FIELD OF INVENTION :
The present invention relates to a method for optimization of ferroalloys
addition to the steel made by basic oxygen furnace (BOF) process. The
invention uses a mathematical model together with the associated
instrumentation for acquiring several kinds of inputs from different
implem entation platforms. This model can be used to obtain optimal
ferroalloys quantities that are to be added to the liquid steel. These
ferroalloys are added to steel to achieve aim grade specificaions. The
invention provides a method for adding proper quantities of ferroalloys so as
to achieve all the requirements of the grade specification, while
simultaneously minimizing the cost of the ferroalloys.
DESCRIPTION OF THE PRIOR ART
The steel-making process in BOF converters involves addition of certain
ferroalloys during tapping to the steel so as to achieve the desired grade
specifications. Each ferroalloy has a composition spanning several elements.
The ferroalloy addition is governed by the desired elemental composition
required at the end of steel making process but the ferroalloys addition
keeping one element in mind can lead to imbalances in the specification of
other elements.
The known art of ferroalloy calculation predicts the ferroalloy weights based
on the Arrhenius rate equation involving thermodynamic considerations. It
selects a combination of ferroalloys, based on a fixed priority of selection
from a set of ferroalloys, to achieve the aim analysis of steel. The cost of the
ferroalloys is not considered, and the known technique does not provided a
least cost ferroalloy addition plan. Moreover, as the sequence of selection of
the ferroalloys is fixed apriori, sometimes the grade specifications cannot be
met, and the steel mat is produced has to be down graded.
LD vessels are used for converting hot metal into steel, by blowing oxygen
and ferroalloys are added during tapping. The existing LD Level 2 systems
have a ferroalloy calculation model, which predicts the alloying materials
that are necessary to reach the desired aim analysis, based on the following
factors :
• Ferroalloys available in the shop for alloying
• Input data like, aim analysis, pre-alloy analysis etc
• The existing model simulates the burning of C, Si, Mn and Al. For
each element targeted in the steel grade specification, priorities are
assigned by manually incorporating in a database, based on the
availability of ferroalloys, and the material specification.
The existing models calculate the ferroalloy addition required to reach the
aim element analysis. A certain proportion of the ferroalloys would
combine with the available oxygen. This has to be compensated by adding
an equivalent excess amount of the ferroalloy. The models calculate the final
composition, by using the standard free energy relationship of.......
the oxidation reactions i.e. thermodynamically, and also estimate the tap end
temperature considering the effects of alloying and burning.
During the steel making process in the LD vessels, there is an excess amount
of oxygen left over after the burning off of the carbon. The thermodynamic
calculation involves determining the equilibrium of the alloying elements
with the oxygen in the liquid steel.
In order to reach a certain aim composition of a particular alloying element,
the amount that is going to burn with excess oxygen in the steel, has to be
estimated. A compensating excess amount of the alloying element can then
be added. The existing thermodynamic ferroalloy model performs this
calculation in an iterative manner and has a convergence limit. The
algorithm in the known model, converges in most cases. But occasionally it
fails to converge and terminates after the maximum specified number of
iterations.
SUMMARY OF THE INVENTION
The root cause of the higher ferroalloy cost, occurs because the existing
models work on the basis of fixed priorities, which is supplied in the form of
a database. It does not consider the prices of the ferroalloys to derive a least
cost solution.
The objective of this invention is to achieve the desired aim specification of
tapped steel by the addition of an optimal combination of ferroallyg. This
will reduce costs by Gutting down on the consumption of ferroalloys.
A sizable quantity of ferroalloys is consumed at the LD plants, and this
accounts for approximately 2.5% of the sale price of the steel produced. The
present invention seeks to modify the procedure, so as to provide an optimal
combination of ferroalloy weights to achieve the aim grade specification of
steel, at lowest cost. The technique of Linear Optimization (Simplex
algorithm) is used, which considers the cost of the ferroalloys in the
objective function, under the constraints of steel grade specifications for the
various elements.
The invention uses a mathematical model and the method of its
implementation in the BOF steel making process. The mathematical model
is a constituent of the Automation system, at the level 2, for the BOF shop.
