1598:
2230:
2330:
1727:. A variable (or system) is observable if the models and sensor measurements can be used to uniquely determine its value (system state). A sensor is redundant if its removal causes no loss of observability. Rigorous definitions of observability, calculability, and redundancy, along with criteria for determining it, were established by Stanley and Mah, for these cases with set constraints such as algebraic equations and inequalities. Next, we illustrate some special cases:
2211:. The level of total redundancy is the sum of sensor redundancy and topological redundancy. We speak of positive redundancy if the system is calculable and the total redundancy is positive. One can see that the level of topological redundancy merely depends on the number of equations (the more equations the higher the redundancy) and the number of unmeasured variables (the more unmeasured variables, the lower the redundancy) and not on the number of measured variables.
1586:
1597:
178:
166:
847:
3143:
2229:
501:
enterprise system. This conversion may require manual, or intelligent reconciliation of the converted values . Systems must be set up to ensure that accurate data are sent to production and from production. Inadvertent operator or clerical errors may result in too much production, too little production, the wrong production, incorrect inventory, or missing inventory.
2794:
seen in the second case above, where streams a and b are in a cycle of unmeasured streams. Redundancy classification follows, by testing for a path of unmeasured streams, since that would lead to an unmeasured cycle if the measurement was removed. Measurements c and d are redundant in the second case above, even though part of the system is unobservable.
598:
2329:
2879:
Result validation is the set of validation or verification actions taken after the reconciliation process and it takes into account measured and unmeasured variables as well as reconciled values. Result validation covers, but is not limited to, penalty analysis for determining the reliability of the
3177:
It is important to note that the remediation of gross errors reduces the quality of the reconciliation, either the redundancy decreases (elimination) or the uncertainty of the measured data increases (relaxation). Therefore, it can only be applied when the initial level of redundancy is high enough
3173:
Gross error relaxation targets at relaxing the estimate for the uncertainty of suspicious measurements so that the reconciled value is in the 95% confidence interval. Relaxation typically finds application when it is not possible to determine which measurement around one unit is responsible for the
2793:
In 1981, observability and redundancy criteria were proven for these sorts of flow networks involving only mass and energy balance constraints. After combining all the plant inputs and outputs into an "environment node", loss of observability corresponds to cycles of unmeasured streams. That is
3170:
reconciled values. Once the gross errors are detected they are discarded from the measurements and the reconciliation can be done without these faulty measurements that spoil the reconciliation process. If needed, the elimination is repeated until no gross error exists in the set of measurements.
2866:
Data filtering denotes the process of treating measured data such that the values become meaningful and lie within the range of expected values. Data filtering is necessary before the reconciliation process in order to increase robustness of the reconciliation step. There are several ways of data
2214:
Simple counts of variables, equations, and measurements are inadequate for many systems, breaking down for several reasons: (a) Portions of a system might have redundancy, while others do not, and some portions might not even be possible to calculate, and (b) Nonlinearities can lead to different
472:
Other sources of errors when calculating plant balances include process faults such as leaks, unmodeled heat losses, incorrect physical properties or other physical parameters used in equations, and incorrect structure such as unmodeled bypass lines. Other errors include unmodeled plant dynamics
1720:, where sensors are duplicated in order to have more than one measurement of the same quantity. Redundancy also arises when a single variable can be estimated in several independent ways from separate sets of measurements at a given time or time averaging period, using the algebraic constraints.
522:
identification and elimination has been presented. In the late 1960s and 1970s unmeasured variables were taken into account in the data reconciliation process., PDR also became more mature by considering general nonlinear equation systems coming from thermodynamic models., , Quasi steady state
500:
Data reconciliation is a serious issue for enterprise-control integration. The data have to be valid to be useful for the enterprise system. The data must often be determined from physical measurements that have associated error factors. This must usually be converted into exact values for the
3169:
Gross error elimination determines one measurement that is biased by a systematic error and discards this measurement from the data set. The determination of the measurement to be discarded is based on different kinds of penalty terms that express how much the measured values deviate from the
3241:
As PDR enables to calculate estimates even for unmeasured variables in a reliable way, the German
Engineering Society (VDI Gesellschaft Energie und Umwelt) has accepted the technology of PDR as a means to replace expensive sensors in the nuclear power industry (see VDI norm 2048,).
3029:, then no gross errors exist with 95% probability. The chi square test gives only a rough indication about the existence of gross errors, and it is easy to conduct: one only has to compare the value of the objective function with the critical value of the chi square distribution.
192:
taken at different places throughout the industrial site, for example temperature, pressure, volumetric flow rate measurements etc. To understand the basic principles of PDR, it is important to first recognize that plant measurements are never 100% correct, i.e. raw measurement
1795:. One needs to know the value of two of the 3 variables in order to calculate the third one. The degrees of freedom for the model in that case is equal to 2. At least 2 measurements are needed to estimate all the variables, and 3 would be needed for redundancy.
842:{\displaystyle {\begin{aligned}\min _{x,y^{*}}&\sum _{i=1}^{n}\left({\frac {y_{i}^{*}-y_{i}}{\sigma _{i}}}\right)^{2}\\{\text{subject to }}&F(x,y^{*})=0\\&y_{\min }\leq y^{*}\leq y_{\max }\\&x_{\min }\leq x\leq x_{\max },\end{aligned}}\,\!}
3429:
VDI-Gesellschaft
Energie und Umwelt, "Guidelines - VDI 2048 Blatt 1 - “Control and quality improvement of process data and their uncertainties by means of correction calculation for operation and acceptance tests”; VDI 2048 Part 1; September 2017",
3133:
is the number of equations. Including energy balances means adding equations to the system, which results in a higher level of redundancy (provided that enough measurements are available, or equivalently, not too many variables are unmeasured).
53:
Models can have different levels of detail, for example one can incorporate simple mass or compound conservation balances, or more advanced thermodynamic models including energy conservation laws. Mathematically the model can be expressed by a
1558:
1762:) of a mathematical system, i.e. the minimum number of pieces of information (i.e. measurements) that are required in order to calculate all of the system variables. For instance, in the example above the flow conservation requires that
473:
such as holdup changes, and other instabilities in plant operations that violate steady state (algebraic) models. Additional dynamic errors arise when measurements and samples are not taken at the same time, especially lab analyses.
2846:. Based on information redundancy, estimates for these unmeasured variables can be calculated along with their accuracies. In industrial processes these unmeasured variables that data reconciliation provides are referred to as
2039:, i.e. the number of additional measurements that are at hand on top of those measurements which are required in order to just calculate the system. Another way of viewing the level of redundancy is to use the definition of
1431:
476:
The normal practice of using time averages for the data input partly reduces the dynamic problems. However, that does not completely resolve timing inconsistencies for infrequently-sampled data like lab analyses.
2164:
1713:. Instead, redundancy arises from combining sensor data with the model (algebraic constraints), sometimes more specifically called "spatial redundancy", "analytical redundancy", or "topological redundancy".
1708:
Data reconciliation relies strongly on the concept of redundancy to correct the measurements as little as possible in order to satisfy the process constraints. Here, redundancy is defined differently from
523:
dynamics for filtering and simultaneous parameter estimation over time were introduced in 1977 by
Stanley and Mah. Dynamic PDR was formulated as a nonlinear optimization problem by Liebman et al. in 1992.
151:, which incorporates all the above-mentioned system constraints (for example the mass or heat balances around a unit). A variable could be the temperature or the pressure at a certain place in the plant.
1350:
30:
and reconciliation by correcting measurements in industrial processes. The use of PDR allows for extracting accurate and reliable information about the state of industry processes from raw measurement
42:
Industrial processes, for example chemical or thermodynamic processes in chemical plants, refineries, oil or gas production sites, or power plants, are often represented by two fundamental means:
1571:
of the corresponding measurement. The standard deviation is related to the accuracy of the measurement. For example, at a 95% confidence level, the standard deviation is about half the accuracy.
2077:
603:
1585:
149:
3158:
can be applied that indicate whether or not a gross error does exist somewhere in the set of measurements. These techniques of gross error remediation are based on two concepts:
3027:
1210:
1121:
1032:
943:
1260:
510:
PDR has become more and more important due to industrial processes that are becoming more and more complex. PDR started in the early 1960s with applications aiming at closing
1150:
488:, so high frequency noise is mostly eliminated. The result is that, in practice, data reconciliation is mainly making adjustments to correct systematic errors like biases.
431:
2978:
2543:
2037:
883:
2947:
2509:
1970:
1883:
2980:
of the probability density function of a chi-square distribution (e.g. the 95th percentile for a 95% confidence) gives an indication of whether a gross error exists: If
249:
2438:
1635:
1061:
972:
371:
342:
2824:
and increase their accuracy and precision: on the one hand they reconciled
Further, the data reconciliation problem presented above also includes unmeasured variables
91:
3494:
Alexander, Dave, Tannar, Dave & Wasik, Larry "Mill
Information System uses Dynamic Data Reconciliation for Accurate Energy Accounting" TAPPI Fall Conference 2007.
2466:
1793:
1760:
459:
1444:
2844:
2822:
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2766:
2744:
2722:
2700:
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2590:
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1927:
1905:
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1818:
1701:
1679:
1657:
1282:
587:
397:
306:
3111:
2209:
2188:
1991:
212:
3458:
Modelling of a Crude Oil
Distillation Unit in Term of Data Reconciliation with ASTM or TBP Curves as Direct Input – Application : Crude Oil Preheating Train
3131:
3060:
Advanced process data reconciliation (PDR) is an integrated approach of combining data reconciliation and data validation techniques, which is characterized by
3050:
1170:
1081:
992:
903:
560:
3154:
the reconciliation results. Therefore, it is important to identify and eliminate these gross errors from the reconciliation process. After the reconciliation
3052:-th penalty term is outside the 95% confidence interval of the normal distribution, then there is reason to believe that this measurement has a gross error.
2880:
reconciliation, or bound checks to ensure that the reconciled values lie in a certain range, e.g. the temperature has to be within some reasonable bounds.
1731:
2907:
that is normally distributed with mean equal to 0 and variance equal to 1. By consequence, the objective function is a random variable which follows a
461:. For ease in deriving and implementing an optimal estimation solution, and based on arguments that errors are the sum of many factors (so that the
1358:
1798:
When speaking about topological redundancy we have to distinguish between measured and unmeasured variables. In the following let us denote by
2072:
3499:
Rankin, J. & Wasik, L. "Dynamic Data
Reconciliation of Batch Pulping Processes (for On-Line Prediction)" PAPTAC Spring Conference 2009.
3222:
PDR finds application mainly in industry sectors where either measurements are not accurate or even non-existing, like for example in the
1563:
In other words, one wants to minimize the overall correction (measured in the least squares term) that is needed in order to satisfy the
3312:
Studies on System
Engineering I. On the Application of the Calculus of the Observations of Calculations of Chemical Engineering Balances
3064:
complex models incorporating besides mass balances also thermodynamics, momentum balances, equilibria constraints, hydrodynamics etc.
