When seeking to improve a sensor network, operators can install additional instruments, replace old devices with new ones, or assign existing instruments to better locations
Access to good data in process plants is essential for process and quality control, production accounting, fault detection, and a variety of maintenance-related activities, such as planning for scheduled shutdowns. Many times, existing process data are no longer of the quality required to meet new production and safety standards. In addition, when techniques like online optimization are implemented, parameters need to be estimated to feed simulation models. As a result of these and other new needs, facilities often need to upgrade their existing instrumentation.
Techniques such as data reconciliation have helped to improve the quality of plant data. Indeed, when data reconciliation is used, one often obtains estimates of process variables that are more precise than measurements. In addition, non-measured variables and process parameters can be calculated, and biases in instrumentation can be identified. However, there is, a limit to these improvements, beyond which new instrumentation is required.
In this article, we briefly review new methodologies and concepts that allow process plant engineers and operators to conduct a systematic study to evaluate both the need for, and the feasibility of, a potential instrumentation upgrade.(1)
When evaluating any potential instrumentation upgrade, the primary goal is often to minimize costs (subject to performance constraints), while ensuring the system's ability to:
* Calculate estimates of measured and unmeasured variables with certain precision and accuracy (without bias)
* Identify malfunctioning instrumentation
* Detect and identify unsafe operating conditions
When data acquisition relies solely on instrument measurements, values for each variable are directly associated to the instrument that measures them. In this case, installing very precise and failure-proof instrumentation -- while costly -- would suffice to ensure the availability of precise and reliable estimates of process variables.
Since sensors often fail, some process operators seek to ensure data availability with redundancy. However, with redundancy comes potential data discrepancy. If the noise associated with the signal is small enough, these discrepancies can be ignored, but in most cases, these readings have to be reconciled.
Redundancy can be accomplished in several ways:
* Hardware redundancy. When two or more sensors are used to measure the same variable
* Software or analytical redundancy. When aside from measuring the variables, other estimate(s) can be obtained using mathematical model equations; the simplest of these are material-balance equations, as illustrated in Figure 1.
[ILLUSTRATION OMITTED]
Data reconciliation and gross-error detection are techniques that help engineers to obtain precise estimates, and at the same time identify instrument malfunction. Several books that clarify the many aspects of data reconciliation are available [24, 22, 32, 28, 26].(2) In addition, hundreds of articles have been devoted to the problem.
Instrument malfunction relates to situations that range from miscalibration or bias to total failure. In the absence of redundancy, miscalibration or bias cannot be detected (unless the deviation is so large that it becomes obvious). Redundancy, and especially software or analytical redundancy, is the only way to contrast data and determine possible malfunctions of this sort. This approach is named gross-error detection and is reviewed in the aforementioned references.
Instrumentation upgrade goals
Several goals exist for all designers of instrumentation networks for chemical process plants or refineries:
* Low cost (installation and maintenance costs)
* Improved precision
* Improved reliability and availability of the estimates of the key variables, which are a function of the reliability and availability of instrument measurements; these are, in turn, determined by instrument failure rates
* Better gross-error robustness [6], which encompasses three properties:
1. Gross-error detectability -- the ability to detect gross errors (usually biases) larger than a pre-specified size
2. Residual precision -- the precision of estimates left when a certain number of instruments fail
3. Gross-error resilience -- which controls the smearing effect of undetected gross errors
* The ability to perform process fault diagnosis and identification
* The ability to distinguish sensor failure from process failure
While all these goals seem equally important, cost is often the traditional objective function during sensor-network design. The first cost-based design procedure developed by the author [6] and solved using mathematical optimization can be summarized as follows:
Minimize total cost, including maintenance cost, subject to:
* Precision of key variables
* Reliability of key variables
* Gross-error robustness
This procedure does not take into account the ability of a sensor network to detect process faults and a logic for alarm systems. Even though important attempts are being made to address this issue [31], a specific model incorporating cost-effective alarm design has yet to be produced. Likewise, the direct incorporation of control-performance measures as additional constraints to this cost-optimal model has not yet been fully investigated.
From the exclusive point of view of fault detection, the procedure for instrumentation design is:
Minimize total cost subject to:
* Desired observability of faults
* Desired level of fault resolution
* Desired level of reliability of fault observation
* Desired level of gross-error robustness in sensor network
The combination of both goals -- that is, the design of a sensor network that is capable of performing estimation of key variables for monitoring, production accounting, and parameter estimation for online optimization, as well as fault detection, diagnosis and alarm -- is becoming a reality. In the rest of this article, existing methods to design and upgrade instrumentation for monitoring purposes, mainly focusing on the minimization of cost, are discussed. Methods to design sensor networks for fault detection and observation are also briefly reviewed.
In most cases, a cost-benefit analysis needs to be performed to determine the thresholds of all the properties that are required from the upgraded network. For example, in the case of production accounting, precision and gross-error robustness can be easily related to revenue, while in the case of quality control, precision, reliability and gross-error robustness can also be related to quality standards and, thus, ultimately, to lost revenue. A connection of this sort can be established for just about all scenarios of instrumentation and sensor design and upgrade.
Design for cost minimization
Consider just the constraints of precision in the following instrumentation-design problem:
Minimize total cost subject to:
* Desired precision of key variables The conceptual intricacies of this problem are illustrated in the box entitled "Optimal Location of Flowmeters" (p. 96-I&C-2).
To solve such a problem, integer variable are used to denote whether a variable is measured ([q.sub.i] = 1) or not ([q.sub.i] = 0). Thus, the investment cost can be represented by a summation of the type
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [c.sub.i] is the cost of the instrument in variable i.
The precision constraint is mathematically represented by requiring the standard deviation of the estimate, [[Sigma].sub.j] (which will be obtained using data reconciliation), to be smaller than a threshold value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This problem was studied by the author [6], and it is assumed that there is only one potential measuring device with associated cost [c.sub.i] for each variable (that is, no hardware redundancy), but this condition can be relaxed [6, 7]
To solve this problem, Reference [6] proposed a tree-enumeration procedure. Recently, Reference [16] proposed an alternative formulation and Reference [8] presented a mixed-integer, linear programming formulation. Finally, a series of techniques to introduce constraints that can force redundancy were introduced by Reference [11] for this case, and were extended to bilinear systems by Reference [7].
Design for maximum precision
Reference [23] proposed to design sensor networks that minimize the mean square error of the required quantities via:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[Sigma].sub.i] is the standard deviation of the i-th required quantity, and [n.sub.M] is the total number of measurements. This problem was efficiently solved using graph theory [22]. A problem maximizing the precision of only one variable was proposed by a team of the BP and University College, London [1, 2, 3]. The generalized maximum-precision problem [7] considers the minimization of a weighted sum of the precision of the parameters.
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [c.sub.T] is the total resource allocated to all sensors and [a.sub.j] are weights. A mathematical connection between the maximum-precision and the minimum-cost representations of the problem exists [10]. The connection states that the problems render the same solutions when the appropriate bounds for the constraints are used. This is why we concentrate on the minimum-cost model.
Parameter estimation and precision upgrade
Considerable attention is being paid these days to the issue of parameter estimation (heat exchanger heat transfer coefficients, flash and column separation efficiencies, and so on), especially in the context of the increasing popularity of online process optimization. For example, flowrate and temperature measurements are often used to predict the rate of fouling in heat exchangers, by simply calculating the heat transfer coefficients. These heat transfer coefficients are then used in a simulation model to schedule periodic cleaning and maintenance. Similarly, parameter-estimation techniques are often used to determine column efficiencies or other reactor parameters.
However, the establishment of sound parameter-estimation algorithms is challenging if the data contains too many gross errors. Therefore, data reconciliation and bias detection are a must if this task is to be done properly. In many cases, an instrumentation upgrade is needed to make this possible, or to improve the precision of these parametric estimates.
There are three possible ways to perform the upgrade of a sensor network:
* By adding new instruments
* By substituting new instruments for existing ones
* By relocating existing instruments to better locations
Engineers typically consider adding new instruments first. One of the first approaches to solve this problem was to combinatorially add measurements until the desired goals are met [17]. This aproach, however, does not use constraints, in particular cost, which is an important issue to consider.
