**Project Heads**

Falk Hante, Michael Hintermüller, Sebastian Pokutta

**Project Members**

Paulina Bock de Barillas

**Project Duration**

01.06.2021 − 15.06.2025

**Located at**

HU Berlin

The transition to sustainable energy networks provides complexities arising from the integration of rather unsteady renewables (figure below). The unsteady network dynamics cause operational challenges for network providers. Mathematical control theory that includes modeling, simulation and optimization can support dispatching processes, but is facing challenges of coupling large scale dynamics from physical models such as partial differential equations (PDE) with discrete decision models such as 0-1 switching for connecting and disconnecting capacitors. These problems are to be solved fast and repeatedly while gathering data to adapt. The research agenda of this project tackles fundamental questions concerning control and optimization in context of this application.

Looking at power networks, our basis is the model used in [10], where the infrastructure is modeled as a graph with the set of nodes , which may describe interlinked power stations, generators, consumers (power demand), transformers (that boost or lower voltage) and the set of edges which are the transmission lines between them. A transmission line is mathematically described by the telegraph equations, a coupled system of equations that predict the voltage and current distributions on this single electrical transmission line with space coordinate and time coordinate .

(1)

The so called characteristic variables , with represent right traveling components (i.e. and left traveling components (i.e. ). Inductance, capacity, resistance and the admittance per unit length of the conductor are represented by the constants . The voltage and the current can be computed out of the characteristic variables.

These equations are defined on the transmission lines (edges), we parametrize an edge by the interval and couple them at the nodes by Kirchhoff’s conditions.

for all incoming edges and outgoing edges at node v. In words, incoming power at a node equals power that leaves the node at the same time and all potentials (voltage) connected at a node must be equal.

The well known alternating current optimal power flow equations (ACOPF) are related to this model in the sense that any solution to the power flow model yields a solution to the telegraph equations with sinusoidal boundary conditions.

Similar models are available for natural gas or hydrogen distribution networks, water networks, and the like; see [Hante at al. 2017],[P1],[P2].

In order to break the imminent computational complexity of Mixed-Integer Optimal Control Problems coming with energy network models such as above, we developed time-domain decomposition techniques and proved convergence of these methods for the important subclass of linear-quadratic instances [P3]. These techniques allow a sequential approach to mixed-integer programming problems by solving suitably chosen subproblems on smaller time horizons. The approach is highly scalable due to the freedom on the number of subintervals and by the possibility to solving all subproblems fully in parallel.

In one direction of our research we consider a partial linearization of the mixed integer optimal control problem, with binary constraints on the control for turning generators on and off. Relaxing these constraints in the sense, that also rational solutions in the interval are becoming feasible, allows us to realize the minimization via first order optimality conditions. In our research we want to utilize an exact relaxation approach as in [Burger et al. 2012], which means that there is no relaxation gap, i.e., the optimal value of the relaxed problem is the same as the optimal value of the original, non-relaxed problem. The main advantage of this is that we can work with the relaxed problem, which is computationally much easier, and we can directly transfer the results to the original problem. The main technical obstacle is handling appropriate trust-region constraints in order to make the linearization meaningful.

We want to emphasize that this approach is contrary to the partial outer convexification and Sum Up Rounding technique going back to [Sager 2005]. It is a strategy to round the solution of the relaxed problem to obtain an integer solution and delivers an upper bound for the integral over the difference of the relaxed and rounded control (see Fig. 2 for an example). This bound depends on the discretization of the control, meaning that the finer the grid, the smaller bound gets and the optimal objective function value is reached up to an arbitrarily small -gap.

Usually we want to avoid very short successive changes of the integer control. This requirement can be expressed by additional constraints on the integer control. Such combinatorial (or dwell time) contraints describe the necessity of activation/deactivation of an integer control for at least a given minimal duration.

A discretization of such problems leads to mixed integer nonlinear problems, that become computationally intractable when the discretization stepsize tends to zero.

We consider another solution approach based on the idea of alternating direction methods, with a clear separation of the combinatorial aspects from the continuous control aspects. In one direction we use partial outer convexification (POC) and combinatiorial integral approximation (CIAP) or the exact relaxation approach mentioned above and in the other direction we solve an mixed integer problem. These two directions are weakly coupled by a penalty parameter. Based on the exactness of the penalty parameter, we can derive a convergence result for this method [P1].