3032:
The individual test compares each penalty term in the objective function with the critical values of the normal distribution. If the
177:
165:
3446:
Stanley G.M., and Mah R.S.H., "Observability and
Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981)
539:
exist in the set of measurements, since they may bias the reconciliation results and reduce the robustness of the reconciliation.
3230:
are difficult or expensive to position (see ); or where accurate data is of high importance, for example for security reasons in
1287:
3223:
2066:, which is the difference between the number of variables (measured and unmeasured) and the number of equations. Then one gets
1710:
3445:
2911:, since it is the sum of the square of normally distributed random variables. Comparing the value of the objective function
3235:
2888:
Result validation may include statistical tests to validate the reliability of the reconciled values, by checking whether
531:
Data reconciliation is a technique that targets at correcting measurement errors that are due to measurement noise, i.e.
3414:
Stanley G.M. and Mah, R.S.H., "Observability and
Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)
3340:
3174:
gross error (equivalence of gross errors). Then measurement uncertainties of the measurements involved are increased.
3080:
2215:
conclusions at different operating points. As an example, consider the following system with 4 streams and 2 units.
3373:
3532:
1717:
1564:
2903:
If no gross errors exist in the set of measured values, then each penalty term in the objective function is a
3413:
251:. When using measurements without correction to generate plant balances, it is common to have incoherencies.
96:
2908:
496:
ISA-95 is the international standard for the integration of enterprise and control systems It asserts that:
3495:
3484:
Plant Monitoring and Fault Detection: Synergy between Data Reconciliation and Principal Component Analysis
2858:
Data validation denotes all validation and verification actions before and after the reconciliation step.
515:
26:, is a technology that uses process information and mathematical methods in order to automatically ensure
3402:
Efficient Data Reconciliation and Estimation for Dynamic Processes Using Nonlinear Programming Techniques
2983:
1591:
Sensor redundancy arising from multiple sensors of the same quantity at the same time at the same place.
462:
1175:
1086:
997:
908:
1215:
3266:
3261:
3231:
1126:
590:
3460:, Proceedings of ESCAPE-9 conference, Budapest, May 31-June 2, 1999, supplementary volume, p. 17-20.
3214:
The result of an advanced PDR procedure is a coherent set of validated and reconciled process data.
405:
2955:
2904:
2514:
1996:
854:
466:
252:
2914:
2471:
1932:
1845:
3251:
1568:
1441:. The objective function is the sum of the penalties, which will be denoted in the following by
217:
2409:
1606:
1553:{\displaystyle f(y^{*})=\sum _{i=1}^{n}\left({\frac {y_{i}^{*}-y_{i}}{\sigma _{i}}}\right)^{2}}
1037:
948:
347:
318:
3336:
3155:
2802:
Redundancy can be used as a source of information to cross-check and correct the measurements
1603:
Topological redundancy arising from model information, using the mass conservation constraint
536:
272:
60:
3325:
Selection of Measurements Necessary to Achieve Multicomponent Mass Balances in Chemical Plant
2443:
1765:
1737:
436:
34:
and produces a single consistent set of data representing the most likely process operation.
3227:
2889:
519:
374:
55:
2827:
2805:
2771:
2749:
2727:
2705:
2683:
2661:
2639:
2617:
2595:
2573:
2551:
2382:
2360:
2338:
2304:
2282:
2260:
2238:
2042:
1910:
1888:
1823:
1801:
1684:
1662:
1640:
1265:
565:
380:
289:
3286:"ISA-95: the international standard for the integration of enterprise and control systems"
3084:
485:
309:
27:
3090:
2193:
2172:
1975:
196:
3079:
Simple models include mass balances only. When adding thermodynamic constraints such as
3256:
3116:
3035:
1155:
1066:
977:
888:
545:
481:
3142:
3526:
3067:
gross error remediation techniques to ensure meaningfulness of the reconciled values,
1724:
532:
511:
283:
259:
189:
3504:
Data reconciliation and gross error detection: an intelligent use of process data
3087:
increases. Indeed, as we have seen above, the level of redundancy is defined as
2847:
276:
3509:
V. Veverka, F. Madron, 'Material and Energy Balancing in the Process Industries
3186:
Advanced PDR solutions offer an integration of the techniques mentioned above:
2950:
3178:
to ensure that the data reconciliation can still be done (see Section 2,).
1993:
measurements given, then the level of topological redundancy is defined as
1842:
the measured variables. Then the system of the process constraints becomes
1426:{\displaystyle \left({\frac {y_{i}^{*}-y_{i}}{\sigma _{i}}}\right)^{2}\,\!}
266:
3146:
The workflow of an advanced data validation and reconciliation process.
2868:
49:
Data that reflects the state of the processes at a given point in time.
3471:
Finding Megawatts in nuclear power plants with process data validation
2169:
i.e. the redundancy is the difference between the number of equations
2159:{\displaystyle {\begin{aligned}red=n-dof=n-(n+m-p)=p-m,\end{aligned}}}
514:
in production processes where raw measurements were available for all
263:
2892:
exist in the set of measured values. These tests can be for example
3342:
Reconciliation and Rectification of Process Flow and Inventory Data
3517:
Data processing and reconciliation for chemical process operations
535:. From a statistical point of view the main assumption is that no
3210:
result storage (raw measurements together with reconciled values)
3190:
data acquisition from data historian, data base or manual inputs
3151:
400:
313:
31:
3456:
P. Delava, E. Maréchal, B. Vrielynck, B. Kalitventzeff (1999),
3432:
465:
has some effect), data reconciliation assumes these errors are
2850:
or virtual sensors, where hardware sensors are not installed.
