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**Day 1: Optimal Power Flow**

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**8:45 – 9:00: Welcome with Coffee: Jean-Bernard Lasserre & Patrick Panciatici**

**9:00 – 9:45: Dan Molzahn (Argonne ****National Lab****)**

*Title: Quickly Certifying Global Optimality of a Candidate Solution to a Polynomial Optimization Problem using a Moment Relaxation Hierarchy *

*Authors: Daniel Molzahn, Hassan Hijazi, Cedric Josz*

*Abstract: Despite being non-convex and NP-Hard, local solvers obtain the global optima of some practically relevant polynomial optimization problems, including certain optimal power flow problems. Certifying that the solution from a local solver is, in fact, globally optimal is traditionally achieved by solving a convex relaxation, which can be computationally burdensome. To bypass this computational burden, this presentation describes an approach that leverages higher-order moment relaxations to formulate sufficient conditions that ensure global optimality. These sufficient conditions take the form of linear feasibility problems that have the potential to be computationally tractable for large problems.*

**9:45 – 10:30: Andy Sun (Georgia Tech, Atlanta)**

*Title: Minor formulation and relaxation for the optimal power flow problem *

*Abstract: I will present recent advances in solving large-scale Optimal Power Flow (OPF) problems. I will present a new reformulation of matrix rank constraint using 2-by-2 minors of the matrix and its application in forming strong convex relaxations based on second-order cone programming (SOCP) for OPF. Extensive experiments show the proposed SOCP relaxations produce very high quality solutions compared to the traditional SDP relaxations and can be orders of magnitude faster. This is joint work with Santanu Dey and Burak Kocuk.*

**Break**

**10:45 – 11:30: Pascal Van Hentenryck (University of Michigan)**

*Title: When is AC-OPF hard?*

*Abstract: Although AC-OPF is NP-hard even for simple radial networks, its difficulty on real networks has remained an intriguing and elusive question. This talk presents a systematic evaluation of AC-OPF on a number of real and synthetic test cases. The computational results consider approximations, relaxations, and global optimization, and reveal some illuminating findings on the behavior of these algorithms.*

**11:30 – 12:15: Manuel Ruiz and Jean Maeght (RTE - R&D)**

*Title : Application of optimization problems in complex variable with a AC-OPF modelling tool.*

*Abstract: Thanks to extensive scientific research, newly developed methods are able to provide good solutions for the non-convex AC-OPF problems. Computational results can be easily reproduced on academic datasets and for some kinds of AC-OPF (minimizing losses, with or without thermal limit, unit commitment etc). In order to experiment on these methods in an industrial context, the time spent in implementing an AC-OPF needs to be reduced. The R&D department of RTE will present the key components of an AC-OPF modeler implemented in Julia, which stores the optimization problem with polynomials in complex variables while keeping information on the network structure. At the moment, the tool can build OPF problems from Matpower and the GridOptimizationCompetition input format. State-of-art relaxations (SDP, SOCP, …) or B&B methods can then be applied in a generic a way. *

**Lunch**

**13:30 – 14:15: Line Roald (Los Alamos National Lab)**

*Title: AC Optimal Power Flow with Robust Feasibility Guarantees - Caveats and Solutions*

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*Abstract: The AC Optimal Power Flow problem is hard in its deterministic form, and the consideration of power injection uncertainty due to e.g. fluctuations in renewable generation complicates the problem further. In this talk, we discuss a method to guarantee robust feasibility for a set of continuous uncertainty realizations. We first discuss the challenges involved in guaranteeing the satisfaction of both engineering constraints and power flow solvability, before providing details on how we address these challenges using moment-based relaxations.*

** 14:15 – 15:00: T. Weisser (LAAS-CNRS, Toulouse) & Sidhant (Los Alamos National Lab)**

*Title: Chance-Constrained Optimization for Non-Linear Network Flow Problems*

*Abstract: Many engineered systems, such as energy infrastructures, are networks governed by non-linear physical flows. Optimizing the operation of such systems tend to bring the system close to the limits of admissible operation, making it vulnerable to uncertainty in the input parameters. To address this problem, we formulate the network flow problem as a chance-constrained optimization problem with non-linear equality and inequality constraints. To obtain tractable representations of the chance constraints, we extend existing approaches to polynomial chance-constrained approximation towards conservative inner approximations.* *Further, we propose a new two-step procedure to improve computational efficiency of the solution procedure. While the method is applicable to general network flow problems with polynomial constraints, we use the AC optimal power flow problem for electric grids as an example to demonstrate the method numerically.*

**15:00 – 15:45: Steven Low (Caltech, Pasadena)**

*Title: Real-time Optimal Power Flow *

*Abstract: We formulate real-time optimal power flow (OPF) as an online solutio to a time-varying AC OPF problem. We describe first and second-order algorithms, bound the error of tracking local solutions of the time-varying problem, and present simulation results to demonstrate their effectiveness (Joint work with Yujie Tang (Caltech), Emiliano Dall’Anese (NREL))*

**Dinner: 19h**

**Day 2: Stability assessment**

**8:45 – 9:00: Coffee**

**9:00 – 9:45: Florian Dörfler & Adrian Hauswirth (ETH Zurich)**

*Title: Online Optimization in Closed Loop on the Power Flow Manifold*

*Authors: Adrian Hauswirth, Alessandro Zanardi, Saverio Bolognani, Florian Dorfler, and Gabriela Hug*

