Decision-Making Strategy and Tool for Sensor Network Design and

Ind. Eng. Chem. Res. 2004, 43, 1711-1722

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Decision-Making Strategy and Tool for Sensor Network Design and Retrofit C. Benqlilou, M. Graells, and L. Puigjaner* Chemical Engineering Department, Universitat Polite´ cnica de Catalunya, Avenida Diagonal 647, 08028-Barcelona, Spain

This work presents a decision-making framework for the design and retrofit of sensor networks, as well as a general strategy for the correlation of the cost and performance of the different sensor arrangements (number, placement) that can be systematically analyzed. The analysis of the steps required to solve the sensor placement problem for design or retrofitting cases implies the identification of the information flows involved. Such a conception allows for the modular design of a tool for computer-aided sensor network design/retrofitting. An interface specification is proposed for this tool, and a prototype is developed for validation purposes. The decisionmaking strategy adopted is independent of the particular algorithms and procedures used. Hardware redundancy, steady-state/dynamic processes, design/retrofitting, and a catalog of different available sensors are all taken into account, thus leading to a generic framework that is able to follow the future trends of sensor placement. Seeking the synergy given by reusability and standardization, the sensor placement tool has been developed following the CAPE-OPEN (CO) guidelines, allowing for the integration of other software modules such as those for process modeling and data reconciliation. 1. Introduction 1.1. Sensor Placement. The design of a sensor network, or the retrofitting of the instrumentation structure of an existing plant, is a necessary step in improving the capacity of the monitoring, controlling, and optimization of the process. The accuracy of the estimations of process variables has a direct effect on control performance and subsequent tasks such as plant optimization and safety. Thus, this problem must be addressed in a way that allows for the exploration of the widest range of opportunities. The problem of sensor network design using the concept of observability and the estimation accuracy of process variables has also been addressed by several research groups. Vaclaveck and Loucka29 were the first to develop a sensor placement strategy for steady-state systems so as to ensure the observability of a specified set of important variables in a pure mass-flow or multicomponent process by using graph theory. Determination of the number of measuring devices is straightforward once the number of process variables and the number of equations relating them are known; from here, the sensor placement is addressed. Ragot and coworkers26 presented a procedure that ensured the observability of all variables in a bilinear process. Kretsovalis and Mah21 quantified the effect of sensor placement on the accuracy of estimated variables for mass-flow processes and used the results to develop a combinatorial search algorithm for sensor network design. In 1992, Madron and Veverka22 extended the work of Vaclaveck and Loucka29 to include the overall cost of the sensors. Their method makes use of GaussJordan elimination to identify a minimum set of variables that need to be measured for all required process * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +34-93-401.66.78. Fax: +34-93401.09.79.

variables to be observed while simultaneously minimizing the overall cost of the sensors. Along this line, Meyer and co-workers23 also developed an algorithm for the minimum-cost design of sensors for linear processes based on a graph-oriented approach. They used a branch-and-bound-type strategy to solve the optimization problem formulated. Ali and Narasimhan5 addressed the issue of sensor failure and its effect on observability variables and took this issue into account in sensor placement strategies. They went on to tackle the problem of sensor placement strategies for steady-state linear processes when sensors are likely to fail. They also proposed the concept of the reliability of estimation of a variable, which gives the probability of being able of estimating a variable value for any given sensor network and specified sensor failure probability. In 1995, Ali and Narasimhan6 extended their work for the optimal design of redundant sensor networks for linear processes. In 1997, Bagajewicz8 posed the sensor network problem as an optimization problem with minimization of cost as the objective function and requirements of error detectability, resilience, and residual precision as the constraints of the optimization problem. Sen and coworkers28 integrated genetic algorithms with graphtheory concepts to solve the problem of the optimal design of a sensor network for linear processes. Using genetic algorithms, they could solve the problem to optimize different objectives simultaneously such as cost, estimation accuracy, and system reliability. Additionally, the encoding procedure they propose is quite intuitive, that is, in the string, the bit represents the characteristics of the solution. Later, Bagajewicz and Sańchez9 merged the concepts of degree of redundancy and degree of observability for variable measurements into a single concept: degree of estimability of a variable. They presented a formulation for the design of a sensor network to achieve a

10.1021/ie034037e CCC: $27.50 © 2004 American Chemical Society Published on Web 02/19/2004

1712 Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004

required degree of estimability. They also demonstrated a minimum overall cost model and a generalized model for the design of a reliable sensor network to be equivalent.10 Reallocation and upgrading of the existing instruments to achieve maximum precision of selected variables has also been considered.10,14 The methods cited above do not address the issue of sensor placement for dynamic systems. Chmielewki and co-workers18 extended the static sensor placement problem to linear dynamic processes. They gave a procedure to make the NLP problem independent of the decision variables. Additionally, in their contribution, they transformed the NLP problem into a convex program through a linear matrix inequality. Otherwise, a software tool for sensor placement decision making is still missing, although some authors have attempted to develop a tool for sensor placement design.2,20,27 Recently, Heyen and co-workers19 proposed a general mathematical formulation of the sensor selection and location problem, to reduce the cost of the measurement system while providing estimates of all specified key process parameters within a prescribed accuracy. Narasimhan and Rengaswamy25 also recognized the need for the development of an integrated approach to sensor network design. This work extends the current sensor placement paradigm to address the design and retrofit problems and to consider dynamic processes by introducing a general framework following the guidelines presented by Benqlilou and co-workers.16 The general framework proposed allows these problems to be addressed from a general decision-making point of view by integrating tools to deal with the necessary cost-performance tradeoff as has been recently proposed by Bagajewicz and Cabrera.11 Moreover, an open system architecture based on CAPE-OPEN standards is developed, allowing for the systematic and rigorous evaluation of different sensor design and retrofit options. The software tool includes the possibility of selecting different measuring devices from a catalog. 1.2. CAPE-OPEN Standard. Industrial users of CAPE tools are no longer developing their own proprietary software; rather, the present trend is to use modular and customized programs provided and maintained by specialized vendors. In a similar way, software engineering has undergone a significant transformation during the past decade. As the hardware standard has moved from large supercomputers to networked PCs, software developers have produced huge monolithic programs to fully exploit the benefits of distributed computing, such as modularity, maintainability, and code reusability. These converging changes motivated the CAPEOPEN and global CAPE-OPEN EU-funded projects, which have provided the mechanisms for ensuring CAPE module interoperability across networks. Thus, CAPE-OPEN standards for communication interfaces were produced and published. Hence, the term COcompliant was coined to designate software modules that correctly implement CO interfaces, allowing such modules to interact with other CO-compliant software. Such compliance and the current status of the standard were recently reviewed by Belaud and Pons.12 The CAPE-OPEN and global CAPE-OPEN (GCO) legacy includes a state-of-the-art monograph on software

architectures and tools for CAPE17 and the CAPEOPEN Laboratories Network (CO-LaN), the internationally recognized user-driven organization for the testing and management of the CAPE-OPEN standard.1 The development of the CO standards was carried out on the basis of a technical decision made for the formal description of the interfaces.13 The Unified Modeling Language (UML) already developed, published and maintained by the Object Management Group4 was adopted as a convenient tool for this purpose because it allows for the specification, visualization, and documentation of models of software systems. This work also uses UML’s standard diagram types for defining the architecture and interfaces proposed. Among the different CO standard interfaces, it is worth noting those related to the architecture presented in this work, which allowed for the incorporation of some already-developed modules such as mathematical models (CO-ESO, equation set object) and solvers (COMINLP). Particularly, the architecture requirements for the parameter estimation and data reconciliation (PEDR) modules were addressed by the GCO project,7 and the prototype implementation was also presented.15 However, the sensor placement problem and the architecture of a standardized sensor placement tool, despite being strongly connected to the previous aspects, were not addressed by the GCO project. The work presented here is also intended to cover this gap. 2. Problem Formulation The sensor placement problem can be regarded as a constrained optimization problem for minimizing the sensor network cost. The constraints are inequalities defining the upper and lower bounds on system performance such as accuracy and/or reliability for each process variable measurement. Thus, given the costs, cj, of the sensors required for measuring each process variable j, the sensor placement problem can be formulated as

∑j cjqj

(1)

A(j) e A(j)*

(2)

Rj(t) g Rj(t)*

(3)

min subject to

where the accuracy A(j) ) A(qj) and reliability Rj(t) ) Rqj(t) are both bounded functions [A(j)* and Rj(t)*] of the sensor assignment

qj )

{

1 if variable j is measured 0 otherwise

(4)

Alternatively, it is possible to choose to minimize the weighted estimated accuracy (the estimation accuracy can be evaluated through data reconciliation) of the involved process variables subject to an upper bound on the cost of the sensor network

min

∑j RA(j)

(5)

Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004 1713

subject to

∑j cjqj e Cmax

(6)

Nevertheless, the sensor placement problem presents a particular feature with the difficulty of determining the threshold value of the inequality constraints in eqs 2, 3, and 6. A partial way to overcome this drawback is to consider the problem as a multiobjective unconstrained optimization problem as follows

min

∑j RA(j) + βcjqj

(7)

However, this formulation poses a tradeoff between the performance (e.g., accuracy) of each possible sensor network and its cost given by the values of the weight parameters R and β in eq 7, which are also difficult to determine. This work addresses this fundamental tradeoff by proposing an information framework to aid an expert user in the decision-making procedure. Additionally, the system exhibits the following characteristics: (1) The system can be implemented in a modular application of multicriteria functions for evaluating performance. The sensor placement constraints (satisfying the desired accuracy or reliability) are usually solved algorithmically. Therefore, using the corresponding analytical equations for optimization purposes can be very arduous and results in a rigid solution approach to the sensor placement problem because it is strongly problem dependent. (2) The system allows for the merging of different sensor placement criteria into a unique one by developing more generic sensor placement constraints. For example, this work shows that increasing the network reliability indirectly increases the network accuracy. (3) The system allows for the selection of a sensor network from among a set of possible networks by using the profile of the system performance versus the corresponding cost. This point is useful and practical for evaluating the necessary investment required to achieve a desired performance. Thus, this information is very valuable for the decisionmaking process. Additionally, this work also considers (1) different measuring device types (cost and performance) (this consideration is the basis for a more generic sensor placement problem including the reallocation, new purchase, and design sensor placement problems); multiple observations (more that one measuring device per measurement point), which is undertaken in a novel way by including additional inequality constraints in the sensor placement problem formulation; and (3) an extension for dealing with dynamic cases. Thus, this work presents a specification for this strategy and a tool that has been developed following the CO guidelines.1 Additionally, the unified modeling language24 description of sequence, interface, and component diagrams for design and retrofitting were produced and validated providing the corresponding prototypes. Several CAPE-OPEN standard interface specifications, such as data reconciliation and optimization, have been adopted because the sensor placement problem shares several features with these techniques in terms of the information flows. 3. Input 3.1. Information Flows. 3.1.1. Input. The process model is a set of equations that correlate and bound

