Industrial Application of MILP-Based Simultaneous Compensation to a

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Ind. Eng. Chem. Res. 2004, 43, 119-126

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Industrial Application of MILP-Based Simultaneous Compensation to a Large-Scale Byproduct Gases Network in an Iron- and Steel-Making Plant Heui-Seok Yi and Chonghun Han* Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja, Pohang, Kyungbuk 790-784, Korea

An extensively long computation time makes it difficult to apply mixed-integer linear programming (MILP)-based simultaneous compensation to large industrial processes. This paper shows successful application of the MILP-based simultaneous compensation to a large industrial process of a byproduct gases network. To reduce the computation time, the set of gross error candidates is calculated before MILP formulation, and then a simultaneous compensation problem is formulated to identify the existence of gross errors from the members of the gross error candidate set. A significant reduction in the computation time makes it possible to apply the MILP-based simultaneous method to hourly management of byproduct gases. Gross errors are identified in measuring instruments for byproduct gas generation and are corrected by the MILP with the gross error candidate method. Variances of the rectified flow rates are smaller than those of measurements, which shows that random and gross errors are corrected by the MILP with the gross error candidate method. 1. Introduction Iron- and steel-making plants generally consume a considerable amount of energy, and energy management between supply and demand is very important for economical operation. Energy consumed in the iron- and steel-making plants can be purchased from other companies in the form of liquefied natural gas, coal, heavy oil, and electricity and be generated as byproducts in the form of gas during the processing of iron ore as well. There are four kinds of byproduct gases in the iron- and steel-making plants: blast furnace gas (BFG), COREX furnace gas (CFG), coke oven gas (COG), and LinzeDonawitz gas (LDG). BFG and CFG are byproducts from the operations of blast furnaces and COREX furnaces during the production of pig iron, respectively. COG is a byproduct from the operation of coke plants during the production of cokes, and LDG is a byproduct from the operation of basic oxygen furnaces during the production of steel. The error-free measurements between generation and consumption for byproduct gases are essential for the production cost estimation, energy savings evaluation, and operational improvement. However, the measurements of generation and consumption for byproduct gases are inherently inaccurate so that conservation laws are not satisfied because of their stochastic properties and possible gross errors. Data reconciliation and gross error identification must be applied to adjust the measurements of byproduct gas flow rates optimally so that they satisfy the conservation laws. In industrial plants, process measurements contain multiple gross errors, and many research works had been implemented to correct the random and gross errors. Compensation methods for multiple gross errors are classified into serial elimination, serial compensation, and simultaneous compensation1. The serial elimi* To whom correspondence should be addressed. Tel.: (+82)54-279-2279. Fax: (+82)54-279-8478.

nation method2,3 considers the measured flows that are identified as gross errors as unmeasured flows, thus losing redundancy during algorithm implementation. Therefore, the serial elimination method may not be applied to gross errors when the removal of the identified gross errors makes the identified flows unobservable. The serial compensation method4 estimates the magnitude of a gross error serially and keeps the redundancy during algorithm implementation. The simultaneous compensation method5-7 estimates the gross error sizes and reconciled values simultaneously. The mixed-integer linear programming (MILP) method for multiple gross errors was also proposed for simultaneous data reconciliation and gross error identification.8 It was reported that the MILP-based simultaneous compensation method is more accurate than the linear combination technique for multiple gross errors. However, an extensively long computation time makes it difficult to apply the MILP-based method to large industrial processes, and the application of MILP-based simultaneous compensation to large industrial processes has not been implemented. In this paper, the simultaneous compensation based on MILP is successfully applied to the large networks of byproduct gases in an iron- and steel-making plant. MILP formulation to identify the existence of gross errors from all measured flows gives a long computation time, and it is impractical to apply the MILP-based simultaneous method to large industrial processes. Therefore, the set of gross error candidates, GC, is calculated by a measurement test before MILP formulation, and then a simultaneous compensation problem is formulated to identify the gross error existence from the members of GC. The computation time by the simultaneous compensation based on MILP with GC (MILP-GC) gives significant reduction compared with the previous MILP-based simultaneous method. Gross errors are identified in the measurements of byproduct gas generation, and their biased values are estimated by MILP-GC method.

