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Design and Retrofit of Reliable Sensor Networks C. Benqlilou, M. Graells, E. Musulin, and L. Puigjaner* Universitat Polite` cnica de Catalunya, Chemical Engineering Department, ETSEIB, Avenida Diagonal 647, E-08028 Barcelona, Spain
This work proposes an approach to sensor network design and retrofitting that combines quantitative process knowledge and fault tree analysis into a new methodology for evaluating the reliability of process variable estimation, taking into account both hardware and functional redundancy. The reliability of estimating process variables is used to determine the sensor network reliability, which, in turn, is used for the design and retrofit of the network. Thus, a reliable sensor network design and retrofit problem is formulated as a mathematical programming optimization problem and solved using genetic algorithms. The performance of this proposed methodology is compared with the current approaches using different case studies and handling several scenarios. Introduction The design of sensor networks is the basis upon which the performance of monitoring systems,1,2 fault diagnosis systems,3-5 and/or optimization systems6 relies. The design of sensor networks includes the determination of sensor characteristics such as type, number, reliability, placement, and so on while minimizing an objective function such as the instrumentation cost. The design of a sensor network that allows for the observation of all process variables was first addressed by Vaclaveck and Loucka.7 Later, this problem was solved by Madron and Veverka8 regarding the minimum total cost. However, sensor failure can lead to a reduction of measurements, thereby seriously affecting control, monitoring, and optimization systems and, hence, the whole process performance. Thus, it is necessary to ensure that it is still possible to observe key process variables even if one or more sensors fail. The observation of a process variable can be expressed mathematically as a nonnull probability of estimating this variable at a given time t (reliability). The evaluation of this probability is closely related to the different ways of estimating a process variable given a sensor failure probability and a specific sensor network. On the basis of these concepts, a method for optimal sensor location in a pure flow process was developed using graph theory by Ali and Narasimhan.9,10 In their proposal, these authors first determined the minimum number of sensors that ensures system observability (or system redundancy). Using this number, the complete set of networks that ensure system observability is obtained. Finally, the network that maximizes the system reliability is obtained from this set. However, this approach does not directly consider the network cost or the reliability of the individual process variables. These points were later addressed by Bagajewicz and Sa´nchez.11 Additionally, these authors transformed the problem presented by Ali and Narasimhan9,10 into a mathematical programming problem. Similarly, by analyzing the cycles of the process graph and taking into * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +34-93-401.66.78. Fax: +34-93401.09.79.
account the observability of the variables and the reliability of the sensors,12 they determined the optimal measurement system according to reliability and cost analysis in conjunction with observability constraints. In this work, an approach is proposed for evaluating the reliability of process variable estimation taking into account all of the redundancies offered by the system in terms of either functionality or hardware. In this evaluation, both quantitative process knowledge and fault tree analysis are considered and combined, which leads to a more suitable and practical evaluation of reliability. The reliability of estimating each one of the key process variables is then used to determine sensor network reliability, which, in turn, is used as a set of sensor placement constraints in the design and retrofitting procedure. Thus, a general sensor placement formulation is first proposed that considers the number (hardware redundancy or multiplicity) of sensors of a given type (reliability) that are to be assigned to a given process variable while satisfying the reliability requirements at the minimum total cost. This proposal can be applied for network design as well as for retrofitting. Nevertheless, an analysis of the minimum reliability value that allows both the observability and the redundancy of the system is performed, along with a determination of the best cost/system reliability tradeoff. The general sensor placement optimization problem formulated is successfully solved using genetic algorithms. The performance of this proposed methodology is compared with that of the current approaches on some motivating case studies. Reliability Evaluation Sensor Reliability. The sensor reliability is an intrinsic quality of the sensor that can be defined as the probability, rsk(t), of the nonfailure of sensor k at time t. Poisson’s law, for instance, can be used to represent a monotonically decreasing probability function as reported by Luong and co-workers.12 More complex probability functions could also be used to describe the system reliability, such as the mean time before failure (MTBF). The sensor reliability, rsk(t), is an important parameter for the sensor network design problem, which
10.1021/ie049605k CCC: $27.50 © 2004 American Chemical Society Published on Web 11/09/2004
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Figure 1. Process variable estimation reliability. Table 1. Reliabilities rF1(t) for Different Sensor Networks reliability equation rF1(t)
value
reliability type
rsS1(t) rsS3(t) or rsS4(t) ) rsS3(t) + rsS4(t) - rsS3(t) rsS4(t) rsS5(t) and rsS6(t) ) rsS5(t)rsS6(t)
0.80 0.96 0.64
sensor hardware functional
consists of determining the number, njk, of each specific sensor type k to be assigned to each process variable j so that the reliability of estimating j at a time t, rj(t), is greater than a minimum allowable reliability. Process Variable Estimation Reliability. The probability, rj(t), of estimating process variable j at time t must simultaneously consider sensor reliability, hardware reliability, and functional reliability. For the sake of simplicity, the time reference is omitted from here on. As an example, consider a simple process unit with one feed stream 1 and two product streams 2 and 3, as shown in Figure 1. For this process unit, the three flow variables (F1, F2, and F3) are related to each other through the pure mass balance expressed by
F1 ) F2 + F3
(1)
Assume that all sensor failures occur randomly and independently (although this assumption should be revised when considering some dependent failure mechanisms such as power surges) and that the mass flow of all streams can be measured using sensors k ) S1, S2, S3, S4, S5, S6, whose sensor reliabilities are all assumed to be 0.80 at time t. Therefore, using the property that, if P1 and P2 are the probabilities of two dependent events then (P1) or (P2) ) P1 + P2 - P1 × P2, the reliability of estimating variable F1 at time t is summarized in Table 1 for each of the cases in Figure 1. Table 1 shows that, in these three illustrating cases (sensor reliability, hardware reliability, and functional reliability), rF1 > 0, so variable F1 is observable. Moreover, in the case of hardware reliability, the reliability of estimating variable F1 is greater than the reliability of each one of the sensors measuring it, showing the existence of more than one way to estimate F1. Thus, rF1 is directly related to the degree of redundancy and can still be estimated even if one sensor fails. However, in the cases of sensor reliability and functional reliability, no sensor failure is permitted. These results illustrate the effect of sensor placement on the reliability of estimating process variables. Hardware Reliability. If more than one sensor is placed at a given measuring point, the resulting reli-
ability is calculated by using a union of disjoint products. This is possible because the sensors operate independently. For example, if three similar sensors S1, S2, and S3 are placed at the same location, the resulting reliability is evaluated by analyzing the disjoint events: {(three sensors are operational) or (only two sensors are operational and one fails) or (one sensor is operational and two sensors fail)}. A general expression for the hardware reliability evaluation is given by K
rhj ) 1 -
[1 - rsk(t)]n ∏ k)1
jk
(2)
where rhj is the reliability of estimating j at time t by using njk sensors of type k ) 1, ..., K, available in the catalog. This set K contains sensors with different “sensor reliabilities” and cost. If no sensor is assigned to j (njk ) 0), then rhj ) 0, whereas if only one sensor is assigned, rhj ) rsk. In the sensor network design and retrofit procedure presented in this work, the hardware reliability is first calculated using eq 2. The resulting reliability is associated with a new virtual sensor to be used for the subsequent calculation of functional reliability. The cost of this virtual sensor is equal to the sum of the costs of all sensors k used for the hardware reliability. Functional Reliability. When the relationships between process variables are considered, the value of a process variable j can be indirectly estimated through the measurements of other process variables and the process model. Accordingly, the reliability of indirectly estimating such a variable is given by the reliability of measuring the other variables on which depends. To determine this functional reliability, it is convenient to consider two cases regarding the dependency of the equations in the model. A simple and particular expression can be derived for the case of independent equations, whereas a more complex procedure is required for dependent equation sets. Independent Equations. If the equations used for estimating process variable j are linearly independent, the reliability of estimating the variable j at time t is calculated by Vi
∏
rj′h(t)] rj ) ∪ [ i∈Qj j′*j)1
(3)
where Qj is the set of independent equations that can be used to estimate the value of process variable j. Each equation i belonging to Qj has a sensor assigned to each one of its variables except j. Vi is the number of variables
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the columns represent the process variables and the rows represent the process equations. Step 2. Generate the matrix A* that contains all of the possible combinations of process equations (the rows of matrix A). The number of such combinations is equal to 2I, where I is the number of equations
A* ) BA
Figure 2. Proposed algorithm for variable estimation reliability.
in the equation i minus 1. This equation can also be applied if the probabilities of sensor failure are not equal. Dependent Equations. Independent equation sets is not the general case. Generally, systems present an interdependency of process variables that generates additional complexity. Therefore, an analytical expression for the functional reliability when dependent equations are involved is quite difficult to generate even though the logic behind it is clear. Thus, the reliability rj is a function not only of the sensor placement, Σ ) {njk}, but also of the process structure. In the case of linear or linearized process models, the quantitative process knowledge can be represented by the incidence matrix A ) {aij}, defining which variables j ) 1, ..., J, occur in which equations i ) 1, ..., I, for a set of equations defining the process
rj ) f(Σ,A)
(4)
For that reason, the reliability evaluation is obtained algorithmically. This work presents a practical and efficient algorithm for obtaining an analytical expression for process variable estimation. The different steps involved are as follows (see Figure 2): Step 1. Identify the incidence matrix A of the digraph corresponding to the process under consideration, where
(5)
where B is a [2I, I] matrix whose elements are given by the series of binary numbers from 1 to 2I. Step 3. Determine the set of equations allowing for the estimation of each variable of interest j (i.e., variables with nonnull minimum required reliabilities, > 0. This set of equations is given by matrix A/j rmin j that is obtained by discarding the rows (equations) in A* having null elements at column j (variable of interest). Step 4. Considering each row of A/j (equation) as a set of variables, remove from A/j any row that is a superset of any other row. Hence, the resulting matrix A// j contains the minimum number of equations (rows) for estimating the variable j (i.e., the degree of redundancy of j). Step 5. Generate the fault tree for each matrix A// j . Fault trees provide a logical modeling framework for analyzing and representing the interactions between component reliabilities. Different software is available for creating and supporting fault trees. This step allows for the generation of a formula for determining the different ways of estimating j. Step 6. Generate the analytical expression for rj from the logical expression. It is worth mentioning that this analytical expression has to be provided prior to the sensor network design and retrofit procedure. If a process variable is not measured, the sensor associated with it has a null reliability value. It is worth noting also that the compact form of the matrix approach provided can be much easier to use than the associated graphs in a certain number of cases.13 For an illustration of the steps of the previous algorithm, consider the plant represented schematically in Figure 3. This case study14 corresponds to a petrochemical plant consisting of a train of two distillation columns in which a group of n-paraffins is separated from kerosene. Step 1. For this plant, the pure flow balance is expressed as
F1 ) F2 + F3
(6)
F3 ) F4 + F5
(7)
that, in matrix form, is given by eq 8, where the columns represent the flow rate variables and the rows represent the mass balance equations around the stripper and the redistillation units
A)
(
1 -1 -1 0 0 0 0 1 -1 -1
)
(8)
Step 2. The number of combinations that can be generated from the above matrix A is 22 - 1, and the corresponding matrix B is
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Figure 3. Simplified petrochemical plant.
