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Ind. Eng. Chem. Res. 1998, 37, 3980-3984
On Predicting Closed-Loop Variability Scott E. Hurowitz,† James B. Riggs,*,† and William J. B. Oldham‡ Departments of Chemical Engineering and Computer Science, Texas Tech University, Lubbock, Texas 79409
A technique is developed which uses industrial feed composition data to predict distillation column closed-loop product variabilities. Signal processing techniques are used to obtain the frequency components of a feed composition signal. These frequency components are then combined with closed-loop product variability versus frequency information to predict the closedloop product variabilities of the column. The prediction technique is illustrated using industrial propylene/propane (C3) splitter feed composition data. The results of the prediction technique are compared with results obtained from a detailed dynamic simulation of a C3 splitter with good correspondence. Introduction Product variability, which can be measured industrially using statistical process control (SPC) charts, is a direct measure of control performance. Improved control reduces product variability. This allows the process to be operated closer to the product specification, which results in lower utility usage or greater processing rates, whichever is appropriate. Additionally, improved control results in products whose composition is more uniform. This is particularly important when the product is a feedstock to a polymerization process since polymers made from more uniform feedstocks are generally higher in quality than polymers produced from less uniform feedstocks. In essence, reduced product variability translates to significant economic savings for the U.S. chemical processing industries (CPI). Unfortunately, it is not known a priori what percentage of reduction in product variability will occur for a given improvement in control performance. This can complicate the economic justification for advanced distillation control decisions. This paper proposes a technique for predicting closedloop product variabilities for a process using measured disturbance data. The potential of this approach is evaluated for a propylene/propane (C3) splitter using industrial feed composition data. Approach Figure 1 shows a schematic for the proposed method for using measured disturbances to calculate product variability. Signal processing techniques are applied to the measured disturbance data to extract the amplitude and frequency content of the measured disturbance. These results are combined with the closed-loop Bode plot for the disturbance to yield the product variability prediction. The closed-loop Bode plot is generated by combining the controller tuning criteria with a linear dynamic model of the process. The linear dynamic model is developed using the steady-state process operating conditions along with its dynamic characteristics. This is a general approach that is not limited merely to predicting product variabilities for distillation columns. * To whom correspondence should be addressed. Fax: (806) 742-3552. E-mail:
[email protected]. † Department of Chemical Engineering. ‡ Department of Computer Science.
Figure 1. Schematic for proposed approach for predicting product variability.
Figure 2. Schematic of method for testing variability prediction approach.
Figure 2 shows a schematic for the methodology that was applied to a distillation column and was tested in this paper. Note that Figure 2 is identical to Figure 1, except that for Figure 2 the closed-loop Bode plot is generated directly by using a tray-to-tray column simulator, instead of using linear model approximations combined with the controller tuning criteria. This approach is used to evaluate the procedure for combining the closed-loop Bode plot with the results of the signal processing analysis to predict closed-loop product variability. To test the approach shown in Figure 2, a test case using a C3 splitter was considered. A dynamic tray-totray simulator of a C3 splitter (Gokhale et al., 1995a) was used along with industrial feed data for a C3 splitter. The dynamic C3 splitter model was used to
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Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 3981
generate the closed-loop Bode plot and was also used to generate the product variability directly. That is, in the latter case, the industrial C3 splitter feed composition data was introduced to the dynamic simulator with product composition control loops in place. The composition controllers were diagonal PI controllers that were applied using the [L,B] configuration and were tuned for setpoint changes (Gokhale et al., 1995b). The product variabilities predicted using the approach in Figure 2 are compared with the results obtained by inputing the feed composition data to the dynamic C3 splitter model with feedback controls in place. Feed composition disturbances were chosen since they represent the most challenging disturbance to distillation control and are sometimes measured on-line. Signal Processing Techniques Industrial feed composition data does not exhibit sinusoidal variation in the explicit sense; however, signal processing techniques can be used to determine the amplitude and frequency component(s) of a given feed composition signal. The feed composition transient may then be represented by a series of sine waves passing through a linear system, each of which yields one output sine wave. The representation of a distillation column as a linear system is merely an assumption that distillation processes exhibit linear responses with respect to sinusoidal amplitude modulation near some operating/closed-loop setpoint. For simulation purposes, if a pure sine wave was input to a process under this assumption, then the above treatment would assume the form of
