Multi-model system parameter estimation

In this paper, the set-membership state and parameter estimation problems are considered for linear time-varying systems. The noises in the system are unknown but bounded and the variation of parameters is considered as a bounded expansion. The entire estimator is an interactive estimation of parameters and states. In parameter estimation, the exact set of parameter estimates may be non-convex because of the uncertain states. A polytopic linear parameter varying (LPV) enclosure of the system regression expression is constructed as a convex relaxation. The result of parameter estimation is a set to include the true parameter of the model. Meanwhile, in the state estimation, a similar LPV enclosure of the state-space model is also used to bound the state matrix uncertainties. This LPV enclosure in the state estimation is narrowed by the parameter estimation, and then the set of state estimates is computed to include the true state. The constrained zonotope (CZ) is used to describe sets of interest and the necessary set operations are expanded to LPV mapping. Because of the highly adjustable accuracy of CZ, the proposed algorithm can make a better trade-off between computational accuracy and efficiency. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Power system state estimation is one of the most important functions of power system control centers. In recent years, the complexity of power system state estimation has significantly increased due to the growing number of distributed, including renewable energy sources, electric vehicles, the demand response technologies, and the increased risk of cyber-attacks. Under these conditions, state estimation methods, which consider information about the time-correlation of the power system states have a great potential. The correlation is described by a transition model. The well-known state estimation methods usually use one single model. However, in case of stochastic behavior of the load and generation, it is impossible to assert the adequacy of the chosen model over the entire observation interval. Therefore, the paper proposes a multi-model forecasting power system state estimation method, which has lower errors in comparison with a single-model assessment at the moments of the lowest accuracy of the latter. Multi-model parameter estimation is used based on three procedures of the single-model Kalman filtering estimation and various transition models based on autoregressive and vector autoregressive analyzes, as well as Holt's exponential smoothing. Uniting the single-model estimation has been carried out according to the criterion of the minimum of variance of the resulting estimate. An algorithm of multi-model power system state estimation has been developed. Its version has been analyzed using a three-model forecasting-aided estimation using linear transition models. The state of the IEEE 30-bus test power system has been assessed by the means of simulation modeling. The maximum accuracy increase of the multi-model estimation in comparison with the single model is set up. 26,1 % is for autoregressive analysis; 16,9 % is for vector regression analysis and 37,7 % is for Holt’s exponential smoothing. The proposed method of multi-model state estimation has higher robustness and accuracy at the moments of the lowest accuracy of single-model estimation. It is advisable to use the method to solve control tasks of power systems with rapidly changing dynamic modes.

This manuscript deals with parametric and state estimation problem of discrete-time nonlinear polynomials systems. In this respect, the aim is to design a Recursive State and Parametric Estimation (RSPE) algorithm, based on least square principle, polynomial adjustable models and Kalman filter theory. Our simulation results show the validity of the proposed algorithm.

A kernel functional representation of linear time-varying systems is employed in conjunction with B-spline functional approximation techniques to construct non-asymptotic state and parameter estimators for LTV systems. Total observability of the estimated system must be assumed. Practical identifiability conditions for parametric estimation are also stated. In the absence of output measurement noise the observer provides almost exact reconstruction of the system state and delivers high fidelity functional estimates of the time varying system parameters. It also shares the usual superior features of algebraic observers such as independence of the initial conditions of the system and good noise attenuation properties. Other advantages of the kernel and B-spline based identification of linear time-varying systems are elucidated.

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.

