Development of a heuristic methodology for precise sensor network design
Introduction
The network of measurements (instruments, sensors, sampling points) in a process flowsheet is termed a ‘sensor or measurement network’. Sensor network design is concerned with the selection and placement of measurement points in a process flowsheet in order to achieve a specified objective. Examples of sensor network design objectives include design for cost, estimability, reliability, robustness, maintenance, fault diagnosis and precision (Bagajewicz, 2001). Design for achieving specified/optimal levels of precision is sometimes termed ‘precise sensor network design’ and is of particular relevance to closing material balances in process accounting (Bagajewicz, 2001). The design of precise sensor networks is generally based on the principle of data reconciliation (Gerkens & Heyen, 2005; Narasimhan & Jordache, 2000). Data reconciliation closes the various balance equations and improves the precision of measured data by reducing variance. The choice of the sensor/measurement network and measurement precisions affects the extent to which the variance of process streams is reduced after data reconciliation. However, even for simple process flowsheets the nature of the reduction in variance on individual streams is unpredictable and is only established after data reconciliation has been performed, i.e. after the fact. Consequently, it is not possible for a design engineer to make a priori design decisions as to which sensor/measurement networks will most improve the precision of process streams of interest, i.e. there are no design principles/heuristics. It is the objective of this paper to address this issue. The remainder of this section reviews literature of relevance to data reconciliation and precise sensor network design, following which the purpose of the study is presented.
Steady-state data reconciliation (SSDR) was first addressed in the pioneering work of Kuehn and Davidson (1961). The SSDR problem was formulated as a weighted least square (WLS) optimisation problem, Eq. (1) subject to mass balance constraints, Eq. (2):
The authors derived the adjusted/reconciled estimates of the process variables xa using the method of Lagrange multipliers, for the case where all process variables xm were measured and the measurement did not contain any gross errors, Eq. (3). Furthermore, the variance–covariance matrix of the reconciled variables Σa was deduced from the theory of propagation of variance in a linear system, Eq. (4). Note that both the adjusted estimates xa and their adjusted variances Σa depend on: (i) the structure of the flowsheet as specified by its connectivity matrix, A and (ii) the measured variance–covariance, Σm of the process variables.
Subsequent to the derivation of the analytical solutions for steady-state linear data reconciliation, researchers focused their efforts into three major directions:
- (i)
Non-linear systems and dynamic data reconciliation. Non-linear data reconciliation has been addressed by several authors, ranging from classical Lagrange multiplier methods (Britt & Luecke, 1973; Knepper & Gorman, 1980), to a family of Newton-based algorithms with Broyden updates for estimating Jacobian matrices (Pai & Fisher, 1988; Kelly, 2004a). Tjoa and Biegler (1991) developed an efficient hybrid successive quadratic programming method (SQP) for solving the combined reconciliation and gross error problem. Stochastic search techniques like Genetic Algorithms (Wongrat, Srinophakun, & Srinophakun, 2005) have also been applied to overcome the difficulty of gradient based methods in handling discontinuities and non-convex properties. A range of special techniques have been developed to tackle bi-linear problems (Crowe, 1989; Simpson, Voller, & Everett, 1991), which constitute a wide-variety of problems relevant to chemical engineering (Hodouin & Makni, 1996; Kelly, 2004b; Narasimhan & Jordache, 2000). Dynamic data reconciliation has also been an active field of research (Albuquerque & Biegler, 1996; Liebman, Edgar, & Lasdon, 1992; Mingfang, Bingzhen, & Bo, 2000). However, the application is still in its infancy. Recently some interesting developments in both these areas have been reported for certain industrial applications (de Andrade Lima, 2006; Barbosa, Wolf, & Maciel Fo, 2000; Eksteen, Frank, & Reuter, 2002). A variety of commercial software packages for data reconciliation have been developed over the past two decades (e.g. Makni & Hodouin, 1994; Morrison & Grimes, 2001; Ravikumar, Singh, Garg, & Narasimhan, 1994).
- (ii)
Bias and gross error detection. The problem of identifying gross errors and its importance in data reconciliation was first noted by Ripps (1965). Subsequently, several standardized statistical tests have been proposed for gross error detection (Almasy & Sztano, 1975; Mah, Stanley, & Downing, 1976; Narasimhan & Mah, 1987; Romagnoli & Stephanopoulos, 1980; Rosenberg, Mah, & Jordache, 1987; Tamhane, Jordache, & Mah, 1988; Tong & Crowe, 1995). Recently, some interesting research has been done in the area of simultaneous data reconciliation and gross error detection using mathematical programming (Soderstrom, Himmelblau, & Edgar, 2003).
- (iii)
Sensor network design. To date, research done in the area of (precise) sensor network design has involved the use of one of the following three methods (Narasimhan & Jordache, 2000):
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Matrix algebra. The extension of the linear data reconciliation problem to a case where all process streams are not measured was first proposed by Vaclavek and Loucka (1976). Analytical expressions, similar to Eq. (3), have been developed for the case of unmeasured streams using the projection matrix method (Crowe, Campos, & Hrymak, 1989) and QR factorization to estimate the projection matrix (Sanchez & Romagnoli, 1996).
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Graph theory. Graph theoretic concepts have been used to determine observability and redundancy in sensor networks and thereby ascertain which streams to measure (Kretsovalis and Mah, 1988a, Kretsovalis and Mah, 1988b; Madron, 1992, Mah et al., 1976; Meyer, Koehret, & Enjalbert, 1993).
