Integrating quantum and classical computing for multi-energy system optimization using Benders decomposition

During recent years, quantum computers have received increasing attention, primarily due to their ability to significantly increase computational performance for specific problems. Computational performance could be improved for mathematical optimization by quantum annealers. This special type of quantum computer can solve quadratic unconstrained binary optimization problems. However, multi-energy systems optimization commonly involves integer and continuous decision variables. Due to their mixed-integer problem structure, quantum annealers cannot be directly used for multi-energy system optimization. To solve multi-energy system optimization problems, we present a hybrid Benders decomposition approach combining optimization on quantum and classical computers. In our approach, the quantum computer solves the master problem, which involves only the integer variables from the original energy system optimization problem. The subproblem includes the continuous variables and is solved by a classical computer. For better performance, we apply improvement techniques to the Benders decomposition. We test the approach on a case study to design a cost-optimal multi-energy system. While we provide a proof of concept that our Benders decomposition approach is applicable for the design of multi-energy systems, the computational time is still higher than for approaches using classical computers only. We therefore estimate the potential improvement of our approach to be expected for larger and fault-tolerant quantum computers

Modeling flocculation processes: continuous particle size distribution method

The flocculation process of cohesive sediment suspended in water consists of aggregation of the fine particles and breakup of the large flocs. The population balance equation (PBE) is a statement of continuity for particulate systems, and it is used to model the flocculation process and predict the particle size distribution (PSD). Different numerical methods are developed to solve the PBE, however most of the methods have difficulties in representing the continuous PSD or improving computational efficiency. In this research, the B-spline FEM and Galerkin FEM are studied to simulate the continuous PSD. The B-spline FEM solves the PBE over the whole domain, which is truncated to finite domain; the open non-uniform B-splines are used as basis function to approximate the PSD; the curve of PSD is required to be smooth enough. The Galerkin FEM discretizes the PBE on each sub-domain (the whole domain is split to several sub-domains), and it is used to solve less-smooth problems. The adaptive technique is applied to readjust the computational grid (particle size domain) to improve computational efficiency and the accuracy, and it is also applied in varied time step to get suitable time step to improve the stability. The analytical solutions of the PBE in special conditions and the experimental data are used to validate both B-spline FEM and Galerkin FEM, and the results are compared with that of the classical DPBE method. It shows that both B-spline FEM and Galerkin FEM can solve the PBE and simulate continuous PSD accurately and efficiently

Using ADMM for Hybrid System MPC

Model Predictive control (MPC) has been studied extensively because of its ability to handle constraints and its great properties in terms of stability and performance [Mayne et al., 2000]. We have in this thesis focused on MPC of Hybrid Systems, i.e. systems with both continuous and discrete dynamics. More specifically, we look at problems that can be cast as Mixed Integer Quadratic Programming (MIQP) problems which we are solving using a Branch and Bound technique. The problem is in this way reduced to solving a large number of constrained quadratic problems. However, the use in real time systems puts a requirement on the speed and efficiency of the optimization methods used. Because of its low computational cost, there have recently been a rising interest in the Alternating Direction Method of Multiplies (ADMM) for solving constrained optimization problems. We are in this thesis looking at how the different properties of ADMM can be used and improved for these problems, as well as how the Branch and Bound solver can be tailored to accompany ADMM. We have two main contributions to ADMM that mitigate some of the downsides with the often ill-conditioned problems that arise from Hybrid Systems. Firstly, a technique for greatly improving the conditioning of the problems, and secondly, a method to perform fast line search within the solver. We show that these methods are very efficient and can be used to solve problems that are otherwise hard or impossible to precondition properly

Value iteration with deep neural networks for optimal control of input-affine nonlinear systems

This paper proposes a new algorithm with deep neural networks to solve optimal control problems for continuous-time input nonlinear systems based on a value iteration algorithm. The proposed algorithm applies the networks to approximating the value functions and control inputs in the iterations. Consequently, the partial differential equations of the original algorithm reduce to the optimization problems for the parameters of the networks. Although the conventional algorithm can obtain the optimal control with iterative computations, each of the computations needs to be completed precisely, and it is hard to achieve sufficient precision in practice. Instead, the proposed method provides a practical method using deep neural networks and overcomes the difficulty based on a property of the networks, under which our convergence analysis shows that the proposed algorithm can achieve the minimum of the value function and the corresponding optimal controller. The effectiveness of the proposed method even with reasonable computational resources is demonstrated in two numerical simulations

A novel design of an isochronous integration [iIntegration] framework for first/second order multidisciplinary transient systems

Of fundamental interest are multidisciplinary interactions encompassing: (1) first order systems such as those encountered in parabolic heat conduction, first order hyperbolic systems such as fluid flow, and so on, and (2) second order systems such as those encountered in hyperbolic heat conduction, hyperbolic second order systems such as elastodynamics and wave propagation, and so on. After space discretization using methods such as finite differences, finite volumes, finite elements, and the like, the consequent proper integration of the time continuous ordinary differential equations is extremely important. In particular, the physical quantities of interest may need to be mostly preserved and/or the equations should be optimally integrated so that there is minimal numerical dissipation, dispersion, algorithm overshoot, capture shocks without too much dissipation, solve stiff problems and enable the completion of the analysis, and so on. To-date, practical methods in most commercial and research software include the trapezoidal family (Euler forward/backward, Galerkin, and Crank Nicholson) for first order systems and the other counterpart trapezoidal family (Newmark family and variants with controllable numerical dissipation) for second order systems. For the respective first/second order systems, they are totally separate families of algorithms and are derived from altogether totally different numerical approximation techniques. Focusing on the class of the linear multistep (LMS) methods, algorithms by design was first utilized to develop GS4-2 framework for time integration of second order systems. We have also recently developed the GS4-1 framework for integrating first order systems. In contrast to all past efforts over the past 50 years or so, we present the formalism of a generalized unified framework, termed GS4 (generalized single step single solve), that unifies GS4-1 (first-order systems) and GS4-2 (second-order systems) frameworks for simultaneous use in both first and second order systems with optimal algorithms, numerical and order preserving attributes (in particular, second-order time accuracy) as well. The principal contribution emanating from such an integrated framework is the practicality and convenience of using the same computational framework and implementation when solving first and/or second order systems without having to resort to the individual framework. All that is needed is a single novel GS4-2 framework for either second- and/or first-order systems, and we show how to switch from one to the other for illustrative applications to thermo-mechanical problems influenced by first/second order systems, respectively

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature