uncertainty analysis of meter volume measurements - part 3, applications to systems

8/12/2019 Uncertainty Analysis of Meter Volume Measurements - Part 3, Applications to Systems

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Uncertainty Analysis of Meter Volume Measurements - Part 3, Applications to Systems

T.M Kegel, Colorado Engineering Experiment Station (CEESI)

ABSTRACT

This paper is a continuation of two previous papers on uncertainty analysis1,2. The first paper presented

a simplified approach to the uncertainty analysis of a volume measurement based on an ultrasonic orturbine meter. The various components that contribute uncertainty were characterized based on

manufacturers specifications. The second paper illustrated the interpretation of calibration results for

inclusion into the uncertainty analysis. The development incorporated considerable discussion of the

impact of systematic and random effects.

This paper expands previously developed uncertainty analysis techniques from single to multiple meter

based volume measurements. The development includes the concept of correlation and illustrates the

effect on the overall measurement uncertainty. A statistical simulation technique is applied to implement

the uncertainty analysis.

PROBLEM SCOPE

Suppose an ultrasonic or turbine meter records a volume of 1 MMCF with a standard uncertainty of 10

MCF. The percent standard uncertainty can be calculated from:

%1%100MMCF1

MCF10u% == [1]

Now suppose several identical ultrasonic meters are installed in parallel. What is the uncertainty of a

total volume recorded by all the meters? This calculation includes several components of uncertainty that

need to be combined. This paper describes the details of how to combine the uncertainty components.

PROPAGATING UNCERTAINTY COMPONENTS

The basic steps in an uncertainty analysis will be reviewed before returning to the question of multiple

parallel meters. In general, an output y is a function of multiple inputs x:

)x,...,x,x,x(fy n321= [2]

The simplified equation for propagating uncertainty is given by:

[3]( )

==

n

1i

2

ii2c uSu

where ui and uc are the individual component and combined standard uncertainties. The sensitivity

coefficient of component i, Si, is defined as:

i

ix

yS

= [4]


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For multiple parallel meters Equations 2 and 4 are written as:

n321

n

1i

it q...qqqqq ++++== =

[5]

Si= 1 [6]

where qiand qtare the individual meter and total volumes in MCF.

The review of uncertainty analysis concludes with the interpretation of the sensitivity coefficient. The

value Si= 1 (Equation 6) means that a change of 1 MCF in qiwill result in a 1 MCF change in qt. It is

noted that qtand qimust be expressed in units of MCF for Si= 1.

The discussion returns to ultrasonic meters. If two identical meters are installed in parallel total volume is

2 MMCF with the uncertainty of each meter being 10 MCF. By applying Equation 3, the uncertainty is

calculated:

MCF1.141010u 22c =+= [7]

which is 0.71% of 2 MMCF. Adding a third identical meter in parallel increases the total volume to 3

MMCF. The uncertainty is calculated:

MCF3.17101010u 222c =++= [8]

which is 0.58% of 3 MMCF.

In general the combined uncertainty, expressed as a percent of the total volume is calculated from:

n

u

u,%i

,%c = [9]

From Equation 9 it would appear that the uncertainty decreases as additional meters are combined in

parallel. While initially appealing as a technique to reduce the uncertainty, our intuition causes doubt to

be raised. In fact, Equation 9 fails in the limit; we cannot eliminate the uncertainty as more meters are

added. The presence of correlation between uncertainty components invalidates the use of Equation 9 in

most applications.

MONTE CARLO SIMULATION

Monte Carlo simulation3 is an old simulation technique that is being applied to complex uncertainty

analysis problems. The technique is based on simulating individual uncertainty components with largesets of random numbers. The equations for the system under investigation are calculated repeatedly while

the inputs are varied based on the random number sets. The resulting variation in the output is interpreted

to represent the combined uncertainty. Referring to Equation 5, each qiis simulated by a series of random

values with standard deviation equal to ui. The standard deviation of the resulting qtvalues is interpreted

to represent uc. The present analysis consists of three cases with independent variables and four cases with

varying degrees of correlation between random number distributions.

Case 1


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This first case is based on eight distributions (a1, a2, a3, , a8) each made up of 10,000 random values.