( )
0 1 B) 1 0 1 1
(9)
The matrix A* can be obtained by multiplying
( )(
)
0 1 1 -1 -1 0 0 A* ) BA ) 1 0 ) 0 0 1 -1 -1 1 1 1 -1 -1 0 0 0 0 1 -1 -1 1 -1 0 -1 -1
(
)
(10)
Step 3. If the reliabilities of process variables F1 and F3 are required, their corresponding A/j matrices are
A/1 )
(
1 -1 -1 0 0 1 -1 0 -1 -1
)
A/3 )
(
1 -1 -1 0 0 0 0 1 -1 -1
)
(11)
Step 4. There is no need to remove any row. Step 5. The corresponding fault trees are shown in Figure 4. The mapping between the generated trees and the probabilities allows for the reliability of estimating F1 and F2 at a time t to be represented analytically as
rF1 ) {rFh 2 and [rFh 3 or (rFh 4 and rFh 5)]} or (rFh 1) (12) rF3 ) (rFh 1 and rFh 2) or (rFh 4 and rFh 5) or (rFh 3)
(13)
Step 6. The analytical expressions for estimating F1 and F3, respectively, are
r1 ) rh1 + rh2 rh3 - rh1 rh2 rh3 + rh2 rh4 rh5 - rh2 rh3 rh4 rh5 rh1 rh2 rh4 rh5 + rh1 rh2 rh3 rh4 rh5 (14) r3 ) rh3 + rh1 rh2 + rh4 rh5 - rh1 rh2 rh3 - rh3 rh4 rh5 rh1
rh2
rh4
rh5
+
rh1
rh2
rh3
Figure 4. Fault trees for process variable estimation reliability.
rh4
rh5
(15)
Considering the sensor networks given in Table 2 for the plant presented in Figure 3 and taking into account the fact that the equations estimating F3 are independent, the reliability of estimating F3 at time t can be obtained using eq 15 and is equal to 0.87.
Finally, any attempt to maximize the reliability of a particular variable can result in other variables becoming unobservable. Indeed, eliminating a sensor that belongs to several equations impacts more negatively on the sensor network reliability than eliminating a sensor that appears in only one unique equation.
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Table 2. Effect of Sensor Network on rF3(t) for Functional Reliability process variable
sensor reliability at time t
F1 F2 F3 F4 F5
0.8 0.8 0.5 0.8 0.8
Table 3. Comparison of System Reliability Evaluation Approaches
Consequently, it is necessary to determine the sensor network reliability R. For this purpose, the sensor reliability and variable estimation reliability should undoubtedly be considered. Sensor Network Reliability. Luong and co-workers12 proposed an analytical expression for evaluating the sensor network reliability. Considering that the reliabilities, rsk, of all sensors are all equal to p, the reliability of the sensor network is given by d
R)
Rn(1 - p)npd-n ∑ n)0
(16)
where the coefficients Rn give the number of configurations allowing for n sensor breakdowns while ensuring system observability. d is the minimum degree of redundancy of all variables, with a redundant variable of degree g being a variable that can still be estimated in the case of the simultaneous failure of any g sensors. This reliability expression is based on the minimum degree of redundancy of all redundant process variables, so it might discard some sensor failures even though the system’s redundancy allows them. This procedure also presents the impossibility of evaluating the reliability of a subset of the total variables of interest. Alternatively,9 the sensor network reliability can be given by the minimum of the process variables reliabilities
R ) min(rj) j
(17)
This expression is based on the philosophy that a chain cannot be stronger than its weakest link. This conclusion is somewhat similar to the proposal by Luong and co-workers12 that network reliability is fixed by the minimum degree of redundancy of the process variables. Ali and Narasimhan9 exploit all the system redundancy by using graph theory; however, their proposal is not applicable for a subset of all variables and requires a greater computational effort when considered from a mathematical programming perspective (the optimization of the sensor network design problem). In our proposal, the network reliability is also defined by the minimum estimation reliability of all variables. The evaluation of rj is given in the previous section. Consider the plant depicted in Figure 3, where the reliabilities of process flow variables F1 and F3 are of interest, and assume that all variables are measured with similar sensors having rhj ) 0.8. 1. If the approach of Luong and co-workers is applied,12 only two scenarios are considered. The first considers that all the five sensors are operational, leading to a reliability of 0.327. The second considers that one sensor fails while the four remaining sensors are still operating, thus leading to a reliability of 0.409.
