C Can Bikçora

Lens Heating Induced Aberration Prediction via Nonlinear Kalman Filters

Lens heating induced aberrations rank among the dominant causes of image deteriorations in photolithography. In order to accurately counteract them via the available manipulators within the projection lens, it is crucial to employ a predictive model that is identified with relatively small errors. In this paper, parameters of a phenomenological model are recursively updated with respect to the measurements taken at the end of a wafer and are subsequently utilized in aberration predictions for the dies of the next wafer. To serve this purpose, two suboptimal Bayesian strategies, namely, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are applied to the nonlinear system at hand. In addition, the classical Kalman filter is tested on an approximate linear model. Filter performances are evaluated using both synthetic and experimental data and compared with respect to the posterior Cramér-Rao lower bound. When synthetic measurements are in use, the UKF moderately outperforms the EKF. Moreover, they both perform significantly better than the classical Kalman filter. However, due to model imperfections, these gains decrease and may even vanish when real measurements are processed. If the computational costs are also considered, then the EKF becomes more preferable over the other options.

Thermal Deformation Prediction in Reticles for Extreme Ultraviolet Lithography Based on a Measurement-Dependent Low-Order Model

In extreme ultraviolet lithography, imaging errors due to thermal deformation of reticles are becoming progressively intolerable as the source power increases. Despite this trend, such errors can be mitigated by adjusting the wafer and reticle stages based on a set of predicted deformation-induced displacements. Since this control scheme operates online, an accurate low-order model is necessary. However, finite element modeling of the reticle and its adjacent components leads to a large-scale thermo-mechanical model that should be simplified. First, parameters of the models initial thermal condition are reduced to only a few from which numerous initial conditions can be accurately reconstructed. This entails placement of temperature sensors at the corresponding locations, and for this purpose, the discrete empirical interpolation method (DEIM) is utilized. Then, linear and nonlinear model reductions are performed via the proper orthogonal decomposition method and DEIM, respectively. The resultant model is employed in the Kalman filter to estimate the parameters of the reticles temperature-dependent coefficient of thermal expansion from several displacement measurements and to subsequently predict the displacements that are used for control. By processing the outputs from the simulated large-scale model, this filter is shown to perform successfully, even in the presence of an unexpected initial condition.

Reduced-order modeling of thermally induced deformations on reticles for extreme ultraviolet lithography

Ranking already among the dominant causes of imaging errors in photolithography, deformation of reticles due to inevitable heating is becoming progressively more crucial in extreme ultraviolet (EUV) lithography as the source power continually increases, leading to higher levels of absorption of the EUV light by reticles. In order to mitigate its impact on exposed layers, accurate predictions to be the inputs of a control scheme are essential. To serve this purpose, a large-scale thermo-mechanical model in partially linear state-space form is derived by using the finite element method (FEM). The temperature-dependent coefficient of thermal expansion of materials produces the only nonlinearity in the model that is present in the static output equations. Since only low-order models are feasible for real-time use, this model is undergone several model reduction techniques to arrive at the best compact model with respect to its prediction performance, compaction rate, and easiness of computation. Treating the simulation outputs from a FEM software as the benchmark, the proper orthogonal decomposition approach combined with the discrete empirical interpolation method is selected as the most suitable route for the studied application.

Semiparametric density forecasting of electricity load for smart charging of electric vehicles

To be utilized in the smart charging of plug-in electric vehicles, this paper proposes semiparametric conditional mean and variance models for the daily density forecasting of the electricity load. The mean is modeled by means of an autoregressive moving average model with exogenous inputs (ARMAX), whereas several options for the variance evolution are investigated, starting with modeling the variance as a power of the conditional mean, then as a piecewise constant function, and finally as a generalized autoregressive conditional heteroskedasticity (GARCH) model. Due to the possible non-Gaussianity of the distribution of the stochastic components, a quasi-maximum likelihood estimation (QMLE) with the Pearson type IV (P-IV) distribution is also considered, apart from the estimation with the Gaussianity assumption. Moreover, the daily density forecasts are generated in a non-parametric manner by propagating samples from the stochastic components iteratively through the available model. These strategies, involving different options for variance modeling and estimation, are compared in terms of their forecast performances on two representative phase currents from the low voltage cables of medium-to-low voltage transformers in the Netherlands. The results indicate that the QMLE with the Gaussianity assumption performs better due to the additional complexity of the P-IV distribution on estimation. Concerning the variance models, the piecewise constant and GARCH models are more preferable when processing phase currents exhibiting only daily seasonality, and the model as a power of the conditional mean outperforms the others if both daily and weekly seasonality and hence more complexity is present.

