24 research outputs found

    Holistic approach for microgrid planning and operation for e-mobility infrastructure under consideration of multi-type uncertainties

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    Integrating renewable energys ources in sectors such as electricity, heat, and transportation must be structured in an economic, technological, and emission- efficient manner to address global environmental issues.Microgrids appear to be the solution for large-scale renewable energy integration in these sectors.The microgrid components must be optimally planned and operated to prevent high costs, technical issues, and emissions. Existing approaches for optimal microgrid planning and operation in the literature do not include a solution for e-mobility infrastructure. As a consequence, a compact e-mobility infrastructure metho- dology is provided.The development of e-mobility infrastructure has as sociated uncertainties (short and long-term). As a result, a new stochastic method re- ferred to as IGDM-DRO is proposed in this dissertation.The proposed method provides a risk-averse strategy for microgrid planning and operation by including long-term and short-term uncertainty related to e-mobility.The multi-cut ben- der decomposition is applied for IGDM-DRO to prevent the suggested method’s intractability.Finally, the deterministic and stochastic methodologies are com bined in an ovelholistic approach for microgrid design and operation in terms of cost and robustness.The proposed method ist ested on a new settlement area in Magdeburg, Germany, under three different EV development scenarios (nega- tive, trend, andpositive).The share for the number of electric vehicles reached 31 percent of conventional vehicles by the end of the planned horizon. As a result, the microgrid’s overall cost has been increased by 2.3 to 2.9 percent per electric vehicle.Three public electric vehicle charging stations will be required in the investigated settlement are a intrend 2031.The investigated settlement area will require a total cost of 127,029 € in the trend scenario.To achieve full robustness against long-term uncertainties,the cost of the microgrid needs to be increased by 80 percent

    Inner Parallel Sets in Mixed-Integer Optimization

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    This thesis contains an extensive study of inner parallel sets in mixed-integer optimization. Inner parallel sets are a recent idea in this context and offer a possibility to relax the difficulties imposed by integrality constraints by guaranteeing feasibility of roundings of their (continuous) elements. To be able to use inner parallel sets algorithmically, various modifications, such as their enlargements and inner and outer approximations, are helpful and sometimes even necessary. Such ideas are introduced and investigated in this thesis, both theoretically as well as computationally. From our theoretical study of inner parallel sets emerge a number of feasible rounding approaches which mainly focus on the computation of good feasible points for mixed-integer linear and nonlinear minimization problems. Good feasible points are useful in the context of solving these problems by providing tight upper bounds on the objective value. In especially difficult cases, feasible rounding approaches may also be considered as an alternative to solving a problem. The contributions of this thesis include a thorough discussion of possibilities to enlarge inner parallel sets in the linear as well as in the nonlinear setting. Moreover, we introduce a novel cutting plane method based on inner parallel sets for mixed-integer convex minimization problems. This method, in addition to computing a good feasible point, also provides a lower bound on the objective value which is another important ingredient for solving such minimization problems. We study the possibility of dealing with equality constraints on integer variables which at first glance seem to prevent a nonempty inner parallel set. Under the occurrence of such constraints, we show that inner parallel sets can be nonempty in a reduced variable space, which allows the application of feasible rounding approaches. Finally, we investigate the behavior of inner parallel sets when integrated into search trees. Our study gives rise to a novel diving method which turns out to be a major improvement over standalone feasible rounding approaches. We test the introduced methods on standard libraries for mixed-integer linear, convex and nonconvex minimization problems separately in several computational studies. The computational results illustrate the potential of our ideas

    Invariant manifold theory for impulsive functional differential equations with applications