sensor network reliability at time t
R(t) value
Luong et al.12 (eq 16) Ali et al.9 proposed approach (eqs 12, 13, and 17)
0.73 0.96 0.96
Therefore, the reliability R is equal to 0.73, resulting from the summation of the reliabilities of the two scenarios. 2. In this work, more scenarios are considered that allow for more than two sensor breakdowns. These scenarios can be easily deduced from the analysis of the trees illustrated in Figure 4. Alternatively, by using eqs 12 and 13, it is possible to get rF1 ) 0.96 and rF3 ) 0.97, and their minimum is R ) 0.96. Table 3 summarizes the results of network reliability values. It can be seen that this work provides values for the network reliability that are more accurate than those obtained following Luong and co-workers12 and equal to that of Ali and Narasimhan’s work.9 Nevertheless, in the latter case, Ali and Narasimhan’s approach9 is harder to incorporate into an optimization procedure because of the inherent difficulty of handling pure graph theory concepts. Generic Design of Reliable Sensor Networks Sensor Network Model. The design of measurement systems addressed with the goal of reliability was earlier considered by Ali and Narasimhan.9 In that work, given a minimum number of sensors N*, all sensor networks that ensure system observability are first determined using graph theory. Next, among all the generated networks, the one that offers the maximum system reliability R is selected. This approach does not directly consider the sensor cost and does not guarantee desired reliability levels on specific variables. These points were addressed by Bagajewicz and Sa´nchez.11 They transformed the model proposed by Ali and Narasimhan9 into a mathematical programming model as follows
max R
(18)
qj ) N* ∑ ∀j
(19)
subject to
Ej(q) ) 1
∀j
(20)
qj ) (0, 1)
∀j
(21)
In the above sensor placement model, the first constraint fixes the number of sensors. If observability is considered, N* is equal to the number of process variables minus the number of independent equations describing the process flowsheet, and if redundancy is considered, a number greater than N* has to be selected. The observability requirement on variable j is mathematically expressed by imposing the condition that the number of different ways to estimate j, the degree of estimability Ej(q), is equal to 1 (see eq 20). The extension of this model to include cost and individual reliability requirements is proposed11 as follows
Ind. Eng. Chem. Res., Vol. 43, No. 25, 2004 8031
min
cjqj ∑ ∀j
(22)
subject to
rj g rmin j qj ) (0, 1)
∀j ∈ MR ∀j ∈ MJ
(23) (24)
where MJ is the set of locations where sensors can be placed, MR is the set of variables whose reliabilities are to be constrained, and cj is the cost of measuring j. However, this model allows for the specification only of the possibility of placing a sensor in a given location (qj ) 0, 1), but it does not consider hardware redundancy, and it assumes that the reliability of the system is implicitly included in the reliability of variable estimation. These considerations have been included in this work. are selected according Additionally, the values of rmin j to the redundancy requirements. If the value of this , is nonnull, then the variable j is minimum value, rmin j observable, and if its value is larger than rhj , then the variable j is redundant. The minimum value of the system reliability, Rmin, is selected using a Pareto analysis of the cost/reliability tradeoff. Furthermore, decisions are no longer represented by binary variables but by integer variables for selecting among a catalog the number njk of sensors (multiplicity) of a certain type k and reliability rsk to be assigned to a given process variable/measuring point j
min
∑∑cknjk
(25)
njk ∀k ∀j
subject to
R g Rmin
(26)
rj g rmin j
(27)
e njk e mmax mmin j j
(28)
A generic objective function that considers both the design and retrofit proposed Benqlilou et al.1 can also be used instead of eq 25 to consider the retrofitting case, namely
∑k cckΘ(∑j njk - ∑j n0jk) + ∑k ∑j cik|njk - n0jk|)
min[ njk
(29)
where n0jk is the number of already installed sensors of type k at measuring point j; cck and cik are the capital and installation costs, respectively, of sensor type k; and Θ is the Heaviside function. The first part of eq 29 evaluates the capital cost taking into account the fact that, for each sensor type k and for all measuring points j, the sum of allocated sensors of type k is subtracted from the number of already installed sensors of the same type (if n0jk ) 0, ∀j ∀k). The second part of eq 29 calculates the installation cost, and the absolute value is applied to allow inclusion of the uninstallation cost. Equation 29 allows for the consideration of both physical and inferential sensors. Inferential sensors have proven to be applicable and have been successfully incorporated into industrial applications.14 For the case of inferential sensors, the reliability of the inferred
variable is calculated using the methodology presented earlier, and the corresponding installation and capital costs are null. This MINLP problem is difficult to address with mathematical programming techniques because of the combinatorial tree (type, multiplicity), the form of the objective function, and the evaluation of the constraints. This makes very attractive the use of metaheuristic techniques such as genetic algorithms (GAs). The next section describes and discusses the use of GAs to solve the sensor network design and retrofit problem just formulated. Given the proposed formulation, the problem size (i.e., the number of different sensor networks) is an important aspect to consider. Assume a process with J measuring points and a catalog allowing for tj different sensors for each location j (j ) 1, ..., J). Finally, suppose that, at each measuring point, up to mj sensors can be located. Therefore, the number of different sensor combinations pj at each measuring point j can be calculated as the number of combinations of tj + 1 elements taken by mj points with repetition
pj ) C′(tj + 1, mj)
(30)
Because C′(s, k) ) C(s + k - 1, k), where C denotes the combination operator without repetition, eq 30 can be reformulated as
pj ) C(tj + mj, mj) )
(tj + mj)! tj!mj!
(31)
The total number of sensor networks can be calculated as the product of the sensor combinations at each measuring point J
P)
∏ j)1
J
pj )
∏ j)1
(tj + mj)!
(32)
tj!mj!
To simplify, suppose that tj ) t and mj ) m for j ) 1, 2, ..., J. Then J
P)
∏ j)1
(tj + mj)!
)
tj!mj!
[
]
(t + m)! t!m!
