**Prognostic Algorithms Design based on Predictive Bayesian Cramér-Rao Lower Bounds**

In the area of failure prognosis, system states typically correspond to critical variables whose future evolution in time might significantly affect the health condition of the process, thus yielding into a critical failure at a particular time instant typically referred to as the Time-of-Failure (ToF). Although prognosis frameworks based on Bayesian processors, such as particle filtering, have demonstrated their efficiency when trying to estimate the probability of failure in nonlinear, non-Gaussian systems undergoing uncertain operating profiles, it is still not clear how to measure the efficacy of the method. For this purpose, it is first necessary to establish adequate performance metrics, and the PHM community has not found a convincing theory that could help to provide adequate performance indicators yet. We have work on a rigorous mathematical definition of the prognostic problem, defining novel performance metrics based on Bayesian Cramér-Rao Lower Bounds for the variance of the predicted state probability density function, conditional to measurement data and model dynamics. We propose a step-by-step methodology to tune the parameters of prognostic algorithms that ensures that the precision guaranteed by those algorithms does not violate these fundamental bounds.

**Novel Performance Measures for Real-time Prognostics Algorithms and Uncertainty Characterization**

This line studies the development of novel prognostic methods that allow proper characterization of the uncertainty associated with the evolution in time of nonlinear dynamical systems. Methods assume state-space representation of the system. We work on a rigorous mathematical formulation for the expression that is used for the computation of the Time-ofFailure (ToF) probability mass function in the context of online monitoring schemes. Also, we focus on the development of methods for improved characterization of the tails of the ToF probability mass function via sequential propagation of sigma-points and the computation of Gaussian Mixture Models (GMMs).

**Contributions:**

- Acuña, D.; and Orchard, M., "Particle-Filtering-Based Failure Prognosis via Sigma-Points: Application to Lithium-Ion Battery State-of-Charge Monitoring", Mechanical Systems and Signal Processing, vol. 85. pp. 827-848, 2017.
- Ley, C. and Orchard, M., "Chi-squared smoothed adaptive particle-filtering based prognosis," Mechanical Systems and Signal Processing, vol. 82, pp. 148–165, 2017.
- Orchard, M.; Lacalle, M.; Olivares, B.; Silva, J.; Palma, R.; Estévez, P.; Severino, B.; Calderon-Muñoz, W.; and Cortés M., "Information-Theoretic Measures and Sequential Monte Carlo Methods for Detection of Regeneration Phenomena in the Degradation of Lithium-Ion Battery Cells," IEEE Transactions on Reliability, vol. 64, Issue 2, pp. 710-720, June 2015.

**Battery State of Health (SoH) and State of Charge (SoC) estimation and prognosis**

Battery Energy Storage Systems (BESS) are important for applications related to both microgrids and electric vehicles. If BESS are used as the main energy source, then it is required to include adequate procedures for the estimation of critical variables such as the State of Charge (SoC) and the State of Health (SoH) in the design of Battery Management Systems (BMS). Furthermore, in applications where batteries are exposed to high charge and discharge rates it is also desirable to estimate the State of Maximum Power Available (SoMPA). We study the implementation of particle-filtering-based prognostic frameworks that allows estimating the state-of-health (SOH) and predicting the remaining useful life (RUL) of energy storage devices, and more specifically lithium-ion batteries, while simultaneously detecting and isolating the effect of self-recharge phenomena within the life cycle model. The proposed scheme provides a characterization of capacity regeneration phenomena, which is validated through experimental data from an accelerated battery degradation test and a set of ad-hoc performance measures to quantify the precision and accuracy of the RUL estimates. In addition, we have developed a novel approach to the estimation of SoMPA in Lithium-Ion batteries. This method formulates an optimization problem for the battery power based on a non-linear dynamic model, where the resulting solutions are functions of the SoC. In the battery model, the polarization resistance is modeled using fuzzy rules that are function of both SoC and the discharge (charge) current. Particle filtering algorithms are used as an online estimation technique, mainly because these algorithms allow approximating the probability density functions of the SoC and SoMPA even in the case of non-Gaussian sources of uncertainty.

**Contributions:**

- Acuña, D.; and Orchard, M., "Particle-Filtering-Based Failure Prognosis via Sigma-Points: Application to Lithium-Ion Battery State-of-Charge Monitoring", Mechanical Systems and Signal Processing, vol. 85. pp. 827-848, 2017.
- Perez, A.; Moreno, R.; Moreira, R.; Orchard, M.; and Strbac, G., "Effect of Battery Degradation on Multi-Service Portfolios of Energy Storage", IEEE Transactions on Sustainable Energy, vol. 7, Issue 4, pp. 1718-1729, 2016.
- Reyes-Marambio, J.; Moser, F.; Gana, F.; Severino, B.; Calderón-Muñoz, W.; Palma-Behnke, R.; Estevez, P.; Orchard, M.; Cortés, M., "A fractal time thermal model for predicting the surface temperature of air-cooled cylindrical Li-ion cells based on experimental measurements," Journal of Power Sources, 161:349-363, 2016.

**Wind-power Generation Forecasting**

We address the problem of wind-power forecasting from the perspective of statistical signal processing. We are currently studying techniques for both the short and long term horizon prediction using tools such as Seasonal Auto Regressive Integrating Moving Average (SARIMA) models, Kalman filters and particle filters (for the short term) and Grey models (for the long term).

**Contributions:**

- Perez, A.; Orchard, M.; Silva, J.; and Cornejo, F., "Dynamic Vector Model Applied to Wind Speed Prognosis for Eolic Generation," Third European Conference of the Prognostics and Health Management Society 2016 - PHME16, July 5th-8th, 2016, Bilbao, Spain.

**Stock Market Forecasting**

In this project we address the problem of stochastic modeling of financial returns and their volatility. We define a variant on GARCH models called uGARCH (unobserver GARCH) that serves as a foundation for applying a particle filter to compute forecasts on a given financial index or stock. We also have developed novel online early detectors of high-volatility clusters based on uGARCH models, risk-sensitive particle-filtering-based estimators, and hypothesis testing procedures. The proposed detector utilizes Risk-Sensitive Particle Filters (RSPF) to generate an estimate of the volatility probability density function (PDF) that offers better resolution in the areas of the state-space that are associated with the incipient appearance of high-volatility clusters. This is achieved using the Generalized Pareto Distribution for the generation of particles. Risk-sensitive estimates are used by a detector that evaluates changes between prior and posterior probability densities via asymmetric hypothesis tests, allowing early detection of sudden volatility increments (typically associated with early stages of high-volatility clusters).

**Contributions:**

- Mundnich, K. and Orchard, M., "Early online detection of high volatility clusters using Particle Filters", Expert Systems with Applications, 54: 228–240, 2016.
- Mundnich, K.; Orchard, M.; Silva, J.; Parada, P., "Volatility Estimation of Financial Returns using Risk-Sensitive Particle Filters," Studies in Informatics and Control, Vol. 22, No. 3, pp. 297-306, September 2013.
- Tobar, F. and Orchard, M., "Study of Financial Systems Volatility Using Suboptimal Estimation Algorithms," Studies in Informatics and Control, vol. 21, Issue 1, pp. 59‑66, March 2012.