Electricity Production Planning

Introduction

This project analyzes and optimizes daily electricity generation planning using Mixed-Integer Linear Programming (MILP).

This work was carried out as part of the course by Sophie DEMASSEY at École des Mines de Paris (PSL).

The project studies different configurations of power plants (thermal and hydroelectric) in order to minimize production costs while respecting demand and reserve constraints.

The main objective is to determine the optimal number of plants to activate and their power outputs at each period of the day, taking into account operating costs, start-up costs, and operational constraints.

Note that this project simplifies the real-world problem by assuming that demand is known and fixed before planning. In reality, demand is stochastic, and planning must consider reserve margins to ensure grid stability.

Theoretical Background on Linear Optimization

Linear Programming (LP) consists of minimizing (or maximizing) a linear function subject to a set of linear constraints. In standard form, a primal problem is written as:

Associated with this problem is a dual problem, expressed as:

Primal feasibility means that there exists at least one vector \(x\) satisfying \(A x = b\) and \(x \ge 0\), while dual feasibility requires a vector \(y\) such that \(A^T y \le c\).
When a problem has feasible solutions for both the primal and dual, strong duality ensures that their optimal values coincide:

This property underpins much of the economic theory of optimization: the Lagrange multipliers \(y_i\) associated with the dual can be interpreted as marginal costs or shadow prices, representing the economic value of an additional unit of resource \(b_i\).

In practice, complementary slackness conditions link primal and dual solutions at the optimum:

These relationships show that:

• A strictly positive variable \(x_j\) corresponds to a saturated dual constraint.
• A saturated primal constraint implies a non-zero dual multiplier.

Thus, studying the primal and dual together allows not only the identification of the optimal solution but also an understanding of the system’s sensitivity to changes in demand, costs, or production capacities.

Problem Data

Thermal Power Plant Characteristics

Daily Demand Profile

Model Evolution

Model 1.5: Base Model with Minimum Power

The first model includes the minimum power constraint for each active plant:

Subject to:

where:

• \(P_{tp}\): total power from type \(t\) during period \(p\)
• \(N_{tp}\): number of active plants of type \(t\) during period \(p\)
• \(C_t\): marginal cost (€/MWh)
• \(T_p\): duration of period \(p\)
• \(D_p\): demand during period \(p\)

Result: Optimal cost = 869 k€

Gurobi variable declaration:

# Variables
N = {}  # Number of active plants (integer)
P = {}  # Produced power (continuous)

for t in PLANTS.keys():
    for p in HOURS.keys():
        N[f"{t},{p}"] = m.addVar(lb=0, name=f"N_{t},{p}", vtype=gp.GRB.INTEGER)
        P[f"{t},{p}"] = m.addVar(lb=0, name=f"P_{t},{p}", vtype=gp.GRB.CONTINUOUS)

Model 2.1: Integration of Fixed Costs (Cbase)

The model evolves to include fixed operating costs:

This formulation separates:

• the variable cost proportional to production above the minimum
• the fixed cost proportional to the number of active plants

Result: Optimal cost = 979 k€ (+12.7% compared to model 1.5)

Model 2.2: Adding Startup Costs (Cstart)

The introduction of the variable \(X_{tp}\) (number of plants started) requires a new consistency constraint:

The objective function becomes:

Result: Optimal cost = 1014 k€ (+3.6% compared to model 2.1)

# Startup constraint
for p in HOURS.keys():
    for t in PLANTS.keys():
        left_side = X[f"{t},{p}"] + Ntp(t, p, N, cyclic=False)
        right_side = N[f"{t},{p}"]
        m.addConstr(left_side >= right_side, name=f"start_{t}_{p}")

Model 3.1: Reserve Constraint (15%)

To ensure grid reliability, a 15% reserve margin is enforced:

Result: Optimal cost = 1015 k€ (+0.1% compared to model 2.2)

Model 4.1: Cyclic Planning

This model enforces that the final state (period 5) is consistent with the initial state (period 1):

This approach represents a 24-hour cyclic schedule.

Result: Optimal cost = 989 k€ (-2.6% compared to model 3.1)

Model 5.1: Adding Hydroelectric Plants

Two hydroelectric plants are available:

These plants consume no fuel (Cmwh = 0) but reduce the reservoir level.

Result: Optimal cost = 890 k€ (-10.0% compared to model 4.1)

Model 5.2: Pumping Constraint

The reservoir level must remain constant over 24 hours. Pumping requires 3000 MWh per meter of elevation:

where \(M_t\) is the reservoir lowering speed (m/h).

Result: Optimal cost = 987 k€ (+10.9% compared to model 5.1)

Theoretical Analysis: Sensitivity and Duality

Merit Order and Optimal Solution

The Merit Order method activates plants in ascending order of marginal cost.

For \(p=1\) (demand = 15 GW): the numerical solution matches Gurobi results.

Duality and Marginal Costs

The primal model in standard form:

Its dual form:

Comparative Results

Economic Interpretation

1. Dominant fixed costs: Moving from model 1.5 to 2.1 (+12.7%) highlights the significance of fixed costs.
2. Moderate startup costs: Model 2.2 adds only +3.6% extra cost.
3. Cyclic planning advantage: Model 4.1 reduces costs by 2.6% by optimizing transitions.
4. Hydropower competitiveness: Model 5.1 reduces costs by 10% thanks to zero variable cost.
5. Pumping costs: Model 5.2 shows reservoir neutrality adds +10.9% in cost.

Mathematical Properties

Total Unimodularity

The constraint matrix of model 1.5 exhibits total unimodularity:

• All coefficients belong to \(\{-1, 0, 1\}\)
• Each column contains at most two nonzero elements

Consequence: The continuous relaxation yields integer solutions, explaining the fast resolution (1 node explored).

Complementarity Gaps

At the optimum, Karush–Kuhn–Tucker (KKT) conditions are:

Conclusion

This project demonstrates the effectiveness of Mixed-Integer Linear Programming for solving complex energy planning problems. The progressive model analysis reveals:

• The importance of modeling all cost components (fixed, variable, startup)
• The benefit of cyclic planning for smoother transitions
• The economic interest of hydropower, despite pumping constraints
• Numerical stability due to the unimodular structure of the problem

Results obtained with Gurobi show very short computation times (<0.2s), this is due to the small size of the problem, in real life there are thousands of variables, here we just have hundreds.

Source Code

Try the Python Streamlit application using Gurobi and Pandas here:
👉 App