The invention relates to capturing of online data, processing it to obtain
optimal material quantity for grade specification and then delivering the
output to the BOF shop.
The steps involved are :
• Modification of the thermodynamic model to achieve iterative
convergence (burning procedure)
• Tuning and incorporation of a Simplex Optimization module in the
model. The Simplex tableau is used in a novel way, by including a
column vector to represent the materials that are consumed by
oxidation (Burning vector). A separate basic variable has been
introduced called the burning variable. The parameters for this vector
serve to incorporate the thermodynamic calculations, to account for the losses
of the alloying elements by oxidation.
Thus, the present invention provides A method for optimization of ferroalloys
addition to the steel made by basic oxygen furnace (BOF) process, said process
comprising the steps of taking steel analysis after basic oxygen furnace (BOF)
operation, a list of ferroalloys currently available in the bins and their costs;
calculating the burning amounts of various elements considering the
thermodynamic rate equations ensuring convergence to accurate values;
estimating optimal quantities of ferroalloys to be added on the basis of element-
wise grade specification; the estimation being based on a model of iterative
linear programming implementation; and adding the estimated optimal
ferroalloy quantity to the steel under constraints of grade specification.
DETAILED DESCRIPTION OF THE INVENTION;
The BOF uses liquid hot metal plus a percentage of scrap (10-15%) as metallic
inputs. A BOF converter is a pear-shaped vessel holding upto 100-200 tons of
liquid hot metal, into which oxygen is blow at high pressure. This reacts with the
carbon in the iron charge, which ranges from 4% to 5%. This oxidation reaction
takes around 10-20 minutes at high flow rates of oxygen to get completed,
generating extensive heat and forming carbon monoxide (CO) and carbon
dioxide (CO2). Various fluxes such as lime, limestone are added, which foam and
trap impurities of the metal in slag. This process is a modification of the earlier
Bessemer process and is called the LD process or the BOF process. A certain
level of dissolved oxygen remains in the steel after the process, which has to be
removed. For this...............
purpose, aluminium is added to the steel, at the time of tapping it into the
transfer ladle.
Since the reactions are vigorous and are completed in a relatively short time,
the chemical reactions do not have sufficient time to proceed to equilibrium.
The dissolved excess oxygen that remains in solution can serve to slowly
combine with the remaining carbon in the steel. The difficulty is that when
alloying elements are added, the dissolved oxygen can combine with them in
preference to carbon. The equilibrium concentrations can be estimated from
the Arrhenius rate equation.
The steel that is made has various physical and chemical properties that
depend upon the content of various elements present in it. These "grades" of
steel contain different elemental composition, like carbon between certain
limits of percentage, aluminium between certain limits of percentage etc.
To achieve different composition for each element, in accordance with the
specification for that grade becomes a difficult task, as none of the elements
is available in isolation.These elements are all available in various
ferroalloys which have a composition spanning several elements,aad
targeting one element can lead to imbalances in the other elements. For
example, addition of Ferro-Manganese-High-Carbon (FeMn HC) to target
manganese might achieve the specification for manganese but increase
carbon content beyond desirable lim its, thus making the steel unfit for that
grade. In such a situation the selection of a proper mix of the ferroalloys for
alloying is very important.
The steel making process works under a high level of automation, using
control strategies and mathem atical models to derive the process parameters
online. The complexity of the steel making process cannot be modeled using
a single set of equations. There are a series of models governing each
subprocess, from the processing station for desulphurisation to the purging
station before casting into slabs. These mathematical models generate
"setpoints" for the process, use communication protocols to download it to
the Distributed Control Systems, which automatically operate the associated
equipment. The invention here comprises of the mathematical model for
alloying additions and the associated system for acquiring data for the
implem entation or generation of set points for the additions.
Various sensors are required for uploading the status of the charging bins,
such as full or empty conditions, and to update the material availability
tables. Temperature sensors are required for validating the estim ated values.
These temperatures are used in turn to evaluate the Reaction Rate constants
for use in the thermodynamic model.
The inputs that are required for the procedure are :
• Aim value of different elements from the steel grade specification
• Plant constraints for the model, depending on the steel-making route to
be followed
• Steel grade specification (Min, Max and Aim analysis) constraints for the
linear optim ization module (Simplex Module).
