Dissertation (PhD)--University of Stellenbosch, 1995. / Bibliography. Also available via the Internet.

Optimal power flow (OPF) is the choice tool for determining the optimal operating status of the power system by managing controllable devices. The importance of the OPF approach has increased due to increasing energy prices and availability of more control devices. Existing OPF approaches exhibit shortcomings. Current OPF algorithms can be classified into (a) nonlinear programming, (b) intelligent search methods, and (c) sequential algorithms. Nonlinear programming algorithms focus on the solution of the Kuhn-Tucker conditions; they require a starting feasible solution and the model includes all constraints; these characteristics limit the robustness and efficiency of these methods. Intelligent search methods are first-order methods and are totally inefficient for large-scale systems. Traditional sequential algorithms require a starting feasible solution, a requirement that limits their robustness. Present implementations of sequential algorithms use traditional modeling that result in inefficient algorithms. The research described in this thesis has overcome the shortcomings by developing a robust and highly efficient algorithm. Robustness is defined as the ability to provide a solution for any system; the proposed approach achieves robustness by operating on suboptimal points and moving toward feasible, it stops at a suboptimal solution if an optimum does not exist. Efficiency is achieved by (a) converting the nonlinear OPF problem to a quadratic problem (b) and limiting the size of the model; the quadratic model enables fast convergence and the algorithm that identifies the active constraints, limits the size of the model by only including the active constraints. A concise description of the method is as follows: The proposed method starts from an arbitrary state which may be infeasible; model equations and system constraints are satisfied by introducing artificial mismatch variables at each bus. Mathematically this is an optimal but infeasible point. At each iteration, the artificial mismatches are reduced while the solution point maintains optimality. When mismatches reach zero, the solution becomes feasible and the optimum has been found; otherwise, the mismatch residuals are converted to load shedding and the algorithm provides a suboptimal but feasible solution. Therefore, the algorithm operates on infeasible but optimal points and moves towards feasibility. The proposed algorithm maximizes efficiency with two innovations: (a) quadratization that converts the nonlinear model to quadratic with excellent convergence properties and (b) minimization of model size by identifying active constraints, which are the only constraints included in the model. Finally sparsity technique is utilized that provide the best computational efficiency for large systems. This dissertation work demonstrates the proposed OPF algorithm using various systems up to three hundred buses and compares it with several well-known OPF software packages. The results show that the proposed algorithm converges fast and its runtime is competitive. Furthermore, the proposed method is extended to a three-phase OPF (TOPF) algorithm for unbalanced networks using the quadratized three-phase power system model. An example application of the TOPF is presented. Specifically, TOPF is utilized to address the problem of fault induced delayed voltage recovery (FIDVR) phenomena, which lead to unwanted relay operations, stalling of motors and load disruptions. This thesis presents a methodology that will optimally enhance the distribution system to mitigate/eliminate the onset of FIDVR. The time domain simulation method has been integrated with a TOPF model and a dynamic programming optimization algorithm to provide the optimal reinforcing strategy for the circuits.

Two important topics in the study of Quadratically Constrained Quadratic Programming (QCQP) are how to exactly solve a QCQP with few constraints in polynomial time and how to find an inexpensive and strong relaxation bound for a QCQP with many constraints. In this thesis, we first review some important results on QCQP, like the S-Procedure, and the strength of Lagrangian Relaxation and the semidefinite relaxation. Then we focus on two special classes of QCQP, whose objective and constraint functions take the form trace(X^TQX + 2C^T X) + β, and trace(X^TQX + XPX^T + 2C^T X)+ β respectively, where X is an n by r real matrix. For each class of problems, we proposed different semidefinite relaxation formulations and compared their strength. The theoretical results obtained in this thesis have found interesting applications, e.g., solving the Quadratic Assignment Problem.

This thesis evaluates decommissioning strategies for the International Space Station ISS. A permanent solution is attempted by employing energy efficient invariant manifolds that arise in the circular restricted three body problem CRTBP to transport the ISS from its low Earth orbit LEO to a lunar orbit. Although the invariant manifolds provide efficient transport, getting the the ISS onto the manifolds proves quite expensive, and the trajectories take too long to complete. Therefore a more practical, although temporary, solution consisting of an optimal re-boost maneuver with the European Space Agency's automated transfer vehicle ATV is proposed. The optimal re-boost trajectory is found using control parameterization and the sequential quadratic programming SQP algorithm. The model used for optimization takes into account the affects of atmospheric drag and gravity perturbations. The optimal re-boost maneuver produces a satellite lifetime of approximately ninety-five years using a two ATV strategy.

Two important topics in the study of Quadratically Constrained Quadratic Programming (QCQP) are how to exactly solve a QCQP with few constraints in polynomial time and how to find an inexpensive and strong relaxation bound for a QCQP with many constraints. In this thesis, we first review some important results on QCQP, like the S-Procedure, and the strength of Lagrangian Relaxation and the semidefinite relaxation. Then we focus on two special classes of QCQP, whose objective and constraint functions take the form trace(X^TQX + 2C^T X) + β, and trace(X^TQX + XPX^T + 2C^T X)+ β respectively, where X is an n by r real matrix. For each class of problems, we proposed different semidefinite relaxation formulations and compared their strength. The theoretical results obtained in this thesis have found interesting applications, e.g., solving the Quadratic Assignment Problem.