J
(33)
Finally, assume that the only decisions to be made involve whether each process variable j is measured. Hence, t ) 1 and m ) 1, which leads to the solution space corresponding to the problem presented by Bagajewicz and Sa´nchez11
P ) 2J
(34)
Sensor Network Solution Based on Genetic Algorithms. Recently, different researchers2,15 have shown the ability of GAs to solve sensor network optimization problems. The most important aspect for the successful application of the GA technique is the codification of the individuals (i.e., possible solutions, chromosomes). This codification contains the necessary information required to evaluate the objective function (fitness function) as well as check the feasibility of the individuals. The fundamental aspects to consider in this codification are the selection and placement of measuring
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Figure 5. Genetic algorithm codification. Table 4. Design of Reliable Sensor Networks Using a Unique Sensor Type njk S1 S2 S3 S4 S5
F1 1 -
rmin (t) 0.81 j rj(t) 0.9 cja
devices. Concretely, the codification includes aspects of sensor networks: reliability, cost, location, and multiplicity. Additionally, the constraints on the allowable number of sensors per location as well as the technical feasibility of placing a particular sensor at a given point are also managed in this codification. Therefore, (i) each chromosome (individual) represents a sensor network, (ii) each gene in a chromosome corresponds to a measuring point, (iii) the value of each allele in each gene reflects the multiplicity of a specific sensor type (see Figure 5), and (iv) each sensor type is associated with a sensor cost and reliability. On the basis of this codification, an initial population is generated randomly. Then, by means of the selection, crossover, and mutation operators, new generations are produced. For each iteration, the population will contain a fraction (1/10) of the best individuals of the preceding generation. The population size is set at Nind ) 100 using the roulette wheel operator, and nine out of 10 are selected to be crossed. Two-point crossover is applied with a probability of Pc ) 0.7, and the rate of mutation is Pm ) Pc/JK, where JK is the length of the individual. The feasibility function is evaluated for each individual of the generated population. This function verifies three constraints: network reliability, variable estimation reliability, and input parameter consistency. If one of these constraints is not satisfied, a very high value is assigned to the objective function (the total cost). The GA implementation was performed in Matlab using the toolbox developed at the University of Sheffield.17 Ammonia Plant Case Study. A simplified ammonia synthesis plant, taken from the work by Kretsovalis and Mah,16 is considered for illustration of the performance of the solution methodology proposed in addressing different design and retrofit situations. This case study (Figure 6) has been used in several sensor network design works addressing reliability.9-11 This plant consists of six units and eight streams (j ) 1-8), with node 6 representing the plant mass
0 -
F3 0 -
F4 0
F5 1 -
0.81 0.81 0.81 0.81 0.9 0.9 0.81 0.9
1500 0 a
Figure 6. Simplified ammonia plant network.
F2
0
0
F6
F7
F8
1 -
0 -
0.81 0.9
0.81 0.81 0.81 0.81
1700 2000 0
0
rsk(t) 0.9 0.9 0.9 0.9 0.9
0
Total cost ) 5200 euros.
Table 5. Flexibility-Based Sensor Type for Designing Reliable Sensor Networks njk S1 S2 S3 S4 S5
F1 1 -
rmin (t) 0.81 j rj(t) 0.84 cja
F2 0 -
0 -
F4 0
F5 1 -
0.81 0.81 0.81 0.81 0.84 0.84 0.82 0.86
1500 0 a
F3
0
0
F6 0 -
F7 1 -
0.81 0.81 0.82 0.84
1700 0
F8 1
rsk(t) 0.70 0.75 0.80 0.85 0.90
0.81 0.93
1500 2800
Total cost ) 7500 euros.
balance. Additionally, the presence of different cycles makes the case more attractive for sensor network design and retrofit purposes. Steady-State Case. Following the case study in Bagajewicz and Sa´nchez,11 consider a catalog of five specific sensors (Sn, n ) 1-5) having the costs 1500, 1700, 2000, 2300, and 2800 euros, respectively. It is assumed that all of the available sensors have the same reliability, rsk ) 0.9.10 Additionally, assume that some sensor/variable assignments are prohibited as indicated by dashes in the above tables. Finally, assume that the , for the eight variminimum allowable reliability, rmin j ables (Fj, j ) 1-8) is set to 0.81. This problem was successfully solved, and results are presented in Table 4. The same values for the total cost and for rj were obtained as shown in the penultimate rows of Table 4. The values of njk obtained by the proposed solution methodology are also equal to the results reported for the minimization of the cost function given by eq 29. The GA performance is of little interest in this case because of the reduced solution space (P ) 28 ) 256) that allows for the solution to be reached in very few iterations. Next, different sensor types with lower reliability, rsk, are considered (Table 5). To satisfy reliability constraints, F7 and F8 are now measured instead of F6, and additional sensors are required, as should be expected.
Ind. Eng. Chem. Res., Vol. 43, No. 25, 2004 8033 Table 6. Flexibility Based on Selecting Sensor Type njk S1 S2 S3 S4 S5
F1 0 -
F2 1 -
rmin (t) 0.81 0.81 j rj(t) 0.85 0.85 a
cj
0 a
F3 0 -
F4 0
F5 1 -
0.81 0.81 0.81 0.85 0.85 0.92
2000 0
0
F6
F7
F8
1 -
1 -
0
0.81 0.94
0.81 0.92
0.81 0.86
rsk(t) 0.75 0.80 0.85 0.90 0.95
Table 8. Hardware Redundancy and Reliable Design of Sensor Networks njk S1 S2 S3 S4 S5
1700 2000 1500 0
Table 7. System Reliability and Reliable Design of Sensor Network F2
S1 S2 S3 S4 S5
1 -
1 -
rmin j (t) rj(t)
0.90
0.90 0.90 0.85
cja
0.93 1500
a
F3 0 -
F4 0
0.93 0.93 0.93 2000
0
0
F5
F6
F7
1 -
1 -
1 -
0.85
0.85
0.85 0.85
0.89 1700
0.89 2000
F8 0
F4
0 -
0
F5 1 -
0.90 0.90 0.85 0.85 0.93 0.93 0.86 0.89
3000 0
0
0
F6
F7
F8
1 -
1 -
0
0.85 0.89
0.85 0.89
0.85 0.86
rsk(t) 0.75 0.75 0.75 0.75 0.75
1700 2000 1500 0
Total cost ) 8200 euros.
Table 9. Retrofitting of a Reliable Sensor Networks rsk(t) 0.75 0.75 0.75 0.75 0.75
0.89 0.93 1500
F3
0 -
2 -
cja a
F1
F2
rmin (t) 0.90 j rj(t) 0.93
Total cost ) 7200 euros.
njk
F1
0
Total cost ) 8700 euros.