The level 2 automation at the LD converter uses a mathematical model
called the static charge calculation (SCC) model. While this model is not the
subject of this invention, the results from this model form an important input
to the ferroalloy calculations.
The inputs to the SCC model consist of the hot metal analysis, the hot metal
weight and temperature as well as the lime, iron oxide, scrap and coke
compositions. In addition the aim steel temperature, aim analysis must be
specified. The model accesses a know-how database for information about
the basicity ranges.
The SCC model proceeds on a heat and mass balances analysis. In the heat
balance, the enthalpies of the inputs (hot metal, heating and cooling agents)
are balanced with the enthalpies of the output (slag, steel)
The outputs from the SCC model consist of the required weights of the slag
building agents (lime+limestone), as well as the weights of the cooling (ore)
and heating (coke) agents, and the volume of oxygen to,be blown, since the
equations of chemical kinetics are not involved,the estimates are inaccurate.
Tuning factors are used to calibrate the estim ates with experim ental resu lts
(cellox readings) which finally provides estim aes of the end blow
temperature within 15 degC with an 80% confidence level.
Since the actual additions might vary from that recommended by the model,
the actual additions are supplied to the SCC model in a final iteration, when
the model estimates the final analysis of the steel-This is referred to as the
end blow analysis, and forms an important input to the ferroalloy
calculations, which is the subject of this invention. The end blow analysis is
a calculated estimate. In addition, facilities are available to determine the
steel analysis from the rapid analysis center (RAC Lab). When such data is
available it overwrites the estimated data.
The complete ferroalloy addition procedure has been shown in Table 1. This
invention claims priority in specific portions of the procedure viz: the
section on Linear Programming (Rtsp lxc) where a novel method of
incorporating the combustion of the individual elements is described
(burning vector), and the procedure for computing the thermodynamic
equilibrium condition (burning).
There are 15 ferroalloys (FAL) that are commonly added (COKE, FESI,
FEMNHC, FEMNLC, FECR, COPPER, FENI, FEMO, AL, SIMN, FETI,
FENB, FEV, FEZR and LIME). Some of these might not be availabe
because of commercial considerations. The availability of the ferroalloys is
updated into a database (FT_MAN_MAT_AVAIL), which is accessed by
the procedure (MatBunkerAvail in Table 1).
There are 15 ferroalloys (FAL) that are commonly added. In addition,
provision has been kept for 5 extra ferroalloy types, to be included
subsequently. This makes a total of 20 FAL types that have been considered
in the model. There are 2 additional basic variables, viz; EndBlow
composition, and the burning vector, making a total of 22 basic variables.
The number of constraints is 54 (upper and lower limits on 27 elements: C,
Mn, P, S, Cr, Ti. Cu, Ni, Mo, Al, H, N, 0, Nb, V, Ca, B, Ta, W, Fe, AS, BS,
Mg, TS, El, E2. The objective function comprises the costs of the alloys, the
aim being to minimize the cost of the additives.
The different grades of ferroalloys have different costs, which influences the
price of the fmal product. The problem is to achieve an optimal mix of
ferroalloys, to meet specifications of the final steel, at the least cost. The
solution that has been implemented addresses the situation by formulating
the problem as a Linear Programming problem. A separate basic variable
has been introduced called the burning variable. The parameters for this
vector serve to incorporate the thermodynamic calculations, to account for
losses of the alloying elements by oxidation.
To achieve aimed specifications for any grade, a formulation using linear
equations was done. This formulation consists of a set of inequalities, two
for each of the elements, representing the upper and lower limiting values.
The variables for this formulation are the quantity of each ferroalloy that
should be added in a heat for achieving the grade specifications. To
construct this model, all elements that are a constituent of the chemistry of
hot metal or of steel were considered to represent a set of two equations, one
for the lower limit constraint and other for the upper limit constraint. Then,
all available materials were taken into consideration which can be added to
the liquid steel to achieve the elemental composition, for example, coke as
well as FeMnHC were considered as additions for carbon. The shop floor
constraints for other process variables, such as addition of coke in case of
temperature after blow is less than the target, or making Silicon constraints
more stringent to include the possibility of Silicon pickup at later stations
were incorporated to make the model robust. As the process of alloying is
not a satic process, and the elements that are added react with the oxygen
remaining in the steel after the blow, the oxidation losses were also to be
considered in the model. The objective of the linear program was to identify
a least cost combination of ferroalloys that is capable of achieving the aim-
grade specifications. In such a case a model that does not take into account
losses, would violate the lower limit constraint for certain elements.
