In this work, a new method of approximating the Maximum-likelihood estimate has been presented. The method consists of first using the Viterbi algorithm to estimate the log likelihood of the state, and then approximating that log likelihood to keep the computational complexity down. Various methods for approximating the log likelihood are introduced, most of these using linear regression and feature vectors. The methods were compared to a Kalman filter or Extended Kalman filter (depending on wether the system was linear or nonlinear) as well as a Particle filter modified to return a maximum likelihood estimate. Two systems were used for testing, one very simple linear system as well as a complex nonlinear system. Both of these were 1-dimensional. When applied to the simple system, the presented method outperformed both the Kalman filter and the Particle filter. While many approximation methods gave a good results the best one was using a cubic spline. For the more complex system, the method presented here could not outperform the particle filter. The most promising approximation method for this system was a Chebyshev approximation.

Various sensors distributed across different parts of the electric power grid provide measurements to the control center operator for situational awareness of the system. Voltage transformer, current transformer, relay and phasor measurement units (PMU) are types of sensors for power system monitoring. The utilities monitor the operating condition of their system by processing the measurements received from these various sensors using a state estimator. A state estimator refines these measurements, compensates for any lost data and provides a snapshot of the power system. The operator at the control center does further analysis using energy management system tools based on the most recent data and required state of the system. The electric power grid is vulnerable to blackouts caused by physical disturbances, human errors and external disasters. These disturbances can also cause loss of data, sensor failure or communication link failure. This research work focuses on comparing state estimation algorithms with loss of measurement data. The measurements are assumed to be lost as clustered and scattered data sets. Weighted Least Square (WLS), Least Absolute Value (LAV) and Iteratively Reweighted Least Squares (IRLS) implementation of Weighted Least Absolute Value (WLAV) algorithms are compared for state estimation with clustered and scattered loss of data. These algorithms are tested on a six bus, I 30 bus and 137 bus utility test cases. The test results indicate the best possible algorithm in several considered scenarios based on an error index. Additionally, phasor measurements data are included in two of the state estimation algorithms to study their ability to mitigate the loss of measurement data.

Phasor Measurement Units (PMUs) are widely acknowledged as one of the most promising developments in the field of real-time monitoring of power systems. By aligning the time stamps of voltage and current phasor measurements that are consistent with Coordinated Universal Time (UTC), a coherent picture of the power system state can be achieved through either direct measurements or simple linear calculations. With the growing number of PMUs planned for installation in the near future, both utilities and research institutions are looking for the best solutions to the placement of units as well as to the applications that make the most of phasor measurements. This dissertation explores a method for optimal PMU placement as well as two applications of synchronized phasor measurements in state estimation. The pre-processing PMU placement method prepares the system data for placement optimization and reduces the size of the optimization problem. It is adaptive to most of the optimal placement methods and can save a large amount of computational effort. Depth of un-observability is one of the criteria to allow the most benefit out of a staged placement of the units. PMUs installed in the system provide synchronized phasor measurements that are highly beneficial to power system state estimations. Two related applications are proposed in the dissertation. First, a post-processing inclusion of phasor measurements in state estimators is introduced. This method avoids the revision of the existing estimators and is able to realize similar results as mixing phasor data with traditional SCADA with a linear afterwards step. The second application is a method to calibrate instrument transformers remotely using phasor measurements. Several scans of phasor measurements are used to accomplish estimating system states in conjunction with complex instrument transformer correction factors. Numerical simulation results are provided for evaluation of the calibration performance with respect to the number of scans and load conditions. Conducting theoretical and numerical analysis, the methods and algorithms developed in this dissertation are aimed to strategically place PMUs and to incorporate phasor measurements into state estimators effectively and extensively for better system state monitoring. Simulation results show that the proposed placement method facilitates approaching the exact optimal placement while keep the computational effort low. Simulation also shows that the use of phasor measurement with the proposed instrument transformer correction factors and proposed state estimation enhancement largely improves the quality of state estimations. / Ph. D.