In statistics, regression analysis is a technique, used to understand and model the relationship between a dependent variable and one or more independent variables. Multiple Adaptive Regression Spline (MARS) is a form of regression analysis. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models non-linearities and interactions. MARS is very important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. In our study, we analyzed Generalized Partial Linear Models (GPLMs), which are particular semiparametric models. GPLMs separate input variables into two parts and additively integrates classical linear models with nonlinear model part. In order to smooth this nonparametric part, we use Conic Multiple Adaptive Regression Spline (CMARS), which is a modified form of MARS. MARS is very benefical for high dimensional problems and does not require any particular class of relationship between the regressor variables and outcome variable of interest. This technique offers a great advantage for fitting nonlinear multivariate functions. Also, the contribution of the basis functions can be estimated by MARS, so that both the additive and interaction effects of the regressors are allowed to determine the dependent variable. There are two steps in the MARS algorithm: the forward and backward stepwise algorithms. In the first step, the model is constructed by adding basis functions until a maximum level of complexity is reached. Conversely, in the second step, the backward stepwise algorithm reduces the complexity by throwing the least significant basis functions from the model. In this thesis, we suggest not using backward stepwise algorithm, instead, we employ a Penalized Residual Sum of Squares (PRSS). We construct PRSS for MARS as a Tikhonov Regularization Problem. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and alternative to the concept of the backward stepwise algorithm. Especially, we apply the elegant framework of Conic Quadratic Programming (CQP) an area of convex optimization that is very well-structured, hereby, resembling linear programming and, therefore, permitting the use of interior point methods. At the end of this study, we compare CQP with Tikhonov Regularization problem for two different data sets, which are with and without interaction effects. Moreover, by using two another data sets, we make a comparison between CMARS and two other classification methods which are Infinite Kernel Learning (IKL) and Tikhonov Regularization whose results are obtained from the thesis, which is on progress.

Thesis (M.S.)--Mississippi State University. Department of Aerospace Engineering. / Title from title screen. Includes bibliographical references.

kernlab is an extensible package for kernel-based machine learning methods in R. It takes advantage of R's new S4 object model and provides a framework for creating and using kernel-based algorithms. The package contains dot product primitives (kernels), implementations of support vector machines and the relevance vector machine, Gaussian processes, a ranking algorithm, kernel PCA, kernel CCA, and a spectral clustering algorithm. Moreover it provides a general purpose quadratic programming solver, and an incomplete Cholesky decomposition method. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

As a special case of the static state estimation problem, the load-flow problem is studied in this thesis. It is demonstrated that the non-linear load-flow formulation may be solved by real-coded genetic algorithms. Due to its global optimisation ability, the proposed method can be useful for off-line studies where multiple solutions are suspected. This thesis presents two methods for estimating the uncertainty interval in power system state estimation due to uncertainty in the measurements. The proposed formulations are based on a parametric approach which takes in account the meter inaccuracies. A nonlinear and a linear formulation are proposed to estimate the tightest possible upper and lower bounds on the states. The uncertainty analysis, in power system state estimation, is also extended to other physical quantities such as the network parameters. The uncertainty is then assumed to be present in both measurements and network parameters. To find the tightest possible upper and lower bounds of any state variable, the problem is solved by a Sequential Quadratic Programming (SQP) technique. A new robust estimator based on the concept of uncertainty in the measurements is developed here. This estimator is known as Maximum Constraints Satisfaction (MCS). Robustness and performance of the proposed estimator is analysed via simulation of simple regression examples, D.C. and A.C. power system models.

In modern aerospace engineering, the physics-based computational design method is becoming more important. However, high-fidelity models require longer computational time, so the advantage of efficiency is partially lost. This problem has been overcome with the development of the approximation management framework (AMF). In the AMF, objective and constraint functions of a low-fidelity model are scaled at a design point so that the scaled functions, referred to as gsurrogate functions, h match those of a high-fidelity model. Since scaling functions and the low-fidelity model constitutes surrogate functions, evaluating the surrogate functions is faster than evaluating the high-fidelity model. Therefore, in the optimization process of the AMF, the surrogate functions are used to obtain a new design point. However, the author found that 1) the AMF is very vulnerable when the computational analysis models have numerical noise, and that 2) the AMF terminates optimization prematurely when the optimization problems have constraints. In order to solve the first problem, automatic differentiation (AD) technique is applied. If derivatives are computed with the generated derivative code, they are analytical, and the computational time is independent of the number of design variables. However, if analysis models implement iterative computations such as computational fluid dynamics (CFD), computing derivatives through the AD requires a massive memory size. The author solved this deficiency by modifying the AD approach and developing a more efficient implementation with CFD. In order to solve the second problem, the governing equation of the trust region ratio is modified so that it can accept the violation of constraints within some tolerance. By accepting violations of constraints during the optimization process, the AMF can continue optimization without terminating immaturely and eventually find the true optimum design point. With these modifications, the AMF is referred to as gRobust AMF, h and it is applied to airfoil and wing designs using Euler CFD software. The proposed AD method computes derivatives more accurately and faster than the finite differentiation method, and the Robust AMF successfully optimizes shapes of the airfoil and the wing in a much shorter time than the sequential quadratic programming with only high-fidelity models.