Figure 1. Scheme of an adiabatic process.

process variables, and it is a prerequisite for placing sensors in a given process plant. It can be given by an explicit set of mathematical equations [for instance, through a standard CO interface, the equation set object (ESO)] or as black box model (e.g., the artificial neural network model). This process model allows for both the generation of possible sensor networks and the evaluation of the system performance. The number of possible sensor networks might be limited according to the set of parameters given by

Mmin ) {mmin } j

(8)

} Mmax ) {mmax j

(9)

Vectors Mmin and Mmax, both of size J, permit the setting of the minimum and maximum numbers of measuring devices that can be allocated to the j ) 1, ..., J, process variables involved in a given process model (multiplicity or hardware redundancy). Addressing multiplicity requires a new integer variable nj instead of the binary variables qj for describing the sensor assignment

qj ) {0, 1} f nj ∈ N

(10)

Thus, sensor assignment is limited by maximum and minimum multiplicity values

e nj e mmax ∀j mmin j j

(11)

As an illustrating example, consider the adiabatic process in Figure 1, where F1 and F2 are mass flow rates; L denotes the tank level; and T1, T2, and T are temperatures (the heat capacity; density, F; and tank section, a, are assumed to be constants). The process model, given by the mass balance and enthalpy balance, consists of the following equations

F1 - F2 ) Fa T1F1 - T2F2 ) FaL

dL dt

dL dT + FaT dt dt

(12) (13)

A possible mapping of process variables can be set by the following vector

variable mapping w {L, F1, F2, T, T1, T2}


and the corresponding multiplicity bounds could be

Table 1. Matrix R Allowing Sensor/Process Variable Allocation

Mmin ) {1, 0, 0, 0, 0, 0} Mmax ) {1, 2, 1, 1, 1, 1} Vectors Mmin and Mmax impose the requirements that the measurement of level L is obligatory whereas process variables T, T1, T2, and F2 could be either measured once or not. However, flow F1 can be measured at most twice. Manipulation of the lower and upper bounds of eq 11 allows for the introduction of information about feasible, infeasible, and obligatory measurement points and their maximum multiple observations. In general, the two ) mmax ) p, where main cases are as follow: (1) If mmin j j p ∈ N, then the measurements of process variable j is obligatory and must be done p times, (for safety or controllability reasons). In the special case where p ) ) mmax ) 0, the measurement of the process 0, mmin j j variable j is infeasible (because of cost or operation ) p and mmax ) q, where {p, q} limitations). (2) If mmin j j ∈ N, q > p, then the number of sensors used for measuring the jth process variable is not known a priori; rather, it is a decision variable whose value pertain to the {p, ..., q} set. In general, industrial practice (including safety aspects) requires only a couple of sensors per measurement point. The number of decisions associated with each process variable (i.e., the number of possible sensors per measurement point) is given by

∆M ) {δmj}

(14)

- mmin + 1, ∀j ) 1, .., J δmj ) mmax j j

(15)

F1 F2 L T T1 T2

FMa

FMb

TM

LM

1 1 0 0 0 0

1 1 0 0 0 0

0 0 0 1 1 1

0 0 1 0 0 0

R ) {rjk} whose elements are binary variables defined as follows

rjk )

{

1 if sensor k can measure process variable j 0 otherwise (19)

Assuming there are two types of flowmeters, FMa and FMb, and one type of measuring device for each temperature (TM) and level (LM), matrix R is obtained as shown in Table 1. It is important to check the consistency between the information contained in R, Mmin, and Mmax. Hence

{

mmin e mmax j j mmax e r j jkU1

(20)

where U1 is a very large number (e.g., U1 g 10 should be large enough). The multiple observations are taken into account by adding the following inequalities

e mmin j

, ∑nijk e mmax j

∀k

njk e rjkU2

(21)

where

The a priori calculation of δmj provides an idea of the dimension of the problem, as the number of “positional” networks, P, that can be generated is given by

P)

∏j δmj

(16)

Positional network i is referred to the assignment of a certain number of sensors of the same type (i.e., assuming that all sensors have the same characteristics and cost) to a process variable j. Thus, each positional network i involves a number of sensors of the same type given by the sum

∑j nij

(17)

When the consideration that different sensors k can be used to measure the same process variable j is incorporated, each sensor network i is characterized by a means of si, and the superset Σ defines all of the networks considered

si ) {nijk} ∈ N,

∑ ) {si}

(18)

The set of potential sensors K for measuring j is an information part of the catalog. This information that should be carefully introduced in to the formulation of the sensor placement problem is in the form of matrix

where U2 is a very large number; in this case, it is assumed to be equal to mmax . Finally, the criteria for j placing measuring devices (e.g., investment cost, eq 22; and/or controllability; and/or accuracy; etc.) have to be specified to evaluate and rank the different alternatives considered and to control the combinatorial explosion if possible. It is important to note that the proposed framework can deal with multiobjective criteria by constructing diverse performance functions

cost(si) )

∑j ∑k cknijk

(22)

per f(si) ) Φ(nijk,process model,...)

