Linear Programming Mini Example

Linear Programming Mini Example#

Linear programming (linear optimization) is a mathematical technique that uses a set of linear equations to find the optimal solution to a problem. It’s an effective type of mathematical programming that’s used to solve complex real-world problems, e.g. in energy systems.

1. Load packages

  • We are using the gurobipy package to formulate a mathematical model and solve it.

import gurobipy as gp
from gurobipy import GRB
import numpy as np

2. Define parameters

  • G: Number of generators 2

  • \(\overline{P}_i\): Production limit [6,4] in MW

  • \(c_i\): Cost [0.015,0.03] in $/kWh

  • D: Load demand 8 in MW

n_generators = 2 # G
p_lim = np.array([6,4]) # MW
cost = np.array([0.015,0.03]) # $/kWh
D = 8 # MW

3. Define mathematical model

Objective function:

  • Minimize the sum cost \(c_i\), which depends on the production \(p_i\) amount of each generator i.

Decision variables:

  • \(p_i\) output of generator i.

Constraints:

  • The sum of generation \(p_i\) for each generator i needs to equal the load D.

  • The output of each generator \(p_i\) has to greater equal to \(0\) and cannot exceed \(\overline{P}_i\).

(1)#\[\begin{align} \min \quad & \sum_{i\in[G]}c_i p_i \\ \text{s.t.} \quad & \sum_{i\in[G]}p_i = D \\ & 0 \le p_i \le \overline{P}_i && \forall i \in [G] \end{align}\]
# Create model object
m = gp.Model()
m.setParam("OutputFlag", 0)

# Create the variables
p = m.addVars(n_generators, lb=0, ub=GRB.INFINITY, name="p")

# Add constraints:
# Total production
m.addConstr(p.sum() == D)

# Generator limits
for i in range(n_generators):
    m.addConstr(p[i] <= p_lim[i])

# Add objective
m.setObjective(sum(p[i]*cost[i]*1000 for i in range(n_generators)), GRB.MINIMIZE)

# Solve
m.optimize()

Set parameter Username

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4. Inspect the solution

# Check the status of the solver
status = m.Status
if status == GRB.OPTIMAL:
    print("The model is optimal.")
elif status == GRB.INFEASIBLE:
    print("The model is infeasible.")
elif status == GRB.UNBOUNDED:
    print("The model is unbounded.")
else:
    print(f"Optimization ended with status {status}.")
print()

# Objective value
objective = m.ObjVal
print(f"Objective value {objective:.1f}.\n")

print("Variables values:")
# Print the values of all variables
for v in m.getVars():
    print(f"{v.VarName} = {v.X}")

The model is optimal.

Objective value 150.0.

Variables values:
p[0] = 6.0
p[1] = 2.0