Each distribution is normally distributed with mean value of 10 and standard deviation of 0.1 which

corresponds to a standard uncertainty of 1%. The distributions are used to calculate:

[10]=

=

j

1i

ij at

which corresponds to the following eight terms:

t1= a1t2= a1+ a2t3= a1+ a2+ a3

t8= a1+ a2+ + a8

For each term the standard deviation, sj, is calculated and expressed as a percent of tj. This case is

intended to confirm Equation 9.

Case 2

This case generates eight additional distributions (b1, b2, b3, , b8) made up of 10,000 random values.

Each distribution is normally distributed with mean value of 10, standard deviation of 0.1. The 16

distributions are used to calculate:

[11]i

j

1i

ij bat =

=

which corresponds to eight terms:

t1= a1b1t2= a1b1+ a2b2t3= a1b1+ a2b2+ a3b3

t8= a1b1+ a2b2+ + a8b8

For each term the standard deviation, sj, is calculated and expressed as a percent of tj. This case isintended to confirm Equation 9 with distributions based on products of random values. The use of

products will prove valuable in later cases.

Case 3

This case is the same as Case 2 except that the random values generated with standard deviation of 1.0.

The corresponding standard uncertainty is 10%. The objective of this case is to identify any effects

associated with the standard deviation of the random data sets.

Summary


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The results from the first three cases are summarized in Table 1. The first column indicates the number

of parallel meters. The second column contains corresponding uc,%values from Equation 9 based on ui,%=

1%. The last three columns show the normalized results for the three cases. The data are normalized such

that the single meter condition has standard uncertainty u = 1%.

Table 1: Summary of Cases 1-3

Parallel

Meters n/1 Case 1 Case 2 Case 3

1 1.000 1.000 1.000 1.000

2 0.707 0.707 0.716 0.707

3 0.577 0.583 0.593 0.577

4 0.500 0.499 0.513 0.500

5 0.447 0.448 0.458 0.447

6 0.408 0.409 0.418 0.407

7 0.378 0.381 0.386 0.377

8 0.354 0.353 0.361 0.354

The results show good agreement with the predictions of Equation 9. There appear to be no significant

effects of either the use of products (Case 2) or the magnitude of the standard deviation (Case 3).

CORRELATED UNCERTAINTY COMPONENTS

The next three cases incorporate the effects of correlate components of uncertainty. Equation 2 is a

simplified form of the propagation equation, the complete equation is:

[12]( ) = ==

+=

n

1j

n

1k

kkjjk,j

n

1i

2

ii2c )kj(uSuSr2uSu

The second term accounts for correlation, rj,k is called the correlation coefficient. It takes on values

between -1 and +1 depending on the correlation between the two components. If uj and uk are

independent, rj,k= 0. If ujand ukare perfectly correlated, rj,k= 1. The effect of correlation is illustrated

based on the application of the Monte Carlo technique to additional cases.

Case 4

This case is based on nine distributions (a1, a2, a3, , a8, c) each with mean value of 10 and standard

deviation of 1.0. The distributions are used to calculate:

c [13]at

j

1i

ij ==

which corresponds to eight terms:

t1= a1c

t2= a1c + a2c

t3= a1c + a2c + a3c


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t8= a1c + a2c + + a8c

The a terms are independent while the c terms are fully correlated because they are identical. This

case is intended to investigate the effect of correlation.

Cases 5 and 6

These cases are the same as Case 4 except that c distributions have standard deviations of 0.5 (Case 5)

and 2.0 (Case 6). These cases are intended to investigate the effect of varying degrees of correlation.

Summary

The results from the next three cases are summarized in Table 2 using the same format as Table 1. It is

clear that the uncertainty increases in proportion to the degree of correlation.