**Related Publications
**

[1] A. Bärmann, A. Heidt, A. Martin, S. Pokutta and C. Thurner: Polyhedral Approximation of Ellipsoidal Uncertainty Sets via Extended Formulations – a computational case study. Computational Management Science, Vol. 13, Nr. 2, 2016.

[2] G. Braun, A. Roy and S. Pokutta. Stronger Reductions for Extended Formulations. arXiv preprint arXiv:1512.04932 (2018).

[3] M. Burger, Y. Dong, and M. Hintermüller: Exact relaxation for classes of minimization problems with binary constraints, arXiv preprint arXiv:1210.7507 (2012).

[4] S. Fiorini, S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf: Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds. Proceedings of STOC 2012. arXiv:1111.0837 (2012).

[5] Hante, F.M. et al. Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications. In: Industrial Mathematics and Complex Systems, P. Manchanda, R. Lozi, A.H. Siddiqi (Eds.), Springer Singapore, pp. 77–122, 2017.

[6] F.M. Hante, and S. Sager: Relaxation methods for mixed-integer optimal control of partial differential equations, Computational Optimization and Applications, 2013.

[7] Hante, F.M.: Relaxation methods for hyperbolic PDE mixed-integer optimal control problems. Optimal Control Applications and Methods. Vol. 38, Nr. 6, pp. 1103–1110, 2017.

[8] F.M. Hante: Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization. In: Industrial and Applied Mathematics, P. Manchanda, R. Lozi, A.H. Siddiqi (Eds.), pp. 157–171 Springer Singapore, 2020.

[9] M. Hintermüller, K. Ito, and K. Kunisch: The primal-dual active set strategy as a semismooth Newton method. SIAM Journal on Optimization, Vol 13, Nr. 3, 2002.

[10] S. Göttlich, M. Herty, and P. Schillen. Electric transmission lines: control and numerical discretization. Optimal Control Appl. Methods, 37(5):980–995, 2016.

[11] M. Burger, Y. Dong, and M. Hintermüller. Exact Relaxation for Classes of Minimization Problems with Binary Constraints, Oct. 2012.

[12] P. Manns and C. Kirches. Multidimensional sum-up rounding for elliptic control systems. SIAM Journal on Numerical Analysis, 58(6):3427–3447, 2020.

[13] S. Sager. Numerical methods for mixed–integer optimal control problems. Der andere Verlag, Tönning, Lübeck, Marburg, 2005. ISBN 3-89959-416-9.

[14] S. Sager, H. G. Bock, and M. Diehl. Solving Mixed–Integer Control Problems By Sum Up Rounding With Guaranteed Integer Gap.(2007)

**Completed Work
**

**Publications**

[P1] S. Göttlich, F.M. Hante, A. Potschka, and L. Schewe: Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints. Mathematical Programming, Vol. 188, pp. 599–619 (2021).

[P2] F.M. Hante: Relaxation methods for optimal switching control of PDE-dynamical systems. Handbook of Numerical Analysis, Numerical Control: Part B, Enrique Zuazua and E. Trelat (Eds.), Volume 24, 2023, Pages 61-76 (2022).

[P3] F.M. Hante, R. Krug, M. Schmidt: Time-Domain Decomposition for Mixed-Integer Optimal Control Problems. Applied Mathematics and Optimization, Volume 87, Number 36 (2023).

[P4] F.M. Hante, M. Schmidt: Gas Transport Network Optimization: Mixed-Integer Nonlinear Models. To appear in Encyclopedia of Optimization, 3rd ed. (Panos M. Pardalos and Oleg Prokopyev, Eds.), Springer Nature, 2024.

[P5] F.M. Hante, M. Schmidt: Gas Transport Network Optimization: PDE-Constrained Models. To appear in Encyclopedia of Optimization, 3rd ed. (Panos M. Pardalos and Oleg Prokopyev, Eds.), Springer Nature, 2024.

**Master thesis**

[M1] Till Schäfer, Regularization of Mixed-Integer Control Approximations based on Relaxation and Optimization, 24.03.2023

[M2] Xinglei Liu, An Augmented Lagrangian Decomposition Approach for Power Grid Optimization, 03.07.2023

**Related Graphics**