Since upgrading requires capital expenditure, it must be done on the basis of a cost-benefit analysis. The costs of the instrumentation are straightforward to obtain. However, the benefits need to be somehow quantified. In the case of data-accuracy needs for accounting purposes, the benefit of adding new instruments can be quantified in terms of a decrease in lost revenue that would result from imprecise data. Describing it in simple terms, the larger the uncertainty in the assessment of the amount of raw materials purchased or products sold, the larger the probability of lost revenue. Thus, for example, a monetary value could be assigned to every percent of added accuracy in a given flow measurement.
In the case of process monitoring, one can associate a revenue loss for off-spec product, and in the case of online optimization, one can rely on measures of loss of economic performance that result a process whose operating conditions have deviated from optimum conditions due to plant-model mismatches [21].
In other cases, companies are looking less at the benefit of an instrumentation upgrade in monetary terms, and they simply plot the increased precision as a function of investment [3]. This approach is intuitive and allows the visualization of the effect of instrumentation cost on precision.
References [18, 19] rely on a three-step screening procedure to determine the best location for instrumentation:
1.Disregard measurements that have little or no effect on the parameters. This can be done by using a mathematical technique called singular value decomposition
2. Disregard measurements whose contribution to the precision of the parameters is small
3. Choose the set of measurements that exhibits good precision and leads to low interaction between the parameters
Unfortunately, this method does not take into account cost and does not offer a systematic procedure to make a final selection of the "best" set. In contrast, Reference [7] discusses linearization techniques in the context of cost-based minimization design procedures.
As noted above, aside from adding instrumentation, sensor networks can be upgraded by substituting new instruments for existing ones, or by relocating existing instruments. These options are sometimes cheaper. One example is the replacement of thermocouples by thermoresistance devices, or the relocation of existing thermocouples. Finally, relocation of concentration measurements only requires changes in the location of sampling, and is therefore inexpensive.
The minimum-cost model for upgrading a sensor network is shown in Reference [13]:
Minimize total cost of new instrumentation subject to:
* Desired precision of key variables [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
* Limitations on the number of new instruments per variable
* Limitations on the relocations that can be made
The mathematical model for this approach makes use of binary variables and is linear [13]. The second constraint establishes an upper bound on the number of sensors used to measure each variable. This number is usually one for the case of flowrate, but can be larger in the case of laboratory measurements of concentrations. A simple example to illustrate the technique is shown in the box entitled "Adding and Relocating Instruments" on p. 96-I&C-6.
One can of course try to solve a maximum-precision model by constraining the capital expenditure. As it was pointed out above, the two problems are equivalent [10, 7].
An example of a precision upgrade, using the maximum-precision ,model is shown in the box entitled "Adding Instruments to Improve Precision in Parameter Estimation," on p. 96-I&C-4.
Robust sensor networks
As it was described above, a robust sensor network provides meaningful values of precision, residual precision, variable availability, error detectability and resilience. These five properties encompass all the most desired features of a network. It is expected that more properties will be added to define robustness in the future. For example, when neural networks, wavelet analysis, principal component analysis (PCA), partial least squares (PLS) and other techniques for process monitoring are used, different definitions of robustness are warranted. A conceptual example showing the effect of robustness constraints are illustrated in the box entitled "Designing Robust Networks" on p. 96-I&C-10.
Reliable, repairable sensor networks
Reference [7] discusses the concepts of availability and reliability for the estimates of key variables (called estimation reliability and availability), and distinguishes them from the same concepts applied to instruments (called service reliability and availability). Reference [4] proposes to use an objective function to be maximized defined as the minimum estimation reliability throughout the whole network. In other words, the reliability of the system is maximized by maximizing its weakest element, which is the variable with the smallest reliability. This representation does not include cost. Therefore, to limit the cost, these authors only propose to explore the class of sensor networks that contain the minimum possible number of sensors, such that estimates for all the variables can be obtained. This obviously excludes redundant networks.
Although their procedure does not guarantee global optimality, it produces good results. Bilinear networks are also discussed in detail in Reference [5]. While this work used methods based on graph theory, genetic algorithms were successfully used by Reference [30], not only for reliable networks, but also a variety of other objective functions.
The cost-based representation for the design of the sensor network subject to reliability constraints is [12]:
Minimize the total cost of new instrumentation subject to:
* Desired reliability of key variables
The reliability of each variable is calculated using the failure probabilities of all the sensors participating in the corresponding material balances. If all sensors have the same cost, one obtains a problem where the number of sensors is minimized. Finally, the representation due to Reference [4] can be put in the form of a minimum cost problem. The details of such equivalency can be found in [12], where examples are shown. Finally, a single model containing precision and reliability constraints can be constructed.