10.1021/ie030301s CCC: $27.50 © 2004 American Chemical Society Published on Web 12/03/2003

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Figure 1. Flow network for BFG, CFG, and LDG.

2. Scope and Formulation Figures 1 and 2 show the flow networks of byproduct gases in the iron- and steel-making plant of POSCO.

Figure 2. Flow network for COG and mixed gas.

In Figure 1, the solid line shows BFG flow, the dotted line CFG flow, and the dash-dotted line LDG flow. BFG is generated as the byproduct in five blast furnaces and consumed in four coke plants and six power plants. The remaining BFG is pressurized and then mixed with COG and LDG in the first and second mixers. CFG is generated as the byproduct in a COREX furnace and is consumed in the fifth and sixth power plants. The remaining amount of CFG is mixed with BFG directly. LDG is generated as the byproduct in basic oxygen furnaces at steel-making plants. LDG generated in the first steel making plant is consumed in the first and second power plants. LDG generated in the second steel making plant is pressurized and then consumed in the third and fourth power plants and a low-pressure boiler. The remaining amount of LDG is mixed with BFG and COG in the first and second mixers. Figure 2 shows the schematic diagram of COG and the mixed gas flows. Solid lines represent the flow network for COG, and dotted lines represent the flow network for the mixed gas of BFG, COG, LDG, and CFG. COG is generated as the byproduct in four coke plants and consumed in five blast furnaces, six power plants, four chemical plants, two steel-making plants, etc. The remaining COG is pressurized and mixed with BFG and LDG. The mixed gas is consumed in plate rolling mills (PMs), wire-rod rolling mills (WRMs), hot strip mills (HSMs), etc. The measurements for the flow rates of byproduct gases show discrepancies between the amount of generation and consumption, which makes it difficult to estimate the consumption amount of byproduct gas and the production cost of each plant. The discrepancies of byproduct gases between supplies and demands result

Ind. Eng. Chem. Res., Vol. 43, No. 1, 2004 121

Figure 4. Structure of the MILP-GC application to byproduct gas networks in the iron- and steel-making plants.

as follows. Figure 3. Procedure of simultaneous compensation for multiple gross errors by the MILP-GC method.

from random and gross errors, and the errors can be corrected by data reconciliation and gross error identification. The most accurate method for the correction of random and gross errors is the simultaneous compensation method. However, much computation time is required for the simultaneous method to give the solution for data reconciliation and gross error identification. The flow networks for byproduct gases in the iron- and steel-making plant for this study are very large, and the simultaneous compensation method is difficult apply because of an extensively long computation time. The MILP-GC method is applied to the flow networks of byproduct gases to reduce the computation time in this study, and the procedure of the MILP-GC method is shown in Figure 3. The MILP-GC method is composed of five steps: Step 0. Include all members of the set of measured flows, MF, to the set of nongross error candidates, NGC. Currently, the set of gross error candidates, GC, is empty. Step 1. Implement data reconciliation using the measured flow rates of byproduct gases and incidence matrix of the network of byproduct gases. Step 2. Implement the measurement test to identify the location of gross errors using the reconciled values and the measured values that are members of NGC. If the measurement test finds a gross error, go to step 3; otherwise, go to step 4. Step 3. The measurements that are identified as gross errors become the members of GC and are removed from NGC. Implement the simultaneous gross error estimation and data reconciliation using MILP formulation. In this step, only measured flow rates that are members of GC are formulated as binary variables to represent the existence of gross errors. Go to step 2. Step 4. Stop. Step 1 can be removed from the procedure when a nodal test is used for gross error identification in step 2 because it does not require the reconciled values. In step 3, the problem for simultaneous gross error estimation and data reconciliation for flow networks of byproduct gases in iron- and steel-making plants is formulated

min Φ ) gF ˆ ,gB,gδ

[

g g g | Fj - F ˆ j - gδj| + ∑ ∑ σj g)BFG,COG, j∈GC CFG,LDG, mixed gas

1

]

g g g | Fk - F ˆ k| + ∑ wjgBj ∑ σj k∈NGC j∈GC

1

g

s.t.

