Abstract

Abstract

We consider a farm of Kite Power Systems (KPS) in the field of Airborne Wind Energy (AWE), in which each kite is connected to an electric ground generator by a tether. In particular, we address the problem of selecting the best layout of such farm in a given land area such that the total electrical power generated is maximized. The kites, typically, fly at high altitudes, sweep a greater area than that of traditional wind turbines, and move within a conic shaped volume with vertex on the ground station. Therefore, constraints concerning kite collision avoidance and terrain boundaries must be considered. The efficient use of a given land area by a set of KPS depends on the location of each unit, on its tether length and on the elevation angle. In this work, we formulate the KPS farm layout optimization problem. Considering a specific KPS and wind characteristics of the given location, we study the power curve as a function of the tether length and elevation angle. Combining these results with an area with specified length and width, we develop and implement a heuristic optimization procedure to devise the layout of a KPS farm that maximizes wind power generation.

Abstract

This article addresses the problem of controlling a constrained, continuous-time, nonlinear system through Model Predictive Control (MPC). In particular, we focus on methods to efficiently and accurately solve the underlying optimal control problem (OCP). In the numerical solution of a nonlinear OCP, some form of discretization must be used at some stage. There are, however, benefits in postponing the discretization process and maintain a continuous-time model until a later stage. This is because that way we can exploit additional freedom to select the number and the location of the discretization node points. We propose an adaptive time-mesh refinement (AMR) algorithm that iteratively finds an adequate time-mesh satisfying a pre-defined bound on the local error estimate of the obtained trajectories. The algorithm provides a time-dependent stopping criterion, enabling us to impose higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. Additionally, we analyze the conditions to guarantee closed-loop stability of the MPC framework using the AMR algorithm. The numerical results show that the proposed AMR strategy can obtain solutions as fast as methods using a coarse equidistant-spaced mesh and, on the other hand, as accurate as methods using a fine equidistant-spaced mesh. Therefore, the OCP can be solved, and the MPC law obtained, faster and/or more accurately than with discrete-time MPC schemes using equidistant-spaced meshes.

Abstract

In the context of continuous-time control systems, we address the problem of guaranteeing that the constraints imposed along the trajectory are in fact satisfied for all times. The problem is relevant and non-trivial in situations in which a continuous-time internal representation of the system is used with a digital device, such as in sampled-data model-based control, in an optimal control solver, or in sampled-data model predictive control. In this letter, we establish a condition that when verified on a finite set of time instants (using limited computational power) can guarantee that the trajectory constraints are satisfied on an uncountable set of times. The case of constrained optimal control problems is further explored here. We develop an algorithm for the numerical solution of constrained nonlinear optimal control problems that combines a guaranteed constraint satisfaction strategy with an adaptive mesh refinement strategy.

Abstract

In this article we investigate the problem of generating electricity through an underwater kite power system (UKPS). For this problem, we develop the dynamical model for the UKPS and we formulate an optimal control problem to devise the trajectories and controls of the kite that maximize the total energy produced in a given time interval. This is a highly nonlinear problem for which the optimization is challenging. We also develop a numerical solution scheme for the optimal control problem based on direct methods and on adaptive time-mesh refinement. We report results that show that the problem can be quickly solved with a high level of accuracy when using our adaptive mesh refinement strategy. The results provide a set of output power values for different design choices and confirm that electrical energy that can be produced with such device.

Abstract

This article addresses the problem of optimizing electrical power generation using kite power systems (KPSs). KPSs are airborne wind energy systems that aim to harvest the power of strong and steady high-altitude winds. With the aim of maximizing the total energy produced in a given time interval, we numerically solve an optimal control problem and thereby obtain trajectories and controls for kites. Efficiently solving these optimal control problems is crucial when the results are used in real-time control schemes, such as model predictive control. For this highly nonlinear problem, we derive continuous-time models — in 2D and 3D —a nd implement an adaptive time-mesh refinement algorithm. By solving the optimal control problem with such an adaptive refinement strategy, we generate a block-structured adapted mesh which gives results as accurate as those computed using fine mesh, yet with much less computing effort and high savings in memory and computing time.

Abstract

When using direct methods to solve nonlinear optimal control, regular time meshes having equidistant spacing are most frequently used. However, in some cases, these meshes cannot cope accurately with nonlinear behaviour and increasing uniformly the number of mesh nodes may lead to a more complex problem, resulting in an incoherent solution. We propose an adaptive time--mesh refinement algorithm, considering different levels of refinement and several mesh refinement criteria. This technique is applied to solve an optimal control problem involving nonholonomic vehicles with state constraints which is characterized by presenting strong nonlinearities and by having discontinuous controls. The presented strategy leads to results with higher accuracy and yet with lower overall computational time, when compared to results obtained by meshes having equidistant spacing. In addition, the algorithm was able to obtain a solution when the traditional approach starting with a very large number of mesh nodes failed to do it, proving its robustness. Afterwards, we apply the necessary condition of optimality in the form of the Maximum Principle of Pontryagin to characterize the solution and to validate the numerical results.

Abstract

The relation between wind speed and electrical power—the power curve—is essential in the design, management and power forecasting of a wind farm. The power curve is the main characteristic of a wind turbine, and a procedure is presented for its determination, after the wind turbine is installed and in operation. The procedure is based on both computational and statistical techniques, in situ measurements, nacelle anemometry and operational data. This can be an alternative or a complement to procedures fully based on field measurements as in the International Electrotechnical Commission standards, reducing the time and costs of such practices. The impact of a more accurate power curve was measured in terms of the prediction error of a wind power forecasting system over 1 year of operation, whereby the methodology for numerical site calibration was presented and the concepts of ideal power curve and nacelle power curve introduced. The validation was based on data from wind turbines installed at a wind farm in complex topography, in Portugal, providing a real test of the technique presented here. The contribution of the power curve to the wind power forecasting uncertainty was found to be from 10% to 15% of the root mean square error.

Abstract

Optimal control can be of help to test and compare different vaccination strategies of a certain disease. In this paper we propose the introduction of constraints involving state variables on an optimal control problem applied to a compartmental SEIR (Susceptible, Exposed, Infectious and Recovered) model. We study the solution of such problems when mixed state control constraints are used to impose upper bounds on the available vaccines at each instant of time. We also explore the possibility of imposing upper bounds on the number of susceptible individuals with and without limitations on the number of vaccines available. In the case of mere mixed constraints a numerical and analytical study is conducted while in the other two situations only numerical results are presented.

Abstract

Slip-tendency analysis is a valuable tool in fault reactivation evaluation and seismic hazard assessment as it provides a means of quantifying the slip potential on mapped or suspected faults in a known or inferred stress field. We developed an interactive graphic tool to perform slip-tendency analysis. The application is written in MATLAB in the form of plug-ins for COULOMB, a graphic-rich deformation and stress change open-source software. In addition to identifying the faults most prone to reactivation, we compute and plot synthetic focal mechanisms from the direction and sense of likely slip. This allows compatibility between focal mechanisms and geological structures to be verified. Both individual faults and fault networks can be considered in three dimensions. The potential for slip depends on the prevailing stress field, the fault surface orientation and the coefficient of friction. These parameters are given interactively in a Windows environment. The application thus offers an easy-to-use graphical interface with the possibility of fast parameter modification and 3D visualization of the results.