(23)

3.1.2. Output. The information that allows a technician (or an optimization algorithm) to make reliable decisions when designing plant instrumentation is the cost (Γ) and performance (Π) of each sensor network (Σ), defined by the type and number of measuring devices per each measuring point

() ( ) ( )

s1 s Σ) 2 , Γ) l si

cost(s1) cost(s2) , Π) l cost(si)

per f(s1) per f(s2) l per f(si)

(24)

If the type of process variable j to be measured is known, then one can obtain the corresponding possible range of sensors available for its measurement from a database. This range includes information related to the sensor accuracy{Ak}, reliability {Rk(t)}, and cost {Ck}.


Figure 2. Component diagram of the proposed sensor placement tool.

3.2. Modular Structure and Module Interactions. 3.2.1. Modules Involved. The analysis of the information flows of the sensor placement problem follows the existing trends. The specification proposed for the sensor placement module can be easily plugged into or use other standardized software components, if required. The modules required by an open and flexible architecture can be classified according to their functionality. Mainly, five components have to be considered, as illustrated in the component diagram (see Figure 2), namely, (1) sensor network generator, (2) system performance calculator, (3) total cost calculator, (4) catalog, and (5) process model. In the UML notation, component diagrams address the static implementation view of a system.24 The physical aspects of the sensor placement system are illustrated from the component view in Figure 2. Additionally, the dependency relations between the different software components are shown. Σ ) {si}. Module I is responsible for generating a set of feasible sensor networks Σ ) {si} from bounds given by Mmin and Mmax, the process model, the sensor characteristics {Ak, Rk(t), Ck}, and the sensor placement criteria. This task can be performed by an elaborate algorithm or by a simple enumeration. Despite the efficiency of these or future algorithms, this work highlights the necessity of such functionality. The networks generated might not necessarily pertain to the feasible space, so it is up to the end user to remove the infeasible subset as their performances will be unacceptable. Furthermore, the end user can add additional networks or remove some of the generated networks to evaluate their performance and/or cost. Π ) {per f(si)}. The objective of module II is the evaluation of the system performance of networks generated in the above functionality or directly introduced by the end user. This module offers the list of performances or combination of performances that it has evaluated and also the evaluation results. These performances might be mostly related to data monitoring such as reliability, accuracy, and gross-error detectability, but sensor placement for control and optimization could also be covered by replacing the performance evaluation module. Moreover, the performance should merge different criteria or offer a multiobjective solution. Γ ) {cost(si)}. Evaluating the total cost of the set of networks introduced is the objective of module III. Additionally, this module can also evaluate the cost corresponding to a retrofitting case. Catalog. The catalog component, module IV acts as a database of measuring devices, providing their char-

Figure 3. Interface diagram for the proposed sensor placement tool.

acteristics. These characteristics include the cost and a list of the sensor’s qualities, mainly accuracy and reliability. Therefore, this module has the process variable type (e.g., level, flow) as the input and the range of measurement device characteristics [i.e., vector of sensor accuracy Ak and reliability Rk(t) for a given type of process variables] as the output. It is important to note that this component is needed for evaluating the system performance and system cost, as well as for generating the sensor networks. Process Model. Module V allows for the introduction of the process model that is required for the performance of this application, as is this model that represents the plant considered for sensor placement. In principle, the interaction of the specified modules will be initiated by a client that could be either an end user or a software application. The component diagram is supplemented by the interface diagram (Figure 3). This diagram gives a static representation of the interfaces that the sensor placement components are required to expose, as well as their interoperability. The interface diagram is of major significance for designing, specifying, and implementing the sensor placement architecture as it declares the methods that are externally exposed. In the next section, the dynamic interactions of the components involved are analyzed. 3.2.2. Module Interactions. First, the input information presented in section 3.1.1 is introduced into the sensor network tool. This information is used to generate a set of networks Σ. The set of networks is then available to the end user, who decides whether to reduce or increment it prior to its evaluation. The performance evaluation module returns to the end user the system performance corresponding to a given set of networks. To fulfill this task, the component needs to interact with the process model. The last functionality is the evaluation of the cost of a set of networks (design/retrofit). For this task, the total cost calculator component interacts with the catalog to calculate the cost. The details of the dynamic interaction are described by means of the sequence diagrams in Figures 4 and 5. A sequence diagram captures the time-oriented dynamic behavior and interactions arround sensor place-


Figure 6. PROCEL pilot plant.

