International Journal of Photoenergy

Volume 2017, Article ID 1825931, 8 pages

https://doi.org/10.1155/2017/1825931

## The PV Corrosion Fault Prognosis Based on Ensemble Kalman Filter

^{1}Université Internationale de Rabat, Technopolis Rabat-Shore, 11 100 Sala Al Jadida, Morocco^{2}Aix-Marseille University, Aix-en-Provence, France

Correspondence should be addressed to Radouane Ouladsine; am.ca.riu@enisdaluo.enauodar

Received 10 March 2017; Revised 2 June 2017; Accepted 17 August 2017; Published 31 October 2017

Academic Editor: Mark van Der Auweraer

Copyright © 2017 Radouane Ouladsine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The degradation of photovoltaic (PV) modules remains a major concern on the control and the development of the photovoltaic field, particularly, in regions with difficult climatic conditions. The main degradation modes of the PV modules are corrosion, discoloration, glass breaks, and cracks of cells. However, corrosion and discoloration remain the predominant degradation modes that still require further investigations. In this paper, a model-based PV corrosion prognostic approach, based on an ensemble Kalman filter (EnKF), is introduced to identify the PV corrosion parameters and then estimate the remaining useful life (). Simulations have been conducted using measured data set, and results are reported to show the efficiency of the proposed approach.

#### 1. Introduction

In order to maximize the system availability and reduce the maintenance cost, the concept of fault prognosis has been proposed to estimate the damage propagation and then predict the fault appearance [1, 2]. More precisely, several approaches have been proposed to tackle this issue and could be categorized into three main families [3, 4]. The first category uses prognosis-based models. These models consider the damage as a continuous variable in which the evolution is defined by a deterministic or stochastic law (see [3, 5]). The second category of models uses measured data without requiring an equipment behavior (e.g., see [6, 7]). The third category is experience-based prognostic and requires, essentially, expert experience together with rigorous stochastic and probabilistic modelling.

This paper focuses on the first category that uses prognosis-based models that attracted the intention of several authors (see, e.g., [8–11]) and can be further categorized into two types. The first one concerns deterministic models (see, e.g., [9, 12, 13]). In this case, the techniques used to estimate degradation are based either on observers or on the Interacting Multiple Model. For example, the conventional observer approach that can be envisaged for studying determinist systems prognosis is highlighted in [14]. The second type is dedicated to the stochastic models (see, e.g., [15, 16]) and Bayesian filters [17, 18]. For stochastic models, which attract our attention, the prediction techniques can be classified according to their natures or uncertainties. For example, when the models are linear with Gaussian uncertainties, techniques based on the ordinary Kalman filter can be used, but, when models are nonlinear, the filtering may be performed with an extended Kalman filter [18, 19]. On the other hand, for systems with non-Gaussian uncertainty, classical filtering techniques are not adapted and present many convergence difficulties. In this case, several types of filters based on the Monte Carlo method could be used (see, e.g., [15, 16]).

This paper puts more emphasis on PV corrosion propagation prognostic and investigates further Bayesian method using EnKF. Previous studies have shown that this filter is more suitable for systems with complex phenomenon and have been applied in many fields [20]. For example, it has been used to predict the flow of fluids in porous environment [21] or to predict the weather [22, 23]. In these two cases, the models that describe the systems’ evolution are either highly nonlinear or having important dimensions.

It is worth noting that the performance study between the EnKF and other filters, mainly the PF, has been performed by many authors in different fields especially for nonlinear stochastic filtering problems (e.g., [24, 25]). For instance, the results of these studies show that the PF is computationally expensive and needs big sampling size to converge, especially, for systems with high dimensions. However, the EnKF gives more robust and promising results even if the sampling size is too small. For example, in the assumption of Gaussian uncertainties and nonlinear model (e.g., [25]), similar to our PV corrosion study, the prediction is better with EnKF. Indeed, the covariance matrix estimation is based on limited data assimilation component of ensemble forecasting. Besides, from this limited sampling size, the time of the algorithm convergence is still fast and consequently good for the purpose of prognosis. The obtained results from our numerical study, presented in Section 3, confirm also that EnKF outperforms the PF for nonlinear system with Gaussian uncertainties.

Throughout this paper, the general mathematical model that describes the evolution of the PV corrosion can be written as follows: where is the degradation variable. The function and are smooth functions. is the input vector of the system, is the output vector, and is an unknown parameter defining the damage speed. and denote the modelling and measurement uncertainties where and represent the covariance matrices.

Generally speaking, this mathematical model is used to describe the nominal system operation. Besides, the estimations consist in analyzing the difference between the sensed values and model outputs (i.e., residual). In our case, the state evolution is modelled and combined with a hidden parameter which describes the damage speed, random noises, and observed measurements. In this case, the residual value is used to eliminate noise and then to restore the actual signal and unknown parameters. To summarize, the main contributions of this paper are the following: (i)Introduce the PV corrosion parametric state model(ii)Estimate the unknown parameter using a Bayesian filter(iii)Analyze the damage propagation instead of estimating the

This reminder of this paper is organized as follows. Section 2 describes the prognostic methodology and presents the filtering technique. In Section 3, the prognostic and numerical results are given. The conclusions and future work are presented in Section 4.

#### 2. Problem Statement and Prognosis Methodology

##### 2.1. Problem Statement

In this work, it is assumed that the structure of is polynomial (see e.g., [8, 26]) and depends on an unknown parameter ([27]). In general, the corrosion of photovoltaic module is assessed by measuring its power loss during its lifetime compared to its initial values. Currently, there are a few corrosion PV models in the literature and further investigations are still required. For example, in [28], the authors propose a degradation model of the PV model given by where and are the unknown parameters of the degradation model. In our work, we further investigate this model using prognostic-based Bayesian filter. The measurements are assumed to be available during a finite time horizon where is the sampling period of the time horizon and .

The aim of the prognosis strategy is to estimate the parameter and analyze the degradation trajectories in order to estimate the RUL. It is worth noting that this strategy was originally used for continuous deterministic model based on observers’ design (see [14]). Despite the relevance of the results obtained in prognostic, uncertainty was not taken into consideration. In our work, (2) is adapted as follows by taking into account the following inherent uncertainties:

In order to overcome nonlinearity and uncertainties, we propose the EnKF filtering technique. Despite its similarity with particle filters, the EnKF combines the Monte Carlo method and the Kalman filter technique in order to estimate the parameters and to compute their covariance matrix even if the models are highly nonlinear [20]. However, this type of filters is limited to the estimation of moments of order 1 and order 2. Therefore, in this study, we deal only with Gaussian uncertainties.

This EnKF reconstruction consists on minimizing the error between the model (2) and the measurement set obtained during a time horizon Then, the corrosion trajectory analysis can be performed for the prognosis and the PV estimation. Furthermore, the parameter estimation is carried out in a recurrent way, that is, at each time . Indeed, when a new measurement is available, can be adjusted by the filtering techniques. Thus, the parameter will be modelled by a variable which changes over time. More precisely, and because of the inherent uncertainties, the parameter’s dynamic is supposed to be given by a forced random path as follows: where is the additive noise and is the matrix of covariance associated to the uncertain parameter. Finally, taking into account (4) and the damage behavior (3), the PV state model can be written as follows:

##### 2.2. Bayesian Filtering

In a Bayesian framework, the estimation of consists in approximating the conditional expectation For , this probability law can be calculated into two steps as shown in Figure 1.