ODE-based Modelling of Spain's Electricity Generation Mix

This project simulates the long-term evolution of Spain's electricity generation mix using a system of ordinary differential equations (ODEs) based on replicator dynamics from evolutionary game theory.

The model captures how five energy technologies—fossil fuels, nuclear, wind, solar, and hydroelectric—compete for market share over a time horizon under different policy scenarios.

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Mathematical Model

Replicator Dynamics

The evolution of market shares is governed by the replicator equation:

dx_i/dt = x_i * (f_i - φ)

where:

• x_i is the market share of technology i (0 ≤ x_i ≤ 1, Σx_i = 1)
• f_i = R_i * g_i is the "fitness" of technology i
• φ = Σ x_j * f_j is the average fitness across all technologies
• R_i is the resource efficiency factor
• g_i is the intrinsic growth rate

Key insight: Technologies with above-average fitness (f_i > φ) grow, while those below average decline. This creates a competitive dynamic where the "fittest" technologies gradually dominate.

Parameters

Index	Technology	R (Efficiency)	g (Baseline Growth)	Notes
0	Fossil	1.0	0.02	Mature infrastructure
1	Nuclear	1.0	0.015	High upfront costs
2	Wind	0.9	0.03	Growing rapidly
3	Solar	0.9	0.035	Fastest cost decline
4	Hydro	0.6	0.01	Geographically limited

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Installation

Prerequisites

• Python 3.9 or higher
• pip (Python package manager)

Setup

1. Clone the repository:

   git clone https://github.com/crisfdezs/ODE-Final-Project.git
   cd ODE-Final-Project

2. Create a virtual environment (recommended):

   python3 -m venv venv
   source venv/bin/activate  # On Windows: venv\Scripts\activate

3. Install dependencies:

   pip install -r requirements.txt

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Usage

Basic Usage

Run all scenarios with interactive plots:

cd examples
python3 run_scenarios.py

Available Scenarios

Scenario	Description
baseline	Business-as-usual: current trends continue
renewable	Strong renewable support: enhanced wind/solar growth rates
nuclear_phaseout	Nuclear phase-out: negative nuclear growth after year 40

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Repository Structure

ODE-Final-Project/
├── src/
│   ├── model.py              # Defines the ODE right-hand side (replicator_rhs)
│   ├── solver.py             # Implements RK4 integrator with normalization
│   ├── simulations.py        # Functions for different scenarios:
│   │                            # baseline_scenario(), renewable_policy_scenario(), nuclear_phaseout_scenario()
│   └── plots.py              # Utilities for plotting energy shares
├── examples/
│   └── run_scenarios.py      # Script that runs all scenarios and generates plots
├── README.md
└── requirements.txt

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Module Descriptions

• model.py: Implements the replicator dynamics equation dx/dt = x * (f - φ)
• solver.py: Fourth-order Runge-Kutta (RK4) integrator with normalization enforcement
• simulations.py: Defines scenario parameters (R, g values) and initial conditions
• plots.py: Creates line plots of market share evolution over time
• run_scenarios.py: CLI interface for running simulations and exporting results

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Example Output

Running the baseline scenario produces a plot showing:

• Solar gaining the largest market share over time (highest growth rate)
• Wind growing steadily
• Fossil and nuclear declining relatively
• Hydro remaining stable but small (geographical constraints)

The renewable policy scenario shows accelerated growth of wind and solar, with fossil fuels declining faster due to carbon pricing effects.

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Interpreting Results

Market Shares

• Values represent the fraction of total electricity generation (0.0 to 1.0)
• All shares sum to 1.0 at every timestep (conservation)

Limitations

• Simplified model: does not capture intermittency, storage, or grid constraints
• Linear fitness function: real-world dynamics may be nonlinear
• Policy shocks are abrupt: real policies phase in gradually
• No feedback loops: electricity prices, supply chains, etc. not modeled

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Extending the Model

Adding New Scenarios

Edit simulations.py to add new scenarios:

def my_custom_scenario():
    """My custom energy policy scenario."""
    params = {
        "R": np.array([...]),  # Resource efficiency
        "g": np.array([...])   # Growth rates
    }
    return integrate(replicator_rhs, x0_spain, 0.0, 100.0, 0.1, params)

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Notes

All Python scripts include detailed docstrings explaining inputs, outputs, and assumptions.

The model assumes normalized shares for all energy sources (sum of shares = 1) and enforces non-negativity.

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Author

Cristina Fernández-Simal Bernard. Final Project – Introduction to Ordinary Differential Equations, Fall 2025.