Grid Load Forecasting, Demand Response, and Optimization

Module: Sustainable Tech | Difficulty: Premium

Optimal Power Flow

The AC optimal power flow problem is:

subject to:

Demand Response Model

The price elasticity of demand is:

Comparison

| Technique | Convergence | Scalability | Accuracy | |-----------|-------------|-------------|----------| | Linear Programming | O(n^3) | High | Optimal | | Genetic Algorithm | Variable | Medium | Near-optimal | | Deep RL | Fast | High | 95-98% | | Model Predictive Control | Medium | Medium | 92-95% |

Python Implementation

import numpy as np
from scipy.optimize import minimize

class GridOptimizer:
    def __init__(self, n_generators, n_buses):
        self.n_gen = n_generators
        self.cost_coeffs = np.random.uniform(0.1, 10, (n_generators, 3))

    def cost_function(self, p):
        return np.sum(self.cost_coeffs[:, 0] + self.cost_coeffs[:, 1] * p + self.cost_coeffs[:, 2] * p**2)

    def optimize(self, demand, renewable_gen=0):
        def objective(p):
            return self.cost_function(p)
        constraints = [{'type': 'eq', 'fun': lambda p: np.sum(p) - demand + renewable_gen}]
        bounds = [(10, 200)] * self.n_gen
        p0 = np.ones(self.n_gen) * demand / self.n_gen
        return minimize(objective, p0, bounds=bounds, constraints=constraints).x

    def demand_response(self, base_demand, price_signal, elasticity=-0.5):
        return base_demand * (1 + elasticity * (price_signal - 1.0))

Research Insight: Real-time demand response using deep reinforcement learning can reduce peak load by 15-25% while maintaining user comfort constraints.