by Lorenzo Baldi (ESR 9 from INRIA)
My secondment at ARTELYS started at the end of January 2022, and it is planned to last three months. Due to the pandemic and the remote working conditions in the company, it has been decided to start it online, but a future physical meeting is in the plan.
My work at INRIA mostly deals with theoretical aspects of the celebrated Lasserre's hierarchies, used to approximately solve Polynomial Optimization problems. The theory behind them is itself very interesting, since it is a combination of ideas from different mathematical areas: Optimization, Real Algebraic Geometry, Functional Analysis,...But another important point on the application side is the modelling power of Polynomial Optimization, and more generally of the Generalized Moment Problems.
Indeed a great number of Optimization problems can be described in this way: for instance discrete optimization, control problems, option pricing in finance, and many more.In particular, during my secondment in ARTELYS I am investigating possible advantages of the use of the Lasserre's hierarchies for Optimal Power Flow problems in electricity systems. My work is strongly connected with the one of Edgar Fuentes, former PhD student at ARTELYS and co-advised by my supervisor, Prof. B. Mourrain, and my supervisor at ARTELYS is Michael Gabay.
In Optimal Power Flow problems great importance is played by two aspects:
- recovering a the global minimum (and not a local one!);
- recovering of the minimizer(s).
These two important properties are naturally studied in the context of the Lasserre's hierarchies, and therefore Optimal Power Flow problems are relevant examples when one wants to apply the hierarchies to some real world problems. One of the main drawbacks of these hierarchies is the size of the matrices involved in the computation, that grows exponentially with the accuracy required: therefore one has to exploit different strategies to reduce their size.
The main contribution in this direction is the Julia package TSSOS, developed at LAAS, Toulouse, a partner of POEMA. This package exploits the structure and the sparsity of the problem to generate smaller matrices (compared with the dense, standard hierarchy), and thus we can use the package to tackle problems that would be out of reach for standard methods, for instance the Optimal Power Flow ones.
What I am trying concretely to do is to use TSSOS to solve Optimal Power Flow standard test cases. In particular our new procedure has two steps:
- solve a low order relaxation of the problem using TSSOS, and extract an approximate global minimizer;
- use the ARTELYS local solver, KNITRO, to compute exactly the global optimizer.
This procedure is interesting, since also with TSSOS it is impossible to solve high-order and thus high-precision relaxations. Therefore the global minimizer that is computed in the first step will be only approximate, and therefore it is necessary to go on with the second step. Although simple to state, there are some challenges in the above procedure.
First, minimizer extraction is not well developed in TSSOS: to solve this issue, my advisor and I are collaborating with the developers of the package (Prof. Victor Magron, part of POEMA, and Jie Wang) to improve this extraction and make it more robust. Second, the local optimization has to be adjusted carefully, and the approach needs to be verified with easier test cases.
In the coming months we plan to proceed in these directions, with the goal of having some concrete results on the Optimal Power Flow test cases to validate the new optimization procedure.