2406:
We incorporate only flow conservation constraints and obtain
3473:, Proceedings of ICONE12, Arlington, USA, April 25–29, 2004.
2871:
of several measured values over a well-defined time period.
589:, data reconciliation can mathematically be expressed as an
16:
A technology to correct measurements in industrial processes
3391:, Proc. CEF’87: Use Comput. Chem. Eng., Italy, 41–46, 1987.
1567:. Additionally, each least squares term is weighted by the
1345:{\displaystyle x_{\min },x_{\max },y_{\min },y_{\max }\,\!}
46:
Models that express the general structure of the processes,
3344:, Ind. & Eng. Chem. Proc. Des. Dev. 15: 175–183, 1976.
3285:
3070:
robust algorithms for solving the reconciliation problem.
1352:
are the bounds on the measured and unmeasured variables.
3375:
Estimation of Flows and Temperatures in Process Networks
3150:
Gross errors are measurement systematic errors that may
1730:
Topological redundancy is intimately linked with the
3119:
3093:
3038:
2986:
2958:
2917:
2830:
2808:
2774:
2752:
2730:
2708:
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2554:
2517:
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2446:
2412:
2385:
2363:
2341:
2307:
2285:
2263:
2241:
2196:
2175:
2075:
2045:
1999:
1978:
1935:
1913:
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1826:
1804:
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1740:
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321:
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220:
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99:
63:
20:
Industrial process data validation and reconciliation
3238:(see ) in oil refining or in the chemical industry.
377:
on the other hand is characterized by a measurement
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3364:
3125:
3105:
3044:
3021:
2972:
2941:
2838:
2816:
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2672:
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2628:
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2584:
2562:
2537:
2503:
2460:
2432:
2393:
2371:
2349:
2315:
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2271:
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2203:
2182:
2158:
2058:
2031:
1985:
1964:
1921:
1899:
1877:
1834:
1812:
1787:
1754:
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1673:
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1629:
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85:
3193:data validation and filtering of raw measurements
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2735:
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2625:
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2368:
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1961:
1918:
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1626:
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1341:
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1201:
1141:
1112:
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1023:
963:
934:
874:
838:
422:
388:
373:is the true value that is typically not known. A
362:
333:
297:
240:
2636:, then the system cannot be calculated (knowing
2219:Example of calculable and non-calculable systems
1335:
1322:
1309:
1296:
825:
806:
791:
765:
607:
3205:gross error remediation (and go back to step 3)
171:Normally distributed measurements without bias.
3469:M. Langenstein, J. Jansky, B. Laipple (2004),
3356:Statistical Analysis of Constrained Data Sets
8:
3486:, Comp. and Chem, Eng. 25, p. 501-507, 2001.
3389:Process measurements analysis and validation
3196:data reconciliation of filtered measurements
183:Normally distributed measurements with bias.
3314:, Coll. Czech Chem. Commun. 34: 3653, 1968.
3299:Computer Control II. Mathematics of Control
214:is not a solution of the nonlinear system
3506:, Golf Publishing Company, Houston, 2000.
3404:, Computers Chem. Eng. 16: 963–986, 1992.
3118:
3092:
3083:to the model, its scope and the level of
3037:
3013:
2997:
2985:
2969:
2963:
2957:
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2707:
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2473:
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2411:
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2384:
2367:
2362:
2345:
2340:
2311:
2306:
2289:
2284:
2267:
2262:
2245:
2240:
2200:
2195:
2179:
2174:
2076:
2074:
2055:
2044:
2027:
1998:
1982:
1977:
1960:
1934:
1917:
1912:
1895:
1890:
1873:
1847:
1830:
1825:
1808:
1803:
1784:
1767:
1750:
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1669:
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1521:
1508:
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1128:
1111:
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1039:
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962:
956:
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837:
824:
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777:
764:
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695:
684:
671:
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659:
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621:
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421:
410:
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361:
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349:
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296:
291:
255:can be categorized into two basic types:
239:
219:
203:
198:
132:
113:
98:
82:
62:
3234:(see ). Another field of application is
3141:
492:Necessity of removing measurement errors
3511:, Elsevier Science BV, Amsterdam, 1997.
3482:Th. Amand, G. Heyen, B. Kalitventzeff,
3400:M.J. Liebman, T.F. Edgar, L.S. Lasdon,
3278:
2222:
2190:and the number of unmeasured variables
1723:Redundancy is linked to the concept of
1578:
433:, which is not equal to the true value
158:
144:{\displaystyle y=(y_{1},\ldots ,y_{n})}
3425:
3423:
3421:
3327:, Chem. Eng. Sci. 31: 1199–1205, 1976.
3301:, Chem. Eng. Process 57: 44–47, 1961.
2790:, then the system can be calculated.
2224:Calculable and non-calculable systems
7:
3056:Advanced process data reconciliation
3022:{\displaystyle f(y^{*})\leq P_{95}}
480:This use of average values, like a
38:Models, data and measurement errors
3377:, AIChE Journal 23: 642–650, 1977.
3358:, AiChE Journal 26: 260–164, 1961.
