Journal of Natural Gas Science and Engineering 31 (2016) 538e545
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Data reconciliation with application to a natural gas processing plant Ahmad Rafiee a, *, Flor Behrouzshad b a b
Sharif Engineering Process Development Company, Tehran, Iran South Pars Gas Company (SPGC), Asalouyeh, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 December 2015 Received in revised form 19 March 2016 Accepted 22 March 2016 Available online 25 March 2016
Data reconciliation is a mathematical technique that uses process information to fulfill material and energy conservation laws. This technique adjusts random errors in the measured data in weighted least squares to satisfy mass and energy balance constraints. In this paper, data reconciliation is applied to a natural gas processing plant. Based on the measured data, the mass flow of output streams is greater than the mass flow of the raw feed to the plant which leads to an imbalance of about 7%. The global test is employed to check the presence of systematic errors (gross errors) in the measured data. The results indicate that there is no gross error in the measurements. © 2016 Elsevier B.V. All rights reserved.
Keywords: Mass and energy balance Data reconciliation Natural gas processing plant Gross error detection
1. Introduction Chemical plants consist of different unit operations (nodes) such as reactors, separators, storage tanks, pumps, compressors, and so forth. The nodes are connected among themselves and with the environment by streams. Measurements of flows, temperatures, pressures, concentrations, etc. at different points of the nodes and streams are taken to control and evaluate process performance. In some cases not all process variables are measured due to cost or technical considerations. A well known problem is that measuring devices in the same location show different values and the mass and energy balances are not fulfilled over the nodes. We need to know the most accurate values (reconciled values) of measured variables and if possible, estimate the unmeasured ones to satisfy the balance equations. Data reconciliation is not a new idea and has been applied to different processes. Sarabia et al. (2012) dealt with the data reconciliation technique on a petrol refinery and the optimal management of hydrogen networks was presented. Islam et al. (1994) developed a reconciliation package for an industrial pyrolysis reactor with nonlinear mass and energy balance equations. There are 11 equations and 36 variables. The results of data reconciliation indicate that a gross error is present in the measurements. Pierucci et al. (1996) performed online data
* Corresponding author. E-mail address: a.rafi
[email protected] (A. Rafiee). http://dx.doi.org/10.1016/j.jngse.2016.03.071 1875-5100/© 2016 Elsevier B.V. All rights reserved.
reconciliation and optimization of a large scale olefin plant. Placido and Loureiro (1998) reconciled the measurements for different units of Fafen ammonia plant in Brazil with 55 mass balance equations, 25 unmeasured and 51 measured variables. The results show that gross errors are present in the measurements. Lida and Skogestad (2008) applied the data reconciliation method on a catalytic naphtha reformer. Their model has 501 variables and 442 equations. Meyer et al. (1993) used the data reconciliation method to treat raw data of an industrial food plant. The process consists of 14 unit operations, 12 components and 34 streams. Lida and Skogestad (2001) developed a real time optimization system of heat exchange network in Statoil Mongstad refinery. There are 85 streams, 20 heat exchanger and totally 210 variables. The results show that several flow measurements have poor performance. Nonlinear steady state data reconciliation approach was applied on a gold processing plant by Lima (2006). Bazin et al. (1998) applied the data reconciliation to a rotary dryer. The results show that there are gross errors in the measurements. Knopf (2012) reconciled the measured data of a gas turbine cogeneration system for electricity and steam generation. There are 23 variables and 8 mass and energy balances. The problem is solved by Excel and the results show that there is no gross error in the system. Bazin et al. (2005) applied the data reconciliation method on the measurements of a copper solvent extraction. There are 2 equations and 30 variables. The problem is solved by Excel. Ijaz et al. (2013) simulated the heat exchange network and reconciled the measured data. The network consists of 9 exchangers and the number of measured and unmeasured variables is 22 and 10, respectively. Jiang et al. (2014)