AgF ˆ )0

(1) (2)

|gδj| e gUjgBj

(3)

|gδj| g gζjgUjgBj

(4)

g

F ˆ i g0

(5)

B ∈ binary variables

(6)

g

∀ i ∈ MF

(7)

∀ j ∈ GC

(8)

∀ k ∈ NGC

(9)

g ∈ BFG, COG, CFG, LDG, mixed gas

(10)

where gFl is the lth measured flow rate of byproduct gas, g, gF ˆ l the reconciled value of the lth measurement for byproduct gas, g, gA the incidence matrix for flow networks of byproduct gas, gδl the biased error magnitude of the lth measurement for byproduct gas, g, gBl the binary variable to represent gross error existence in the lth measurement for byproduct gas, g, gUl a large number of the lth measurement of byproduct gas, g, that can be considered as an upper bound for the gross error size, gζl a small number of the lth measurement for byproduct gas, g, that is related with a lower bound for the gross error size, gσl the standard deviation of the lth measurement for byproduct gas, g, wj the weighting factor for binary variable to represent gross error existence, MF the set of measured flows for byproduct gases, GC the set of gross error candidates, and NGC the set of nongross error candidates. The objective function is composed of the absolute differences between the measured flow rates and the reconciled flow rates for BFG, COG, CFG, LDG, and mixed gas. All measured flow rates in Figures 1 and 2 are included in the objec-

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Figure 5. Rectification results of the first COG generation by the MILP-GC method.

Figure 6. Rectification results of the second COG generation by the MILP-GC method.

tive function. Equation 2 is the vector form for mass balance equations around the byproduct gas network that is described in Figures 1 and 2. The mass balance equation is prepared for each node of a byproduct gases network, which includes the networks of BFG, COG, CFG, LDG, and mixed gas. Incidence matrices are prepared from the input-output information of Figures 1 and 2. Equations 3 and 4 represent the upper and lower bounds for the magnitudes of gross errors, respectively. The formulation of the simultaneous data reconciliation and gross error estimation problem includes the absolute values in the objective function and constraints, which make it difficult to solve the problem. The absolute formulation in the objective function can be removed by the introduction of the second set of binary variables, S, and the absolute formulation in the constraints can be removed by the introduction of the slack variables, p, n, q, and r. The linear formulation

for the simultaneous data reconciliation and gross error estimation is as follows.

1

min gF ˆ ,gB,gS,gδ, gp,gn,gq,gr

Φ)

∑ gσj ( pj + g

g

nj) +

j∈GC

∑

1

k∈NGC

g

g g σk( qk + rk) +

wjgBj ∑ j∈GC g

(11)

AgF ˆ )0

(12)

Fj - gF ˆ j - gδj ) gpj - gnj

(13)

s.t. g

g

Fk - gF ˆ k ) gqk - grk

(14)

δj - gUjgBj e 0

(15)

g

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Figure 7. Rectification results of the third COG generation by the MILP-GC method.

Figure 8. Rectification results of the fourth COG generation by the MILP-GC method.

-gδj - gUjgBj e 0 g

δj - gSj(1 + gζj)gUj + gζjgUjgBj e 0

-gδj + gSj(1 + gζj)gUj - gζjgUjgBj e (1 - gζj)gU g

g

(16) (17) (18)

Sj - gBj e 0

(19)

pj, gnj, gqk, grk, gF ˆi g 0

(20)

g

∀ i ∈ MF

(21)

∀ j ∈ GC

(22)

∀ k ∈ NGC

(23)

δj - gSj ∈ binary

(24)

Binary variable gBj represents the gross error existence of the jth measurement for byproduct gas, g, and gSj determines the sign of the gross error of the jth measurement for byproduct gas, g. If the value of gSj gB is equal to 0 and the value of gB is equal to 1, the j j sign of the gross error for the jth measurement of byproduct gas g is positive. If the value of gSj - gBj is less than 0 and the value of gBj is equal to 1, the sign of the gross error for the jth measurement of byproduct gas g is negative. The number of continuous variables by the simultaneous data reconciliation and gross error estimation for the byproduct gas network in Figures 1 and 2 is about 2100, and the number of binary variables by the MILP-based simultaneous data reconciliation and gross error estimation is 4200, which makes it difficult to solve the rectification problem. However, the proposed method decreases the number of binary variables, which is dependent on the number of identified gross errors in the networks of byproduct gases.

