Puzzling out the coexistence of terrestrial planets and giant exoplanets. The 2/1 resonant periodic orbits

Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Using the Detrended Fast Lyapunov Indicator as a chaotic indicator, we constructed maps of dynamical stability by varying both the eccentricity of the outer giant planet and the semi-major axis of the inner terrestrial planet, and identify the boundaries of the stability domains. Guided by the computation of families of periodic orbits, the phase space is unravelled by meticulously chosen stable periodic orbits, which buttress the stability domains. We provide all possible stability domains for coplanar symmetric configurations and show that a terrestrial planet, either in mean-motion resonance or not, can coexist with a giant planet, when the latter moves on either a circular or an (even highly) eccentric orbit. New families of symmetric and asymmetric periodic orbits are presented for the 2/1 resonance. It is shown that an inner terrestrial planet can survive long time spans with a giant eccentric outer planet on resonant symmetric orbits, even when both orbits are highly eccentric. For 22 detected single-planet systems consisting of a giant planet with high eccentricity, we discuss the possible existence of a terrestrial planet. This study is particularly suitable for the research of companions among the detected systems with giant planets, and could assist with refining observational data.Comment: Accepted for publication in A&

Optimal Energy Scheduling of Grid-connected Microgrids with Battery Energy Storage

The coupling of small-scale renewable-based energy sources, such as photovoltaic systems, with residential battery energy storages forms clusters of local energy resources and customers, which can be represented as controllable entities to the main distribution grid. The operation of these clusters is similar to that of grid-connected microgrids. The future distribution grid of multiple grid-connected microgrids will require proper coordination to ensure that the energy management of the microgrid resources satisfies the targets and constraints of both the microgrids’ and the main grid’s operation. The link between the battery dispatch and the induced battery degradation also needs to be better understood to implement energy management with long-term economic benefits. This thesis contributes to the solution of the above-mentioned issues with an energy management model developed for a grid-connected microgrid that uses battery energy storage as a flexible energy resource. The performance of the model was evaluated in different test cases (simulations and demonstrations) in which the model optimized the schedule of the microgrid resources and the energy exchange with the connected main grid, while satisfying the constraints and operational objectives of the microgrid. Coordination with the distribution system operator was proposed to ensure that the microgrid energy scheduling solution would not violate the constraints of the main grid.Two radial distribution grids were used in simulation studies: the 12-kV electrical distribution grid of the Chalmers University of Technology campus and a 12.6-kV 33-bus test system. Results of the Chalmers’ test case assuming the operation of two grid-connected microgrids with battery energy storage of 100-200 kWh showed that the microgrids’ economic optimization could reduce the cost for the distribution system operator by up to 2%. Coordination with the distribution system operator could achieve an even higher reduction, although it would lead to sub-optimal solutions for the microgrids. Application of decentralized coordination showed the effectiveness of utilizing microgrids as flexible entities, while preserving the privacy of the microgrid data, in the simulations performed with the 33-bus test system. The developed microgrid energy management model was also applied for a building microgrid, where the battery energy storage was modeled considering both degradation and real-life operation characteristics derived from measurements conducted at real residential buildings equipped with stationary battery energy storages. Simulation results of a building microgrid with a 7.2 kWh battery energy storage showed that the annual building energy and battery degradation cost could be reduced by up to 3% compared to when the impact of battery degradation was neglected in the energy scheduling. To demonstrate the model’s practical use, it was integrated in an energy management system of the real buildings, where the buildings’ battery energy storages and, by extent, their energy exchange with the main grid, were dispatched based on the model’s decisions in several test cases.The test cases’ results showed that the model can reduce the energy cost of the microgrid both in short-term and in long-term. Moreover, with the help of this model, the microgrid can be employed as a flexible resource and reduce the operation cost of the main distribution grid

Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronom

Optimal Energy and Flexibility Dispatch of Grid-Connected Microgrids

This thesis proposes an optimization model to efficiently schedule energy and flexibilities of a grid-connected microgrid (MG) with non-dispatchable renewable energy sources and battery energy storages (BESs). The model can also be used to coordinate the MG operation with the connected upstream distribution grid and to assess the MG flexibility considering economic viability, technical feasibility, and BES degradation. The performance of the model was tested for both deterministic and stochastic formulations using two solution approaches i.e., day-ahead and rolling horizon, in different simulation and demonstration test cases. In these test cases, the model optimizes the schedule of the MG resources and the energy exchange with the connected main grid, while satisfying the constraints and operational objectives of the MG. The flexibilities from the MG would also be optimized when the MG provided flexibility services (FSs) to the distribution systems. The coordination with the distribution system operator (DSO) was proposed to ensure that the microgrid operation would not violate the technical constraints of the distribution grid. \ua0Two types of test systems were used for the simulations studies: 1) distribution grids with grid-connected MGs and 2) building MGs (BMGs). The distribution test systems included the 12-kV electrical distribution grid of the Chalmers campus and a 12.6-kV 33-bus standard test system, while the BMGs were based on real residential buildings i.e., the HSB LL building and the Brf Viva buildings. Results of the Chalmers’ test case showed that the MGs’ economic optimization could reduce the annual cost for the DSO by up to 2%. Centralized coordination, where the MG resources were scheduled by the DSO, led to an even higher reduction, although it also led to sub-optimal solutions for the MGs. Decentralized coordination was applied on the 33-bus network with a bilevel optimization framework for energy and flexibility dispatch. Two types of FSs were integrated in the bilevel model i.e., the baseline (FS-B) and the capacity limitation (FS-C). The latter has found to be more promising, as it could offer economic incentives for both the DSO and the MGs. In the studies of the BMGs, the BESs were modeled considering both degradation and real-life operation characteristics derived from measurements conducted at the buildings. Results showed that the annual building energy and BES degradation cost could be reduced by up to 3% compared to when the impact of BES degradation was neglected in the energy scheduling. With the participation of the BMG in FS-C provision, the building’s operation cost could be further reduced depending on the flexibility price. A 24-h simulation of the BMG’s operation yielded an economic value of flexibility of at least 7% of its daily energy and peak power cost, while the DSO could benefit from the FS assuming that the dispatched flexibility could be used to reduce the subscription fee that guarantees a certain power level. For frequent flexibility provision i.e., multiple times within a year, the value of flexibility for the MG operator could be reduced due to the BES degradation.\ua0To demonstrate the practical use of the proposed model, an energy management system was designed to integrate the model and employ it to optimize the energy schedule of the BMGs’ BESs and energy exchange with the main grid. The energy dispatch was performed in real-time based on the model’s decisions in real demonstration cases. The demonstration results showed the benefits of the model in that it helped reduce the energy cost of the BMG both in short term and in long term. The model can also be used by the MG operators to quantify the potential and assess the value of microgrid flexibility. Moreover, with the help of this model, the MG can be employed as a flexible resource and reduce the operation cost of the connected distribution grid

Continuation and stability deduction of resonant periodic orbits in three dimensional systems

In dynamical systems of few degrees of freedom, periodic solutions consist the backbone of the phase space and the determination and computation of their stability is crucial for understanding the global dynamics. In this paper we study the classical three body problem in three dimensions and use its dynamics to assess the long-term evolution of extrasolar systems. We compute periodic orbits, which correspond to exact resonant motion, and determine their linear stability. By computing maps of dynamical stability we show that stable periodic orbits are surrounded in phase space with regular motion even in systems with more than two degrees of freedom, while chaos is apparent close to unstable ones. Therefore, families of stable periodic orbits, indeed, consist backbones of the stability domains in phase space.Comment: Proceedings of the 6th International Conference on Numerical Analysis (NumAn 2014). Published by the Applied Mathematics and Computers Lab, Technical University of Crete (AMCL/TUC), Greec

Resonant planetary dynamics: Periodic orbits and long-term stability

Many exo-solar systems discovered in the last decade consist of planets orbiting in resonant configurations and consequently, their evolution should show long-term stability. However, due to the mutual planetary interactions a multi-planet system shows complicated dynamics with mostly chaotic trajectories. We can determine possible stable configurations by computing resonant periodic trajectories of the general planar three body problem, which can be used for modeling a two-planet system. In this work, we review our model for both the planar and the spatial case. We present families of symmetric periodic trajectories in various resonances and study their linear horizontal and vertical stability. We show that around stable periodic orbits there exist regimes in phase space where regular evolution takes place. Unstable periodic orbits are associated with the existence of chaos and planetary destabilization.Comment: Proceedings of 10th HSTAM International Congress on Mechanics, Chania, Crete, Greece, 25-27 May, 201