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proposed an approach to reconcile the data in a steam turbine. The number of equations is 20 and the number of measured and unmeasured variables is 6 and 13, respectively. Process observers and data reconciliation with application to mineral processing plants with short examples was presented by Hodouin (2010). Coupling of data reconciliation with real-time optimization and automatic control techniques was also covered by Hodouin (2010). Vasebi et al. (2012) introduced a method that takes into account the auto-covariance function of the node imbalance. They proposed an observer and used two benchmark plants from the mineral and metallurgical processing industries. The presented observer gives more precise estimates than steady-state and standard stationary observers. The connection between data reconciliation and principal component analysis was covered by Narasimhan and Bhatt (2015). Principal component analysis is a multivariate data processing technique which is used to reduce the dimensionality of data as well as denoise them. These two techniques can be used together to process data. Process simulators with optimization capabilities such as Aspen HYSYS can be used to carry out data reconciliation and parameter estimation. The advantage of using process simulators is that a library of process modules including reaction, equilibrium equations of multiple components, and multi-phase process units with recycles is available which satisfies the equality constraints (Piccolo et al., 1996). In case of solving data reconciliation optimization problem using a commercial process simulator, it is recommended to follow the following steps (Piccolo et al., 1996): 1) develop the process flow diagram (PFD) to represent the equality constraints. To do this, we need to select a thermodynamic package (fluid package), add process units (nodes), connect the nodes among themselves and with the environment by streams and provide values of flowrates, compositions, temperatures, pressures, etc. of each stream. 2) add an optimization loop to carry out the data reconciliation problem subject to inequality constraints and bounds on variables. We must introduce the decision variables in this step. The objective function can be written in weighted least squares. 3) Once the above mentioned steps are completed, we can change optimization parameters such as convergence parameters, number of iterations, number of function evaluations, optimization algorithm, etc. In case of having recycle streams in the PFD, where down-stream material mixes with up-stream material, we can specify the numerical calculation mode, number of iterations and convergence factors for each of the variables and chemical species. 4) The final step is to solve the problem and analyze the optimization results. In Aspen HYSYS, the flowsheet wide mass/energy balance and the value of imbalance for each unit operation (node) are provided. There are commercial softwares for data reconciliation and detection of gross errors such as BELSIM VALI and SimSci DATACON. In this paper, data reconciliation is applied on a natural gas processing plant and data free of random errors are obtained. 2. Principle of data reconciliation In order to improve the performance of an operating plant, the first step is to measure process variables including mass flow rates, mole fractions, temperature, pressure, etc. The measured process variables (raw data) do not satisfy mass and energy balance equations and need to be reconciled to adjust measurement noises (random errors). The reconciled variables can then be used by process economic optimization and energy management strategies
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to improve energy and carbon efficiencies, and profitability of the plant (Fig. 1). Misusing measurements that are not satisfying mass and energy balance equations will lead to false decision making and may unfavorably influence different nodes of the process. The measured data are subject to errors. These errors give rise to inconsistencies in mass and energy balances and are divided into two groups: random errors due to the accuracy of sensors. These errors are normally distributed around the true value of measurements. systematic biases (or gross errors). These errors are considered to be due to nonrandom events such as process leaks, miscalibrated instruments, malfunctioning of sensors, uncounted losses, modeling errors, etc. In absence of gross errors, the measurement vector which relates measured value, random error and true value (noise-free value) is given by Eqn. (1):
xþ i ¼ xi þ εi i varðεi Þ ¼ E ε2i ¼ s2i
(1)
Where, εi is the vector of random errors. In a steady state system, the reconciled values can be determined by minimizing sum of weighted least squares of measured variables, Eqn. (2). The weighting factors are considered to be the instrument standard deviations, s, and hk(xi,uj) are mass and energy balance constraints.