To account for such an occurrences additional variable was constructed
which contained the estim ated losses for burning,called the burning variable.
The entries to this variable were obtained from the thermodynamic routine.
This calculates the initial amount of element to be added so that the requisite
amount remains in the steel after some part is lost due to oxidation. This
value is obtained after iterative calculations for the solution scheme of the
model, constituting of the Simplex routine and the thermodynamic
optimization routine.
The addition of ferroalloys during tapping is dependent on the route of steel
making. Hence information about the route is essential. The steel making
routes are ;
1. Vessel-O lp-Caster, 2. Vessel-O lp-LF-Caster, 3Vessel-01p-RH-Caster
4. Vessel-O lp-LF-RH-Caster and 5. Vessel-Olp-RH-LF-Caster
The existing art for estimation of the end analysis of steel, starting from an
initial composition is now described. Subsequently, the changed calculation
procedure will be described. During the steel making process in the LD
vessels, there is an excess of oxygen left over after the burning off of the
Carbon. The basic thermodynamic calculation involves determining the
equilibrium of the alloying elements with the oxygen in the liquid steel.
The thermodynamic equation [Arrhenius rate equation] is of the form .
K»E**pe*O**po=l
Where K is a constant [defined as a function of temperature]
For the chemical reaction peE + po 0 -> E pe 0 po [eg.C + 0-> CO;pe=l
po=l]
E is the concentration of the alloying element E
0 is the concentration of oxygen, expressed as mole fractions.
The common alloying reactions involve the burning with nascent Oxygen
C + 0 -»CO;pe= 1 po = 1
Mn + O -»MnO;pe=l po - 1
2A1 + 30 -» A1203; pe - 2 po - 3
Si +20-»SiO2;pe-l po = 2
In order to determine the final composition of a particular alloying element,
the amount that is going to burn with the excess oxygen in the steel, has to
be estimated. A compensating excess amount of the alloying element can
then be added. The technique that is known in the art performs this
calculation in an iterative manner.
Starting from an initial composition, the known technique incrementally
combines the alloying element with oxygen, so that the concentration of
both the element as well as the oxygen is reduced. The resulting
concentrations are then checked against the equilibrium equation, to
determine whether equilibrium is reached within a permissible error limit,at
which stage the calculations are deemed to have converged. Checks are
incorporated to prevent cycling (say a maximum of 50 iterations).
This technique converges in most cases. But occasionally it gives erroneous
results. So an alternative algorithm to perform the same function was
developed. As an example :

The amount of Al consumed should be in the proper stochiometric ratio with
the amount of oxygen.
Al consumed - [.099045-001242] - 0.097803;
Oxygen consumed = [09781- 010817] - 0.086993
Ratio Al consumed to Oxygen consumed = 097803/ 086993=1.124263
Ratio of 2*AtomicWtAl to 3*AtomicWtOxygen=2*26.9815/3*16=l.124229
This satisfies the chemical equation
For the equilibrium reaction we have :
K*E**pe*O**po = l
1.46924503040E + 11*.001242**2*.010817**3 - 0.286852019
This violates the equilibrium equation.
The procedure has been detailed in Table 3, and has been incorporated in
the procedure (Burning in Table 1). This gave the solution as
"oEndE - 0.0021
*oEndO =0.0116
Al consumed = [099045-. 002094] = 0.096951;
Oxygen consumed - [09781-. 011576] = 0.086234
Ratio Al consumed to Oxygen consumed = .096951/.086234 = 1.124278
Ratio of 2* AtomicWt to 3*AtomicWtOxygen=2*26.9815/3*16=1.124229
This qualifies the chemical equation.