Given an increasing tendency towards distributed generation and alternative energy sources, the power grid must be more carefully monitored in order to ensure stability. Phasor Measurement Units (PMUs) provide very good observation of a small area of a network, but their relatively high cost prevents them from being deployed at every point. Therefore, to monitor an entire network, State Estimation is still required. By combining these two techniques, the accuracy and speed of power network monitoring can be improved. This thesis presents a method for achieving this goal from both hardware and computational perspectives. Practical considerations for PMU placement are discussed, such as instrument transformer calibration, and an algorithm is developed to apply this technique to any power system. The resulting method is termed reconfigurable monitoring - computationally isolated areas which may be grouped as necessary to allow for flexibility in power system monitoring. / Master of Science

Wind observations that are critical for understanding meteorological processes occurring inside of the Earth's atmospheric boundary layer (ABL) are sparse due to limitations of conventional atmospheric sensors. In this dissertation, dynamic systems and estimation theory are combined with experimental methods to exploit the flight envelope of multirotor UAS for wind sensing. The parameters of three quadrotor motion models, consisting of a kinematic particle, a dynamic particle, and a dynamic rigid body models are developed to measure wind velocity in hovering flight. Wind tunnel and steady level flight tests are used to characterize kinematic and dynamic particle models. System identification stepwise regression and output error algorithms are used to determine the model structure and parameter estimates of rigid body models. The comparison of all three models demonstrates the rigid body model to have higher performance resolving slow-varying winds based on a frequency response analysis and field experiments conducted next to a 3-D sonic anemometer. The dissertation also presents an extension of the rigid body wind estimation framework to profile the horizontal components of wind velocity in vertical steady ascending flight. The extension employed system identification to characterize five rigid body models for steady-ascending flight speeds increasing from 0 to 2 m/s in intervals of 0.5~m/s. State observers for wind profiling were synthesized using all five rigid body models. Performance assessments employing wind observations from in situ and remote sensors demonstrated model-based wind profiling results to be be in close agreement with ground-truth wind observations. Finally, the rigid body wind sensing framework developed in this dissertations for multirotor UAS is employed to support science objectives for the Advanced Lagrangian Predictions for Hazards Assessment Project. Quadrotor wind measurements sampled at 10 m above sea level were used to characterize the leeway of a person in water for search and rescue scenarios. Leeway values determined from quadrotor wind measurements were found to be in close to leeway parameters previous published in the literature. This results demonstrates the utility of model-based wind sensing for multirotor UAS for providing wind velocity observations in complex environments where conventional wind observations are not readily available. / Doctor of Philosophy / Wind observations that are critical for understanding meteorological processes occurring inside of the Earth's atmospheric boundary layer (ABL) are sparse due to limitations of conventional atmospheric sensors. In this dissertation, dynamic systems and estimation theory are combined with experimental methods to exploit the flight envelope of multirotor UAS for wind sensing. The parameters of three quadrotor motion models, consisting of a kinematic particle model, a dynamic particle model, and a dynamic rigid body model, are characterized to measure wind velocity in hovering flight. Parameter characterizations are realized using data from wind tunnel, steady level flight tests and system identification experiments. Model-based wind estimations algorithms are developed using the kinematic particle model directly and by synthesizing state observers for the dynamic particle and rigid body models separately. For comparison purposes, the frequency response characteristic of the dynamic particle and rigid body models is examined to determine the range of wind fluctuations that each model can resolve. Performance comparisons demonstrate that the rigid body model to resolve higher wind fluctuations and yield more accurate wind estimates. The dissertation extends the rigid body wind estimation algorithm to estimate wind velocity profiles of the horizontal wind vector. The rigid body wind estimation algorithms is used to answer science questions about about the drift of a person in water.