xþ xi i
P
Minimize
i;measured
s:t: hk xi ; uj ¼ 0
!2
si
(2)
k ¼ 1; 2; :::; K xi ¼ measured variables uj ¼ unmeasured variables
In Eqn. (2),
1/s2i
is the weight factor and
xþ xi i
si
!2 is called the
penalty of measurement i. Larger weights are given to more accurate measurements aiming to force their adjustments to be as small as possible. Mass and energy balance equations can be linear, bilinear, or nonlinear. A good survey of available techniques to deal with constraints is given by Narasimhan et al. (2000). Linear constraints arise when only total mass flow measurements are used. Bilinear constraints appear when chemical species measurements are multiplied by total mass flow measurements. In case of chemical reactions, we have high order nonlinear terms. Estimating unmeasured process variables is known as coaptation (Mah, 1990). In case of linear data reconciliation problem, the decoupling of the coaptation and reconciliation problems can be efficiently carried out by Crowe's projection technique (Crowe et al., 1983). In this technique the unmeasured process variables are eliminated from the mass and energy balance constraints by premultiplying these equations by a “projection matrix”. This technique reduces the number of equations. In case of nonlinear problem, the procedure involves the linearization of the process constraints. The resulting linear sub-problem can be solved by using Crowe's projection matrix method. The standard deviation of N data points is calculated by Eqn. (3). We use a limited number of measurements to make an estimate of standard deviation, so s is set to be the s:
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Fig. 1. Online (real-time) optimization strategy (Knopf, 2012).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u PN 2 u i¼1 ðxi xÞ u t s¼ ðN 1Þ
(3)
sys
Gross errors are non-random errors that may bias the optimization results. Therefore it is important to identify and remove these errors from the reconciliation process. The global test (GT) method is used for the detection of nonrandom errors (gross errors) in the measured data, Mah (1990).
The standard deviation is used to calculate the Gaussian or normal distribution function:
1 FðxÞ ¼ pffiffiffiffiffiffi exp s 2p
ðx xÞ2 2s2
! (4)
GT ¼
i;measured variables
Plot of F(x) vs. x is bell-shaped with a single pick at x. The width of the curve is controlled by standard deviation. Small value of standard deviation corresponds to a sharper curve, Knopf (2012). The probability that a measured variable x falls within a range of x1 to x2 is:
1 Pðx1 x x2 Þ ¼ pffiffiffiffiffiffi s 2p
Zx2 exp x1
! ðx xÞ2 dx 2s2
jerrorjmax jaccuracyj ¼ 1:96 1:96
(5)
(6)
xþ xi i
!2 (7)
si
If GT is less than c2ð1aÞ ðyÞ, gross errors are not expected in the process. Here, a is level of significance, and y is degree of freedom. c2 is the chi-square statistical test. The value of chisquare test depends on the number of degrees of freedom and the statistical threshold of the test, typically 95%. If GT is greater than c2ð1aÞ ðyÞ, gross errors are present in the process. In order to locate the most likely suspect measurement, MTi is computed using Eqn. (8) and the highest one is the suspect measurement. The suspect measurement must be removed from the measured variables and placed in the unmeasured data set. New data reconciliation is started and repeated until no more gross errors can be detected using Eqn. (8), Knopf (2012).
The integral of P(x) from ∞ to ∞ is 1 (i.e. 100%). A 95% confidence level is a P(x1 x x2) ¼ 0.95. Since the function is symmetric we have: x1 ¼ x 1:96s and x2 ¼ x þ 1:96s. For most of measuring devices, the uncertainty of s is:
s¼
X
Note that in order to use Eqn. (8), A must involve linear constraints.
Table 1 Inlet and outlet streams to the plant. Name of stream Inlet streams Outlet streams
a b
Flow Measurement. Storage Tank Level Measurement.