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    The primary contribution of this thesis is a development of invariant manifold theory for impulsive functional differential equations. We begin with an in-depth analysis of linear systems, immersed in a nonautonomous dynamical systems framework. We prove a variation-of-constants formula, introduce appropriate generalizations of stable, centre and unstable subspaces, and develop a Floquet theory for periodic systems. Using the Lyapunov-Perron method, we prove the existence of local centre manifolds at a nonhyperbolic equilibrium of nonlinear impulsive functional differential equations. Using a formal differentiation procedure in conjunction with machinery from functional analysis -- specifically, contraction mappings on scales of Banach spaces -- we prove that the centre manifold is smooth in the state space. By introducing a coordinate system, we are able to prove that the coefficients of any Taylor expansion of the local centre manifold are unique and sufficiently regular in the time and lag arguments that they can be computed by solving an impulsive boundary-value problem. After proving a reduction principle, this leads naturally to explorations into bifurcation theory, where we establish generalizations of the classical fold and Hopf bifurcations for impulsive delay differential equations. Aside from the centre manifold, we demonstrate the existence and smoothness of stable and unstable manifolds and prove a linearized stability theorem. One of the applications of the theory above is an analysis of a SIR model with pulsed vaccination and finite temporary immunity modeled by a discrete delay. We determine an analytical stability criteria for the disease-free equilibrium and prove the existence of a transcritical bifurcation of periodic solutions at some critical vaccination coverage level for generic system parameters. Then, using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution until a Hopf point is identifed. A cylinder bifurcation is observed; the periodic orbit expands into a cylinder in the extended phase space before eventually contracting onto a periodic orbit as the vaccination coverage vanishes. The other application is an impulsive stabilization method based on centre manifold reduction and optimization principles. Assuming a cost structure on the impulsive controller and a desired convergence rate target, we prove that under certain conditions there is always an impulsive controller that can stabilize a nonhyperbolic equilibrium with a trivial unstable subspace, robustly with respect to parameter perturbation, while guaranteeing a minimal cost. We then exploit the low-dimensionality of the centre manifold to develop a two-stage program that can be implemented to compute the optimal controller. To demonstrate the effectiveness of the two-stage program, which we call the centre probe method, we use the method to stabilize a complex network of 100 diffusively coupled nodes at a Hopf point. The cost structure is one that assigns higher cost to controlling of nodes that have more neighbours, while the jump functionals are required to be diagonal -- that is, they do not introduce further coupling. We also introduce a secondary goal, which is that the number of nodes that are controlled is minimized

    Optimal Energy Storage Strategies in Microgrids

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    Microgrids are small-scale distribution networks that provide a template for large-scale deployment of renewable energy sources, such as wind and solar power, in close proximity to demand. However, the inherent variability and intermittency of these sources can have a significant impact on power generation and scheduling decisions. Distributed energy resources, such as energy storage systems, can be used to decouple the times of energy consumption and generation, thereby enabling microgrid operators to improve scheduling decisions and exploit arbitrage opportunities in energy markets. The integration of renewable energy sources into the nation's power grid, by way of microgrids, holds great promise for sustainable energy production and delivery; however, operators and consumers both lack effective strategies for optimally using stored energy that is generated by renewable energy sources. This dissertation presents a comprehensive stochastic optimization framework to prescribe optimal strategies for effectively managing stored energy in microgrids, subject to the inherent uncertainty of renewable resources, local demand and electricity prices. First, a Markov decision process model is created to characterize and illustrate structural properties of an optimal storage strategy and to assess the economic value of sharing stored energy between heterogeneous, demand-side entities. Second, a multistage stochastic programming (MSP) model is formulated and solved to determine the optimal storage, procurement, selling and energy flow decisions in a microgrid, subject to storage inefficiencies, distribution line losses and line capacity constraints. Additionally, the well-known stochastic dual dynamic programming (SDDP) algorithm is customized and improved to drastically reduce the computation time and significantly improve solution quality when approximately solving this MSP model. Finally, and more generally, a novel nonconvex regularization scheme is developed to improve the computational performance of the SDDP algorithm for solving high-dimensional MSP models. Specifically, it is shown that these nonconvex regularization problems can be reformulated as mixed-integer programming problems with provable convergence guarantees. The benefits of this regularization scheme are illustrated by way of a computational study that reveals significant improvements in the convergence rate and solution quality over the standard SDDP algorithm and other regularization schemes
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