Figure 4. Design sequence diagram for sensor network generation.

Figure 5. Design sequence diagram for cost and performance evaluations.

ment components and their relationships. Figure 4 proposes a a scenario for generating sensor networks, whereas Figure 5 illustrates the temporal interaction for evaluating cost and performance of the generated sensor networks. The possible information returned to the end user by the sensor placement tool is summarized in section 3.1.2 above. 3.3. Sensor Placement Prototype. A prototype was developed to demonstrate and validate the specifications proposed for a sensor placement tool and the general strategy introduced. The specification of the functionality that each involved component offers and the sequen-

tial interactions among these components were also tested. In a fist step, this prototype was developed in MATLAB.3 However, the implementation into a set of CO-compliant modules working in a distributed way is straightforward, as the interface specification proposed in this work follows CO standards and guidelines. The CAPE-OPEN option enhances the potential of the approach adopted because this standard is obtaining a growing interest and acknowledgment within the academic as well as industrial environments. For illustration purposes, only system accuracy is considered in the examples discussed here. The data reconciliation algorithm given in the Appendix was intended to become the system’s performance evaluation component. This component offers process variable accuracy as the unique criterion to be selected for placing sensors within a given plant. It is important to note that the variance is calculated without any previous knowledge of process measurements. Additionally, because data reconciliation is adopted, observability and redundancy are implicitly included in the objective function. For sake of simplicity, no more than one sensor per potential measurement point is assumed; that is, mmax j ) 1, ∀j. Moreover, redundancy analysis, which is the basis for combining the data reconciliation technique and sensor placement, ensures some degree of observability and required redundancy. This requirement leads to a minimum allowable number of sensors, which, in turn, reduces the space of feasible combinations and decreases the computational effort. An enumerative algorithm is adopted for the generation of feasible combinations. At this level, it is assumed that the sensors involved have the same characteristics and price. The mass balance model is used for sensor placement purposes. 4. Case Study 4.1. Sensor Network Design for PROCEL Pilot Plant. A case study is proposed for validating the decision-making strategy as well as the prototype developed. As a test plant, Process Cell (PROCEL), a pilot plant at the Universitat Politećnica de Catalunya (UPC), was used (additional information on PROCEL can be obtained from the corresponding author). PROCEL consists of three tanks with agitators, heaters, and heat exchangers. The tanks are connected in a highly flexible way so that different configurations are possible. In Figure 6, the continuous operation mode of


this pilot plant in a specific configuration is shown. The design of the sensor network of the PROCEL plant was undertaken, focusing on the tradeoff between the maximization of the accuracy of estimation through data reconciliation and the instrumentation cost. Commonly, the accuracy is found to increase monotonically with the investment. However, different networks might lead to the same performance, which implies a certain degree of degeneracy. This degeneracy is reduced when the process under consideration is further complicated by the inclusion of recycling or the availability of different sensor types. The former case involves the possibility that adding a sensor does not directly imply an improvement in the plant performance. Here, the dynamic case of PROCEL is considered. A dynamic mass balance has to be adopted if the sensor placement considers the dynamic behavior of the plant. Thus, the levels in the two tanks are given by

F5 - F4 ) Fa1

dL1 dt

(25)

F2 - F1 ) Fa2

dL2 dt

(26)

If the hold-ups of the heat exchangers are neglected, the mass balance around them is represented by

0 ) F2 - F3 0 ) F6 - F7 0 ) F3 - F4

(27)

0 ) F5 - F6 Given this model, module I generates the set of sensor networks. This set is then evaluated by module II to produce the corresponding performance set (Π). Accuracy was selected as the performance index, and dynamic data reconciliation is achieved using a Kalman filter16(see the Appendix for further details). Module III was used to obtain the corresponding set of costs (Γ). Figure 7 shows a plot of the performances given by the different sensor networks that can be arranged by using 9 (see Figure 7) and 10 (see Figure 7) sensors. The graph shows the dependence of the performance on the arrangement and clearly indicates that increasing the number of sensors does not necessarily lead to higher performance results. The decision-making process is greatly aided by an information system allowing for the proper management of the data obtained from systematic analysis. Thus, the sensor arrangements evaluated can be sorted by performance to obtain the plots in Figure 8. The jagged performance profile confirms the idea that certain design decisions can have a significant impact on plant operation. In the case of equivalent measuring devices, the cost of the sensor network is given by the number of sensors. This is also plotted in Figure 8, also sorted by performance. This second profile does not run parallel to the performance above, but allows for the determination that certain design decisions can imply a cost reduction simultaneously with a performance rise. Once again, the information management system allows for the acquisition of deeper knowledge of the problem by gathering the previous data into Figure 9, which outlines the best performance that can be obtained up to a fixed cost. Figure 9 provides useful information for deciding the

Figure 7. Performance vs number of measuring devices for systems with (a) 10 and (b) 9 sensors.

Figure 8. Increasing performance profile and related cost for the design case. (Only the range of significant performance is plotted.)

cost-performance tradeoff through the comparative data for the basic design options. Moreover, the information system also allows for further treatment of the data as a formal approach to the multiobjective problem such as Pareto optimal analysis. 4.2. Reliability and Accuracy. The incidence matrix representing the mass balance of PROCEL operat-


Figure 10. Accuracy and reliability trends. Figure 9. Performance vs cost. (The sensor network si producing the best performance is indicated for each investment margin.)

