Operational loss modelling for process facilities using multivariate loss functions
Introduction
Different sources of variations in a process operation, such as feed specifications, wrong settings, control system malfunction and operator error can cause deviation of process variables from the specification limits. The subsequent unprofitable process operation incurs operational loss, which is defined in this work as the loss due to production of sub-quality products and increased energy usage resulting from a deviated process variable. Process facilities possess different characteristics that jointly impact process operational loss. For example, the temperature and differential pressure across a distillation column can be used jointly to monitor the operational loss of the distillation system. Thus, integrated operational loss modelling of process industries requires understanding the joint distribution of all key process characteristics and their correlations.
The loss function approach is widely used to quantify quality loss in the manufacturing industry (Leung and Spiring, 2004, Tahera et al., 2010) by relating a key characteristic of a system (e.g. product composition) to its business performance. More recently, loss functions have been applied to model operational loss for process facilities (Hashemi et al., 2014a). Choosing and estimating a useful form for the marginal loss functions of each process characteristic is often a straightforward task (Hashemi et al., 2014a, Hashemi et al., 2014b), given that enough loss information from the system is available. For multivariate cases, traditionally, the pairwise dependence between loss functions has been described using traditional families of loss functions. The two most common models occurring in this context are the multivariate quadratic loss function (QLF) (Chan and Ibrahim, 2004, Pignatiello, 1993) and the multivariate inverted normal loss function (INLF) (Drain and Gough, 1996, Spiring, 1993). For instance, Spiring (1993) proposed the following equation for bivariate cases with two parameters for which INLF can be used to represent operational loss:where Y and T are 2 × 1 column vectors of key process characteristics under scrutiny and associated target values, respectively. MEL is the maximum estimated loss and Γ is a 2 × 2 scaling matrix (shape parameter) relating deviation from target to loss for both parameters. The main limitation of this approach is that the individual behaviour of the marginal loss functions must then be characterized by the same parametric family of loss functions. This restriction has limited their useful application in practical situations. Moreover, other than the QLF and INLF, loss functions usually do not have a convenient multivariate generalization.
According to a review of the existing literature in the area of multivariate loss functions conducted by Hashemi et al. (2014a), it can be concluded that the existing research challenge is to develop a flexible framework to assign appropriate marginal loss functions to key process characteristics irrespective of their dependence structure. Copula models, which provide this flexibility, have begun to make their way into process engineering literature (Hashemi et al., 2015c, Meel and Seider, 2008, Pariyani et al., 2012). Copulas are used to describe the joint distribution of dependent random variables with any marginal distribution. While the theoretical properties of copula functions are now fairly well understood, inference for copula models is, to an extent, still under development (Genest and Favre, 2007).
The contributions of this paper are twofold. First, a new methodology is provided to construct multivariate loss functions using copulas. Second, methodologies are provided to estimate copula parameters and choose the best copula for a specific application. The main objective of this paper is to present the successive steps required to use copulas for modelling the dependent losses and constructing multivariate distributions for specific purposes, including operational loss modelling.
Following the introduction, Section 2 proposes a methodology to develop multivariate loss functions using copula functions. Section 3 reviews the theory of copula functions and Section 4 provides methods to estimate and select copula functions and conduct an uncertainty assessment. A separation column case study is then used in Section 5 to illustrate the practical implementation of copulas, followed by some concluding remarks.
Section snippets
Methodology: copula-based multivariate loss functions
It has been shown in earlier studies that the application of the general class of inverted probability loss functions (IPLFs) is a flexible approach to model loss due to process deviations (Hashemi et al., 2014a, Leung and Spiring, 2004). However, the application of IPLFs for systems with multiple key process variables is an existing research challenge due to the restriction in multivariate generalizations. Copula functions are used in this work to overcome this challenge.
Before developing a
Definition
Copulas are used to describe the joint distribution of dependent random variables. With copula modelling, the marginal distributions from different families can be combined (Genest and Neslehová, 2007). This is the main advantage of copulas compared with alternative methods, such as the use of multivariate distributions, to construct dependencies.
In this study, as shown in Fig. 1, the copula concept is used as a mechanism to develop a joint multivariate loss density function. Considering the
Review of the parameter estimation methods
When modelling the joint density of two random variables using copula functions, care must be taken to correctly and efficiently estimate the copula parameters. Genest and Favre (2007) proposed a nonparametric way of estimating the copula by using the relationship between Kendall's τ and the copula parameter to get an estimate of the latter. However, this method is suitable for explicit copulas, mainly Archimedean copulas, and requires the Kendall's τ to be known.
Methods based on maximizing the
Case study description
The practical application of the proposed multivariate operational loss model is demonstrated using a de-ethanizer column case study. The de-ethanizer simplified process schematic and the feed and product characteristics are depicted in Fig. 3. A liquid feed stream, consisting of a mixture of hydrocarbon components to be separated, is fed into the column. If the top product (ethane) is within specification (≤3% C3), it is fed to a downstream unit for further processing and transportation to the
Conclusions
In this paper, a methodology to construct the multivariate loss functions is proposed using copula functions, which allows selection of any type of inverted probability loss function for the marginal loss functions irrespective of their dependence structure. Although application of copula functions in practical problems is straightforward using the existing computational software, challenges exist in estimation of the copula parameter(s) and selection of the best copulas. To address these
Acknowledgments
The authors gratefully acknowledge the financial support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Atlantic Canada Opportunities Agency (ACOA), the Research and Development Corporation (RDC) Newfoundland and Labrador, and the Vale Research Chair Grant. The first author would like to express his appreciation for the financial support from RDC with the Ocean Industries Student Research Award.
References (32)
- et al.
Modeling dependence with copulas: a useful tool for field development decision process
J. Petrol. Sci. Eng.
(2004) Goodness of fit tests for copulas
J. Multivar. Anal.
(2005)- et al.
Risk-based operational performance analysis using loss functions
Chem. Eng. Sci.
(2014) - et al.
Loss scenario analysis and loss aggregation for process facilities
Chem. Eng. Sci.
(2015) - et al.
Selecting copulas for risk management
J. Bank. Financ.
(2007) - et al.
Real-time risk analysis of safety systems
Comput. Chem. Eng.
(2008) Modelling the Dependence Structure of Financial Assets: A Survey of Four Copulas
(2004)- et al.
Analysis of crude oil and gasoline prices through copulas
Cad. IME Estat.
(2008) - et al.
Using copulas in risk analysis
- et al.
Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach
(2002)