Settings of the algorithm

In order to solve the investment problem defined by an Antares study and its associated candidates.ini file, Antares-Xpansion uses the Benders decomposition algorithm. The simulation options and algorithmic parameters are defined in the settings.ini file, located in the folder user/expansion/ of the Antares study.

The following section lists the configurable parameters. If the user does not specify the value of a parameter, its default value is used.

Overview of the settings.ini file

Name	Default value	Description
optimality_gap	1	Tolerance on absolute gap
relative_gap	1e-6	Tolerance on relative gap
max_iteration	+Inf	Maximum number of Benders iterations
timelimit	1e12	Timelimit (in seconds) of the Benders step
uc_type	expansion_fast	Unit-commitment type used by Antares
master	integer	Resolution mode of the master problem
yearly-weights	None	Path of the Monte-Carlo weights file
solver	Cbc	Name of the solver
log_level	0	Logs severity
additional-constraints	None	Path of the additional constraints file
separation_parameter	0.5	Step size for the in-out separation
relaxed_optimality_gap	1e-5	Threshold to switch from relaxed to integer master
batch_size	0	Number of subproblems per batch

The format is a standard .ini and should follow this template:

uc_type = expansion_fast
master = integer
optimality_gap = 0
max_iteration = 100
timelimit = 300
additional-constraints = constraint.txt
log_level = 0

Details of the parameters

optimality_gap

Positive float. Default value: 1.

The optimality_gap, defined in euros, is the tolerance on the absolute gap for the Antares-Xpansion algorithm.

At each iteration, the algorithm computes upper and lower bounds on the optimal cost. The algorithm stops as soon as the quantity best_upper_bound - best_lower_bound falls below optimality_gap. The solution returned by the algorithm has a cost equal to best_upper_bound, which is guaranteed to be within optimality_gap euros of the optimal cost.

• If optimality_gap = 0, Antares-Xpansion will continue its search until the optimal solution of the investment problem is found.
• If optimality_gap > 0, the search will stop as soon as best_upper_bound - best_lower_bound < optimality_gap.

Figure 10 – Illustration of the optimality gap and the set of solutions that can be returned by the package when the gap is strictly positive.

The interest of a strictly positive optimality_gap is that it speeds up the search by stopping as soon as a "good" solution is found.

The interpretation of this stopping criterion is not always obvious. It certainly guarantees that a solution found by the algorithm has a cost that is close to the optimum, but it does not provide any information on the distance (in MW) between the installed capacities of this solution and those of the optimal solution: if the cost function is relatively flat around the optimum, solutions whose costs are close may have significantly different installed capacities (see Figure 10).

Which settings should I use for the optimality_gap?

1. I have to run several expansion optimizations of different variants of a study and compare them. In that case, if the optimal solutions are not returned by the package, the comparison of several variants can be tricky. The inaccuracy of the method might indeed be of the same order of magnitude as the changes brought by the input variations. It is therefore advised to be as closed as possible from the optimum of the expansion problem. To do so, we advise to set the optimality_gap to zero.

2. I am building one consistent generation/transmission scenario. As the optimal solution is not more realistic than an approximate solution of the modelled expansion problem, the settings can be less constraining with an optimality_gap of a few million euros.

relative_gap

Positive float. Default value: 1e-6.

The relative_gap parameter is the tolerance on the relative gap for the Antares-Xpansion algorithm.

At each iteration, the algorithm computes upper and lower bounds on the optimal cost. The algorithm stops as soon as the quantity (best_upper_bound - best_lower_bound) / max(|best_upper_bound|, |best_lower_bound|) falls below relative_gap. For a relative gap \(\alpha\), the cost of the solution returned by the algorithm satisfies:

$$\frac{{\scriptstyle\texttt{xpansion solution cost}} - {\scriptstyle\texttt{optimal cost}}}{{\scriptstyle\texttt{optimal cost}}} < \alpha .$$

Remark

The algorithm stops as soon as the first criterion among optimality_gap and relative_gap is met. Keep in mind that if either parameter is not specified by the user, the default value is used.

max_iteration

Strictly positive integer or infinite. Default value: +Inf.