A volumetric approach to biased estimation: Demonstration on shrinkage estimators

This work proposes a new approach, named as the volumetric design (VD), of developing biased estimators of deterministic parameters that are known in advance to belong to a compact subset in the parameter space. For analytical tractability, this approach is demonstrated on the choice of the shrinkage parameter of an estimator that scales the celebrated minimum variance unbiased estimator (MVUE) towards zero, where a spherical set is taken as the a priori knowledge on the parameters and the mean-squared error is adopted as the performance measure. Similar to the existing methods of the minimax (MX) and the deepest minimum criterion (DMC) estimators, the VD estimator also belongs to the class of admissible estimators that dominate the MVUE on the provided parameter (spherical) set. However, as its fundamental difference, it corresponds to the estimator that has the largest total relative volume on which it dominates the other estimators in this class, thereby achieving the best volumetric robustness in this manner.

Parameter-oriented reduction strategies for a thermo-mechanical model of extreme ultraviolet reticles

With the demands for shrinkage in feature sizes becoming more than ever, the extreme ultraviolet (EUV) lithography is the route pursued by the industrys leading authorities. Leaving the strive for achieving a high source power aside, we hereby consider one of the most prominent disrupters of imaging quality: heating induced deformation of EUV reticles. To diminish its detrimental effect, a model-based prediction scheme is ideally used for steering certain actuators in an EUV tool. To enable such a solution in practice, computationally fast models are required. Along this direction, we use the POD and DEIM reduction methods on finite-element-based thermal and mechanical equations that relate to a recent geometry where cooling channels are placed nearby the reticle. Several sensor measurements approximating the initial thermal state of this model are treated as parameters, and two parameter-oriented reduction methods are designed which we refer to as local reduction (LR) and decomposed reduction (DR), respectively. The former builds local simplified models based on clustering of the parameters, whereas the latter expands the terms in the model so that the reduction is independent of the parameters. The results indicate that only two sensors are sufficient for an accurate characterization of the initial condition. Furthermore, among the evaluated methods, similar accuracies were observed for various tested scenarios, and therefore, it is concluded that the LR is more suitable for this application due to its less intricate structure.

Prediction of availability and charging rate at charging stations for electric vehicles

To enable better smart charging solutions, this paper investigates the day-ahead probabilistic forecasting of the availability and the charging rate at charging stations for plug-in electric vehicles. Generalized linear models with logistic link functions are at the core of both forecast scenarios. Moreover, the availability forecast at a charging point is simply a binomial problem, whereas the charging rate forecast is handled via an ordered logistic model after categorizing the feasible range of values. These two scenarios are evaluated on real data collected from two representatives of the most occupied charging points in the Netherlands, with the focus of the analysis kept at the selection of essential regressors. Based on the ranked probability scores associated with the day-ahead forecasts generated for the last nine months of 2015, it is concluded that the usefulness of predictive models depends highly on the charging station. When contributing substantially to performance, such models possess a simple structure with a few basic lagged and indicator variables.

Probabilistic short-term load forecasting with conditional mean-variance and quantile regression models

For the day-ahead density forecasting of electricity load, this paper proposes the combination of the autoregressive moving average (ARMA) model and the generalized autoregressive conditional heteroskedasticity (GARCH) model, with both of them admitting exogenous inputs. This composite structure on the conditional mean and variance is referred to as the ARMAX-GARCHX model. As an alternative to its estimation by means of log-likelihood maximization, approaches based on iterative least-squares (ILS) and nonlinear least-squares (NLS) are considered. Apart from the ARMAX-GARCHX model, quantile regression models (QRMs) are also tested in forecasting where a wide range of quantiles are separately modeled to approximate a density. Phase currents of several low voltage transformer cables from the Netherlands are forecasted to compare the performances, and as the probabilistic evaluation criterion, the continuous ranked probability score is used. As an outline of the results, the ARMAX-GARCHX model outperformed QRMs and among its estimation techniques, the likelihood-based approach had the best performance, though the differences in the errors are often minor. Thus, owing to its computational simplicity, the ILS solution can be a valuable option when processing large batches of data in practice.

A min-min approach to non-blind and blind parameter estimation under admissibility and domination constraints

In this work, we develop min-min type of biased estimators for the deterministic parameter vector in a linear regression model under the condition that an ellipsoidal prior knowledge is either available or is inferred from the measurements, leading to the non-blind and blind designs, respectively. As the design specifications for the former, from which the latter is developed, domination over the least-squares (LS) estimator with respect to all weighted mean-squared error (MSE) measures and the corresponding admissibility condition are ensured. The blind counterpart of this solution is then built based on an ellipsoidal set that is estimated from the LS approach. This nonlinear estimator admits a closed-form solution and is proved to outperform the LS estimator under a wide range of conditions. Numerical comparisons of the developed non-blind solution, referred to as the constrained min-min (CMM) estimator, to alternatives demonstrate the superior weighted MSE performance of our solution when the true parameters are not nearby the boundary of the ellipsoid. On the other hand, as the case with all blind approaches, the blind CMM (BCMM) estimator is also more likely to have a major improvement over the LS estimator in problems of relatively high dimension. When compared to existing blind approaches, the BCMM estimator is more preferable when various weighted MSE measures are taken into account in the performance evaluation.