It can be seen that this results in a change in the selection/placement of measuring devices and an increase in the total instrumentation cost. These types of solutions are obtained only when models considering sensor type selection are used, such as the more general model presented in this work. Moreover, if the reliability of these sensors is slightly increased (Table 6), the total cost is reduced by a different assignment, njk, while satisfying reliability constraints. Furthermore, this solution also produces a slight improvement in certain rj values. This analysis demonstrates that permitting the incorporation of different sensor types into the design of a reliable sensor network allows for the reliability requirements to be achieved at a lower cost. From a general point of view, more flexibility is given to the decision maker. To further illustrate the importance of including the hardware redundancy and system reliability R(t), new scenarios are proposed. Let us assume that lower sensor reliability is considered for all available sensors (rsk) 0.75) and that greater minimum sensor network reli) 0.9) than ability is requested for F1, F2, and F3 (rmin j ) 0.85). The results of for the rest of the variables (rmin j such a scenario are presented in Table 7 where the total investment cost to satisfy the reliability constraints is 8700. If multiplicity is considered next (Table 8), the results are significantly improved. That is, to satisfy the same reliability constraints, a lower investment is needed (8200). This is because of the assignment of two sensors of type S1 to F1 instead of assigning S3 to F3. For retrofitting purposes, consider an operating plant whose instrumentation (that given in Table 8) has to be updated to achieve new tighter reliability constraints ). Furthermore, assume that any sensor/location (rmin j assignment is possible (i.e., each sensor k can be assigned to each location j), thus increasing the number of decision variables. Table 9 presents the solution obtained when the installation costs are assumed to be zero. Even in this
njk S1 S2 S3 S4 S5
F1 1 0 0 0 0
F2 0 0 0 0 0
F3 1 0 0 0 0
rmin (t) 0.97 0.97 0.97 j rj(t) 0.98 0.98 0.98 cja
0 a
0
F4 1 0 0 0 0
F5 0 1 0 0 0
F6 0 0 1 0 0
F7 0 0 0 0 0
F8 1 0 0 0 0
rsk(t) 0.75 0.75 0.75 0.75 0.75
0.97 0.97 0.97 0.97 0.97 0.972 0.972 0.972 0.98 0.971
1500 1500
0
0
0
1500
Total cost ) 4500 euros.
case requiring a larger computational effort, the solution is obtained by the GA in few generations. Dynamic Case. In a number of cases, industrial practice can require the consideration of the dynamic behavior of processes; thus, the necessity to design reliable sensor network for dynamic systems. From the point of view of sensor location problem, such a network is needed to ensure that the necessary data that describe the dynamic behavior of the process will be available with a given reliability, regardless of the way in which these data will later be processed. Therefore, the dynamic case is addressed here by assuming that the accumulation terms are measurable variables. From the reliability perspective, the point is getting the information (measuring) rather than the way in which it is used in the process models. Thus, equations relating stream and accumulation terms have to be included in matrix A. Furthermore, adequate sensors for measuring the accumulation have to be added to the catalog. When the ammonia plant case study is addressed dynamically, the design of the sensor network has to be considered at the level of measurements corresponding to the nodes of the process graph (obviously discarding node 6, representing the overall mass balance) (Figure 7). The starting point of this dynamic case is the last solution obtained for the steady-state case. Next, additional assignment opportunities are considered: the assignment to the level measuring points L1-L5 of level meters of types S6-S8, whose reliabilities and costs are given in Table 10. This new dynamic example (Table 11) can be regarded as the retrofitting case of the instrumentation network given in Table 9. This optimization problem considers 13 process variables (8 flows and 5 levels), 8 kinds of sensors in the catalog (5 flowmeters and 3 level meters), and a maximum multiplicity equal to 2 for each measuring
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Figure 7. Ammonia plant network. Dynamic case. Figure 8. Costs of the networks obtained for different reliability requirements. The information can be a useful tool for decision making.
Table 10. Level Meter Characteristics for the Dynamic Case Study S6 S7 S8
cost
reliability
2000 3500 5000
0.60 0.70 0.90
GA with a population of 30 individuals and considering up to 500 generations. The average computational cost for each run was 120 CPU s on a 512-MB RAM PC with an AMD Athlon XP 1600 processor. A most interesting point in the profile is that given by the reliability value of 0.90, which corresponds to an inflection in the cost/performance trend. Table 12 provides the details for this solution. Despite the lower reliability requirement (0.90 instead of 0.97), this solution is more expensive than the starting point because the estimation of more process variables has to be guaranteed. The network designed includes three new flowmeters (S1, S2, and S4) and three new level meters of type S6. The resulting cost is 11500 euros. The last point in the profile in Figure 8, corresponding to the most reliable and expensive point, is examined next. This solution corresponds to a common reliability requirement of 0.98 and is given in Table 13. Four level meters were selected in this case for nodes L2-L5. It is also interesting to note the sensor duplication introduced for flow F1. Thus, satisfying this harder constraint given by Rmin ) 0.98 results in a cost of 20000 euros. To validate the search performance and the solution obtained, the same case study (Rmin ) 0.98) was addressed using 105 generations and 500 individuals, resulting in 20 CPU h of total search time (Table 14). Convergence was reached at a cost of less than 18200 euros, and the solution obtained improved the objective function by less than 10%. This new solution offers a lower-cost sensor assignment at the expense of reliability, which results in being much closer to the problem bounds (rmin ). j
Table 11. Retrofitting Starting Point for the Dynamic Case Study njk F1 F2 F3 F4 F5 F6 F7 F8 L1 L2 L3 L4 L5 rsk(t) S1 S2 S3 S4 S5 S6 S7 S8
1 0 0 0 0 -
0 0 0 0 0 -
1 0 0 0 0 -
1 0 0 0 0 -
0 1 0 0 0 -
0 0 1 0 0 -
0 0 0 0 0 -
1 0 0 0 0 -
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0.70 0.75 0.80 0.85 0.90 0.60 0.70 0.90
point. Thus, the size of the solution space corresponding to this problem is obtained using eq 32 as J
P)
) )
∏ j)1
(tj + mj)! tj!mj!