When repairs are not performed, the service availability of a sensor is equal to its service reliability. In addition, the failure rate has been considered in a simplified way as a constant. However, in the presence of repairs, failure is no longer an event that depends on how many hours the sensor survived from the time it has been put in service. It is also conditioned by the fact that, due to preventive or corrective maintenance, the sensor has been repaired at a certain time after being put in service. These events condition the failure rate. We thus distinguish unconditional from conditional events in failure and repair.
These concepts are important because sensor-maintenance costs account for nearly 20% of all maintenance costs [25]. Its reduction or containment is therefore essential. The connection between failure rate, repair rate and the expected number of repairs, as well as illustrations of the impact of maintenance on sensor-network design, are described by [29]. These are also illustrated in the box entitled "Repairable Sensor Networks" on p. 96-I&C-8.
Design for fault diagnosis
Process faults, which typically are rooted in some unit, propagate throughout the process, altering the readings of instruments that measure pressure, temperature, flowrate and so on. The task of detecting and identifying faults is different from that of gross-error detection, which concentrates on instrument malfunction.
As a consequence, the discrimination between instrument malfunction and process fault is an additional task of the alarm system. And thus, the problem of designing an alarm system consists of determining the cost-optimal position of sensors, such that all process faults, single or multiple and simultaneous, can be detected and distinguished from instrument malfunction (biases). In addition, alarm settings need to be determined. In the text that follows, we will concentrate on the qualitative task of identifying the faults and not discuss the alarm settings.
The first attempt to present a technique to locate sensors was done by Reference [20], where fault trees are used based on failure probabilities. Since fault trees cannot handle cycles, the technique has not been developed further.
Reference [27] proposes an algorithm to obtain the minimum number of sensors that will guarantee that single and multiple faults can be observed and distinguished (resolved) from each other. Those authors use directed graphs (DG), that is, graphs without signs. The arcs of the DG represent a "will cause" relationship; that is, an arc from Node A to Node B implies that A is a sufficient condition for B, which, in general, is not true for a signed DG, where an arc represents a "can cause" relationship.
The strategy used to solve the problem is based on identifying directed paths from root nodes where faults can occur, to nodes where effects can be measured, called the observability set. Of all these paths, the objective is to choose the minimal subset of sensors from the observability set that would have at least one directed path from every root node. Consider the DG shown in Figure 6, where all the nodes corresponding to sensors are in the top row, mad the root nodes are in the bottom row.
[ILLUSTRATION OMITTED]
All the faults are observable if a directed path exists from every fault to at least one active sensor on the top row. For the example above, sensors in [C.sub.7] and [C.sub.8] constitute the minimal set. Indeed, [R.sub.1], [R.sub.2] and [R.sub.3] are observable from [C.sub.7], and the rest from [C.sub.8].
Maximum fault resolution is a property that guarantees that the location and number of faults can be always achieved. Therefore, a sensor network for maximum fault resolution is such that each fault has one and only one set of nodes from which it is observable.
Consider three fault nodes [R.sub.1], [R.sub.2], and [R.sub.3] (Figure 7). Clearly, if only one fault is expected to occur at a time, then the set [[C.sub.1], [C.sub.3]] would be adequate to distinguish between the three faults. A fault in [R.sub.1] is reflected in [C.sub.1], but not in [C.sub.3]. Similarly, a fault in [R.sub.2] is reflected in [C.sub.3], but not in [C.sub.1]. Finally, a fault in [R.sub.3] is reflected in both [C.sub.1] and [C.sub.3] simultaneously.
[ILLUSTRATION OMITTED]
Reference [9] presented a cost-optimal model for fault resolution, which is illustrated using a continuous stirred-tank reactor (CSTR) example in the box entitled ".Design of Instrumentation for Fault Detection" on p. 96-I&C-12. This example shows that different alternative solutions featuring the same number of sensors exists.
More recently, Reference [14] presented a minimum-cost model based on process and sensor failure probabilities. An extension of these problems to include normal monitoring goals simultaneously with fault detection and resolution goals is warranted because of the expected synergistic effects.