3236:performance and process monitoring
2867:filtering, for example taking the
2468:. It is possible that the system
518:. At the same time the problem of
14:
2896:the chi square test (global test)
1885:, which is a nonlinear system in
1580:Sensor and topological redundancy
1284:process equality constraints and
1205:{\displaystyle i=1,\ldots ,n\,\!}
1152:is the standard deviation of the
1116:{\displaystyle j=1,\ldots ,m\,\!}
1027:{\displaystyle i=1,\ldots ,n\,\!}
938:{\displaystyle i=1,\ldots ,n\,\!}
24:process data reconciliation (PDR)
2658:does not give information about
2357:does not give information about
2328:
2228:
1711:redundancy in information theory
1637:, for example one can calculate
1596:
1584:
1255:{\displaystyle F(x,y^{*})=0\,\!}
399:which is a random variable with
275:(or gross errors) due to sensor
176:
164:
3433:Association of German Engineers
2511:is not calculable, even though
2335:non-calculable system, knowing
1145:{\displaystyle \sigma _{i}\,\!}
885:is the reconciled value of the
188:Data originates typically from
3339:, G.M. Stanley, D.W. Downing,
3003:
2990:
2934:
2921:
2490:
2478:
2134:
2116:
1951:
1939:
1864:
1852:
1464:
1451:
1241:
1222:
746:
727:
426:{\displaystyle {\bar {y}}\,\!}
415:
230:
224:
138:
106:
73:
67:
1:
3372:G.M. Stanley and R.S.H. Mah,
2973:{\displaystyle P_{\alpha }\,}
2538:{\displaystyle p-m\geq 0\,\!}
2032:{\displaystyle red=n-dof\,\!}
1820:the unmeasured variables and
974:is the measured value of the
878:{\displaystyle y_{i}^{*}\,\!}
56:nonlinear system of equations
3515:J. Romagnoli, M.C. Sanchez,
3502:S. Narasimhan, C. Jordache,
3387:P. Joris, B. Kalitventzeff,
2942:{\displaystyle f(y^{*})\,\!}
2548:If we have measurements for
2504:{\displaystyle F(x,y)=0\,\!}
1965:{\displaystyle F(x,y)=0\,\!}
1878:{\displaystyle F(x,y)=0\,\!}
279:or faulty data transmission.
160:Random and systematic errors
3354:J.C. Knepper, J.W. Gorman,
286:means that the measurement
3549:
244:{\displaystyle F(y)=0\,\!}
3297:D.R. Kuehn, H. Davidson,
2702:). On the other hand, if
2433:{\displaystyle a+b=c\,\!}
1716:Redundancy can be due to
1630:{\displaystyle a=b+c\,\!}
1083:-th unmeasured variable (
1056:{\displaystyle x_{j}\,\!}
967:{\displaystyle y_{i}\,\!}
366:{\displaystyle y^{*}\,\!}
337:{\displaystyle y^{*}\,\!}
3323:V. Vaclavek, M. Loucka,
2235:Calculable system, from
86:{\displaystyle F(y)=0\,}
3519:, Academic Press, 2000.
3165:gross error relaxation.
3162:gross error elimination
3138:Gross error remediation
2909:chi-square distribution
2461:{\displaystyle c=d\,\!}
1972:is calculable with the
1788:{\displaystyle a=b+c\,}
1755:{\displaystyle dof\,\!}
593:of the following form:
454:{\displaystyle y^{*}\,}
3147:
3127:
3107:
3046:
3023:
2974:
2943:
2840:
2818:
2784:
2762:
2740:
2718:
2696:
2674:
2652:
2630:
2608:
2586:
2564:
2539:
2505:
2462:
2434:
2395:
2373:
2351:
2317:
2295:
2273:
2251:
2205:
2184:
2160:
2060:
2033:
1987:
1966:
1923:
1901:
1879:
1836:
1814:
1789:
1756:
1697:
1675:
1653:
1631:
1554:
1490:
1427:
1346:
1278:
1256:
1206:
1166:
1146:
1117:
1077:
1057:
1028:
988:
968:
939:
899:
879:
843:
653:
583:
556:
503:
455:
427:
393:
367:
338:
302:
245:
208:
145:
87:
3145:
3128:
3108:
3047:
3024:
2975:
2944:
2884:Gross error detection
2841:
2839:{\displaystyle x\,\!}
2819:
2817:{\displaystyle y\,\!}
2785:
2783:{\displaystyle c\,\!}
2763:
2761:{\displaystyle b\,\!}
2741:
2739:{\displaystyle d\,\!}
2719:
2717:{\displaystyle a\,\!}
2697:
2695:{\displaystyle b\,\!}
2675:
2673:{\displaystyle a\,\!}
2653:
2651:{\displaystyle c\,\!}
2631:
2629:{\displaystyle b\,\!}
2609:
2607:{\displaystyle a\,\!}
2587:
2585:{\displaystyle d\,\!}
2565:
2563:{\displaystyle c\,\!}
2540:
2506:
2463:
2435:
2396:
2394:{\displaystyle b\,\!}
2374:
2372:{\displaystyle a\,\!}
2352:
2350:{\displaystyle c\,\!}
2318:
2316:{\displaystyle b\,\!}
2296:
2294:{\displaystyle a\,\!}
2274:
2272:{\displaystyle c\,\!}
2252:
2250:{\displaystyle d\,\!}
2206:
2185:
2161:
2061:
2059:{\displaystyle dof\,}
2034:
1988:
1967:
1924:
1922:{\displaystyle x\,\!}
1902:
1900:{\displaystyle y\,\!}
1880:
1837:
1835:{\displaystyle y\,\!}
1815:
1813:{\displaystyle x\,\!}
1790:
1757:
1698:
1696:{\displaystyle b\,\!}
1676:
1674:{\displaystyle a\,\!}
1654:
1652:{\displaystyle c\,\!}
1632:
1555:
1470:
1428:
1347:
1279:
1277:{\displaystyle p\,\!}
1257:
1207:
1167:
1147:
1118:
1078:
1058:
1029:
989:
969:
940:
900:
880:
844:
633:
584:
582:{\displaystyle y_{i}}
557:
498:
463:Central limit theorem
456:
428:
394:
392:{\displaystyle y\,\!}
368:
339:
303:
301:{\displaystyle y\,\!}
246:
209:
146:
88:
3267:Chemical engineering
3262:Industrial processes
3232:nuclear power plants
3199:result verification
3117:
3091:
3075:Thermodynamic models
3036:
2984:
2956:
2915:
2899:the individual test.