a discrete Fourier transform to N discretely sampled data points requires (N - 1)2 calculations (Walker, 1991). By applying a decimation-in-time, radix 2, fast Fourier transform, the required number of calculations is reduced to 1/2N log2 N (Walker, 1991). It is important to recognize that, to apply a radix 2 fast Fourier transform, N must be an integer power of 2, N ) 2x. A more thorough discussion of fast Fourier transforms is given by Kamen (1990), Proakis and Manolakis (1996), and Walker (1991). One phenomenon that results from Fourier transforming discrete time-series data is the wrap-around effect, which occurs as a result of the convolution theorem (Press et al., 1992; Walker, 1991). Because the convolution theorem assumes that the data is periodic, it falsely pollutes the first output channel with some wrapped-around data from the far end of the data sample: hN, hN-1, hN-2, and so forth. To deal with the wrap-around effect, a buffer zone of zeros is added at the end of the data sample to make the pollution zero. The data sample needs to be padded with a number of zeros equal to the maximum positive duration of the response function (Press et al., 1992). Prediction Technique The Fourier transform of a given function h(t) results in H(ω). If the inverse Fourier transform of H(ω) is taken, then the original function h(t) should be obtained:
h(t) )
∫-∞∞H(ω)e-jωt dω
1 2π
(2)
In polar form, eq 2 may be rewritten as Equation 1 states that a pure sine wave with amplitude A and angular frequency ω passed through a linear system will result in an output sine wave at the same frequency ω, but with a different amplitude A′ and a phase shift φ′. The linear system assumption has been verified by simulation for a C3 splitter. This was done by Fourier transforming a feed composition disturbance and the resulting output transient and confirming that the frequency components of each signal were synchronized. Therefore, the objective of this approach is to characterize the amplitude and frequency components of a feed composition disturbance by a series of sine waves at various frequencies. Each sine wave is then multiplied by its frequency-associated amplitude ratio [linear system, i.e., Bode plot] to yield an output series of sine waves whose summation represents the closed-loop product variability. To characterize the amplitude and frequency components of an industrial feed composition signal, the power, amplitude, and phase spectra for that signal are required. These can be obtained through Fourier transformation of the data. The detailed mathematics of Fourier transforms will not be discussed here; for a more in-depth coverage, the reader is referred to Kamen (1990) and Proakis and Manolakis (1996), both of which give excellent treatments of Fourier mathematics. Since the current discussion deals with discrete data, a discrete Fourier transform of the data will be required. The disadvantage of applying discrete Fourier transforms is that it is computationally intensive. Applying
h(t) )
∫-∞∞|H(ω)|e-j(ωt-φ) dω
1 2π
(3)
Since the Fourier transform and its inverse transform are symmetric over positive and negative frequencies (Kamen, 1990), eq 3 may be rewritten as
∫-∞∞[|H(ω)|e-j(ωt-φ) + |H(ω)|ej(ωt-φ)] dω
1 2π
h(t) )
(4)
Discretized over N frequency components, eq 4 becomes
h(t) )
N/2
1
[H(ω)0) + HN/2+1 +
N
(|H(ω)|e-j(ωt-φ) + ∑ i)2 |H(ω)|ej(ωt-φ))] (5)
which is equivalent to
h(t) )
N/2
1
[H(ω)0) + HN/2+1 +
N
2|H(ω)| cos(ωt-φ)] ∑ i)2
(6)
From the trigonometric reduction,
(
π 2
cos(ωt-φ) ) sin ωt-φ+ Equation 6 becomes
)
(7)
3982 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
h(t) )
1
[
N
H(ω)0) + HN/2+1 +
(
2|H(ω)| sin ωt-φ+ ∑ 2 i)2 N/2
)]
π
(8)
Once the product composition Bode plot for feed composition disturbances and the frequency spectrum of a feed composition disturbance are available, the closedloop product variabilities can be predicted by
output(t) )
1
[
N
|
AR(ω) H(ω)
| |
AR(ω) H(ω)
t)N/2+1 +
ω)0
+
(
2AR(ω) |H(ω)| sin ωt-φ+ ∑ i)2 N/2
π 2
)]
+θ(ω)output
(9)
where N is the number of discretely sampled data points, t is the time, AR(ω) is the product purity (or impurity) amplitude ratio at angular frequency ω (from Bode plot), |H(ω)| is the magnitude of the Fourier transform at angular frequency ω, φ is the angular phase shift at angular frequency w (from Bode plot), and θ(ω)output is the angular phase shift of the output with respect to an input disturbance at angular frequency ω, which is given by
θ(ω)output ) ωτoutput )
2πτoutput T
Figure 3. C3 splitter feed composition signal. Table 1. Signal Processing Procedure Step 1. Step 2.