The literature abounds with the application of optimization methods for estimating model parameters in equation systems. The utility of these methods is frequently demonstrated on pathological examples using simulated data generated from a known model with a random error component and a known statistical distribution. Unfortunately, parameter estimation problems encountered in practice do not have this advantage. The true model is frequently not known. In fact, one is faced with choosing among various candidate models, all of which may be wrong. Moreover, the error structure is generally unknown and must be estimated from the data. Finally, a great deal of mathematical expertise is required to transform the model and select meaningful starting guesses before parameter estimation can be successful.In order to demonstrate the difficulties of parameter estimation in the industrial environment and the limitations of existing methods, a parameter estimation problem formulated by the Dow Chemical Company is presented and solved. This test problem consists of a stiff differential/algebraic (DAE) model that describes complex kinetics and requires the estimation of nine parameters from batch reactor data. Here the model was inadequate to describe the data, the error structure was not specified and the starting guesses led to a nontrivial optimization problem.The Dow parameter estimation problem was distributed in 1981 to 165 researchers as a followup to the 1980 FOCAPD conference. Of those researchers, eleven agreed to apply their methodologies and expertise to this problem. However, only five acceptable solutions were finally submitted. Here we present and compare these results. Each solution was obtained using different strategies. In most cases the form of the model was also changed to accommodate the algorithms used and to ease the solution procedure. Therefore, while this case study does not present a direct numerical comparison of algorithms, it does offer guidelines and insight towards the solution of difficult parameter estimation problems.

To develop an operational methodology for estimating soil moisture and crop biophysical parameters and to generate a crop cover map, backscattering signatures of vegetation canopies are investigated using multitemporal Radarsat synthetic aperture radar (SAR) data over a predominantly cotton-growing area in India during low to peak crop growth stage. A simple parameterization of the water-cloud model with volumetric soil moisture content (m/sub v/) and leaf area idex (LAI) is used to simulate the microwave backscattering coefficient (/spl sigma//sup 0/), as it is found to be a good candidate for operational purposes as demonstrated by several workers in past. The influence of crop height (H), LAI, and m/sub v/ on /spl sigma//sup 0/ is investigated during peak crop growth stage. A linear relationship between LAI and crop height is derived semiempirically, and a linear zone is chosen for analysis during the peak crop-growing stage. Estimation of average volume fraction of leaves (V~/sub l/) and attenuation factor (L) by two different approaches is discussed: 1) using linear relationship between LAI versus crop height and 2) from the water-cloud model parameter (/spl kappa/) estimation by iterative minimum least square error approach. It is observed that model-estimated parameters agree well with the measured values within an acceptable error limit. At lower soil moisture, m/sub v//spl cong/0.02(cm/sup 3//spl middot/cm/sup -3/), the dynamic range of /spl sigma//sup 0/ is found to be about +5 dB for 0-70 cm of crop height but monotonously decreases to null at a transition point, having m/sub v//spl ap/0.38(cm/sup 3//spl middot/cm/sup -3/). A positive correlation is found between backscattering coefficient and crop height till this transition point but shows a negative correlation beyond that, signifying the predominant attenuation by vegetation over soil. Differential moisture sensitivity (d/spl sigma//sup 0//dm/sub v/) of the backscattering coefficient decreases by half from 20.55 dB/(cm/sup 3//spl middot/cm/sup -3/) for dry and bare-field conditions to 10.68 dB/(cm/sup 3//spl middot/cm/sup -3/) for wet and crop-covered fields (m/sub v/=0.38cm/sup 3//spl middot/cm/sup -3/, H=70cm), whereas differential crop height sensitivity (d/spl sigma//sup 0//dH) varies from 0.22-0.03 dB/cm for bare-field conditions to crop-covered fields with crop height 70 cm. It is found that the percentage of relative error is smallest (2.27%) for LAI and attenuation factor estimation using the value of V~/sub l/, from LAI models, whereas it is 4.25% when estimating from the attenuation coefficient (/spl kappa/) from the model.

The problem of parametric estimation in single-phase transformers is addressed in this research from the point of view of metaheuristic optimization. The parameters of interest are the series resistance and reactance as well as the magnetization resistance and reactance. To obtain these parameters considering only the voltage and the currents measured in the terminals of the transformer, a nonlinear optimization model that deals with the minimization of the mean square error among the measured and calculated voltage and current variables is formulated. The nonlinear programming model is solved through the implementation of a simple but efficient metaheuristic optimization technique known as the black-hole optimizer. Numerical simulations demonstrate that the proposed optimization method allows for the reduction in the estimation error among the measured and calculated variables when compared with methods that are well established in the literature such as particle swarm optimization and genetic algorithms, among others. All the simulations were carried out in the MATLAB programming environment.

