Non-smooth Methods for Complementarity Formulations of Switched Advection-Diffusion Processes

有机朗肯循环(ORC)工艺建立在工艺工程中，用于从废气热源中回收能量。他们运行循环运行的对流-扩散过程，使用锅炉将能量从废热转移到工作介质中。蒸汽相的工质被膨胀以利用能量，而现在处于流体相的工质通过冷凝器和泵反馈到锅炉。我们感兴趣的是ORC在排气流量和温度描述的瞬时边界条件下等压相的时间和空间动力学。该问题由一维或二维分布相变介质参数的瞬时偏微分-代数方程模拟的对流-扩散过程描述。由于过程模型的规模很大，且外部负荷点的变化对过程的扰动比过程的时间常数小得多，过程的非光滑行为是非周期的，表现出非平凡的模式。暂态行为的优化操作是使这一概念具有实用价值的关键，对于经典控制概念来说，这是一项极具挑战性的任务。该研究项目发展了具有相变的对流扩散过程的优化理论和有效的数值方法，该理论和方法将结合PDE约束优化的简化方法、利用离散值控制的非光滑建模方法、混合整数最优控制的部分外凸化方法和对问题非光滑部分的约束的分解方法。特别是，我们开发了：-选择一个同时优化框架，其中离散化的模型方程作为非线性约束进入优化问题；-离散和非光滑现象在部分外凸化框架中建模。利用空间和时间上的自适应离散化，这给出了施加在指示控制的单形上的非凸约束结构；-将所产生的MIOCP分解成MPVC并使用SUR算法。应使用该方法将PDE优化任务与识别最佳切换结构的组合任务分离；-将SUR算法的非光滑结果与非光滑问题的解决相关联的平流扩散过程类的理论框架。这种关系应显示在控制空间和偏微分方程状态空间中，并以可行性的有界损失和有界次最优性的形式表示，并应与离散化的选择有关--简化的半光滑牛顿型框架用于解决由模型方程和控制离散化引起的非光滑非线性规划问题。

Organic Rankine Cycle (ORC) processes are established in process engineering for energy recovery from an exhaust heat source. They run a cyclically operated advection-diffusion process to transfer energy from exhaust heat into a working medium using a boiler. The working medium in steam phase is expanded to harness energy, while the working medium, now in fluid phase, is fed back to the boiler by way of condenser and pump.We take interest in the temporal and spatial dynamics of the isobaric phases of an ORC under transient boundary conditions described by exhaust flow and temperature. The problem is described by an advection-diffusion process modeled by instationary partial differential-algebraic equations in 1D or 2D with distributed phase-dependent medium parameters. Due to the large scale of process models and significant perturbations of the process by external load-point changes on a time scale considerably smaller than the process' time constant, the non-smooth behavior of the process is non-periodic and shows nontrivial patterns. Optimal operation of the transient behavior is key to making the concept practically worthwhile, and is a highly challenging task for classical control concepts.The proposed research project develops theory and efficient numerical methods for optimization of advection-diffusion processes with phase changes described by instationary nonlinear PDAEs.Theory and methods shall combine the state of the art in reduced approaches for PDE-constrained optimization, a modeling approach for non-smoothness that makes use of discrete-valued controls and a partial outer convexification approach to mixed-integer optimal control combined with a decomposition approach for constraints on the non-smooth parts of the problem. In particular, we develop:- a simultaneous optimization framework is chosen in which the discretized model equations enter the optimization problem as nonlinear constraints;- discrete and non-smooth phenomena are modeled in a partial outer convexification framework. Using adaptive discretizations in space and time, this gives rise a non-convex constraint structure imposed on the simplex of indicator controls;- a decomposition approach for the resulting MIOCP into an MPVC and the use of a SUR algorithm. The approach shall be used to decouple the PDE optimization task from the combinatorial task of identifying an optimal switching structure;- a theoretical framework for the class of advection diffusion processes that relates the non-smooth result of the SUR algorithm to the to the solution of the non-smooth problem. Such relations shall be shown in the spaces of controls and of PDE states, and in terms of bounded loss of feasibility and bounded suboptimality, and shall be related to the choice of discretizations- a reduced semi-smooth Newton-type framework is used to solve the non-smooth nonlinear programming problems that result from the discretization of model equations and controls.