When the primary measurement sensor is passive in nature--by which we mean that it does not directly measure range or range rate--there are well-documented challenges for target state estimation. Most estimation schemes rely on variations of the Extended Kalman Filter (EKF), which, in certain situations, suffer from divergence and/or covariance collapse. For this and other reasons, we believe that the Kalman filter is fundamentally ill-suited to the problems that are inherent in target state estimation using passive sensors. As an alternative, we propose a bounded-error (or set-membership) approach to the target state estimation problem. Such estimators are nearly as old as the Kalman filter, but have enjoyed much less attention. In this study we develop a practical estimator that bounds the target states, and apply it to the two-dimensional case of a submarine tracking a surface vessel, which is commonly referred to as Target Motion Analysis (TMA). The estimator is robust in the sense that the true target state does not escape the determined bounds; and the estimator is not unduly pessimistic in the sense that the bounds are not wider than the situation dictates. The estimator is--as is the problem itself--nonlinear and geometric in nature. In part, the simplicity of the estimator is maintained by using redundant states to parameterize the target's velocity. These redundant states also simplify the incorporation of other measurements that are frequently available to the system. The estimator's performance is assessed in a series of simulations and the results are analyzed. Extensions of the algorithm are considered.

The continuous growth in power system electric grid by adding new substations lead to construct many new transmission lines, transformers, control devices, and circuit breakers to connect the capacity (generators) to the demand (loads). These components will have a very heavy influence on the performance of the electric grid. The renewable technical solutions for these issues can be found by robust algorithms which can give us a full picture of the current state of the electrical network by monitoring the behavior of phase and voltage magnitude. In this thesis, the major idea is to implement several algorithms including weighted least square, extend kalman filter, and interior point method in three different electrical networks including IEEE 14, 30, and 118 to compare the performance of these algorithms which is represented by the behavior of phases and magnitude voltages as well as minimize the residual of the balance load flow real time measurements to distinguish which one is more robust. Also to have a particular understanding of the comparison between unconstraint and constraint algorithms.

Given the ability of the Phasor Measurement Unit (PMU) to directly measure the system state and the increasing implementation of PMUs across the electric power industry, a natural expansion of state estimation techniques would be one that employed the exclusive use of PMU data. Dominion Virginia Power and the Department of Energy (DOE) are sponsoring a research project which aims to implement a three phase linear tracking state estimator on Dominion's 500kV network that would use only PMU measurements to compute the system state. This thesis represents a portion of the work completed during the initial phase of the research project. This includes the initial development and testing of two applications: the three phase linear state estimator and the topology processor. Also presented is a brief history of state estimation and PMUs, traditional state estimation techniques and techniques with mixed phasor data, a development of the linear state estimation algorithms and a discussion of the future work associate with this research project. / Master of Science

The Defence, Peace, Safety and Security (DPSS) competency area within the Council for Scientific and Industrial Research (CSIR) has identified the need for the development of a robot that can operate in almost any land-based environment. Legged robots, especially hexapod (six-legged) robots present a wide variety of advantages that can be utilised in this environment and is identified as a feasible solution. The biggest advantage and main reason for the development of legged robots over mobile (wheeled) robots, is their ability to navigate in uneven, unstructured terrain. However, due to the complicated control algorithms needed by a legged robot, most literature only focus on navigation in even or relatively even terrains. This is seen as the main limitation with regards to the development of legged robot applications. For navigation in unstructured terrain, postural controllers of legged robots need fast and precise knowledge about the state of the robot they are regulating. The speed and accuracy of the state estimation of a legged robot is therefore very important. Even though state estimation for mobile robots has been studied thoroughly, limited research is available on state estimation with regards to legged robots. Compared to mobile robots, locomotion of legged robots make use of intermitted ground contacts. Therefore, stability is a main concern when navigating in unstructured terrain. In order to control the stability of a legged robot, six degrees of freedom information is needed about the base of the robot platform. This information needs to be estimated using measurements from the robot’s sensory system. A sensory system of a robot usually consist of multiple sensory devices on board of the robot. However, legged robots have limited payload capacities and therefore the amount of sensory devices on a legged robot platform should be kept to a minimum. Furthermore, exteroceptive sensory devices commonly used in state estimation, such as a GPS or cameras, are not suitable when navigating in unstructured and unknown terrain. The control and localisation of a legged robot should therefore only depend on proprioceptive sensors. The need for the development of a reliable state estimation framework (that only relies on proprioceptive information) for a low-cost, commonly available hexapod robot is identified. This will accelerate the process for control algorithm development. In this study this need is addressed. Common proprioceptive sensors are integrated on a commercial low-cost hexapod robot to develop the robot platform used in this study. A state estimation framework for legged robots is used to develop a state estimation methodology for the hexapod platform. A kinematic model is also derived and verified for the platform, and measurement models are derived to address possible errors and noise in sensor measurements. The state estimation methodology makes use of an Extended Kalman filter to fuse the robots kinematics with measurements from an IMU. The needed state estimation equations are also derived and implemented in Matlab®. The state estimation methodology developed is then tested with multiple experiments using the robot platform. In these experiments the robot platform captures the sensory data with a data acquisition method developed while it is being tracked with a Vicon motion capturing system. The sensor data is then used as an input to the state estimation equations in Matlab® and the results are compared to the ground-truth measurement outputs of the Vicon system. The results of these experiments show very accurate estimation of the robot and therefore validate the state estimation methodology and this study. / MIng (Computer and Electronic Engineering), North-West University, Potchefstroom Campus, 2015