Step 3.
Step 4.
Zero-mean the feed composition data samples. Detrend the zero-meaned data by subtracting from the data a straight line connecting the first and last points (i.e., the first and last points should equal zero). Apply zero-padding to the data to solve the wrap-around problem. Remember that to apply a radix 2 fast Fourier transform, N must be an integer power of 2, N ) 2x. 2432 zeros were used in each of samples 1-4, so that N ) 8192 ) 2.13 Apply a fast Fourier transform to obtain the following: 1. power spectral density 2. amplitude spectrum 3. phase spectrum
(10)
where τoutput is the shift in time between the input sinusoidal disturbance wave and the output (product) oscillation wave. As a final note, since the Fourier transform is symmetric over positive and negative frequencies, a Fourier transform of a feed composition sample will essentially provide N/2 + 1 frequency components that can be applied to eq 9. However, in many cases, all of this information is not required. The required number of components necessary to obtain an accurate output prediction can be determined by recognizing over what frequency range most of the signal power lies (determined from the power spectral density, PSD). Test Case A propylene/propane (C3) splitter was chosen to demonstrate the methods that have been presented. The C3 splitter has the following steady-state characteristics: the overhead product contains 0.3 mol % propane impurity, the bottoms product purity is 2 mol % propylene, and the feed composition is 70 mol % propylene. There are 232 trays with a Murphree tray efficiency of 85% and an operating pressure of 211 psia. Complete details of the C3 splitter are given by Gokhale et al. (1995a). Figure 3 presents an industrial feed composition signal for a C3 splitter. The sharp vertical lines in the signal correspond to analyzer failure; these were removed before signal processing was applied. The feed composition sample contains 5760 data measurements with a sampling interval of 1 min. The feed composition data needs to be Fourier-transformed to obtain its power, amplitude, and phase spectrum. The complete
signal processing procedure is given in Table 1. Once the signal processing procedure is complete, the signal’s power spectral density can be analyzed to identify the principal frequency component(s) of the signal. Then the amplitude and phase of each frequency component can be obtained from the amplitude and phase spectra, respectively. Results Analysis of the power spectral density and amplitude spectrum of the feed composition signal, illustrated in Figure 4, revealed several distinct peaks over the first 2 decades of the signal’s power, most of which present at frequencies below 0.01 rad/min. These peaks in signal power are referred to as the principal frequency components of the signal. Once the signal processing analysis was complete, the frequency component content of the feed composition signal was combined with the closed-loop Bode plot using eq 9 to predict the closed-loop product variability for the given feed composition disturbance. Figure 5 provides the closed-loop Bode plot for feed composition disturbances. This plot was generated using a detailed dynamic tray-to-tray simulation of an industrially benchmarked C3 splitter under dual composition PI control (Gokhale et al., 1995a,b). To test the variability prediction technique, the feed composition data was input to the C3 splitter simulation of Gokhale et al. The closed-loop simulator response was then compared to the predicted response using the results from the signal processing analysis combined with the closed-loop Bode plot. The results of the product variability prediction technique are provided in Figure 6. These results were
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 3983
Figure 6. Closed-loop rejection of disturbance using 131 frequency components for prediction.
