tie} + B \Delta f $$ ($B$ = frequency bias constant).
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Controller (ACE signal) → Integral Controller (I) → Reference Power $$\displaystyle \Delta P_{ref} $$ → Governor → Turbine → Change in Generation $$\displaystyle \Delta P_g $$.
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Plant Model: $$\displaystyle \frac{\Delta f(s)}{\Delta P_g(s)} = \frac{1}{M s + D} $$ (swing equation simplified, $M$=inertia, $D$=damping).
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Governor Model: $$\displaystyle \frac{\Delta P_g(s)}{\Delta P_{ref}(s)} = \frac{1}{1 + T_g s} $$ (with speed droop $R$).
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Full Block Diagram: Shows two parallel paths: (1) $\Delta f$ through $1/(Ms+D)$ and (2) $$\displaystyle \Delta P_{tie} $$ through tie-line transfer function $$\displaystyle K_{tie}/s $$. Sum gives ACE.
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[[DIAGRAM: CANVAS: Standard LFC block diagram for a single area. Input: $$\displaystyle \Delta P_{ref} $$. Summing junction: -($$\displaystyle \Delta P_{tie} + B\Delta f $$) = ACE. ACE goes to I-controller (1/s). Output $$\displaystyle \Delta P_{ref} $$ (actually $$\displaystyle \Delta P_c $$) goes to governor (1/(1+T_gs)) with droop R. Governor output $$\displaystyle \Delta P_g $$ goes to plant (1/(Ms+D)). Plant output $\Delta f$ feeds back through B to summing junction. Also, $\Delta f$ goes through tie-line model (K_tie/s) to produce $$\displaystyle \Delta P_{tie} $$.]
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Parameters Influencing Performance:
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Governor Speed Droop $R$: Lower $R$ → better frequency regulation but less sharing of load between units.
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Frequency Bias $B$: Should be chosen as $$\displaystyle B = \frac{1}{R} + D $$ for zero steady-state error in $\Delta f$.
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Integral Gain $$\displaystyle K_I $$: Determines speed of response and steady-state error elimination.
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Inertia $M$ & Damping $D$: Physical system parameters.
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C. Small Signal Stability Analysis
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Stability Concepts: Ability of power system to maintain synchronism after a small disturbance (e.g., small load change). Characterized by electromechanical oscillations (0.1-2.5 Hz) of generator rotors.
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AI Techniques for Stability Improvement & Control:
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Neural Networks: For online stability assessment. Train an MLP/RBFN on simulation data (features: power flows, voltages, generator states) to classify as stable/unstable or predict damping ratio.
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Fuzzy Logic: Design damping controllers (Fuzzy PSS) where rules adjust PSS parameters based on speed/acceleration.
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GA/PSO: Tune parameters of Power System Stabilizers (PSS) or FACTS controllers (e.g., SVC, TCSC) to maximize damping ratio of critical modes.
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SOM: Cluster system operating points to identify vulnerable regions.
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D. Load Forecasting
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Types:
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Short-Term (STLF): Hourly/daily, up to 1 week. For unit commitment, dispatch.
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Medium-Term (MTLF): Weekly/monthly, up to 1 year. For maintenance scheduling, fuel planning.
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Long-Term (LTLF): Yearly, 5-10 years. For generation/transmission expansion planning.
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AI Methodologies:
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Neural Networks: MLP with backpropagation is most common. Inputs: historical load, weather (temp, humidity), day-of-week, hour, holiday flags. RBFN can be faster for certain datasets.
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Fuzzy Logic: Handle uncertainty in weather and human factors. Often hybridized with NNs (Neuro-Fuzzy).
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GA/PSO: Used for feature selection (which past hours matter?) or optimizing NN weights.
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Hybrid Models: e.g., Wavelet-NN (decompose load into components), SVM for regression.
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[!TIP] Key Point: STLF uses similar-day and weather; LTLF uses economic/demographic indicators. AI models are data-hungry; preprocessing is critical.
E. Power System Modeling for Control
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1. Single Area System Model:
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Assumes a coherent, infinite bus. Focuses on frequency control within one control area.
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Components: Speed changer (governor), turbine, generator & load (aggregated as $1/(Ms+D)$), frequency sensor.
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Transfer Function: $$\displaystyle \frac{\Delta f(s)}{\Delta P_{ref}(s)} = \frac{1}{1 + R} \cdot \frac{1}{1 + T_g s} \cdot \frac{1}{Ms + D} $$ (simplified, without ACE feedback loop shown in LFC block diagram).
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Steady-State: $$\displaystyle \Delta f_{ss} = -\frac{\Delta P_{ref}}{1/R + D} $$.
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2. Two Area System Model:
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Interconnections: Two control areas (Area 1, Area 2) connected by a tie-line.
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Model: Each area has its own LFC loop (governor, turbine, $$\displaystyle M_i, D_i, R_i $$). The areas are coupled through the tie-line.
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Tie-Line Power Deviation: $$\displaystyle \Delta P_{tie,1} = T_{12} (\theta_1 - \theta_2) \approx K_{tie} \int (\Delta f_1 - \Delta f_2) dt $$. In s-domain: $$\displaystyle \Delta P_{tie,1}(s) = \frac{K_{tie}}{s} (\Delta f_1(s) - \Delta f_2(s)) $$.
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ACE for Area 1: $$\displaystyle ACE_1 = \Delta P_{tie,1} + B_1 \Delta f_1 $$.
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Control Objective: In steady-state, $$\displaystyle \Delta f_1 = \Delta f_2 = 0 $$ and $$\displaystyle \Delta P_{tie,1} = \Delta P_{tie,2} = 0 $$ for a load change in one area (undisturbed area's generation changes to support the disturbed area).
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[!TIP] Exam Essential: Be able to draw the two-area block diagram and write the ACE equations. Know that $$\displaystyle B_i $$ should be $$\displaystyle 1/R_i + D_i $$ for zero steady-state error.