2828:
2806:
2772:
2750:
2728:
2706:
2684:
2662:
2640:
2618:
2596:
2574:
2552:
2515:
2472:
2444:
2410:
2383:
2361:
2339:
2305:
2283:
2261:
2239:
2194:
2173:
2073:
2043:
1997:
1976:
1933:
1911:
1889:
1846:
1824:
1802:
1766:
1738:
1685:
1663:
1641:
1607:
1445:
1359:
1288:
1266:
1216:
1176:
1156:
1127:
1087:
1067:
1038:
998:
978:
949:
909:
889:
855:
599:
591:optimization problem
566:
546:
467:normally distributed
437:
406:
381:
348:
319:
290:
218:
197:
97:
61:
3106:{\displaystyle p-m}
2746:are known, but not
2204:{\displaystyle m\,}
2183:{\displaystyle p\,}
1986:{\displaystyle n\,}
1513:
1384:
872:
676:
527:Data reconciliation
207:{\displaystyle y\,}
22:, or more briefly,
3252:Process simulation
3148:
3123:
3103:
3042:
3019:
2970:
2939:
2836:
2814:
2780:
2758:
2736:
2714:
2692:
2670:
2648:
2626:
2604:
2582:
2560:
2535:
2501:
2458:
2430:
2391:
2369:
2347:
2313:
2291:
2269:
2247:
2201:
2180:
2156:
2154:
2056:
2029:
1983:
1962:
1919:
1897:
1875:
1832:
1810:
1785:
1752:
1732:degrees of freedom
1693:
1671:
1649:
1627:
1569:standard deviation
1565:system constraints
1550:
1499:
1423:
1370:
1342:
1274:
1252:
1202:
1162:
1142:
1113:
1073:
1053:
1024:
984:
964:
935:
895:
875:
858:
839:
835:
662:
628:
579:
552:
451:
423:
389:
363:
334:
298:
253:Measurement errors
241:
204:
141:
83:
3156:statistical tests
3126:{\displaystyle p}
3045:{\displaystyle i}
2875:Result validation
1718:sensor redundancy
1538:
1409:
1172:-th measurement (
1165:{\displaystyle i}
1076:{\displaystyle j}
994:-th measurement (
987:{\displaystyle i}
905:-th measurement (
898:{\displaystyle i}
720:
701:
606:
555:{\displaystyle n}
537:systematic errors
512:material balances
418:
273:systematic errors
262:due to intrinsic
93:in the variables
3540:
3487:
3480:
3474:
3467:
3461:
3454:
3448:
3443:
3437:
3427:
3416:
3411:
3405:
3398:
3392:
3385:
3379:
3370:
3359:
3352:
3346:
3334:
3328:
3321:
3315:
3308:
3302:
3295:
3289:
3283:
3132:
3130:
3129:
3124:
3112:
3110:
3109:
3104:
3051:
3049:
3048:
3043:
3028:
3026:
3025:
3020:
3018:
3017:
3002:
3001:
2979:
2977:
2976:
2971:
2968:
2967:
2948:
2946:
2945:
2940:
2933:
2932:
2845:
2843:
2842:
2837:
2823:
2821:
2820:
2815:
2789:
2787:
2786:
2781:
2767:
2765:
2764:
2759:
2745:
2743:
2742:
2737:
2723:
2721:
2720:
2715:
2701:
2699:
2698:
2693:
2679:
2677:
2676:
2671:
2657:
2655:
2654:
2649:
2635:
2633:
2632:
2627:
2613:
2611:
2610:
2605:
2591:
2589:
2588:
2583:
2569:
2567:
2566:
2561:
2544:
2542:
2541:
2536:
2510:
2508:
2507:
2502:
2467:
2465:
2464:
2459:
2439:
2437:
2436:
2431:
2400:
2398:
2397:
2392:
2378:
2376:
2375:
2370:
2356:
2354:
2353:
2348:
2332:
2322:
2320:
2319:
2314:
2300:
2298:
2297:
2292:
2278:
2276:
2275:
2270:
2257:one can compute
2256:
2254:
2253:
2248:
2232:
2210:
2208:
2207:
2202:
2189:
2187:
2186:
2181:
2165:
2163:
2162:
2157:
2155:
2065:
2063:
2062:
2057:
2038:
2036:
2035:
2030:
1992:
1990:
1989:
1984:
1971:
1969:
1968:
1963:
1929:. If the system
1928:
1926:
1925:
1920:
1906:
1904:
1903:
1898:
1884:
1882:
1881:
1876:
1841:
1839:
1838:
1833:
1819:
1817:
1816:
1811:
1794:
1792:
1791:
1786:
1761:
1759:
1758:
1753:
1702:
1700:
1699:
1694:
1680:
1678:
1677:
1672:
1658:
1656:
1655:
1650:
1636:
1634:
1633:
1628:
1600:
1588:
1559:
1557:
1556:
1551:
1549:
1548:
1543:
1539:
1537:
1536:
1527:
1526:
1525:
1512:
1507:
1497:
1489:
1484:
1463:
1462:
1432:
1430:
1429:
1424:
1420:
1419:
1414:
1410:
1408:
1407:
1398:
1397:
1396:
1383:
1378:
1368:
1351:
1349:
1348:
1343:
1339:
1338:
1326:
1325:
1313:
1312:
1300:
1299:
1283:
1281:
1280:
1275:
1261:
1259:
1258:
1253:
1240:
1239:
1211:
1209:
1208:
1203:
1171:
1169:
1168:
1163:
1151:
1149:
1148:
1143:
1139:
1138:
1122:
1120:
1119:
1114:
1082:
1080:
1079:
1074:
1062:
1060:
1059:
1054:
1050:
1049:
1033:
1031:
1030:
1025:
993:
991:
990:
985:
973:
971:
970:
965:
961:
960:
944:
942:
941:
936:
904:
902:
901:
896:
884:
882:
881:
876:
871:
866:
848:
846:
845:
840:
836:
829:
828:
810:
809:
799:
795:
794:
782:
781:
769:
768:
758:
745:
744:
721:
719:subject to
718:
712:
711:
706:
702:
700:
699:
690:
689:
688:
675:
670:
660:
652:
647:
627:
626:
625:
588:
586:
585:
580:
578:
577:
561:
559:
558:
553:
460:
458:
457:
452:
449:
448:
432:
430:
429:
424:
420:
419:
411:
398:
396:
395:
390:
375:systematic error
372:
370:
369:
364:
360:
359:
343:
341:
340:
335:
331:
330:
307:
305:
304:
299:
250:
248:
247:
242:
213:
211:
210:
205:
180:
168:
150:
148:
147:
142:
137:
136:
118:
117:
92:
90:
89:
84:
3548:
3547:
3543:
3542:
3541:
3539:
3538:
3537:
3533:Data management
3523:
3522:
3491:
3490:
3481:
3477:
3468:
3464:
3455:
3451:
3444:
3440:
3428:
3419:
3412:
3408:
3399:
3395:
3386:
3382:
3371:
3362:
3353:
3349:
3335:
3331:
3322:
3318:
3309:
3305:
3296:
3292:
3284:
3280:
3275:
3248:
3224:upstream sector
3220:
3184:
3140:
3115:
3114:
3089:
3088:
3081:energy balances
3077:
3058:
3034:
3033:
3009:
2993:
2982:
2981:
2959:
2954:
2953:
2924:
2913:
2912:
2905:random variable
2886:
2877:
2864:
2856:
2854:Data validation
2826:
2825:
2804:
2803:
2800:
2770:
2769:
2748:
2747:
2726:
2725:
2704:
2703:
2682:
2681:
2660:
2659:
2638:
2637:
2616:
2615:
2594:
2593:
2572:
2571:
2550:
2549:
2513:
2512:
2470:
2469:
2442:
2441:
2408:
2407:
2402:
2381:
2380:
2359:
2358:
2337:
2336:
2333:
2324:
2303:
2302:
2281:
2280:
2259:
2258:
2237:
2236:
2233:
2221:
2192:
2191:
2171:
2170:
2153:
2152:
2071:
2070:
2041:
2040:
1995:
1994:
1974:
1973:
1931:
1930:
1909:
1908:
1887:
1886:
1844:
1843:
1822:
1821:
1800:
1799:
1764:
1763:
1736:
1735:
1704:
1683:
1682:
1661:
1660:
1639:
1638:
1605:
1604:
1601:
1592:
1589:
1577:
1528:
1517:
1498:
1492:
1491:
1454:
1443:
1442:
1437:of measurement
1399:
1388:
1369:
1363:
1362:
1357:
1356:
1330:
1317:
1304:
1291:
1286:
1285:
1264:
1263:
1231:
1214:
1213:
1174:
1173:
1154:
1153:
1130:
1125:
1124:
1085:
1084:
1065:
1064:
1041:
1036:
1035:
996:
995:
976:
975:
952:
947:
946:
907:
906:
887:
886:
853:
852:
834:
833:
820:
801:
797:
796:
786:
773:
760:
756:
755:
736:
722:
714:
713:
691:
680:
661:
655:
654:
629:
617:
597:
596:
569:
564:
563:
544:
543:
529:
508:
494:
486:low-pass filter
440:
435:
434:
404:
403:
379:
378:
351:
346:
345:
322:
317:
316:
310:random variable
288:
287:
216:
215:
195:
194:
184:
181:
172:
169:
157:
128:
109:
95:
94:
59:
58:
40:
28:data validation
17:
12:
11:
5:
3546:
3544:
3536:
3535:
3525:
3524:
3521:
3520:
3513:
3507:
3500:
3497:
3489:
3488:
3475:
3462:
3449:
3438:
3417:
3406:
3393:
3380:
3360:
3347:
3329:
3316:
3303:
3290:
3277:
3276:
3274:
3271:
3270:
3269:
3264:
3259:
3257:Pinch analysis
3254:
3247:
3244:
3219:
3216:
3212:
3211:
3208:
3207:
3206:
3203:
3197:
3194:
3191:
3183:
3180:
3167:
3166:
3163:
3139:
3136:
3122:
3102:
3099:
3096:
3076:
3073:
3072:
3071:
3068:
3065:
3057:
3054:
3041:
3016:
3012:
3008:
3005:
3000:
2996:
2992:
2989:
2966:
2962:
2936:
2931:
2927:
2923:
2920:
2901:
2900:
2897:
2885:
2882:
2876:
2873:
2863:
2862:Data filtering
2860:
2855:
2852:
2833:
2811:
2799:
2796:
2777:
2755:
2733:
2711:
2689:
2667:
2645:
2623:
2601:
2592:, but not for
2579:
2557:
2532:
2529:
2526:
2523:
2520:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2455:
2452:
2449:
2427:
2424:
2421:
2418:
2415:
2404:
2403:
2388:
2366:
2344:
2334:
2327:
2325:
2310:
2288:
2279:, and knowing
2266:
2244:
2234:
2227:
2225:
2220:
2217:
2199:
2178:
2167:
2166:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2078:
2054:
2051:
2048:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1981:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1916:
1894:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1829:
1807:
1783:
1780:
1777:
1774:
1771:
1749:
1746:
1743:
1706:
1705:
1690:
1668:
1646:
1624:
1621:
1618:
1615:
1612:
1602:
1595:
1593:
1590:
1583:
1581:
1576:
1573:
1547:
1542:
1535:
1531:
1524:
1520:
1516:
1511:
1506:
1502:
1495:
1488:
1483:
1480:
1477:
1473:
1469:
1466:
1461:
1457:
1453:
1450:
1433:is called the
1418:
1413:
1406:
1402:
1395:
1391:
1387:
1382:
1377:
1373:
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3310:V. Vaclavek,
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3288:. isa-95.com.