(

ing at steady state is given by eq 28

-1 1 0 0 0 0

1 0 0 A) 0 0 0

0 -1 1 0 0 0

0 0 -1 1 0 0

0 0 0 -1 1 0

0 0 0 0 -1 1

0 0 0 0 0 -1

)

(28)

5. Retrofitting Strategy

The rows correspond to mass balances around tanks and heat exchangers, and the columns represent the seven flow rates. It is important to mention that the mass balances around the heat exchangers consider separately the balances of the cooler flow and the heater flow. For accuracy in calculation, the Lagrange multiplier solution of the data reconciliation optimization problem is adopted, and the variance of the estimated process variables in a matrix form is is given by eq 29 when all variables are measured. The variance of the measured variables is given by Q.

Q ˆ ) Q - Q‚AT‚(A‚Q‚AT)-1‚A‚Q

(29)

As expected, not all of the variables are measured. Nevertheless, the actually unmeasured variables will be handled as measured variables with large standard deviations. For reliability in evaluation, because all of the redundant equations are formed by a disjoint set of measurements, eq 30 can be employed. The corresponding reliability is calculated by the probability analysis using the sum of disjoint products. For example, the reliability of F1 when only F1, F2, and F3 are measured is given by eq 30. Moreover, the reliability of actually unmeasured variables is assigned a value of zero. 3

RF1(t) )

RF (t) ) RF (t) ∑ i)1 i

1

the PROCEL case, when modeled according to a steadystate mass balance. Once again, the data are stored by accuracy, and hence, the two graphs confirm for this case the parallel trend of the two performance indexes (i.e., accuracy and reliability). Such information allows for the reduction of a double-objective problem to a single-objective problem.

or RF2(t) or RF3(t)

RF1(t) ) RF1(t) + [1 - RF1(t)]RF2(t) + [1 - RF1(t)][1 - RF2(t)]RF3(t) (30) Figure 10 shows the accuracy and reliability given by the different sensor networks that can be proposed for

Retrofitting is regarded as the procedure for providing the placement of the number and type of measuring devices (considering multiple observations and different initial sensor characteristics as well) that should be added/relocated in the operating plant to increase plant performance within budget limitations. The decisions involved in such a retrofitting strategy can be made according to two main strategies: (1) purchasing new measuring devices or (2) reallocating existing sensors while permitting new purchases. In either case, the starting point for all of these partial approaches is the information related to the current sensor network. Thus, the parameter set S is defined as follows

νjk ) n number of sensors k installed for measuring j 0 otherwise (31)

{

Additionally, because the reallocation of sensors saves acquisition costs but not some costs related to their installation/deinstallation, the cost parameters need to be separated into two contributions, i.e.

ck ) cck + cik where cck and cik are the capital and installation costs, respectively. This information should be provided to the sensor placement system when the retrofitting case is addressed. Additionally, for further system performance evaluation, the characteristics of the installed measuring devices have to be included in the catalog database. In the proposed approach, the set S is considered as consisting of “new” sensors with costs cik and certain performances that can be installed as if in a design problem.


5.1. Purchase. Once this information is set, retrofitting by purchasing new items imposes the condition that already-installed instrumentation is not to be modified, and the goal is to determine the new measuring devices (number, type, and placement) that will satisfy the new performance requirements. This procedure can be supported in a straightforward way by the design procedure presented in the previous sections just by indicating to the system the set of sensors that are already installed (S). Additionally, the set of already-installed sensors needs to be prevented from reallocation, which can be easily achieved by properly adjusting the multiplicity , mmax ) and assignment (rjk) parameters. (mmin j j 5.2. Reallocation. Retrofitting by means of reallocation is another particular case of retrofitting where the placements of some or all of the installed sensors can be changed. The goal is to determine the set of sensors to be reallocated, as well as their new placements. The old sensors can be considered as new sensors with a price corresponding uniquely to their installation cost cik (old features can also be assigned to these old sensors by tuning the corresponding parameters). This point is very important as it favors the reallocation of already-installed sensors before the acquisition of new sensors with higher costs (cck + cik). Despite this intrinsic priority, additional constraints could be added to force the reinstallation of already-existing sensors. Thus, new purchase strategy can be seen as a special case of reallocation for which the installed sensor network S does not admit any changes. 5.3. Information Flows. Once the information given by the parameter set νjk, which provides cost as well as sensors characteristics, solution of the retrofitting problem requires the same components and information as needed for the design case. At this point, the interfaces designed prove to be robust and general enough to undertake a wide range of sensor placement problems. Given the sensor network already installed, the current system performance (ΠS) can be obtained by calling the method EvaluatePerf() in the interface INetworkPerfCalculation presented in the interface diagram. [See Figure 3. A reference or desired system performance (Πd) can be set by assuming the design of a new sensor network for the same plant (νjk ) {0}, ∀j).] Then, the investment needed to lead the current system performance to the target (Πd) can be evaluated by invoking the method EvaluateCost(). To evaluate the investment accurately, direct costs (as instrument purchase and installation) as well as benefits resulting from additional measuring devices have to be considered; therefore, module III must include an algorithm for this task. Thus, the extension of the sensor placement framework presented in previous sections to deal with retrofitting is straightforward, although additional information should be provided to the system. Additionally, cost evaluation can be extended to consider the acquisition of new items and the reallocation of old ones through the related cost parameters. Such a change does not require any revision of the framework and interfaces proposed, but necessitates only the use of a new evaluation function (eq 32) instead of eq 22. Equation 32 considers capital and installation costs, and when νjk ) 0, ∀j, the function becomes