For the equilibrium reaction we have : K*E**pe*O**po - 1
1.46924503040E+11*.002094**2*.011576**3=0.99936
This satisfies the equilibrium equation, within the acceptable error limit.
The new algorithm is called, when the conventional method gives an
erroneous solution. This improved the accuracy of the program. It has been
included in the procedure (Burning) in Table 1. A solved example with this
algorithm has been shown in Table 3. This algorithm calculates the end
burning compositon, in case the iterative conventional technique fails. The
criterion for a successful calculation is determined form the difference
between the last two iterations, which has to be less than a predetermined
limit. In case the difference is larger than the permissible limit, the
conventional iterative procedure is considered to have failed, and the new
algorithm is invoked. This is one of the items in this invention. This
algorithm is accessed each time that the thermodynamic calculations are
invoked in the procedures (AlBurning, Churning, SiBurning,
MnBurning.SiMnBurning ; Table 1),
The solution procedure first evaluates a starting solution. Since the
Arrhenius equations contain exponents, the solution changes sharply near
the solution. That is why the starting point has to be chosen carefully,
otherwise the iterations might converge slowly or in some cases not at all.
The starting value is checked against certain limiting values, and the one,
which meets all the requirements, is taken. Next an iterative procedure is
adopted. In the iterative stage a multiplying factor continuously reduces the
value of the amount of material being consumed, and the calculations are
checked for convergence.
There are 15 ferroalloys (FAL) that are commonly added. These are Coke,
FerroSilicon (FeSi), FerroManganeseHighCarbon(FeMnHC)
FerroManganeseLowCarbon(FeMnLC), FerroChromium(FeCr), Copper(Cu)
FerroNickel (FeNi), FerroMolybdenum (FeMo), Aluminium (Al),
SilicoManganese (SiMn), FerroTitanium (FeTi), FerroNiobium (FeNb),
FerroVanadium (FeVa), FerroZirconium (FeZr) and Lime (CaO). In addition
provision has been kept for 5 extra ferroalloy types, to be included
subsequently. ThiB makes a total of 20 FAL types that have been considered
in the program. There are two additional basic variables, viz. EndBlow
composition, and the burning amount, making a total of 22 basic variables.
The composition of each ferroalloy spans across several elements. A total of
25 elements have been considered, whose values affect the steel grade being
produced. These are carbon, silicon, manganese, phosphorus, sulfur,
chromium, titanium, copper, nickel, molybdenum, aluminium, hydrogen,
nitrogen, oxygen, niobium, vanadium, calcium, boron, tantalum, tungsten,
iron, aluminium soluble, boron soluble, magnesium and titanium soluble. In
additio, provision has been kept for 2 extra elements, which might be
included subsequently. This makes a total of 27 element types that have been
considered in the program. The composition of the various FALs is kept in a
separate file (RT_MAT_FAL_SPECS), and the values are downloaded into
the procedure during execution. This data is accessed in the procedures
GetAlloyHeatEquData and ReadAnalyses in Table 1. An edited portion of
the FAL composition table has been shown in Table 2. The 4 columns at the
right provide additional information.
The End Blow Composition is estimated by a separate model (SCC Model)
and is stored in a database (RT_HEAT_DATA). This database is updated in
case an analysis is available from the rapid analysis center. This database is
accessed in the procedures GetAlloyHeatEquData and PutAlloyModOutput
(Table 1).
The thermodynamic module calculates the amount of element that is burnt.
The Grade Specific Lower and Upper Limits for each element is obtained
from a database in the file RT_STL_GRD-DATA. This database is accessed
in the procedure GetAlloyHeatEquData (Table 1). The tabular representation
of the data in this form is known as the Simplex Tableau. This has been
shown in Table 2.
The flux material properties are kept in a data file
RT_MAT_FLUX_SPECS, and the scrap steel properties are kept in a data
file RT_MAT_SCP_SPECS. The information from these files is accessed
by the procedure ReadAnalyses (Table 1).
It will be appreciated that the sequence of data retrieval can be changed by
anyone conversant with the art. What this invention claims is the various
types of information that are required for estimation of the optimal ferroalloy
additons, and the way that this information is pfocessed. The input
information is compiled from numerous sensors and laboratory analyses as
well as from static databases. The calculation procedure is described in the
following sections.