Number of measuring instruments Natural gas feed (FMa) Export gas to pipeline (FM) Ethane (FM) C3 (FM and/or LMb) C4 (FM and/or LM) Condensate (LM) Sulfur (FM) Fuel gas (FM) Flaring (FM)
4 6 2 4 4 4 4 2 1
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Export gas to pipeline F5+F6+F7+F8+F9+F10 Ethane export (C2) F11+F12
Natural gas processing plant Natural gas feed F1+F2+F3+F4
LPG Export F13+F14+F15+F16 Sulfur F17+F18+F19+F20 Fuel gas F21+F22 Flaring F23 LPG storage F24+F25+F26+F27 Condensate storage F28+F29+F30+F31
Fig. 2. Inlet and outlet streams of the natural gas processing plant.
on daily performance of the gas processing plant. Figure (2) shows the overall process, inlet and outlet streams. The equality constraint of the optimization problem is the total mass balance of the plant, Eqn. (9).
jai j ffi MTi ¼ pffiffiffiffiffi Vii ai ¼ xi xþ i Vii are the diagonals of the covariance of a 1 AQ V ¼ covðaÞ ¼ Q AT AQAT 3 2 2 6 6 6 Q ¼6 6 6 4
s1
(8)
7 7 7 7 7 7 5
s22 s23 s2n
FNatural Gas Feed FExport Gas FC2 FLPG Export FSulfur FFuel Gas FFlaring FLPG Storage FCondensate Storage ¼0 (9) Or
3. Data reconciliation problem formulation and solution In this study, the data reconciliation method is applied on a natural gas processing plant. The process data including flowrate, level of storage tanks, temperature, pressure, etc. are measured and saved in an Excel file every minute (for each variable over 1400 measurements per day). These data are used to monitor the performance of the plant. For each measured variable, Eqn. (3) is used to calculate the standard deviation (N ¼ 345,000). The following Table 1 gives the inlet and outlet streams to the plant: The total number of streams is 31. LPG and Condensate products are sent to the storage tanks. The incoming mass flow into the storage tanks depends on the level of the tanks and can be computed using the linearized empirical correlations given by the tank manufacturer. The objective function in Eqn. (2) is formulated based on the average of measured variables in 1 day and the standard deviation of each variable. The reason of using the average of measured variables of 1 day is that the inventory reports are prepared based
F1 þ F2 þ ::: þ F31 ¼ 0 Where:
FNatural Gas Feed ¼ F1 þ F2 þ F3 þ F4 FExport Gas ¼ F5 þ F6 þ F7 þ F8 þ F9 þ F10 FC2 ¼ F11 þ F12 FLPG Export ¼ F13 þ F14 þ F15 þ F16 FSulfur ¼ F17 þ F18 þ F19 þ F20 FFuel Gas ¼ F21 þ F22 FFlaring ¼ F23 FLPG Storage ¼ F24 þ F25 þ F26 þ F27 FCondensate Storage ¼ F28 þ F29 þ F30 þ F31
(10)
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F17 F18 F19 F20 F24 F25 F26 F27 F28 F29 F30 F31
0
¼ 0:145 F17 0 ¼ 0:145 F18 0 ¼ 0:145 F19 0 ¼ 0:145 F20 ¼ 43:10 L1 1:027 106 ¼ 43:17 L2 0:724 106 ¼ 34:52 L3 0:500 106 ¼ 31:53 L4 0:526 106 ¼ 122:75 L5 1:837 106 ¼ 122:57 L6 0:648 106 ¼ 123:01 L7 1:121 106 ¼ 122:93 L8 1:086 106
Substituting Eqn. (10) in Eqn. (9) yields:
F1 þ F2 þ F3 þ F4 þ F5 þ F6 þ F7 þ F8 þ F9 þ F10 þ F11 þ F12 þ F13 þ F14 þ F15 þ F16 þ 0:145 F17 þ 0:145 F18 þ 0:145 F19 þ 0:145 F20 þ F21 þ F22 þ F23 þ 43:10 L1 þ 43:17 L2 þ 34:52 L3 þ 31:53 L4 þ 122:75 L5 þ 122:57 L6 þ 123:01 L7 þ 122:93 L8
Matrix of coefficients, A, is:
1 1 1 1 1 1
43:17 34:52 31:53
4. Results and discussion The first case considered here is the reconciliation of the daily average of the measured variables. The mass imbalance of the measured data is equal to þ6.70% (mass flow rate of products is greater than the natural gas feed). This is due to the presence of random and/or nonrandom errors in the system. Such inaccurate
¼ 7:471 106
A ¼ ½1 1 1 1