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Figure 9. Rectification results of the CFG generation by the MILP-GC method. Table 1. Comparison of Computation Times (min) for the Worst Case by the MILP-GC Method and the Previous MILP-Based Method

computation time for the worst case

MILP-GC method

MILP-based method

3.4

129.0

Table 2. Magnitudes of the Estimated Biases by the MILP-GC Method generation

bias magnitude

first COG

second COG

third COG

fourth COG

CFG

13.6

11.7

12.8

3.9

13.2

3. Implementation Results and Discussion Simultaneous compensation by the MILP-GC method is applied to the flow network of byproduct gases in the iron- and steel-making plants shown in Figures 1 and 2. Management of byproduct gases is implemented at intervals of 1 and 24 h in the iron- and steel-making plants; therefore, data rectification must be also implemented at intervals of 1 and 24 h. Figure 4 shows the application method of MILP-GC to byproduct gas networks. It is composed of four modules: data input, data reconciliation, gross error identification, and gross error estimation. Each module is developed by Excel VBA and executed at intervals of 1 and 24 h. What’s Best (Lindo Systems Inc.) is used to solve the optimization problem in the data reconciliation and gross error estimation modules. To show a shorter calculation time by the MILP-GC method than that by the previous MILP-based simultaneous compensation method, operational data of byproduct gases are collected from the plant information system every hour, and then the MILP-GC method and the previous MILP-based simultaneous method are implemented. Table 1 shows the computation times for the worst case by the MILP-GC method and the previous MILP-based method. Computation times result from solving the simultaneous gross error compensation problem on Intel Pentium IV 1.6 GHz with 512 MB RAM, and What’s Best (Lindo Systems Inc.) is used to

Table 3. Variances for Measurements and the Rectified Flow Rates

generation

variances for measured flow rates

variances for rectified flow rates

relative reduction of variancea

first COG secnd COG third COG fourth COG CFG

10.65 27.34 8.39 4.99 165.07

2.33 10.60 3.69 3.89 131.71

0.781 0.612 0.560 0.220 0.202

a The relative reduction of variance is defined as (σ 2 - σ ˆ 12)/ 1 σ12, where σ12 is the variance of the lth measurement and σˆ 12 is the variance of the lth reconciled value.

solve MILP problems for simultaneous gross error compensation. The MILP-GC method requires 3.4 min to give the solution of reconciled values and gross error sizes for byproduct gases. However, the previous simultaneous method based on MILP requires 129 min to give the reconciled values, which is insufficient time for hourly management of byproduct gases. In the iron- and steel-making plants, byproduct gases are consumed as energy sources in many downstream processes, and the assessment of operational improvement and the estimation of energy cost must be calculated for each individual downstream plant using the rectified flow rates of byproduct gases at intervals of 1 and 24 h. The computation time of 129 min for data rectification is insufficient to implement the hourly energy management of byproduct gases. Therefore, the previous MILP-based simultaneous compensation can be used for daily energy management but not hourly energy management because of long computation time. However, the MILPGC method can be used for both daily and hourly energy management of byproduct gases. Only the MILP-GC method is applied to the flow network of byproduct gases in the iron- and steelmaking plant every hour in real time. The MILP-GC method identifies the flow rates of first, second, third, and fourth COG generations and CFG generation as gross errors, and their magnitudes are shown in Table 2. In general, the measured value by the flowmeter designed for large flow rates is not more accurate than that by the flowmeter designed for small flow rates. No

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Figure 10. Rectification results of COG consumption in the first coke plant by the MILP-GC method.