∏ j∈flows
[
(5 + 2)!
(3 + 2)!
5!2!
3!2!
][
(5 + 2)! 5!2!
8
∏ j∈levels
]
(3 + 2)! 3!2!
5
) 3.78‚1015
(35)
To address the cost/reliability tradeoff, the problem was solved for a series of minimum system reliability (Rmin) in order to obtain useful information for the decision-making process.1 Figure 8 shows the cost of the solution found for each of the problems obtained for a given value of Rmin. The corresponding optimization runs were performed using
Table 12. Best Instrumentation Network Obtained for Rmin ) 0.90a njk
F1
F2
F3
F4
F5
F6
F7
F8
L1
L2
L3
L4
L5
S1 S2 S3 S4 S5 S6 S7 S8
1 0 0 0 0 -
1 0 0 1 0 -
1 0 0 0 0 -
1 0 0 0 0 -
0 1 0 0 0 -
0 0 1 0 0 -
0 1 0 0 0 -
1 0 0 0 0 -
0 0 0
0 0 0
1 0 0
1 0 0
1 0 0
rmin (t) j rj(t)
0.900 0.923
0.900 0.955
0.900 0.919
0.900 0.951
0.900 0.970
0.900 0.973
0.900 0.971
0.900 0.952
0.900 0.910
0.900 0.904
0.900 0.958
0.900 0.940
0.900 0.917
cjb
0
3800
0
0
0
0
1700
0
0
0
2000
2000
2000
a
500 generations and 30 individuals. b Total cost ) 11500 euros.
rsk(t) 0.70 0.75 0.80 0.85 0.90 0.60 0.70 0.90
Ind. Eng. Chem. Res., Vol. 43, No. 25, 2004 8035 Table 13. Best Instrumentation Network Obtained for Rmin ) 0.98a njk
F1
F2
F3
F4
F5
F6
F7
F8
L1
L2
L3
L4
L5
S1 S2 S3 S4 S5 S6 S7 S8
2 0 0 0 0 -
1 0 0 1 0 -
1 0 0 0 0 -
1 0 0 0 0 -
0 1 0 0 0 -
0 0 1 0 0 -
0 1 0 0 0 -
1 0 1 0 0 -
0 0 0
1 0 0
1 0 0
0 0 1
1 0 0
rmin (t) j rj(t)
0.980 0.991
0.980 0.992
0.980 0.984
0.980 0.995
0.980 0.994
0.980 0.995
0.980 0.995
0.980 0.998
0.980 0.981
0.980 0.981
0.980 0.997
0.980 0.997
0.980 0.992
cjb
1500
3800
0
0
0
0
1700
2000
0
2000
2000
5000
2000
L2
L3
L4
L5
a
rsk(t) 0.70 0.75 0.80 0.85 0.90 0.60 0.70 0.90
500 generations and 30 individuals. b Total cost ) 20000 euros.
Table 14. Best Instrumentation Network Obtained for Rmin ) 0.98a njk
F1
F2
F3
F4
F5
F6
F7
F8
L1
S1 S2 S3 S4 S5 S6 S7 S8
2 0 0 0 0 -
0 1 1 0 0 -
2 0 0 0 0 -
1 0 0 0 0 -
0 1 0 0 0 -
0 0 1 0 0 -
1 0 0 0 0 -
1 0 0 0 0 -
0 0 0
1 0 0
1 0 0
1 0 0
2 0 0
rmin (t) j rj(t)
0.980 0.992
0.980 0.990
0.980 0.991
0.980 0.987
0.980 0.987
0.980 0.990
0.980 0.990
0.980 0.990
0.980 0.980
0.980 0.987
0.980 0.994
0.980 0.985
0.980 0.988
cjb
1500
3700
1500
0
0
0
1500
0
0
2000
2000
2000
4000
a
105
rsk(t) 0.70 0.75 0.80 0.85 0.90 0.60 0.70 0.90
generations and 500 individuals. Total cost ) 18200 euros. b
Conclusions In this work, the sensor network design problem is addressed at two levels: the evaluation of the reliability and the optimization of the sensor network. First, a practical and efficient method for evaluating the reliability of variable estimation is presented that is based on a combination of both quantitative process knowledge and fault tree analysis. Second, once the system reliability is defined, a mathematical programming model for plant instrumentation is formulated and solved using genetic algorithms. The solution of this problem provides the assignment of measuring devices (quantity and type) that have to be added and/or reallocated in an operating plant in order to lead its performance to the desired reliability target at a minimum cost. The approach of this work provides the decision maker with the flexibility to analyze the different alternatives that could be considered when designing a reliable sensor network. This flexibility is illustrated in the different examples presented, including sensor type selection, hardware reliability, reliability constraints on both network and individual variables, and retrofitting capabilities. The dynamic case was also demonstrated to be addressable in terms of reliability in a more complex case study. Accumulation terms were included in the process model, and adequate accumulation sensors were incorporated into the sensor catalog. The approach presented is also quite close to dealing with the nonlinear case. Designing reliable sensor networks for nonlinear systems is an important aspect of our present work and is the subject of a new paper that is underway. Acknowledgment Financial support for this research received from the Generalitat de Catalunya FI programs and from the