(x0). fmincon (fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) returns the minimum of the objective function (fun) subject to the linear equality (Aeq*x ¼ beq) and inequality constraints (A*x b). Furthermore, nonlinear inequality c(x) or equality ceq(x) constraints are defined in nonlcon. The optimal solution is always in the range lb x ub. fmincon uses one of these algorithms: interior-point, active-set, trust-region-reflective, or SQP. SQP is the most successful method for solving nonlinear constrained problems. The size and complexity of the problem to be examined and nature of the mathematical methods will influence the computational requirements for optimization problem. As the number of streams grows, the number of measured flow-rates and temperature will grow as well which has a direct effect on the size of A. Scaling of process variables and equations improves the numerical solution of data reconciliation problem. In case of linear constraints, it is possible to reduce Eqn. (2) to an unconstrained Quadratic Problem (QP) that can be solved analytically. The solution is obtained by means of Lagrangian multipliers (Knopf, 2012).
1 1 1 1 1 1 122:75
122:57
0:145 0:145 0:145 0:145 1 1 123:01
143:10
122:93
A is 1-by-31 matrix. The optimization problem (Eqn. (2)) is solved by fmincom in Matlab™. fmincon is used as a general nonlinear solver to find the minimum of a constrained function starting at an initial estimate
raw data can lead to poor decisions, and may unfavorably influence different nodes of the process. Note that the mass flow-rate of 12 streams out of 31 is equal to zero in the time period which the data are reconciled. Based on the measured data, the mass flow of outlet streams
Fig. 3. The difference between daily averages of measured and reconciled mass flows.
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Fig. 4. The difference between averages of measured and reconciled variables. Case 1 to Case 24: hourly averages, Case 25: daily averages (Case 1: 0e1 AM, Case 2: 1e2 AM,…). Note that the mass flow-rate of 12 streams out of 31 is equal to zero in the time period which the data are reconciled.
compared to the mass flow-rate of natural gas feed in % is:
Refined gas to sales gas pipeline (export gas): 71.91%, Ethane (C2): 3.64%, LPG: 7.01%, Condensate: 19.72%, Fuel gas: 3.65%, Sulfur: 0.33%, Flaring: 0.44%. The mass imbalance of the reconciled data will be zero.
The results of data reconciliation based on daily average indicate that 66.14% of the feed is sent to sales gas pipeline, 26.05% is sent to the storage tanks, 3.53% is ethane, 3.54% is used as fuel gas, 0.32% is solid sulfur and the rest is routed to flaring system. Figure (3) shows the difference between the reconciled and measured data. The reconciled flow of natural gas feed is 2.55% higher than the measured flow. The reconciled mass of outlet streams is less than the measured mass flow. For example, the reconciled flow of export gas is 5.68% less than the measured flow. In this case the value of objective function is equal to 0.182. Since the reconciled values of the process variables deviate from the
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Fig. 5 represents the value of the objective function for the entire time period considered here. In all cases, y ¼ 1 and c2ð0:95Þ ð1Þ ¼ 3.84. This means that there is no gross error in the process. The penalty of the measurements on export gas has the highest impact on the objective function. Since the measured data deviates from the true value (reconciled value), process performance is probably far from optimum condition. Reconciled data can be used for simulation and optimization (Fig. 1). 5. Conclusions
Fig. 5. The value of the objective function for the entire time period considered here. Case 1 to Case 24: hourly averages, Case 25: daily averages (Case 1: 0e1 AM, Case 2: 1e2 AM,…).