C)

∑k cckΘ(∑j njk - ∑j νjk) + ∑k ∑j cik|njk - νjk|

(32)

where

Θ(x) )

{

1 if x g 0 0 if x < 0

(33)

It is important to note that the retrofitting cost evaluation presented above can handle the situations in which the already-installed sensors are similar to but different from the sensors provided in the catalog (e.g., because of decreassing efficiency). This can be done in a parametric way by providing the sensor performances (i.e., the set K is increased by the inclusion of these sensor types). Additionally, once a general problem definition and solution framework have been set, the design case can be considered as a special retrofitting case that is parametrically described by setting S ) {νjk} to zero, thus meaning that no sensors are already installed. A case study was used to validate the information framework and the interfaces proposed. Some enumerative algorithms were implemented to systematically analyze the problem and to provide useful graphical information for the user to make her/his own decisions. This is of paramount significance in the case that no suitable algorithm is available for solving the problem. Following the same interface specification, specific solution modules could be included when available, and more sophisticated tools for the management of the output information, such as those proposed by Bagajewicz and Cabrera,11 could be incorporated to enhance the decision-making procedure. 6. Case Study Assume that the PROCEL pilot plant considered before already contains two flow-meters for measuring flows F1 and F2. Assume also that the model is given by the set of eqs 27. We next generate the profile of the system performance and cost for the possible sensor networks satisfying the constraint set of the initial condition. A hypothesis adopted in this case study is that the already-installed sensors will be kept in the plant. Even though it does not lead to any performance improvement, the cost is increased (deinstallation cost). Moreover, it is assumed that the existing sensors are of the same type and that the new ones are of a single different type. The procedure for generating Σ is more complicated for the retrofitting case: Each time an additional sensor is added to the existing ones, the plant instrumentation combinations that fulfill the sensor placement goals are generated. One difficulty that arises is the identification of the set of already-allocated sensors. The cost of a new sensor is set to 1.25, and the cost of installation or deinstallation is set to 0.25. Thus, Figures 11 and 12 show plots of the costs obtained by this way and the resulting performances. Figure 11 shows the cost and performance profiles of the different sensor networks generated as instrumentation-upgrading opportunities. Once again, information management allows these alternatives to be plotted in order of increasing performance, which, in turn, permits the observation that some retrofitting options might result in higher performance as well as lower cost. Figure 12 summarizes the previous information in a graph relating the best performance that can be obtained up to a given cost. It is worth noting that, in this case, the cost is not only related to the number of


Figure 11. Increasing performance profile and related cost for the retrofit case.

considers dynamic as well as steady-state systems and considers multicriteria sensor placement, hardware redundancy, and a catalog of available measuring devices. Additionally, in this work, the solution approach merges in a unique formulation “all” the possible sensor placement problems (design, reallocation, new purchase). As a result, beyond the specific chances given by the diverse optimization techniques, a decision-making framework for instrumentation design and retrofitting is presented. This framework is described in terms of the UML, and thus, the interface, sequence, and component diagrams showing the strategy proposed are also presented. The resulting interface specification was implemented in a software prototype and validated using some illustrative examples. The specifications were developed following the CAPE-OPEN guidelines, which permits easy adaption to specific scenarios and accepts further integration with other chemical engineering tools such as process simulators and data reconciliation packages. The case studies presented also demonstrate the usefulness and potential of a computer-aided decisionmaking tool allowing for the comparative analysis of several instrumentation alternatives (networks) and the management of the information required to solve the cost-performance tradeoff. The “what-if” analysis provides deeper insight into the sensor placement problem and provides a learning procedure that supports the decision-making process through more valuable knowledge of the problem. Acknowledgment Financial support received from the Spanish MCyT (Project DPI2002-00856), from the “Generalitat de Catalunya” (Project I0353), and from the European Community (Projects G1RD-CT-2001-00466 and GRD12000-25172) is fully appreciated.