In the example shown in Tabe 2, the initial condition
(EndBlowComposition), violates the lower limit requirement of Manganese
(.02% against a minimum requirement of 0.2%), and aluminium (0 against a
requirement of 0.021%). In addition there is an excess of dissolved oxygen
in the liquid steel (.1078%) that has to be removed. To reach equilibrium
condition within these specified limits would involve combustion of a
certain proportion of these elements.
The Simplex Optimization technique is a standard Linear Programming
technique. The novelty that is claimed pertains to the way in which the
combustion process of the alloying elements is incorporated in the
technique. This is described at length in the following section on the Burning
vector.
The Simplex technique is performed in the procedure (Rtsplxc). This
accesses the procedure (Wralan) for the equilibrium alloying values, which
refers to (Cmpstn) for the compositions, which refers to (CalcDeOx) for the
quantities being burned, which refers to several procedures (AlBurning,
Churning, SiBurning, MnBurning, SiMnBurning) for the various elements
being burned. All these burning calculations refer to the procedure
(Burning), which is one of the items claimed in this invention. All these
procedures have been enumerated in the overall structure of the ferroalloy
addition model (Table 1).
The calculation of the Burning vector (in Table 2), is the other claim of this
invention. The proportion of combustion is calculated form the
thermodynamic model (5.5.4.3) and the values have been indicated in the
BURNING column. Thus in order to reach a final aluminium composition of
.021%, the amount of Al that would combine with the excess Oxygen would
be .10782% (the -ve sign indicates deduction). This would being down the
amount of free oxygen to a level that would be in equilibrium with the
.021% Al level required in the final composition. The tableau has been
shown in Table 2.
Burning is considered only for the elements (Al, Si, C, Mn, and the element
combination SiMn). This burning acts to reduce the end blow content of
these elements. For other elements, if the end blow content exceeds the
upper limit, there is no way to reduce the elemental composition in the BOF.
The heat must be separately treated in the ladle furnace to decrease these
elements. Hence the upper limit is modified to 100% in case the end blow
composition of the other elements exceeds the upper limit. Otherwise, the
Simplex procedure tries to put in FAL with low compositions of these
elem ents.
The Simplex procedure (Rtsplxc Table 1) accesses the procedure
(CalcDeOx) through several intermediate procedures (Wralan, Cmpstn). The
procedure (CalcDeOx Table 1)Calculates the actual analysis in the first
stage, and the so-called fictitious aim analysis, i.e.the analysis the steel
should have prior to oxygen burning, in the second stage.
In the first stage the equilibrium composition (actual analysis) is derived
from the initial analysis by oxidation of an element (-combination). These
are taken in turn as follows (Al, Si,C, Mn, and the element combination
SiMn). The reaction rate constants are temperature dependant, but some
typical values at 1700 degC are (1.46ell for Al,5077 for Si, 454 for C, and 6
for Mn.
Since aluminium is several orders of magnitude more reactive with oxygen
than the other elements, it is assumed that the combustion with Al proceeds
essentially to completion. A sample of such a calculation has been shown in
Table 3. The other elements get a chance of being oxidized only when there
is a shortfall with aluminium. But the calculation procedure is similar, with
the specific value of the rate constant for that element being substituted.
After carrying out this procedure with each element in turn (Al, SiC, Mn,
and the element combination SiMn), the remaining oxygen compostiion is
compared with previous remaining oxygen estimate. In case the new
estimate is lower, the oxygen composition is assigned this new value. This
completes the first stage, and provides the estimate of the actual analysis of
the steel. The calculation is similar to that detailed in Table 3.
The actual analysis estimate, calcuilated in the first stage, is used to calculate
the "fictitious aim analysis", i.e. the analysis the steel should have prior to
oxygen burning, in the second stage. This comprises of the elemental
compositions of the various items, and indicates that if the steel contains
these elements in the calculated proportions, then after combining with
oxygen, the target composition of the steel grade will be met. Since the
target composition is not specified as a single value for an element but rather
as bounding upper and lower limits, the target values are taken to lie in the
range by a simple linear interpolation. The mix of ferroalloys that will
provide the requisite amounts of the elements is then determined from the
Simplex procedure (Rtsplxc in Table 1). The "fictitious aim analysis" is
incorporated into the Simplex tableau, by formulating a special column
called the "Burning vector". The variable representing this column is forced
to enter the basis in the Simplex procedure, and is constrained to a unity
value. The values in the column comprise of the difference between the
linear interpolation target composition, and the "fictitious aim analysis".
