Continuous and discrete-time Successive Backward Sweep (SBS) methods for solving nonlinear optimal control problems involving terminal and control constraints are proposed in this dissertation. They closely resemble the Neighboring Extremals and Differential Dynamic Programming algorithms, which are based on the successive solutions to a series of linear control problems with quadratic performance indices. The SBS methods are relatively insensitive to the initial guesses of the state and control histories, which are not required to satisfy the system dynamics. Hessian modifications are utilized, especially for non-convex problems, to avoid singularities during the backward integration of the gain equations. The SBS method requires the satisfaction of the Jacobi no-conjugate point condition and hence, produces optimal solutions. The standard implementation of the SBS method for continuous-time systems incurs terminal boundary condition errors due to an algorithmic singularity as well as numerical inaccuracies in the computation of the gain matrices. Alternatives for boundary error reduction are proposed, notably the aiming point and the switching between two forms of the sweep expansion formulae. Modification of the sweep formula expands the domain of convergence of the SBS method and allows for a rigorous testing for the existence of conjugate points. Numerical accuracy of the continuous-time formulation of the optimal control problem can be improved with the use of symplectic integrators, which generally are implicit schemes in time. A time-explicit group preserving method based on the Magnus series representation of the state transition is implemented in the SBS setting and is shown to outperform a non-symplectic integrator of the same order. Discrete-time formulations of the optimal control problem, directly accounting for a specific time-stepping method, lead to consistent systems of equations, whose solutions satisfy the boundary conditions of the discretized problem accurately. In this regard, the second-order, implicit mid-point averaging scheme, a symplectic integrator, is adapted for use with the SBS method. The performance of the mid-point averaging scheme is compared with other methods of equal and higher-order non-symplectic schemes to show its advantages. The SBS method is augmented with a homotopy- continuation procedure to isolate and regulate certain nonlinear effects for difficult problems, in order to extend its domain of convergence. The discrete-time SBS method is also extended to solve problems where the controls are approximated to be impulsive and to handle waypoint constraints as well. A variety of highly nonlinear optimal control problems involving orbit transfer, atmospheric reentry, and the restricted three-body problem are treated to demonstrate the performance of the methods developed in this dissertation.

Existing low-voltage networks may not accommodate high penetrations of low-carbon technologies. The topic of this thesis is unbalance, which if minimised can delay or avoid the constraining of these technologies or the replacing of still-useful network assets. Most of the discussion on unbalance, as seen in the standards and the literature, centres on the effects of voltage unbalance on consumer equipment. Its effects on the network are not equally reported. This thesis recognises fundamental differences between the consumer and network perspectives. It can inform distribution network operators on the interpretation of measurements taken on low-voltage networks and guide research on unbalance due to high penetrations of low-carbon technologies. Much of the work involved simulations of LV networks. Initially, existing 3 x 3 or 5 x 5 approaches to the forward-backward sweep method were thought suitable. After a review of these approaches however, there were doubts as to how accurately they accounted for the shared neutral-earth return path on which the out-of-balance current flows. This led to the derivation of a new 5 x 5 approach using only Kirchhoff s voltage (KVL) and current laws (KCL). Its results are validated thoroughly in the thesis. In addition to satisfying KVL and KCL, they match Matlab SimPowerSystems exactly and are in close agreement with measurements taken on a very unbalanced rural feeder. This thesis also investigates the mitigation of unbalance using the static balancer. This is a transformer with a single interconnected-star winding. It was used in 1930-1950s to correct unbalance. Contributions are made for its possible re-introduction as a retrofit option. They include a model for use in the forward-backward sweep method, validated by laboratory and field measurements, and the quantification of the static balancer s strengths and weaknesses as this can help identify when it should be used.

In recent years, photovoltaic (PV) power production have seen an increase and the PV power systems are often located in the distribution grids close to the consumers. Since the distributions grids rarely are designed for power production, investigation of its effects is needed. It is seen in this thesis that PV power production will cause voltages to rise, potentially to levels exceeding the limits that grid owners have to abide by. A model of a distribution grid is developed in MathWorks MATLAB. The model contains a transformer, cables, households, energy storage systems (ESS:s) and photovoltaic power systems. The system is simulated by implementing a numerical Forward Backward Sweep Method, solving for powers, currents and voltages in the grid. PV power systems are added in different configurations along with different configurations of ESS:s. The results are analysed, primarily concerning voltages and voltage limits. It is concluded that addition of PV power production in the distribution grid affects voltages, more or less depending on where in the grid the systems are placed and what peak power they have. It is also concluded that having energy storage systems in the grid, changing the power factor of the inverter for the PV systems or lowering the transformer secondary-side voltage can bring the voltages down. / På senare tid har det skett en ökning i antalet solcellsanläggningar som installeras i elnätet och dessa är ofta placerade i distributionsnäten nära hushållen. Eftersom distributionsnäten sällan är dimensionerade för produktion så behöver man utreda effekten av det. I det här arbetet visas det att solcellsproduktion kommer att öka spänningen i elnätet, potentiellt så mycket att de gränser elnätsägarna måste hålla nätet inom överstigs. En modell över lågspänningsnätet skapas i MathWorks MATLAB. Modellen innehåller transformator, kablar, hushåll, energilager och solcellsanläggningar. Systemet simuleras med hjälp av en numerisk Forward Backward Sweep-lösare som beräknar effekter, strömmar och spänningar i elnätet. Solcellanläggningarna placeras ut i elnätet i olika konfigurationer tillsammans med olika konfigurationer av energilager. Resultaten från simuleringarna analyseras främst med avseende på spänningen i elnätet utifrån dess gränser. De slutsatser som dras i arbetet är att solcellsproduktion kommer att påverka spänningen, mycket beroende på var i elnätet anläggningarna placeras och storleken hos dem. Det visas också att energilager, justering av effektfaktor hos solcellsanläggningarna eller en spänningssänkning på transformatorns lågspänningssida kan få ner spänningen i elnätet. / <p>LiTH-ISY-EX--19/5194--SE</p>