Conclusion
This article illustrates techniques for the design and upgrade of networks with several requirements such as precision and reliability in key variables, and the ability to handle gross errors. Sensor networks that are able to detect process faults can also be designed and upgraded. This emerging technology will soon find its direct application in practice.
OPTIMAL LOCATION OF FLOWMETERS
Consider the process flow diagram shown in Figure 2. Assume that flowmeters of precision 3%, 2% and 1% are available at costs $800, $1,500 and $2,500, respectively. When only one of these can be located in each stream and when precision is only required for variables [F.sub.1] and [F.sub.4], with standard deviation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], two solutions are obtained with a total cost of $3,000 (see table below).
[ILLUSTRATION OMITTED]
Solutions [F.sub.2] = 2% [F.sub.3] = 2% Cost = $53,000 [F.sub.2] = 2% [F.sub.4] = 2% Cost = $3,000
Although precision is achieved using, in this case, a non-redundant network, this solution does not offer bias-detection capabilities. This can only be achieved with analytical redundancy. Therefore, if at least one degree of redundancy is requested -- that is, the use of at least two ways of estimating each key variable -- then there are two solutions with a cost of C = $3,100, just $100 higher. These solutions are: ([F.sub.1] = 3%, [F.sub.2] = 3%, [F.sub.3] = 2%) and ([F.sub.1] = 3%, [F.sub.1] = 3%, [F.sub.1] = 2%). In Reference [11], a detailed description of how this can be formally requested through a mathematical model is provided. An example is discussed in the main text, on p. 96-1&C-10.
ADDING INSTRUMENTS TO IMPROVE PRECISION IN PARAMETER ESTIMATION
Figure 3 shows a set of heat exchangers where crude oil is heated using hot gas-oil coming from a column. The heat transfer coefficients for the heat exchangers are estimated using temperature and flowrate measurements.
[ILLUSTRATION OMITTED]
The location of the existing instrumentation is shown in the figure; their corresponding precision is 3% for flowmeters, and 2 [degrees] F for the thermocouples. The standard deviations of heat transfer coefficients calculated using the installed set of instruments are [12.27, 2.96, 3.06] (Btu/h [ft.sup.2] [degrees] F). To obtain these values, all redundant measurements have been used.
In order to enhance the precision of the parametric relationship, new instruments should be added. In this example, hardware redundancy is considered. Furthermore, different types of new instruments are available.
New flowmeters have a 3% precision and a cost of $2,250. Two thermocouples are available for streams [S.sub.1], [S.sub.4] and [S.sub.9]. These thermocouples have a standard deviation of 2 [degrees] F and 0.2 [degrees] F, and a cost of $500 and $1,500, respectively.
In the rest of the streams, only one thermocouple of 2 [degrees] F standard deviation, and a cost of $500, can be installed, and only one flowmeter is allowed to be added in all streams, except [S.sub.6] and [S.sub.8]. No thermocouple can be installed in stream [S.sub.8]; a maximum of two additional thermocouples can be installed in streams [S.sub.8], [S.sub.4] and [S.sub.9]; and a maximum of one in the rest.
Table 1 presents results for the upgrade problem using the minimum-cost approach. When there are two possible instrument options to measure a variable, as in the case of thermocouples for streams [S.sub.1], [S.sub.4] and [S.sub.9], the type of instrument, (1) or (2), is indicated between parentheses in the optimal solution set. For example, [T.sub.4] (1) indicates that the first instrument available to measure this temperature is a thermocouple with a precision of 2 [degrees] F.
Heat Area, [F.sub.T] [Cp.sub.h], exchanger [ft.sup.2] Btu/lb [degrees] F [U.sub.1] 500 0.997 0.6656 [U.sub.2] 1,100 0.991 0.6380 [U.sub.3] 700 0.995 0.6095 Heat [Cp.sub.c], exchanger Btu/lb [degrees] F [U.sub.1] 0.5689 [U.sub.2] 0.5415 [U.sub.3] 0.52
Table 1 (below) shows different cases of precision requirements. Some may exhibit several alternative solutions, such as in Case 3, while others may not be feasible, such as in Case 4.