Figure 4. Frequency spectrum for sample 1: (a) power spectral density; (b) amplitude spectrum.
Figure 5. Closed-loop Bode plot for feed composition disturbances.
generated using only 131 of the 4097 available frequency components (in essence, only the frequency components in the range from 0 to 0.1 rad/min were used in the output prediction). The remaining 3966 frequency components in the range from 0.1 to π rad/ min contained little or no signal power and were neglected. Using 131 frequency components, the product variabilities were very accurately predicted, although there was a relatively small degree of prediction
mismatch that is probably associated with process nonlinearities. Although the prediction mismatch appears to be slightly larger for the overhead product than for the bottoms, it should be noted that the scale of data is much tighter for the overhead product than for the bottoms product; the prediction mismatch is comparable for both column products. It should be noted that the mismatch during the first 500 min of the prediction was a result of the C3 splitter simulator being initialized at steady state, whereas the feed composition disturbance data was not initially at steady state. When these data were input to the simulator, a large initial transient resulted which the proposed technique could not possibly account for. Even with the presence of nonlinearities, an excellent prediction of the closed-loop product variabilities was obtained by the proposed technique. The prediction technique was also applied using only the 9 principal frequency components out of the 4097 available frequency components that were generated by the signal processing analysis. Using only the 9 principal frequency components, a crude estimate of the closed-loop product variabilities was obtained; however, the amount of prediction mismatch was significant. Conclusions A technique has been developed which uses industrial feed composition signals to predict distillation column closed-loop product variabilities. Standard signal processing techniques were used to obtain the frequency components of a feed composition disturbance signal for a C3 splitter. These frequency components were then combined with closed-loop product variability versus
3984 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
frequency information (Bode plot) to predict the closedloop product variabilities of the tower. Industrial feed composition data were input to the C3 splitter simulation, and the resulting product variabilities were compared with that predicted by the proposed technique. The prediction technique accurately predicted the product variabilities that resulted from the feed composition disturbance. The prediction technique developed here could be easily extended to other chemical processes and provides a new tool for quantitatively evaluating control performance. With regard to predicting variability for distillation columns, the approach presented here may have potential for (1) quantitatively evaluating the configuration selection problem, (2) identifying columns operating below their expected potential, and (3) identifying the sources of product variability. Acknowledgment David A. Hokanson and the Exxon Chemical Co. are gratefully acknowledged for providing industrial C3 splitter feed composition data. Nomenclature A ) input amplitude A′ ) output amplitude AR(ω) ) product purity (or impurity) amplitude ratio D.C. ) direct current H(ω) ) Fourier transform h(t) ) continuous time-domain function; inverse Fourier transform f ) frequency fC ) Nyquist critical frequency N ) number of discretely sampled data points t ) time
Greek Letters θ(ω)output ) angular phase shift of the output with respect to an input disturbance ∑ ) summation τoutput ) shift in time between the input sinusoidal disturbance wave and the output (product) oscillation wave φ ) angular phase shift ω ) angular frequency Miscellaneous |H(ω)| ) magnitude of Fourier transform
Literature Cited Gokhale, V.; Hurowitz, S.; Riggs, J. B. A Comparison of Advanced Distillation Control Techniques for a Propylene/Propane Splitter. Ind. Eng. Chem. Res. 1995a, 34, 4413-4419. Gokhale, V.; Hurowitz, S.; Riggs, J. B. A Dynamic Model of a Superfractionator: A Test Case for Comparing Distillation Control Techniques. Proceedings from DYCORD+’95: 4th IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes; 1995b; pp 311-316. Kamen, E. W. Introduction to Signals and Systems, 2nd ed.; Macmillan Publishing Co.: New York, 1990. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRANsThe Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1992. Proakis, J. G.; Manolakis, D. G. Digital Signal Processings Principles, Algorithms, and Applications, 3rd ed.; Prentice Hall, Inc.: Englewood Cliffs, NJ, 1996. Walker, J. S. Fast Fourier Transforms; CRC Press: Boca Raton, FL, 1991.
Received for review April 15, 1998 Revised manuscript received July 20, 1998 Accepted July 21, 1998 IE9802414