Since the value of this variable is constrained to unity, in effect it reduces the
availablity of the elements by a like amount.
The end blow composition is also represented as a column vector in the
Simplex tableau (Table 2). This variable is also forced to enter the basis, and
is constrained by an equality constraint to take the value of unity. In effect,
this operation adds the end blow steel elemental composition to the solution.
The steps in the Simplex procedure have been shown in Table 5.
The simplex algorithm arrives at the basic feasible solution, in the first
phase, by considering an auxiliary objective function, which is evaluated by
assigning a penalty weight to the constraints that have been violated. Then
an entering and a leaving variable are identified, after which the tableau is
pivoted. This process is repeated, till all the constraints are satisfied.
A basic feasible solution (one which satisfies the constraints) would be ;
FEMNHC : .25% and AL.14% of the total steel in the heat (#4732) of 136T.
For the heat, the FAL addition would be FEMNHC335Kg and AL 188Kg, to
meet the constraints.
In the second or optimization phase, the objective function comprising of the
ferroalloy costs is substituted in place of the auxiliary objective, and the
iterative procedure is repeated till optimum. The cost of the various
ferroalloys is accessed by the procedure. A sample of ferroalloy cost has
been shown in Table 4. In the instance that has been shown, the optimal
solution (obtained in the second phase) has been shown in Table 2. The final
composition (slack limits), has also been shown.
This method can be applied in all BOF either online or offline to obtain
optimal ferroalloy material quantities to be added to the steel.
The following changes can be made on the level II system for the
convenience of the operators to enable them to follow the model predictions,
i) Automatic calculation by the model in every heat after blowing end.
ii) Automatic activation of the ferroalloy calculation screen after blowing
end.
Hi) Display of the aluminium weight as Al (ingot) and Al(Notch) on
Level II Screen,
iv) Display of aluminium weight as zero for open tap heats (RH heats)
v) Provision for daily reports of the model performance,
The same procedure may also be extended to secondary steel making
processes such as in Ladle Furnaces.
errmul-0.999
iKeAl = 1.46924503040E+11; ATOM_AI=26.9815; AT0M_WT_0=16; Pe=2; Po=3
Initial----------------------------» Al Burning - Entry
iStartAI= 0.099; iStartO= 0.0978

The smallest value of the starting point [r4=.0018345] provides the initiation for the iterative
convergence procedure. Since the Arrhenius equations contain exponents, the solution changes
sharply near the solution. That is why the starting point has to be chosen carefully, otherwise the
iterations might converge slowly or in some cases not at all.
In the iterative stage the value of r4 is continuously reduced by a multiplying factor of
[errmul-0.999], and the calculations are checked for convergence.
ij=0;r4=r4/errmul;
while(ij<1 ||fabs(r14)>fabs(r18))
{
r4=r4*errmul
[.0018326;.0018308;.0018290;0018271; After 24 iterations:.0017956]

This gives the solution as: r4=.0017956; *oEndE = r9= 0.0021; *oEndO = r10= 0.0116
Table 3: Solved example of the algorithm for the thermodynamic equilibrium to ensure
convergence [Burning procedure].
1. Prepare the Simplex tableau from the data of the Ferroalloy compositions. [Procedure GetAHoyHeatEquData and
ReadAnalyses in Table 1]
2. Access the data for the End Blow composition, and prepare the End Blow vector [PutAlloyModOutput]. Define
Equality constraint with unity value.
3. Access the Steel grade data, and prepare the bounding limits of the elemental compositions [Upper and Lower limits]
[Procedure GetAHoyHeatEquData].
3.1 Modify the Constraint Lower Limits by a linear interpolation to LL+(UL-LL)*0.15 ForC+Si+MN & *0.25 For AL [Wralan]
3.2 Modify UpperLimit To 100% in case EndBlow of other elements exceeds UL.
Otherwise Simplex tries to put in FALS with low compositions of these elements
[Wralan]
4. Access the Ferroalloy cost data and prepare the objective function [Static data in the procedure Rtsplxc]
5. Calculate the items for the Burning vector in the following steps. Define Equality constraint with unity value.
5.1 Assign Oxygen content. In the first stage calculate the the equilbrium composition
[actual analysis]. This is derived from the initial analysis by Oxidation of an element [-
combination] [Procedure CalcDeOx: Stage 1]
5.2 In the second stage calculation of fictitious aim analysis. Now start oxygen value
becomes the last oxygen value achieved. Assign start values of Al, Si.C, and Mn.
Search for that element [-combination] that reduces O content of the steel bath to the
lowest level [Procedure CaJcDeOx Stage 2]
5.2 Try Oxidation with an element [-combination]. The folowing elements [-combination]
are taken in turn [Al, Si.C, Mn, and the element combination SiMn].
5.3.1 If there is an aim composition for one Element and the aim analysis for that
Element is greater than the initial composition, then search for an Element
composition trial value that burns down to the aim value.
5.3.2 Assign start value. Set zero iteration counter. While the difference between the end
Element composition and the aim Element composition is greater than 0.0001,
iterate till convergence or till the maximum cut-off iteration counter [50].
5.3.3 Set the new value for the Element to get near the aim analysis for that Element.
The ElementBuming procedures [Al, Si,C, Mn, and the element combination SiMn]
calculate new end values for the Element under consideration. These refer in turn
to the [Burning] procedure in Table 1, and detaled in Table 3.
5.3.4 Increase iteration counter. Assign the new O [Oxygen] end value if that O end
value is lower than the previous one.
6. Formulate equality constraints fa the End Blow variable and the Burning variable. The variables are constrained to
unity values.
7. Calculate the auxiliary objective function [Procedure Auxobj]
8. Derive a basic feasible solution [Procedures Bigs, Smals, Invs]
9. Derive the optimal solution [Procedures Bigs, Smals, Invs]
10. Write comprehensive actual FAL addition & resulting composition in the database {Procedure Wralan]. The
calculated shadow costs of the non basic variables is also appended.
Table S: The steps in the Simplex procedure
WE CLAIM:
1. A method for optimization of ferroalloys addition to the steel made
by basic oxygen furnace (BOF) process, said process comprises the
steps of:
- taking steel analysis after basic oxygen furnace (BOF)
operation, a list of ferroalloys currently available in the bins
and their costs;
- calculating the burning amounts of various elements
considering the thermodynamic rate equations ensuring
convergence to accurate values;
- estimating optimal quantities of ferroalloys to be added on
the basis of element-wise grade specification; the estimation
being based on a model of iterative linear programming
implementation; and
- adding the estimated optimal ferroalloy quantity to the steel
under constraints of grade specification.
2. A method as claimed in claim 1, wherein said steps of calculating
the burning amounts of various elements for estimating the end
analysis and adding optimal quantities of ferroalloys are performed
in every heat after blowing end.
3. A method as claimed in claims 1 and 2, wherein said step of
calculating can be performed on lone or off line.
4. A method as claimed in the preceding claims, wherein said
process comprises an additional step of displaying the results of the
calculation in a display unit for the basic oxygen furnace (BOF)
operations.
5. A method for optimization of ferroalloys addition to steel made by
basic oxygen furnace (BOF) process substantially as herein
described.

This invention is about adding an optimal ferroalloy quantity to the steel,
under constraints of grade specifications, with an objective of minimizing
the associated cost.
The application describes the development and implementation of a model
to achieve optimal ferroalloy additions during the steel making process in a
BOF shop. The model is an iterative linear programming
implementation,which takes into consideration element-wise grade
specification and gives the optimal material quantities to be added to each
heat. A number of grades of steel are manufactured each of which contain
specific alloying compositions spanning several elements. To meet the
element wise specifications, materials containing those elements are added
to liquid steel. These additions are done either during tapping the steel from
the converter to the transfer ladles or at later processing stations.