TABLE 1. Results for the Minimum-Cost Problem Case [Sigma]*[U.sub.1] [Sigma]*[U.sub.2] [Sigma]*[U.sub.3] 1 4.0 4.0 4.0 2 3.5 2.0 2.5 3 3.0 1.5 2.5 4 3.5 2.0 2.0 Case [Sigma][U.sub.1] [Sigma][U.sub.2] [Sigma][U.sub.3] 1 3.6160 1.9681 2.7112 2 2.7746 1.6892 2.3833 3 2.7230 1.4972 2.2844 4 -- -- -- Case Cost Optimal set 1 $500 [T.sub.6] (1) 2 $1,500 [T.sub.2] (1) [T.sub.4] (1) [T.sub.6] (1) 3 $6,500 [F.sub.2] [F.sub.3] [T.sub.2] (1) [T.sub.4] (1) [T.sub.6] (1) [T.sub.9] (1) [F.sub.2] [F.sub.4] [T.sub.2] (1) [T.sub.4] (1) [T.sub.6] (1) [T.sub.9] (1) [F.sub.3] [F.sub.4] [T.sub.2] (1) [T.sub.4] (1) [T.sub.6] (1) [T.sub.9] (1) 4 -- --
ADDING AND RELOCATING INSTRUMENTS
To improve the estimation of the vaporization efficiency, [Eta], for the flash tank shown in Figure 4, an engineer wants to reallocate the instrumentation to a more advantageous position. The flash tank is well-instrumented and hardware redundancy on the feed flowrate is available.
[ILLUSTRATION OMITTED]
The precision of these flowmeters is 2.5 for both instruments on line [F.sub.1]; 1.515 for the flowmeter on [F.sub.2]; and 1.418 for the device on [F.sub.2]. In turn, the precision of the concentration measurements is 0.015 and 0.01 for [y.sub.1], respectively, and 0.01 for [y.sub.2] and [y.sub.3]. Finally, the pressure gage has a precision of 14 (all precision values in this example are in absolute values, not percentages).
New flowmeters are available. They have a cost of $350, $350, $400, and a precision of 2, 1.48 and 1.38, for [F.sub.1], [F.sub.2] and [F.sub.3], respectively. In turn, devices that measure new composition, with a cost $2,700 and a precision of 0.01, are available. Finally, a new pressure gage of the same precision and cost of $100 can be installed.
Flowmeters can be exchanged from [F.sub.1] and [F.sub.2] with a cost of $80 and vice versa, but not transferred to [F.sub.3], and only the concentration measurement of [y.sub.1] can be transferred to [y.sub.2] at no cost, and to [y.sub.3] at a cost of $50.
The first row of Table 2 shows results that represent the case for the existing instrumentation. The second column indicates the threshold precision required for the vaporization efficiency, [Eta], when solving a minimum-cost model, and the third indicates the precision obtained.
TABLE 2. Sensor Relocation & Upgrade Case [Sigma]* [Sigma] Cost, Reallo- $ cations 1 [degrees] 0.00438 -- -- 2 0.0038 0.00352 100 -- 0.00347 100 [y.sub.1] to [y.sub.2] 3 0.0033 0.00329 2,800 [y.sub.1] to [y.sub.2] Case New instruments 1 -- 2 P P 3 [y.sub.3] P
A reduction of the standard deviation from 0.00438 to 0.00347 results, if the laboratory analysis for the feed stream is relocated to the liquid stream, and a pressure sensor is added (Row 2). The cost for this case is $100. Higher precision (Case 3) is obtained by means of the relocation and the addition of another measurement. When more precision is required ([Sigma]* = 0.0031), no relocation and instrument addition can achieve this goal.
REPAIRABLE SENSOR NETWORKS
In the simplified ammonia loop shown in Figure 5, flowmeters for each stream may be selected from a set of three instruments with different precision, purchase cost and failure rates. Three flowmeters are available at a cost of $350, $250 and $200, respectively. These instruments are referred to as Type 1, 2 and 3, respectively. Their precision and failure rates are 1.5%, 2.5%, 3%; and 0.3, 0.6, 0.7 failures/yr, respectively.
[ILLUSTRATION OMITTED]
Maintenance expenses include spare part and labor costs of $10 and $40, respectively. A life cycle of 5 years, and an annual interest of 6%, are used. The limits on the requirements of precision, residual precision and availability (probability of being running properly) are included in Table 3 for two selected flowrates. The repair rate of instruments, a parameter that is a characteristic of the plant in consideration, has been varied between 1 and 20 repairs/yr. The results of the optimization problem are presented for each case in Table 4.