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Mathematical programming. Madron (1992) used a graph-oriented method to develop a combinatorial optimization scheme, which could be applied to small and medium size problems. Gerkens and Heyen, 2004, Gerkens and Heyen, 2005 developed a method for including both energy balance and non-linear equations by linearising these. Bagajewicz (Bagajewicz, 2001; Bagajewicz & Sanchez, 2000; Bagajewicz, Markowski, & Budek, 2005) addressed the problem of sensor network design for non-linear mass balance equations by using binary (0,1) decision variables for the sensor networks, resulting in MI(N)LP formulations.
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The research discussed above, and the software implementation thereof, has tended to focus on numerical algorithms and optimisation routines for solving problems that render themselves into linear, bi-linear and non-linear systems, and even MI(N)LP formulations as in the case of sensor network design. These numerical approaches are satisfactory for solving problems where one has a given flowsheet and measurement scheme but offer little insight into the a priori design of sensor/measurement networks. Barring a few recent efforts (Bepswa et al., 2006, Chakraborty and Deglon, 2006; Lyman, 2005), very little work has been reported on the use of a priori principles/heuristics for sensor network design.
The aim of this study is to develop a heuristic methodology for designing precise sensor/measurement networks for linear material flow circuits based on the general principle of variance reduction through data reconciliation. Firstly, the paper develops a new ‘flowsheet-independent’ formula for estimating the adjusted variance of process streams on linear material flow circuits (cf. Section 2). Here, the weighted least squares problem is symbolically optimised to obtain a generalized ‘flowsheet-independent’ formula, which depends only on the measured variances of process streams, unlike the famous analytical solution of Kuehn and Davidson (1961). Secondly, this formula is used to develop a set of generic design principles/heuristics for maximising the variance reduction of process streams (cf. Section 3). These design principles/heuristics can be used for making a priori design decisions in selecting optimal flowsheets and/or measurement schemes for reducing the variance of process streams on linear material flow circuits through data reconciliation. The generalized formula and the design principles/heuristics are then tested numerically using an industrial case study (cf. Section 4).
The design principles/heuristics developed in this study are for maximising the reduction in variance of terminal process streams on linear material flow circuits. A precise knowledge of material flow rates for these streams is critical to areas such as process accounting and quality control. This is becoming increasingly important to large process industries, such as the petrochemical and mining industries, as companies are forced to have reliable process accounting (production) information for, amongst others, process efficiency and corporate governance reasons. The generalized formula developed in this study is based on the following simplifying assumptions: (i) all process streams are observable, (ii) all process streams are measured, (iii) there is no bias in measurements, (iv) the random errors follows Gaussian distribution, and (v) all measured variables are independent of one another, i.e. only the diagonal elements of the matrix Σm exists.
These assumptions are commonly made when performing data reconciliation on linear material flow circuits, in particular assumptions (i) and (iii)–(v). Process streams are generally observable for linear material balances but not all streams are necessarily measured. However, a few unmeasured streams does not greatly influence the predictions of the formula or, more importantly, the design principles/heuristics. If necessary, unmeasured streams may be treated as measured streams with large variances. The assumption of minimal bias and Gaussian error distributions is considered to be relatively uncontroversial. The assumption of uncorrelated measurements is routinely made in material flow circuits as covariances are very difficult to determine. This assumption is untrue in general as measurements are always correlated to some extent but is perhaps not overtly simplified for material flow circuits as including covariances does not greatly influenced the quality of data reconciliation (Hodouin, Mirabedini, Makni, & Bazin, 1998). Lastly, the analysis in this study is for linear material flow circuits (i.e. component mass flowrates) which are sometimes in the form of bi-linear systems (i.e. total mass flowrates and compositions). Here, the bi-linear terms may be linearized by propagation of variance.
Section snippets
Nomenclature of the generalized flowsheet
Fig. 1 is a schematic of a generalized process flowsheet with N mass balance nodes. A node is any junction in a flowsheet where a linearly independent mass balance equation can be derived. Physically a node signifies unit operations like mixing, splitting, storage, reaction or even simple junctions in a pipeline where stream splitting occurs.
According to the selected nomenclature, a process stream s associated with any node n is represented as s(n). Process streams are classified into two
Design principles/heuristics for sensor networks based on variance reduction of terminal streams
The reduction in variance rs(n) of a terminal stream s(n) is defined as the change in variance of that stream after data reconciliation per unit measurement variance, i.e. Eq. (10).By substituting Eq. (5) into Eq. (10), the reduction in variance rs(n) of any terminal stream may be obtained in terms of measured variances only, i.e. Eq. (11).
By inspection of Eq. (11), one may infer the following
Industrial case study—numerical validation of generalised formula and heuristics
The industrial case study chosen for consideration is adapted from Lyman (2005) and is typical of a separation circuit used in the minerals industry. Here, linear material flow circuits are common and a precise knowledge of terminal streams is important for process accounting due to the large associated financial values. Structurally, it corresponds to the flowsheet that was employed at Bougainville Copper in their flotation circuit (Hinckfuss, 1976). Fig. 7 is a schematic of the process
Conclusions and significance
The aim of this study was to develop a heuristic methodology for designing sensor/measurement networks for linear material flow circuits based on the general principle of variance reduction through data reconciliation. Firstly, the paper developed a new ‘flowsheet-independent’ formula for estimating the adjusted variance of process streams on linear material flow circuits. The formula is exact for any generic flowsheet but the expressions for the interacting terms are only exact for systems
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