The Defence, Peace, Safety and Security (DPSS) competency area within the Council for Scientific and Industrial Research (CSIR) has identified the need for the development of a robot that can operate in almost any land-based environment. Legged robots, especially hexapod (six-legged) robots present a wide variety of advantages that can be utilised in this environment and is identified as a feasible solution. The biggest advantage and main reason for the development of legged robots over mobile (wheeled) robots, is their ability to navigate in uneven, unstructured terrain. However, due to the complicated control algorithms needed by a legged robot, most literature only focus on navigation in even or relatively even terrains. This is seen as the main limitation with regards to the development of legged robot applications. For navigation in unstructured terrain, postural controllers of legged robots need fast and precise knowledge about the state of the robot they are regulating. The speed and accuracy of the state estimation of a legged robot is therefore very important. Even though state estimation for mobile robots has been studied thoroughly, limited research is available on state estimation with regards to legged robots. Compared to mobile robots, locomotion of legged robots make use of intermitted ground contacts. Therefore, stability is a main concern when navigating in unstructured terrain. In order to control the stability of a legged robot, six degrees of freedom information is needed about the base of the robot platform. This information needs to be estimated using measurements from the robot’s sensory system. A sensory system of a robot usually consist of multiple sensory devices on board of the robot. However, legged robots have limited payload capacities and therefore the amount of sensory devices on a legged robot platform should be kept to a minimum. Furthermore, exteroceptive sensory devices commonly used in state estimation, such as a GPS or cameras, are not suitable when navigating in unstructured and unknown terrain. The control and localisation of a legged robot should therefore only depend on proprioceptive sensors. The need for the development of a reliable state estimation framework (that only relies on proprioceptive information) for a low-cost, commonly available hexapod robot is identified. This will accelerate the process for control algorithm development. In this study this need is addressed. Common proprioceptive sensors are integrated on a commercial low-cost hexapod robot to develop the robot platform used in this study. A state estimation framework for legged robots is used to develop a state estimation methodology for the hexapod platform. A kinematic model is also derived and verified for the platform, and measurement models are derived to address possible errors and noise in sensor measurements. The state estimation methodology makes use of an Extended Kalman filter to fuse the robots kinematics with measurements from an IMU. The needed state estimation equations are also derived and implemented in Matlab®. The state estimation methodology developed is then tested with multiple experiments using the robot platform. In these experiments the robot platform captures the sensory data with a data acquisition method developed while it is being tracked with a Vicon motion capturing system. The sensor data is then used as an input to the state estimation equations in Matlab® and the results are compared to the ground-truth measurement outputs of the Vicon system. The results of these experiments show very accurate estimation of the robot and therefore validate the state estimation methodology and this study. / MIng (Computer and Electronic Engineering), North-West University, Potchefstroom Campus, 2015