TABLE 3. Desian Requirements Flow Precision requirements availability Requirements [F.sub.1] -- 0.9 [F.sub.2] 1.5% -- [F.sub.5] 2.5% -- [F.sub.7] -- 0.9 TABLE 4. Results Repair Measured Instrument Cost, $ rate variables precision 1 [F.sub.1][F.sub.4][F.sub.5] 3 % 1 % 1 % 2,040.2 [F.sub.6][F.sub.7][F.sub.8] 1% 3% 2% 2 [F.sub.4][F.sub.5][F.sub.6] 3 % 3 % 1 % 1,699.8 [F.sub.7][F.sub.8] 3% 1% 4 [F.sub.4][F.sub.5][F.sub.6] 3 % 3 % 1% 1,683.7 [F.sub.7][F.sub.8] 3% 3% 20 [F.sub.4][F.sub.5][F.sub.6] 3 % 3 % 1% 1,775.2 [F.sub.7][F.sub.8] 3% 3% Repair Precision,% Availability rate ([F.sub.2]), ([F.sub.1), ([F.sub.7]) ([F.sub.5]) 1 0.8067 0.9021 1.2893 0.9021 2 0.9283 0.9222 1.9928 0.9062 4 1.2313 0.9636 1.9963 0.9511 20 1.2313 0.9983 1.9963 0.9969
In the first case, the repair rate is comparatively low. Consequently the availability of instruments in the life cycle is also relatively low. To satisfy the availability of key variables, the optimal solution includes a set of six instruments. Three of these instruments are of Type 1, meaning the flowmeter has a relatively low failure rate, high precision and the highest cost. For this reason, precision and residual precision are better than the required values. When the repair rate is 2, an optimal solution exists, consisting of five instruments. Two of these are of Type 1 and the rest are of Type 3. Consequently, the total instrumentation cost decreases.
A lower instrumentation cost is obtained for a repair rate of 4. In this case, although sensors are located on the same streams as in the previous case, one sensor of higher failure rate is installed to measure F8. The results of the last case show that the influence of availability constraints decreases for high repair rates. The cost increases because of the effect of increasing the repair rate (from 4 to 20).
In short, the repair rate has a direct influence on the availability of a variable. If the repair rate is high, the design follows the requirements of precision and residual precision constraints. Thus, the availability of a variable is likely to be driving the design far lower repair rates and cost may increase because it is necessary to incorporate more instruments to calculate the variable by alternative ways.
DESIGNING ROBUST NETWORKS
We now add residual precision capabilities to the example of Figure 2 (p. 96-1&C-2). Consider now that residual precision of order one (precision left after one measurement is deleted from the set) is added to flows F1 and F4 as follows. The requirements are that precision should not drop below 1.5% and 3% respectively, when one measurement is lost. The solution is to put sensors of precision 2%, 3%, 3%, 3% in F1 through F4, respectively at a cost of $3,900. Assume now that residual precision is requested to the same level as precision (1.5% and 2%, respectively). Then two alternative solutions with a cost of $5,500 are obtained, as shown in Table 5.
TABLE 5. Solutions of the Residual Precision-Constrained Problem [F.sub.1] [F.sub.2] [F.sub.3] [F.sub.4] 1% 2% 2% -- 1% 2% -- 2%
Not only is the cost higher, but there is one more degree of redundancy. For more-complex problems, the number of alternatives will increase, requiring new criteria to further screen the potential alternatives.
We now turn to adding error detectability. As the capability of detecting smaller gross errors in the data increases, so does the precision of the sensor network. However, if the requirement is too stringent, no network may be able to satisfy it. For instance, if an error-detectability level of 3.9 is required, that is, that errors higher than 3.9 times the standard deviation will be detected, the resulting network will be able to detect gross errors of 3.9 times the precision of the sensors. Assuming a statistical power of 50% for the detection algorithm, two solutions from a set of only 4 feasible solutions are found, with a cost of $4,800 (Table 6). If an error-detectability level of 3.4 is requested, the problem has only one solution, namely flowmeters with precision of 1%, 3%, 1%, 1% for F1 through F4, respectively, with a cost of $8,300.