*Abstract: The focus of this paper is the online load flow optimization of power systems in closed loop. In contrast to the conventional approach where an AC OPF solution is computed before being applied to the system, our objective is to design an adaptive feedback controller that steers the system in real time to the optimal operating point without explicitly solving an AC OPF problem. Our approach can be used for example to simultaneously regulate voltages, mitigate line congestion, and optimize operating costs under time-varying conditions. In contrast to related work which is mostly focused on distribution grids, we introduce a modeling approach in terms of manifold optimization that is applicable in general scenarios. For this, we treat the power flow equations as implicit constraints that are naturally enforced and hence give rise to the power flow manifold (PFM). Based on our theoretical results for this type of optimization problems, we propose a discrete-time projected gradient descent scheme on the PFM. In this work, we confirm through a detailed simulation study that the algorithm performs well in a more realistic power system setup and reliably tracks the time-varying optimum of the underlying AC OPF problem.*

**9:45 – 10:30: Bachir El Khadir (Princeton)**

*Title: A converse SOS Lyaponuv result for stable homogeneous polynomial vector fields.*

Abstract: The existence of a Lyaponuv function gives a necessary and sufficient condition for systems stability, and has proved to be useful in many areas of science and engineering. Despite the recent progress in the field, much is still unknown about how to construct such a function algorithmically. In this talk, we prove that if a dynamical system given by a homogeneous polynomial vector field is stable, then it admits a rational Lyaponuv function. This generalizes the classical result that if a linear vector field is stable then it admits a quadratic Lyaponuv function. Furthermore, we show that this rational function and its derivative have an SOS certificate of positivity that can be found using semi definite programming.

**Break**

**10:45 – 11:30: Didier Henrion (LAAS-CNRS Univ Toulouse, FR and Czech Tech Univ Prague, CZ)**

*Title: Approximating the region of attraction of a polynomial control system with the Lasserre hierarchy*

Abstract: The region of attraction (ROA) of a given non-linear control system is the set of states that can be steered to a given set after a given time (possibly arbitrarily large) while respecting state and control constraints. When the system dynamics and the constraints are described by polynomials, we show that the ROA can be approximated arbitrary well (in the Lebesgue metric) by level sets of polynomials of increasing degree. These polynomials can be explicitly computed by a Lasserre hierarchy of moment-sum-of-square semidefinite programming problems. In this talk we will survey the essential ideas between this technique, following [D. Henrion, M. Korda. Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control 59(2):297-312, 2014]. Instrumental to these developments are infinite linear programming problems on occupation measures and the duality between the cones of positive polynomials and moments of positive measures. The Lyapunov interpretation of these results follows as an outcome of conic duality theory.

**11:30 – 12:15: Marian Anghel (Los Alamos National Lab)**

*Title: Distributed stability analysis and control design for large scale complex systems*

*Abstract: analysis and control design for large scale complex systems is notoriously difficult. Lyapunov’s direct method has long been used in stability analysis and control design of dynamical systems, but finding a Lyapunov function is generally a non-trivial task. Recent progress in computational methods, especially in sum-of-squares (SOS) and semi-definite programming (SDP), have enabled the algorithmic construction of polynomial Lyapunov functions for sufficiently small systems.*

*However such methods become soon intractable as the system size grows, rendering the analysis of large systems almost impossible via a Lyapunov function approach. For this reasons, a decomposition-agregation approach has been proposed to circumvent these difficulties. In this approach a large-scale systems in decomposed into many interacting subsystems and the stability of the full interconnected system is performed only the properties of the subsystem Lyapunov functions and estimated bounds on the strength of the interactions between the subsystems.*

*In this lecture I describe two scalable and parallel iterative algorithms that decomposes the system into a network of interacting subsystems and certifies the asymptotic stability of the interconnected systems by using the concept of vector Lyapunov functions. By introducing a notion of power and energy flow in a dynamical network, we first show how SOS tools can be used to partition polynomial networks into weakly interacting subsystems. Then, using only the subsystem Lyapunov functions, and minimal communications between the neighbors, under convergence the two algorithms certify the asymptotic stability of the full system under a given disturbance.*

** Lunch**

**13:30 – 14:15: Kostya Turitsyn (MIT)**

*Title: Algorithmic construction of voltage and angle stability certificates.*

*Abstract: Despite decades of research, stability assessment is still one of the computational bottleneck in power grid operation process. The talk will provide an overview of a number of new approaches to power system stability, security and emergency control. The first part of the talk will focus on the problem of voltage stability and introduce novel algorithms for fast construction of inner approximations (convex restrictions) of power flow solvability and feasibility sets. The second part will focus on the problem of transient stability and introduce the Lyapunov Function Family approach provide a computationally tractable means for constructing approximation of operating point basin of attractions. This technique is shown to be applicable to a wide range of problems including synthesis of special protection systems and real-time network reconfiguration*

** 14:15 – 15:00: Matteo Tacchi (LAAS-CNRS & RTE-R&D)**

*Title: Sums Of Squares For Power Systems Stability Analysis*

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*Abstract: RTE needs new tools to assess power networks' stability. A well-known method for stability analysis is the Lyapunov approach, which consists in estimating the Region Of Attraction of an Equilibrium Point as a sublevel set of a Lyapunov function ; however, such a method is difficult to implement in practice. The use of Sums Of Squares theory allows to do so with good results for polynomial systems.*

** 15:00 – 15:30: Patrick Panciatici (RTE-R&D)**

**Challenges, Perspectives and possible links between the two topics.**

** 15:30 – 16:00: All, Brain Storming and Conclusions**