measured ones, the minimum of the quadratic objective function is greater than zero. High values of objective function means that the data discrepancies (reconciled values e measured values) are significant. If GT is greater than c2ð1aÞ ðyÞ, gross errors are present in the process. c2ð1aÞ ðyÞ is the chi-square statistical test. The value of chi-square test depends on the number of degrees of freedom and the statistical threshold of the test, typically 95%. In all cases, y ¼ 1 and c2ð0:95Þ ð1Þ ¼ 3.84. In this case, the penalty of the measurements on export gas (F5eF10) has the highest effect on the objective function. An hourly average of measured data is also reconciled. For example between 1:00e2:00 AM, the imbalance between the average values of measured data is equal to 7.02% (mass flow rate of products is greater than the natural gas feed). In this case, the reconciled flow of export gas is 66.07% of the natural gas feed, 25.92% of the feed is sent to storage tanks, and the reconciled mass flow of other streams is equal to 8.01% of feed. The value of objective function is equal to 0.379. The penalty of the measurements on export gas has the highest effect on the objective function. The results of global test show that there is no gross error in the measurements for the limited amount of time studied. In the absence of any and significant changes in operation of the gas processing plant, the daily or hourly imbalance is approximately the same as the cases considered here. The systematic biases may appear suddenly in the process. The best way to detect and remove the gross errors in the system is to develop on-line data reconciliation tool to detect bias in a number of sensors, process leaks, etc. Fig. 4 depicts the values of data discrepancies of the cases considered. Fig. 4a shows the difference between the measured and reconciled values of the natural gas feed (F1eF4). The reconciled flow of natural gas feed is greater than the measured flow. Fig. 4b shows the discrepancy between the measured and noise free flow of export gas to sales pipeline (F5eF10). The reconciled mass flows are less than the measured data. The export gas accounts for about 65e70% of the outlet streams of the plant on mass basis. The difference between the raw and reconciled values of the ethane product (F11 and F12) is depicted in Fig. 4c. Ethane accounts for about 3.5% of the outlet streams on mass basis. Fig. 4d and e represent the difference between the measured and noise free flow of solid sulfur (F17 and F18), fuel gas and flaring. These streams account for about 5% of the total flow on mass basis. The difference between the measured and true values of LPG and condensate is depicted in Fig. 4e. About 20% of the final product is LPG and condensate.
In this paper, the measured data of a natural gas processing plant is reconciled to satisfy mass balance equations. Since the inventory report of the plant is prepared based on the daily performance, one case can be deemed as the reconciliation of the daily average of the measured variables. Here, the mass imbalance of the measured data is equal to þ6.70% (mass flow rate of products is greater than the natural gas feed). The value of objective function is equal to 0.182. An hourly average of measured data is also reconciled. For example between 1:00e2:00 AM, the imbalance between the average values of measured data is equal to 7.02%. The global test results indicate that there is no gross error in the measured data. Symbols ai ¼ xi xþ i Data discrepancy A Matrix of linear constraints in Eqn. (8) c(x) Nonlinear inequality constraint ceq(x) Equality constraint ! FðxÞ ¼
p1ffiffiffiffiffi exp
s 2p
2
ðxxÞ 2s2
Gaussian or normal distribution
function Measured mass flow of stream i, kg/h. Fi GT Global test hk(xi, ui) Mass or energy balance equation lb Lower bound on x L
Level of storage tanks, m
N Number Pðx1 xþ x2 Þ 2 2 Q 6 6 6 4 s ui ub Vii xþ i xi x x0
s1
of measurements Probability
s22
s23
3 7 7 7 5
s2n
Standard deviation Unmeasured flow
Upper bound on x Diagonal matrix of a
Reconciled variable Measured variable Average of measured variable
Initial estimate of x
Subscripts k
i j
Number of constraints k ¼ 1, 2,...,K Measured variable Unmeasured variable
Greek symbols
a
εi
y
Level of significance Vector of random errors Degree of freedom.
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sys s2i c2
Instrument standard deviations Variance
Chi-square statistical test
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