Figure 12. Best performance vs cost for the retrofit case.

sensors, but also includes the relative costs associated with rearranging plant instrumentation. This cost definition also reduces the problem degeneracy given by the simpler cost definition of the previous case study. Figure 12 provides useful information that can be used to discard some level of investment. For example, the local minimum at cost 11 indicates that it makes no sense to consider arrangements of 11 or 12 sensors, given that better performance can be attained with 10 sensors. Moreover, it shows that the performance cannot be improved unless up to 13 sensors are aquired. This example shows the capability of the framework proposed to address the general sensor placement problem and to provide practical results for the decision maker. Additionally, a main advantage of the framework is its openness and the possibility of incorporating any calculation module (plug-in) for use in solving the problem or part of it, thus allowing for an increase in the system performance with no rewriting of code. 7. Conclusions This work addresses the sensor placement problem through an analysis of the information flows involved in a comprehensive problem definition and a general purpose solution approach. The problem statement

Nomenclature A ) incidence matrix A(j) ) accuracy of process variable j A(j)* ) accuracy upper bound for process variable j Cmax ) maximum cost allowable for instrumentation investment ck ) total cost of sensor k cck ) capital cost of sensor k cik ) installation cost of sensor k Mmax ) vector specifying maximum number of sensors per measuring point j Mmin ) vector specifying minimum number of sensors per measuring point j mmax ) maximum number of measuring devices per meaj suring point j mmin ) minimum number of measuring devices per meaj suring point j njk ) number of sensors of type k measuring variable j Q ) variance-coavariance matrix qj ) binary variable indicating whether the variable j is measured or not Rj(t) ) reliability of process variable j at time t Rj(t)* ) reliability lower bound for process variable j at time t rjk ) binary variable indicating whether sensor type k can measure variable j or not S ) sensor network already installed in an operating plant si ) sensor network i

Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004 1721 Greek Letters δmj ) number of possible (decision) sensors per measurement point j νjk ) number of already-installed sensors of type k per measuring point j Γ ) cost matrix Π ) performance matrix Θ ) Heaviside function Σ ) matrix of generated sensor networks Subscripts i ) sensor network, i ) 1, ..., I j ) measurement point/measured variable, j ) 1, ..., J k ) sensor types, k ) 1, ..., K

Appendix. System Performance Evaluation for the Dynamic Case Kalman filtering is a recursive procedure for estimating process state variables and their associated error variances. The algorithm simultaneously uses a statespace dynamic model (eq 34) and a measurement model (eq 35) to compute the optimal unbiased estimator of a state vector

xk ) Akxk-1 + Bkuk-1 + wk-1

(34)

yk ) Ckxk + vk

(35)

The subscript k represents time instant t ) kT at which the variables are sampled, where T is the sampling period. x is the vector of state variables, u is the vector of manipulated variables (if no control input is considered, u ) 0), and y is the vector of measurements. The matrices Ak, Bk, and Ck are matrices of appropriate dimension, and if their coefficients are time-independent, then the subscript k can be dropped. These matrices must be introduced for each sensor placement problem to define the model, although need only be introduced once at the beginning of the procedure. The vector v represents the random errors in measurements, and w is the vector of random disturbances. The covariance matrix Qk of vk is used to connect a given sensor network with the Kalman filter performance estimation. The Kalman filter first computes the state and the error covariance matrix Pk of the process variable using the model presented in eq 34

xˆ k/k-1 ) Axk-1/k-1 + wk-1

(36)

Pk/k-1 ) APk-1/k-1AT + Rk

(37)

where Rk is the variance matrix of wk. The next phase is the updating of the state estimation and its error covariance matrix using the process measurements y

xˆ k/k ) xˆ k/k-1 + kk(yk - Cxˆ k/k-1)

(38)

Pk/k ) (I - kkC)Pk/k-1

(39)

where kk is the Kalman filter gain, given by

kk ) Pk/k-1CT(CPk/k-1CT + QkT)-1

(40)

It is important to emphasize that the estimation of the error covariance matrix is independent of the availability of process measurements y. This interesting feature leads to the determination of a sensor network

performance at the design stage. Once the accuracy of each process variable is computed, it is important to provide a procedure to estimate the system performance (accuracy), Γ, of the complete process to allow for the comparison of the generated sensor networks in set Σ. The performance kjperf of the process variable j can be calculated by averaging [Pk/k]j over the entire time horizon k ) 0, ..., n as follows

1 n kjperf ) ( [Pk/k]j) n k)1

∑

(41)

Under conditions where Qk and Rk are time-invariant, both the estimations of the Pk/k error covariance and the Kalman gain kk stabilize quickly (in a few iterations) and then remain constant during the rest of the calculations. Therefore, the asymptotic value of Pk/k can also be used as a performance measure (see eq 42). This is the value used within this work. In fact, when the Kalman filter is applied to a system that is continuous and dynamic, the latter is preferred, whereas when conditions reflect short-lived batch systems the former is more appropriate.

kjperf ) lim([Pk/k]j) kfn

(42)

However, it is necessary to choose or develop a global performance measure for the performance of the overall system when given a particular sensor network. This global performance is selected as an elaborate function that relates the individual performance measure given by eq 42. A possible global performance measure can be constructed by comparing the currently evaluated system performance, kperf,E, corresponding to a partial measurement of the process variables involved, with that associated with the same system but in which all of the process variables involved in the process model are completely measured, kperf,C. When only a few process variables are of interest, only these will be considered, and they constitute S, the set of variables of interest

perf ) (k0 +

|kjperf,E - kjperf,C|)-1 ∑ j∈S

(43)

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Received for review August 2, 2003 Revised manuscript received December 17, 2003 Accepted January 21, 2004 IE034037E

Decision-Making Strategy and Tool for Sensor Network Design and

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