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533:random errors
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284:Random errors
278:
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260:random errors
258:
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3218:Applications
3213:
3185:
3176:
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3149:
3078:
3059:
3031:
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2890:gross errors
2887:
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2848:soft sensors
2801:
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2547:
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595:
541:
530:
509:
499:
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484:, acts as a
479:
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471:
282:
190:measurements
187:
52:
41:
23:
19:
18:
3228:flow meters
3202:range check
520:gross error
277:calibration
155:Error types
3337:R.S.H. Mah
3273:References
3085:redundancy
2951:percentile
1703:are known.
1575:Redundancy
3098:−
3007:≤
2999:∗
2965:α
2930:∗
2528:≥
2522:−
2144:−
2129:−
2114:−
2096:−
2016:−
1530:σ
1515:−
1510:∗
1472:∑
1460:∗
1401:σ
1386:−
1381:∗
1355:The term
1237:∗
1192:…
1132:σ
1103:…
1014:…
925:…
869:∗
818:≤
812:≤
784:≤
779:∗
771:≤
742:∗
693:σ
678:−
673:∗
635:∑
623:∗
516:variables
446:∗
416:¯
357:∗
328:∗
123:…
3527:Category
3246:See also
3182:Workflow
3113:, where
2798:Benefits
1262:are the
1123:), and
344:, where
267:accuracy
3436:, 2017.
2869:average
2301:yields
1659:, when
1435:penalty
1063:is the
506:History
3226:where
851:where
542:Given
264:sensor
312:with
308:is a
3152:bias
2768:and
2724:and
2680:and
2614:and
2570:and
2440:and
2379:and
1907:and
1681:and
401:mean
314:mean
32:data
1336:max
1323:min
1310:max
1297:min
1212:),
1034:),
945:),
826:max
807:min
792:max
766:min
608:min
269:and
3529::
3420:^
3363:^
3015:95
2545:.
1560:.
469:.
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2991:(
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2961:P
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2754:b
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2600:a
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2531:0
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2491:)
2488:y
2485:,
2482:x
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2454:d
2451:=
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2423:=
2420:b
2417:+
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2401:.
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2365:a
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2243:d
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2177:p
2150:,
2147:m
2141:p
2138:=
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2126:m
2123:+
2120:n
2117:(
2111:n
2108:=
2105:f
2102:o
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2093:n
2090:=
2087:d
2084:e
2081:r
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2050:o
2047:d
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2022:o
2019:d
2013:n
2010:=
2007:d
2004:e
2001:r
1980:n
1958:0
1955:=
1952:)
1949:y
1946:,
1943:x
1940:(
1937:F
1915:x
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1871:0
1868:=
1865:)
1862:y
1859:,
1856:x
1853:(
1850:F
1828:y
1806:x
1782:c
1779:+
1776:b
1773:=
1770:a
1748:f
1745:o
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1734:(
1689:b
1667:a
1645:c
1623:c
1620:+
1617:b
1614:=
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1546:2
1541:)
1534:i
1523:i
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1505:i
1501:y
1494:(
1487:n
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1468:=
1465:)
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1452:(
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1365:(
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1226:x
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1100:,
1097:1
1094:=
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1071:j
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1011:,
1008:1
1005:=
1002:i
982:i
958:i
954:y
931:n
928:,
922:,
919:1
916:=
913:i
893:i
864:i
860:y
831:,
822:x
815:x
803:x
788:y
775:y
762:y
753:0
750:=
747:)
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734:,
731:x
728:(
725:F
709:2
704:)
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686:i
682:y
668:i
664:y
657:(
650:n
645:1
642:=
639:i
619:y
615:,
612:x
575:i
571:y
550:n
442:y
413:y
385:y
353:y
324:y
294:y
237:0
234:=
231:)
228:y
225:(
222:F
201:y
139:)
134:n
130:y
126:,
120:,
115:1
111:y
107:(
104:=
101:y
80:0
77:=
74:)
71:y
68:(
65:F
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