Maximum number of iterations for the Benders decomposition algorithm. Once this number of iterations is reached, the Antares-Xpansion algorithm ends, regardless of the quality of the solution.

timelimit

Strictly positive integer. Default value: 1e12.

Maximum allowed time in seconds for the execution of the Benders step of Antares-Xpansion (i.e. the time of the initial Antares simulation and for the problem generation step is not accounted for). Once the timelimit is reached, the algorithm finishes the current Benders iteration - which can take several additional seconds or minutes - and terminates.

uc_type

Possible values: expansion_fast and expansion_accurate. By default: expansion_fast.

The uc_type (unit-commitment type) parameter specifies the simulation mode used by Antares to evaluate the operating costs of the electrical system:

• If uc_type = expansion_fast: the fast mode of Antares is used, deactivating the flexibility constraints of the thermal units (Pmin constraints and minimum up and down times), and not taking into account either the start-up costs or the impact of the day-ahead reserve.

• If uc_type = expansion_accurate: the expansion mode of Antares is used. This simulation mode corresponds to the accurate mode of Antares in which the unit-commitment variables are relaxed. The flexibility constraints of the thermal units as well as the start-up costs are taken into account.

master

Possible values: integer and relaxed. By default: integer.

The master parameter provides information on how integer variables are taken into account in the Antares-Xpansion master problem.

• If master = relaxed: the integer variables are relaxed, and the level constraints of the investment candidates (cf. unit-size) will not be necessarily respected. The master problem is linear.

• If master = integer: the investment problem is solved by taking into account unit-size constraints of the candidates. The master problem is a MILP (Mixed-Integer Linear Program).

For problems with several investment candidates with large max-units, using master = relaxed can accelerate the Antares-Xpansion algorithm very significantly.

yearly-weights

String, specifying the name of a file. Default: None.

The parameter yearly-weights allows to assume that the Monte Carlo years simulated in the Antares study are not equally probable. The most representative years may be given greater weight than those that are less representative. The yearly-weights parameter defines a file that stores a vector \(\left( \omega_{1},\ldots,\omega_{n} \right)\), with \(n\) the number of Monte-Carlo years in the study, which is used to evaluate the expected production cost:

$$ \mathbb{E}\left(\text{cost}\right) = \frac{\sum_{i = 1}^{n}{\omega_i\text{cost}_i}}{\sum_{i = 1}^{n}\omega_{i}} \ , $$

with \(\text{cost}_{i}\) the production cost of the \(i\)-th Monte Carlo year and \(\omega_i\) the associated weigth.

The file defined by the yearly-weights parameter must be located in the user/expansion/weights folder of the Antares study. It must contain a column with as many numerical values as there are Monte-Carlo years in the Antares study. The value of the \(i\)-th row is the weight \(\omega_i\) of the \(i\)-th Monte Carlo year (see Figure 11).

user/expansion/settings.ini

uc_type = expansion_accurate
master = integer
optimality_gap = 0
yearly-weights = weights.txt

user/expansion/weights/weights.txt

Figure 11 – Example of a setting of Antares-Xpansion with Monte-Carlo years that are not equally-weighted.

If the yearly-weights parameter is not used, the Monte-Carlo years of the Antares study are considered to be equally-weighted.

The yearly-weights parameter must be set in line with the Antares study playlist by the user: years with zero weights must be removed from the Antares study playlist in order not to be simulated unnecessarily.

solver

String. Default value: Cbc.

Defines the solver that is used to solve the master and the subproblems in the Benders decomposition. The user can either write solver = Cbc or solver = Coin to use the COIN-OR optimization suite. These settings are identical: depending on whether the problem has integer variables, Antares-Xpansion calls either the linear solver (Clp) or the MILP solver (Cbc) of the COIN-OR optimization suite.

Remark

In Antares-Xpansion, the subproblems are always linear. If master = relaxed, the master problem is linear as well, whereas if master = integer, the master problem is a MILP.

