\(\epsilon\)-constraint and the Pareto Front

A possible usecase of Multiobjective Optimization

What is a Pareto Front?

The Pareto Front is a method to achieve a balance between two conflicting optimization goals, commonly used in multiobjective optimization. A typical use case in energy system optimization is balancing costs and emissions. It’s crucial that the two objectives are not correlated to derive a meaningful Pareto Front.

Pareto Front example

To calculate the Pareto Front, follow these steps:

Step 1: Optimize your problem using \(obj1\) as the objective function. Calculate the value \(value_{obj2}\) for \(obj2\) from the results.

Step 2: Re-optimize using \(obj1\) as the objective, adding an \(\epsilon\)-constraint:

\[obj2 \le \epsilon \cdot value_{obj2}\]

where \(\epsilon \in [0,1]\) determines the step size for approximating the Pareto Front..

Step 3: Repeat Steps 1 and 2 until a termination condition is met. Examples for Termination conditions include:

  • The problem becomes infeasible.

  • The optimal value for \(obj2\) is reached. Therefore you first need to optimize with \(obj2\) as the objective and than chose \(\epsilon\) accordingly.

  • A time limit is reached.

The real Pareto Front?

The described method may not yield the exact Pareto Front. A potentially more optimal \(value_{obj2}\) can be found by fixing the optimal value of \(obj1\) and then optimizing for \(obj2\). This is computationally expensive and mainly used for scientific purposes. For more on augmented \(\epsilon\) constraints, see this article .

How to realise it in Code

# create your energy system normally but give the flows the
# to set other key words the just varible costs you could pass one with the flow
... solph.Flow(variable_costs=...,
                keyword_set_on_the_flow=...,
            )
# Optimize your energy system normally using obj1
optimization_model.solve(..)

# read the vaule of obj2 from your results
obj2= ...

# store all results needed

While termination_condition:
    # add the epsilon constraint to you model
    optimisation_model = solph.constraints.generic_integral_limit(
        optimisation_model, "keyword_set_on_the_flow", limit= step_size * obj2
    )

    # solve the model with the epsilon constaint
    solver_results = optimisation_model.solve(
        ...
    )

    #store obj2
    obj2=...

    # store all results needed

    # set termination condition
    # this could be when the model gets infeasible, when the optimal for
    # obj2 is reached (therefore you need to optimize your problem with obj2
    # as objective function once), a time limit,...
    termination_condition=...