Table 2: Summary of Cases 4-6

Parallel

Meters n/1 Case 4 Case 5 Case 6

1 1.000 1.000 1.000 1.000

2 0.707 0.859 0.773 0.947

3 0.577 0.810 0.679 0.925

4 0.500 0.786 0.628 0.915

5 0.447 0.768 0.595 0.910

6 0.408 0.757 0.572 0.907

7 0.378 0.748 0.555 0.904

8 0.354 0.742 0.542 0.901

Correlation Coefficient

The use of Monte Carlo simulation allows for the calculation of correlation coefficients, the current

values are 0.502 (Case 4), 0.203 (Case 5) and 0.795 (Case 6). The interpretation of the correlation

coefficient is described aided by the data contained in Figure 1. Each graph shows 500 data points

selected from the various analyses. The (a) data sets include a significant correlation term while the (b)

data sets are independent. The correlation coefficient for the (a) data is calculated to be 0.965. It is

interpreted to mean that 96.5% of the variation in the ordinate is correlated with the abscissa and 3.5% of

the variation is random. The correlation coefficient for the (b) data is calculated to be 0.009, nearly all the

variation is random. The coefficient values reported above identify correlation between the combined

uncertainties of the individual meters. This definition is slightly different than that associated with

Equation 12 where the coefficients refer to correlation between individual components. The basicinterpretations of both coefficient definitions are the same.


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80

90

100

110

120

80 90 100 110 120

(a)

80

90

100

110

120

80 90 100 110 120

(b)

Figure 1: Typical Correlated (a) and Independent (b) Data

Application to Volume Measurement

In the last three cases the ai and c distributions can be interpreted represent the combine

uncertainties associated with independent and correlated components. A typical multiple meterinstallation has four sources of uncertainty that are generally considered to be correlated. First, a singlegas chromatograph is used to determine the compressibility for each of the meters. Second, the same

equation of state is used for the calculation associated with each meter. Third, it is likely that all the

meters were calibrated against the same standard. The final components are the standards used to calibrate

the pressure and temperature transmitters. The typical independent components are the random effects

associated with the flowmeters, and pressure and temperature transmitters.

Case 7

This case considers the analysis of a hypothetical meter station; uncertainty values are contained in

Table 3. The first column identifies the components considered in this case. The second and third columns

contain numerical values for the uncertainty components. The fourth column and seventh row containtotals calculated based on Equation 3.

Table 3: Hypothetical Meter Station

Correlated Independent Total

Flowmeter 0.20 0.10 0.224

Pressure Transmitter 0.15 0.10 0.180

Temperature Transmitter 0.10 0.05 0.112

Gas Chromatograph 0.25 0.00 0.250

State Equation 0.05 0.00 0.050

Total 0.371 0.150 0.400

The combined standard uncertainties are 0.371% (correlated components) and 0.150% (independent).

These values were used in the Monte Carlo simulation with calculations based on Equation 13. The

multiple meter results are contained in Table 4, the uncertainty values are normalized as was done withthe previous cases. The correlation coefficient value is 0.858. Clearly for this case there is sufficient


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correlation such that the uncertainty is not reduced as additional meters are combined in parallel. Other

cases, with different uncertainty components may yield different results.

Table 4: Summary of Case 7

ParallelMeters

NormalizedUncertainty

ParallelMeters

NormalizedUncertainty

1 1.000 5 0.938

2 0.962 6 0.935

3 0.947 7 0.933

4 0.942 8 0.932

SUMMARY

The gas volume measured by an array of parallel meters is simply the sum of the individual volumes.

The uncertainty in the total volume is calculated by combining the uncertainties of the individual metered

volumes. That process must account for the potential presence of correlation between uncertaintycomponents. This paper has presented a technique for estimating the impact of correlation, that technique

has been applied to a number of cases. The final case is based on numerical values associated with a

hypothetical meter station. The results clearly illustrate that correlation has a direct impact on the

uncertainty in total flowrate.

REFERENCES

1. Kegel, T. M., Uncertainty Analysis of Turbine and Ultrasonic Meter Volume Measurements,AGA Operations Conference, Orlando, FL, May 2003.

2. Kegel, T. M., Uncertainty Analysis of Turbine and Ultrasonic Meter Volume Measurements Part 2, Advanced Topics,AGA Operations Conference, Phoenix, AZ, May 2004.

3. Naylor, T. H, et al, Computer Simulation Techniques, John Wiley & Sons, 1968.

uncertainty analysis of meter volume measurements - part 3, applications to systems

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