To use another solver, you have to build the package with the chosen solver, please contact us. It’s not possible to put it on github for the moment.

log_level

Possible values :{0, 1, 2}, specifying the solver's log severity. Default value: 0.

Logs can be printed both in the console and in a file. There are 3 types of logs:

• Operational: Displays progress information for the investment on candidates and costs,
• Benders: Displays information on the progress of the Benders algorithm,
• Solver: Logs of the solver called for the resolution of each master or subproblem.

The table below details the behavior depending on the log_level.

	Operational	Benders	Solver
File	Always (reportbenders.txt)	Always (benders_solver.log)	2 (solver_log_proc_<proc_num>.txt)
Console	0	>= 1	Never

additional-constraints

String, specifying the name of a file. Default: None.

The file defined by the additional-constraints parameter must be located in the user/expansion/constraints folder of the Antares study. It allows to impose linear constraints between the invested capacities of investment candidates. These linear constraints will be added to the master problem.

The syntax of the additional-constraints file is illustrated on Figure 12. The format is inspired by Antares' binding constraints.

user/expansion/settings.ini

uc_type = expansion_accurate
master = integer
optimality_gap = 0
additional-constraints = constraints.txt

user/expansion/constraints/constraints.txt

Figure 12 – Example of an additional constraints file.

A constraint in the additional-constraints is defined with the following parameters:

• name: String. The constraint name must be unique and must not contain any special symbols or space.

• <investment-candidate-name>: Float. Coefficient of the left-hand side variable corresponding to <investment-candidate-name>.

• sign: Either less_or_equal, equal or greater_or_equal. Defines the relationship between the left-hand side and right-hand side of the constraint.

• rhs: Float. Right-hand side of the constraint.

The user can also optionally use binary constraints to represent, for example, exclusion constraints. In Figure 13, Antares-Xpansion cannot invest in semibase and peak at the same time, but it can invest in neither.

Warning

The use of binary variables is not recommended as it greatly increases the calculation time.

Figure 13 – Example of an additional constraint file.

separation_parameter

Float in \([0,1]\). Default value: 0.5.

Defines the step size for the in-out separation. If \(x_{in}\) is the current best feasible solution and \(x_{out}\) is the master solution at the current iteration, the investment in the subproblems is set to

$$ x_{cut} = {\small \texttt{separation parameter}} * x_{out} + (1 - {\small \texttt{separation parameter}}) * x_{in} .$$

The in-out stabilisation technique is used in order to speed up the Benders decomposition. When separation_parameter < 1, it is necessary to relax the master problem in the first iterations as the cut point \(x_{cut}\) is a convex combination of the best feasible solution and of the solution of the master problem. In the case where master = integer, the algorithm proceeds as follows:

1. Solve the first iterations with the relaxed formulation using the in-out stabilisation technique,
2. Once the gap is sufficiently small (see relaxed_optimality_gap), switch back to the integer formulation and set back separation_parameter = 1 i.e. use the classical Benders algorithm.

relaxed_optimality_gap

Positive float. Default value: 1e-5.

The relaxed_optimality_gap parameter only has effect when master = integer. In this case, the master problem is relaxed in the first iterations. The relaxed_optimality_gap is the threshold from which to switch back from the relaxed to the integer master formulation.

In the first iterations, the algorithm computes upper and lower bounds on the optimal cost of the relaxed master problem. The algorithm switches to the integer formulation as soon as the quantity (best_upper_bound - best_lower_bound) / best_upper_bound falls below relaxed_optimality_gap. For the subsequent iterations, the best upper bound is reset to \(+\infty\) as the solutions of the relaxed problem are not feasible for the integer formulation.

batch_size

Positive integer. Default value: 0.

This parameter is used only when --method is benders_by_batch, otherwise it has no effect. The batch_size specifies the number of subproblems per batch. If set to 0, then all subproblems are in the same batch. In this case, the Benders by batch algorithm is launched with a single batch, which is mathematically equivalent to the classical Benders algorithm.