Measurements with Gross Error

In the following a serial elimination procedure (Romagnoli and Stephanopoulos, 1981 Romagnoli, 1983) is described. This scheme isolates the sources of gross errors by a systematic treatment of the measurements. 

Let us consider the system of g overmeasured (redundant) variables in m balance equations. Assuming that all of the errors are normally distributed with zero mean and variance, it has been shown that the least squares estimate of the measurement errors is given by the solution of the following problem  

If certain measurements have gross biases, this solution is not valid. 

Assume now that c measurements have gross errors while the rest g — c) have only random errors with zero mean. Partitioning the matrix A along these lines, we have 

In order to estimate the vector i in the presence of gross errors, we need to invert the covariance matrix, as Eq. (7.22) indicates. It is possible, though, to relate to (the inverse of the balance residuals in the absence of gross errors) through the simple recursive formula (6.32), which was presented in the previous chapter. In this case we obtain the following relation  

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Recommendation When all measurements were recorded by hand, operators and engineers could use their judgment concerning their validity. Now with most acqmred automatically in enormous numbers, the measurements need to be examined automatically. The goal continues to be to detect correctly the presence or absence of gross errors and isolate which measurements contain those errors. Each of the tests has limitations. The hterature indicates that the measurement test or a composite test where measurements are sequentially added to the measurement set are the most powerful, but their success is limited. If automatic analysis is required, the composite measurement test is the most direct to isolation-specific measurements with gross error. 

It is now clear how we can establish a recursive strategy to identify the measurements with gross errors ... 

Rectification accounts for systematic measurement error. During rectification, measurements that are systematically in error are identified and discarded. Rectification can be done either cyclically or simultaneously with reconciliation, and either intuitively or algorithmically. Simple methods such as data validation and complicated methods using various statistical tests can be used to identify the presence of large systematic (gross) errors in the measurements. Coupled with successive elimination and addition, the measurements with the errors can be identified and discarded. No method is completely reliable. Plant-performance analysts must recognize that rectification is approximate, at best. Frequently, systematic errors go unnoticed, and some bias is likely in the adjusted measurements. 

Unlike the other two tests, this is associated with each measurement. Reconcihation is required before this test is apphed, but no further isolation is required. However, due to the limitations in reconciliation methods, some measurements can be inordinately adjusted because of incorrectly specified random errors. Other adjustments that do contain gross errors may not be adjusted because the selected constraints are not sensitive to these measurements. Therefore, even though the adjustment in each measurement is tested for gross error, rejection of the mill hypothesis for a specific measurement does not necessarily indicate that that measurement contains gross error. 

Table 9 presents a summary of the variables in a typical real-time run. The raw measurements are initially used to run the simulation with PROCESS (therefore, only the simulation switch is activated). The first column of the table shows the raw measurements, and the second indicates the results from PROCESS. It is clear that the results from the simulation are not in agreement with the measurements.1 It can be seen from Table 9 that the measurements of the condenser and reboiler duties are quite different from the simulation results. This suggests that there are gross errors in those measurements. The gross error detection and data reconciliation modules are then activated. The third, fourth, and fifth columns show the rectified and reconciled data. 

There are many known ways of detecting results with gross errors. Each is applied in certain specific conditions [2]. After eliminating results with gross errors, the trueness of the obtained final determination (most often the mean value of the measurement series) is influenced by biases and/or random errors. 

Mean of large number A result which Is an True value of measurements outlier with gross error... 

The strength of radioactive source is to be measured with a counter that has a background of 120 8 counts/min. The approximate gross counting rate is 360 counts/min. How long should one count if the net counting rate is to be measured with an error of 2 percent ... 

The authors test two methods coupled with the measurement test. In one, they sequentially eliminate measurements and rearrange the constraints to isolate the specific measurements that contain gross errors. In the other, streams are added back as the search continues. 

However, given that reconciliation will not always adjust measurements, even when they contain large random and gross error, the adjustments will not necessarily indicate that gross error is present. Further, the constraints may also be incorrect due to simphfications, leaks, and so on. Therefore, for specific model development, scrutiny of the individual measurement adjustments coupled with reconciliation and model building should be used to isolate gross errors. 

Gro.s.s-error-detection methods detect errors when they are not pre.sent and fail to detect the gro.s.s errors when they are. Couphng the aforementioned difficulties of reconciliation with the hmitations of gross-error-detection methods, it is hkely that the adjusted measurements contain unrecognized gross error, further weakening the foundation of the parameter estimation. 

In the previous development it was assumed that only random, normally distributed measurement errors, with zero mean and known covariance, are present in the data. In practice, process data may also contain other types of errors, which are caused by nonrandom events. For instance, instruments may not be adequately compensated, measuring devices may malfunction, or process leaks may be present. These biases are usually referred as gross errors. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. In order to avoid these shortcomings we need to check for the presence of gross systematic errors in the measurement data. 

There are various ways to identify a systematic large error (1) with a theoretical analysis of all the effects leading to a gross error, (2) with measurements of a given process variable by two methods with different precision, or (3) by checking the... 

Following the same procedure, we have considered the deletion of fa and fa. The corresponding values of the least squares objectives were fa = 1.5702 and fa = 8.4374, respectively. The results indicate that measurement fa contains a gross error and has to be deleted, or some corrective action has to be undertaken with the corresponding measurement device. 

Now if the test fails at the i th step, a systematic error has been detected. The source of the gross error will be located in one or more of the new measurements incorporated by the i th balance, and so they constitute a subset of suspect measurements. In practical applications we will be faced with two possible alternatives ... 

The preceding results are applied to develop a strategy that allows us to isolate the source of gross errors from a set of constraints and measurements. Different least squares estimation problems are resolved by adding one equation at a time to the set of process constraints. After each incorporation, the least square objective function value is calculated and compared with the critical value. 

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