gross error is identified in the measurement of the consumption unit, whose design specification for the flow rate is smaller than that in the generation unit. It is also confirmed by plant engineers that the measuring instruments for flow rates of byproduct gases in generation plants have not been repaired for some years because the management or assessment of byproduct gases has been implemented based on the amount of consumption rates for byproduct gases. Therefore, it is feasible that all gross errors are identified in the measurements of generation plants for byproduct gases. Figures 5-9 compare the measured flow rates with the rectified flow rates that are identified as gross errors. Dashed lines represent measurements for flow rates of byproduct gases, and solid lines represent the rectified flow rates for byproduct gases. The rectified flow rates are biased relative to the measured flow rates because of gross errors in measurements. Table 3 shows the values of variances for the measured flow rates and the rectified flow rates by the MILP-GC method. The variances for the rectified flow rates are smaller than those for the measured flow rates, and all of the relative reductions of variances have positive values, which means that the MILP-GC method is successfully implemented on a byproduct gases network and the random and gross errors in the measured flow rates are removed.9 Figure 10 shows the results of simultaneous data reconciliation and gross error estimation for COG consumption rates in the first coke plant by the MILPGC method. The only COG consumption rate in the first coke plant is shown in this paper as one of the flow rates that have no gross error because of the lack of space. The dashed line represents the measured flow rate of COG consumption in the first coke plant, and the solid line represents the rectified flow rate for COG consumption in the first coke plant by the MILP-GC method. Any gross error is not identified in the measured flow rate for COG consumption in the first coke plant. Figure 10 shows that random errors are removed by the MILPGC method, and the rectified flow rate is not biased to the measured flow rate. Variance of the measured flow rate for COG consumption in the first coke plant is

Table 4. Relative Reductions of Variances for Byproduct Gas Flow Rates

no. 1 no. 2 no. 3 no. 4 no. 5 COREX 1 BF 2 BF 3 BF 4 BF 5 BF COREX 1 COKE 2 COKE 3 COKE 4 COKE 1 HSM 2 HSM 1 P.M 2 P.M 2 NMZ 3 P.M BLT.M 1 WRM 2 WRM 3 WRM ESM CT/B 1 P/P 2 P/P 3 P/P 4 P/P 5 P/P 6 P/P

BFG

LDG

0.472 0.740 0.275 0.258 0.411

0.848 0.878

COG

CFG Generation 0.781 0.612 0.560 0.220 0.202

0.962 0.954 0.925 0.726 0.918 0.415 0.745 0.072 0.281

Consumption 0.602 0.947 0.605 0.052 0.040 0.005 0.292 0.939 0.792 0.949 0.893 0.982 0.560

0.655 0.800

0.980 0.976

0.241

0.251

0.134 0.149 0.105 0.201 0.108

0.365 0.146 0.389 0.397 0.974

0.242 0.450 0.738 0.582 0.896 0.919

0.558 0.860 0.913 0.101 0.005 0.109 0.627

0.484 0.241 0.849 0.109 0.367 0.140 0.150 0.218 0.148 0.201 0.775 0.024 0.945 0.610 0.776 0.514 0.944 0.307

1 BY-PRO 2 BY-PRO 3 BY-PRO WMT 1 SINT 2 SINT 3 SINT 4 SINT F SINT 1 CDQ 2 CDQ OLC STS_M PCI 1 BOF 2 BOF 1 CCP 2 CCP 3 CCP 4 CCP 1 BCCP 2 BCCP Lime 1C-PL 1C-BCAL 2 CRM S-1SMP S-1APL H2 S-2SMP

0.162 0.432 0.370 0.712 0.688 0.642 0.946 0.399 0.058 0.952 0.928 0.674 0.587 0.210 0.792 0.466 0.971 0.881 0.936 0.932 0.930 0.730 0.441 0.888 0.108 0.799 0.599 0.234 0.437 0.567

about 0.458, and variance of the rectified flow rate for COG consumption in the first coke plant is about 0.244, which shows that random errors in measurements are removed by the MILP-GC method. Tables 4 and 5 show the relative reductions of variances for the rectified flow

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Table 5. Relative Reductions of Variances for Byproduct Gas Flow Rates by the Previous MILP-Based Method BFG