This paper considers the estimation problem of traffic state in freeway networks in light of the unknown input filtering (UIF) framework. The freeway traffic flow is modeled as a dynamic stochastic nonlinear system and is based on a recently developed speed-extended cell-transmission model of freeway traffic. Since the environmental conditions on a freeway may change over time, model parameters estimation is also considered. It is shown that the dual state and parameter estimation problem can be solved by applying the UIF to a nonlinear system with unknown inputs. Recently, a nonlinear version of the extended recursive three-step filter, named as the NERTSF, was employed to solve the problem. However, a numerical approximation method is used to calculate the model partial derivatives. To relax that restriction, in this paper a derivative-free versions of the NERTSF is further proposed to solve the addressed estimation problem of the freeway traffic flow.

Presently, the problem of parameter estimation is addressed for a more general nonlinear model. In this respect, the proposed nonlinear system is composed of the parallel connection of linear and nonlinear blocks. The interconnected structure of linear and nonlinear blocks systems leads to a highly nonlinear problem involving several unknown parameters and inaccessible internal signals. In this parameter estimation method, the linear and nonlinear block parameters are estimated in one stage by using simple a sine signal or multi-cosine wave.

We present a distributed observer for nonlinear systems, which can be used to estimate the state of a central plant with unknown parameters using a network of sensors that measure noisy linear combinations of the state. Our state estimator is based on the unscented Kalman filter and the dual Kalman filter method, which runs two Kalman filters in parallel for both state and parameter estimation. This allows us to improve estimates of the central plants parameters in order to improve state estimates, and vice versa. Furthermore, using results from stochastic stability analysis, we prove that under certain conditions the expected value of the proposed algorithm's estimation error is bounded. This result also shows us that as t goes to infinity, the error bound scales as a function of the largest eigenvalue of the sensor noise covariance matrix. Lastly, we demonstrate the ability of our state estimator to accurately track the state of nonlinear systems through simulations.

State and parameter estimators are obtained for systems described by nonlinear evolution equations. Linear infinite dimensional observability theory together with a variety of fixed point theorems can be employed to obtain a finite time observer. Moreover, a nonlinear asymptotic observer is produced using stability results. The problem of joint state and parameter estimation is converted to the state estimation case, via an augmented state, so that these observer results can be utilised. Examples and remarks on the generality of the results are given.

Data assimilation is predominantly used for state estimation, combining observational data with model predictions to produce an updated model state that most accurately approximates the true system state whilst keeping the model parameters fixed. This updated model state is then used to initiate the next model forecast. Even with perfect initial data, inaccurate model parameters will lead to the growth of prediction errors. To generate reliable forecasts, we need good estimates of both the current system state and the model parameters. This article presents research into data assimilation methods for morphodynamic model state and parameter estimation. First, we focus on state estimation and describe implementation of a three‐dimensional variational (3D‐Var) data assimilation scheme in a simple 2D morphodynamic model of Morecambe Bay, UK. The assimilation of observations of bathymetry derived from synthetic aperture radar (SAR) satellite imagery and a ship‐borne survey is shown to significantly improve the predictive capability of the model over a 2‐year run. Here, the model parameters are set by manual calibration; this is laborious and is found to produce different parameter values depending on the type and coverage of the validation dataset. The second part of this article considers the problem of model parameter estimation in more detail. We explain how, by employing the technique of state augmentation, it is possible to use data assimilation to estimate uncertain model parameters concurrently with the model state. This approach removes inefficiencies associated with manual calibration and enables more effective use of observational data. We outline the development of a novel hybrid sequential 3D‐Var data assimilation algorithm for joint state‐parameter estimation and demonstrate its efficacy using an idealised 1D sediment transport model. The results of this study are extremely positive and suggest that there is great potential for the use of data assimilation‐based state‐parameter estimation in coastal morphodynamic modelling. Copyright © 2012 Royal Meteorological Society

Guest Editorial: Situational awareness of integrated energy systems

Fast and robust parameter estimation for statistical partial volume models in brain MRI.