TABLE 6. Effect of Error-Detectability Constraints [F.sub.1] [F.sub.2] [F.sub.3] [F.sub.4] 1% 3% -- 2% 1% 3% 2% --
Finally, we illustrate a resilience requirement, which limits the smearing effect of gross errors of a certain size when they are undetected. If a level of 3 times the standard deviation for all measurements is required, then the solution is to put sensors with precision 1%, 3%, 1%, 1% in F1 through F4, respectively, with a cost of $8,300. Relaxing (increasing) the resilience levels and maintaining the error delectability at the same level may actually lead to solutions of higher cost, even to infeasibility. Thus, robustness has a cost.
DESIGN OF INSTRUMENTATION FOR FAULT DETECTION
Consider a continuous stirred-tank reactor (CSTR) example [15], shown in Figure 8. Figure 9 shows the graph that represents the connections between faults and sensors [9].
[ILLUSTRATIONS OMITTED]
The results of running a mathematical programming model for four cases are shown in Table 7. These cases correspond to single and double fault-detection capabilities. The table is organized as follows: The costs used in each case are first given, followed by one or two columns depicting the solutions obtained. An "x" indicates that the corresponding node should be measured. Case 1 uses the same cost for all sensors, which is equivalent to minimize the number of sensors used. This set is an alternative set to the one obtained in Reference [15] (the fourth column).
TABLE 7. Results for Four Cases Node Cost,$ Cost, $ name Case 1 q* q* Case 2 q* CA 100 x x 100 TS 100 100 TC 100 1 x VT 100 x 1 Fc 100 x 1 F4 100 1 Tc 100 x x 1 x N 100 1 x PS 100 100 PC 100 1 VP 100 x x 1 x Fvg 100 1 VS 100 100 VC 100 x 1 x VL 100 1 F3 100 1 F2 100 1 F 100 x 1 # sensors 5 5 5 Node Cost, $ Cost, $ name Case 3 q* Case 4 q* CA 100 100 TS 100 1 x TC 100 100 VT 100 100 Fc 1 x 1 x F4 1 1 Tc 1 x 1 x N 100 100 PS 100 1 x PC 100 100 VP 100 x 100 x Fvg 1 x 1 VS 100 1 VC 100 100 VL 100 100 F3 1 x 1 x F2 1 1 F 1 1 # sensors 5 6
If one assumes that the cost of measuring the outlet concentration (CA) is too high, and one does not want to use the controlled variables sensors as valid sensors for the problem of fault sensors searching, then one can alter the costs accordingly (Case 2), obtaining the result shown in the sixth. This solution suggests that N, the number of moles in the gaseous phase, should be measured. If one wants to avoid this measurement one can assign N a high cost (Case 3). Furthermore, for this case we assigned a high cost to controllers and valves to avoid them, as well. The result is shown in the eighth column. Case 4 is the case where controlled-variable sensors can be used together with process variables.
(1.) A more detailed examination of this subject is offered in a book published by the author; Bagajewicz, M., "Design and Upgrade of Process Plant Instrumentation," Technomic Publishing Co., Lancaster, Pa., www.techpub .com, 2000, ISBN: 1-56676-998-1.
(2.) The References for this article are organized alphabetically, so they are hot called out in numerical order.
References
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[9.] Bagajewicz M., and Fuxman, A., An MILP Model For Cost Optimal Instrumentation Network Design And Upgrade For Fault Detection. Proceedings of the 4th IFAC Workshop on Online Fault Detection & Supervision in the Chemical Process Industries, Seoul, Korea, June 8-9, 2001.
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Miguel Bagajewicz is the Samuel Roberts Noble Foundation Presidential Professor at the ,University of Oklahoma's School of Chemical Engineering (100 East Boyd St., T-335, Norman, OK 73019; Phone: 405-325-5458; Fax: 405-325-5813), where he is also the director of the Center for Engineering Optimization. Prior to that, he worked for Simulation Sciences. Bagajewicz has authored more than 100 publications and a book (Reference [7]). He holds M.S. and PhD degrees in chemical engineering from the Calif. Inst. of Technology (Caltech) in 1984 and 1987, respectively. He is also owner of Ok-Solutions (www.oksolutionsinc.com), a company devoted to implement process technology developed through his efforts at the university.
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