LDG

no. 1 no. 2 no. 3 no. 4 no. 5 COREX

0.475 0.738 0.274 0.259 0.415

0.849 0.874

1 BF 2 BF 3 BF 4 BF 5 BF COREX 1 COKE 2 COKE 3 COKE 4 COKE

0.964 0.953 0.925 0.727 0.919

1 HSM 2 HSM 1 P.M 2 P.M 2 NMZ 3 P.M BLT.M 1 WRM 2 WRM 3 WRM ESM CT/B 1 P/P 2 P/P 3 P/P 4 P/P 5 P/P 6 P/P

0.656 0.718

0.981 0.975

0.240

0.250

0.135 0.151 0.106 0.203 0.110

0.364 0.145 0.391 0.398 0.973

COG

CFG Generation 0.782 0.612 0.561 0.219

Acknowledgment This work was partially supported by the Brain Korea 21 Project and the IMT2000 (project number: 00015993) in 2003.

0.201

0.414 0.744 0.072 0.281

0.241 0.452 0.736 0.580 0.896 0.919

Consumption 0.601 0.946 0.606 0.055 0.041 0.005 0.293 0.938 0.793 0.948 0.894 0.984 0.561

0.560 0.861 0.914 0.100 0.005 0.108 0.625

0.484 0.241 0.849 0.105 0.366 0.141 0.151 0.219 0.144 0.205 0.777 0.024 0.949 0.614 0.776 0.511 0.940 0.300

flow rates, and it shows that the random and gross errors in the flow rates of byproduct gases are removed by the MILP-GC method.

1 BY-PRO 2 BY-PRO 3 BY-PRO WMT 1 SINT 2 SINT 3 SINT 4 SINT F SINT 1 CDQ 2 CDQ OLC STS_M PCI 1 BOF 2 BOF 1 CCP 2 CCP 3 CCP 4 CCP 1 BCCP 2 BCCP lime 1C-PL 1C-BCAL 2 CRM S-1SMP S-1APL H2 S-2SMP

0.165 0.431 0.371 0.711 0.683 0.645 0.946 0.397 0.058 0.959 0.924 0.671 0.583 0.212 0.792 0.464 0.972 0.883 0.931 0.933 0.931 0.730 0.441 0.883 0.108 0.801 0.603 0.232 0.432 0.569

rates to those for the measured flow rates by the MILPGC method and the previous MILP-based method. All of the relative reductions of variances have positive values, which means that random and gross errors are successfully removed by the MILP-GC method and the performances of relative reduction by the MILP-GC method and the previous MILP-based method show little difference. 4. Conclusions MILP-based simultaneous compensation with the set of gross error candidates is applied to the flow networks of byproduct gases in iron- and steel-making plants. The averaged computation time by the MILP-GC method shows a significant reduction compared with that by the previous MILP-based simultaneous compensation method. Byproduct gas management is implemented at intervals of 1 and 24 h, and the previous MILP-based method cannot be used for hourly management because the computation time is larger than 1 h. However, the MILP-GC method can be applied to hourly management for byproduct gases because of the short computation time of 3.4 min. Gross errors are identified in the flow rates for byproduct gas generation and are corrected by the MILP-GC method. All of the variances for the rectified flow rates are smaller than those for measured

Nomenclature A ) incidence matrix of a byproduct gases network B ) vector for the binary variable to represent gross error existence F ) vector for the measured flow rates of byproduct gases F ˆ ) vector for the reconciled flow rates of byproduct gases GC ) set of gross error candidates MF ) set of measured flow rates for byproduct gases NGC ) set of nongross error candidates p, q, r, n ) vectors for the slack variables to remove the absolute formulation in constraints S ) vector for the binary variables to remove the absolute formulation in the objective function Ul ) upper bound of gross error magnitude for the lth measurement Greek Letters δ ) vector for the magnitudes of gross errors σl ) standard deviation for the lth measurement ζl ) small number to represent the lower bound of gross error magnitude for the lth measurement Superscript g ) type of byproduct gases

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Received for review April 7, 2003 Revised manuscript received October 21, 2003 Accepted October 29, 2003 IE030301S