Simulated Annealing for Power Generation Scheduling with Minimum Fuel Cost

Project info

• Field of studyElectrical Engineering and Energy
• LevelDoctoral dissertation (S3)
• Method typeOptimisation
• Estimated build time± 7 working days
• Toolsnumpy (implementasi manual); scipy.optimize.dual_annealing sebagai pembanding
• PriceUS$34 - US$213

Research goal

Optimise total fuel cost by tuning unit 1 output and unit 2 output with Simulated Annealing.

Variables

Example result

• 86.231Best objective
• 83.118Global optimum
• 3.746Gap to optimum %
• 0.552Acceptance ratio
• 500Iterations
• 0.081Final temperature

Simulated annealing (500 iterations) lowers Total fuel cost to 86.23 at Unit 1 output = 66.43 and Unit 2 output = 69.93; 55.2% of moves were accepted and the gap to the optimum is 3.75%.

Computed from illustrative synthetic data, not from a real study.

About Simulated Annealing

Simulated annealing explores the solution space by accepting worse moves with a probability that falls as temperature decreases. It suits global optimisation of functions with many local minima and combinatorial problems.

Steps

1. 1Define the objective, bounds, and starting point
2. 2Set the initial temperature and cooling schedule
3. 3Generate a random neighbour solution
4. 4Accept if better, or with probability exp(-delta/T) if worse
5. 5Lower the temperature and repeat
6. 6Plot the objective trajectory and report the best solution

Assumptions to check

• The cooling schedule is slow enough to avoid local minima
• The neighbour step matches the problem scale
• Several runs with different seeds assess consistency

Reading and reporting the results

Report the initial temperature, cooling rate, best solution and objective value, and their variation across runs. Explain that results are heuristic with no guarantee of the global optimum.

Example Python code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import dual_annealing

def f(p):
    return (p[0] ** 2 + p[1] ** 2) / 10 + 3 * (2 - np.cos(2 * p[0]) - np.cos(2 * p[1]))

rng = np.random.default_rng(32)
unit_1_output = rng.uniform(-5, 5, 2)          # current solution
cur = f(unit_1_output)
best_x, best = unit_1_output.copy(), cur
T, cooling = 5.0, 0.995
total_fuel_cost = []                              # objective trajectory
for it in range(4000):
    cand = np.clip(unit_1_output + rng.normal(0, 0.5, 2), -5, 5)
    fc = f(cand)
    if fc < cur or rng.random() < np.exp(-(fc - cur) / T):
        unit_1_output, cur = cand, fc
        if cur < best:
            best_x, best = unit_1_output.copy(), cur
    total_fuel_cost.append(best)
    T *= cooling

print("SA best x:", best_x.round(4), "f =", round(best, 5))
ref = dual_annealing(f, bounds=[(-5, 5), (-5, 5)], seed=32)
print("scipy dual_annealing:", ref.x.round(4), "f =", round(ref.fun, 5))

plt.plot(total_fuel_cost)
plt.xlabel("Iteration"); plt.ylabel("Best objective")
plt.show()

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions