The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

System Identification I

Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

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Stochastic Processes And Filtering Theory

In a microgrid connected with both intermittent and dispatchable sources, intermittency caused by sources such as solar and wind power plants can be balanced by dispatching hydro power into the grid. Both intermittent generation and consumption are stochastic in nature, not known perfectly, and require future prediction. The stochastic generation and consumption will cause the grid frequency to drift away from a required range. To improve performance, operation should be optimized over some horizon, with the added problem that intermittent power varies randomly into the future. Optimal management of dynamic system over a future horizon with disturbances is often posed as a Model Predictive Control (MPC) problem. In this paper, we have employed an MPC scheme for generating a hydro-turbine valve signal for dispatching necessary hydro power to the intermittent grid and maintaining grid frequency. Parameter sensitivity analysis shows that grid frequency is mostly sensitive to the turbine valve signal. We have found that controller discretization time, grid frequency, and power injection into the grid are interrelated, and play an important role in maintaining the grid frequency within the thresholds. Results also indicate that the fluctuations in grid frequency are insignificant on the turbine valve position during power injection into the grid.

https://www.mdpi.com/1996-1073/14/4/874/pdf

Using MPC to Balance Intermittent Wind and Solar Power with Hydro Power in Microgrids

Systematic Analysis of Parameter Identifiability for Improved Fitting of a Biological Wastewater Model to Experimental Data

/pdf/systematic-analysis-of-parameter-identifiability-for-97ae2lr3j1.pdf

An adaptive control algorithm based upon on-line estimation of the - 180 degree phase shift frequency and the gain margin is investigated. The estimates are found by perturbing the system with a small signal and then correlating the response with this excitation. The control algorithm can be of any form as long as there is a functional relationship between the estimated parameters and the control algorithm parameters. To avoid an excessive perturbation signal, the square of the amplitude of this signal is automatically adjusted to a certain ratio of the variance of the error signal. A mathematical analysis of convergence is presented and the theory is confirmed by experimental results.

/pdf/an-adaptive-controller-based-upon-continuous-estimation-of-41quqelb0g.pdf

An Adaptive Controller Based upon Continuous Estimation of the Closed Loop Frequency Response

The ever-increasing penetration of intermittent energy sources introduces new demanding operating regimes for the bulk power generation in the power grid. During a worst-case power system disturbance scenario, the generator needs to operate beyond its limits to maintain stable operation. Therefore, a new online low-order thermal model of a hydrogenerator has been recently proposed, where the periodic extension of the long-forgotten capability diagram of the machine was in-depth investigated. An increased performance can be obtained if the total thermal capacity of the generator is exploited by applying optimal control theory in the Automatic Voltage Regulator (AVR). This paper proposes a Non-linear Model Predictive Controller (NMPC) combined with an Unscented Kalman Filter (UKF) with a modeling framework geared for use in a supervisory control structure for the conventional control system. The method provides maximum utilization of the machine's thermal capacity by providing the controller with an Enhanced Capability Diagram (ECD). Case studies on a single-machine system and a 16-bus multi-machine system were investigated. The results show that a satisfying controller action during different long-term voltage instability scenarios is realized.

/pdf/enhanced-power-capability-of-generator-units-for-increased-1nkbjcmn6s.pdf

Bernt Lie

Papers

Using MPC to Balance Intermittent Wind and Solar Power with Hydro Power in Microgrids

Systematic Analysis of Parameter Identifiability for Improved Fitting of a Biological Wastewater Model to Experimental Data

An Adaptive Controller Based upon Continuous Estimation of the Closed Loop Frequency Response

Enhanced Power Capability of Generator Units for Increased Operational Security Using NMPC

Development of a Dynamic Model of a Post Combustion CO2 Capture Process