European community (Project G1RD-CT-2001-00466) is acknowledged. Nomenclature A ) incidence matrix A* ) matrix resulted from combining matrix A’s rows A/j ) matrix focusing variable j A// j ) minimum matrix focusing variable j B ) matrix of binary coefficients cj ) total cost of sensing variable j, euro ck ) total cost of sensor k, euro cik ) installation (or uninstallation) cost of sensor k, euro cck ) capital cost of sensor k, euro d ) minimum degree of redundancy of all variables Ej ) number of different ways to estimate variable j J ) set of process variable i ) equation that belong to set I K ) set of available sensors in the catalog MR ) set of variables for which reliability is required MJ ) set of locations where it is possible to place a sensor mj ) multiplicity allowed at each measuring point j mmin ) minimum allowable number of sensors for location j j mmax ) maximum allowable number of sensors for locaj tion j N ) number of sensors N* ) minimum number of sensors n ) number of admitted sensor breakdowns njk ) number of sensors of type k assigned to location j P ) number of sensor networks that can be generated p ) particular value of sensor reliability pj ) number of sensor arrangements that can be located at each position j Qj ) set of independent equations for estimating j qj ) binary variable indicating whether it is possible to assign a sensor to j R(t) ) sensor network (system) reliability
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Rmin(t) ) minimum value of the sensor network reliability rj(t) ) probability of estimating variable j at time t rsk(t) ) probability of nonfailure of sensor k at time t rhj (t) ) reliability resulting from placing a set of sensors at the same location j rmin (t) ) minimum reliability value on the estimation of j variable j t ) time at which the reliability is evaluated tj ) sensors available in the catalog of each measurement point j Vi ) number of variables in equation i Greek Letters Rn ) number of configurations with a value of n that ensure system observability Θ(x) ) heaviside function (equal to x if x g 0 and equal to 0 otherwise) Σ ) set of sensor assignments njk Subscripts i ) process model equation j ) location/variable where it is possible to place a sensor k ) sensor type
Literature Cited (1) Benqlilou, C.; Graells, M.; Puigjaner, L. Decision-Making Strategy and Tool for Sensor Networks Design and Retrofit. Ind. Eng. Chem. Res. 2004, 43, 1711-1722. (2) Musulin, E.; Bagajewicz, M. J.; Nougue´s, J. M.; Puigjaner, L. Design and Upgrade of Sensor Networks for Principal Components Analysis Monitoring. Ind. Eng. Chem. Res. 2004, 43, 21502159. (3) Trave´-Massuye`, L.; Escobet, T.; Milne, R. Model-based Diagnosability and Sensor Placement Application to a Frame 6 Gas Turbine Subsystem. Proceedings of the Seventeenth International Joint Conference on Aritifical Intelligence (IJCAI-01); Seattle, WA, Aug 4-10, 2001; pp 551-556. (4) Hodouin, D.; Mazzour, E.; Berton, A. Le placement strate´gique des capteurs, la de´tection des pannes et la re´conciliation des donne´es pour une utilisation optimale des mesures. In Les techniques de l’industrie mine´ rale; SIM (La Socie´te´ de l’Industrie Mine´rale): Marseille, France, 2001.
(5) Raghuraj, R.; Bhushan, M.; Rengaswamy, R. Locating Sensors in Complex Chemical Plants based on Faults Diagnosis Observability Criteria. AIChE J. 1999, 45, 310-322. (6) Fraleigh, L. M.; Guay, M.; Forbes, J. F. Sensor selection for model-based real-time optimization: Relating design of eperiments and design cost J. Process Control 2003, 13, 667-678. (7) Vaclaveck, V.; Loucka, M. Selection of Measurements Necessary to Achieve Multicomponent Mass Balances in Chemical Plants. Chem. Eng. Sci. 1976, 1199-1205. (8) Madron, F.; Veverka, V. Optimal Selection of Measuring Points in Complex Plants by Linear Models. AIChE J. 1992, 38, 227-236. (9) Ali, Y.; Narasimahan, S. Sensor Network Design For Maximizing Reliability of Linear Processes. AIChE J. 1993, 39, 820-826. (10) Ali, Y.; Narasimhan, S. Redundant Sensor Network Design for Linear Processes. AIChE J. 1995, 41, 2237-2249. (11) Bagajewicz, M. J.; Sa´nchez, M. C. Cost-optimal design of reliable sensor networks. Comput. Chem. Eng. 2000, 23, 17571762. (12) Luong, M.; Maquin, D.; Huynh, C. T.; Ragot, J. Observability, Redundancy, Reliability and Integrated Design of Measurement Systems. In IFAC Symp. on Intelligent Components and Instrument Control Applications, Budapest; 1994. (13) Maquin, D.; Luong, M.; Ragot, J. Some ideas about the design of measurement systems. Presented at the Third European Control Conference (ECC’95), Rome, Italy,, Sep 5-8, 1995. (14) Ruiz, D.; Benqlilou, C.; Nougue´s, J. M.; Ruiz, C.; Puigjaner, L. Proposal To Speed Up the Implementation of an Abnormal Situation Management in the Chemical Process Industry. Ind. Eng. Chem. Res. 2001, 41, 817-824. (15) Sen, M.; Narasimhan, S.; Deb, K. Sensor Network Design of Linear Processes Using Genetic Algorithms. Comput. Chem. Eng. 1998, 385-390. (16) Kretsovalis, A.; Mah, R. S. H. Observability and Redundancy Classification in Generalized Process Network. Comput. Chem. Eng. 1988, 12, 671-703. (17) Genetic Algorithm Toolbox Web Site, http://www.shef. ac.uk/gaipp/ga-toolbox/ (accessed Aug 1, 2003).
Received for review May 11, 2004 Revised manuscript received September 2, 2004 Accepted September 13, 2004 IE049605K
