L6 - Continuous Optimization
Transcript of L6 - Continuous Optimization
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L6—
Continuous Optimization
Davide Manca
Lesson 6 of “Process Systems Engineering” – Master Degree in Chemical Engineering – Politecnico di Milano
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• There are at least three distinct fields that characterize the optimization of industrial processes– Management
• Project assessment• Selecting the optimal product• Deciding whether to invest in research or in production• Investment in new plants• Supervision of multiple production sites
– Design• Process design and Equipment design• Equipment specifications• Nominal operating conditions
– Operation• Plant operation• Process control• Use of raw materials• Minimizing energy consumption• Logistics (storage, shipping, transport) Supply Chain Management
Optimization
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• The optimization problem is characterized by:– Objective function– Equality constraints (optional)– Inequality constraints (optional)
• The constraints may be:– Linear– Nonlinear
– Violable– Not violable
– Real constraints– Lower and upper bounds of the degrees of freedom
• The optimization variables are defined as: degrees of freedom (dof)
• Mathematically we have: Min
. . ( )( )
f
s t
xx
h x 0g x 0
Definition
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Linear function and constraints
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Linear function and constraints
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Nonlinear function and constraints
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Nonlinear function and constraints
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Nonlinear function and constraints
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Nonlinear constraints + lower/upper bounds
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Infeasible region
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• The equality and inequality constraints may also include the model of the processto be optimized and the law limits, process specifications and degrees of freedom.
• The constraints identify a “feasibility” area where the degrees of freedom can bemodified to seek for the optimum.
• The constraints have to be consistent in order to define a “feasible” searchingarea.
• There is no theoretical limit to the number of inequality constraints.
• If the number of equality constraints is equal to the number of degrees offreedom the only solution is with the optimal point. If there are multiple solutionsof the nonlinear system, in order to obtain the absolute optimum, we will need toidentify all the solutions and evaluate the objective function at each point, andeventually selecting the point that produces the best result.
Constraints
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• If there are more variables than equality constraints then the problem is
UNDERDETERMINED and we must proceed to the effective search of the optimum
point of the objective function.
• If there are more equality constraints than degrees of freedom then the problem
is OVERDETERMINED and there is NOT a solution that satisfies all the constraints.
This is a typical example of data reconciliation.
Constraints
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• If both the objective function and constraints are linear, the problem is called LINEAR PROGRAMMING (LP)
• If the objective function and/or the constraints are NOT linear with respect to the degrees of freedom, the problem is called NOT linear (NLP)
• A NLP is more complicated than a LP
• A LP has a unique solution only if it is feasible
• A NLP may have multiple local minima
• The research for the absolute optimum can be quite complicated
• Often we are NOT interested in the absolute optimum, especially if we are performing an online process optimization
• The research of the optimum point is influenced by the possible discontinuities of the objective function and/or constraints
• If there is a functional dependency among the dof, the optimization is strongly affected and the numerical method can fail. For example:
23121 3, xxxxfobj
Features
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• Usually the objective function is based on an economic assessment of the involved problem. For instance:
(revenues – costs),
• Also, the objective function may be based on other criteria such as:
• pollutant minimization,
• conversion maximization,
• yield, reliability, response time, efficiency,
• energy production
• environmental impact
• With reference to the process, if consider only the operating costs and the investment costs are neglected, then we have to solve the so called SUPERVISION problems (also CONTROL in SUPERVISION)
Structure of the objective function
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• If we consider both operating and investment costs then we fall in the field of
“Conceptual Design” and “Dynamic Conceptual Design”.
• Since in CD and DCD the CAPEX terms [€] and OPEX terms [€/y] are not directly
comparable (due to the different units of measure) a suitable comparing basis
must be found. This can be the discounted back approach together with the
annualized approach to CAPEX assessment where the depreciation period
allows transforming the CAPEX contribute from [€] into [€/y].
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Structure of the objective function
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Introductory examples
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PROCESS DATA 1) A + B E
2) A + B F
3) 3A + 2B + C G
RAW MATERIALSComponent Availability kg/d Cost €/kg
A 40,000 1.5
B 30,000 2.0
C 25,000 2.5
PRODUCTSProcess Product Reactant required
for [kg] of productProcessing costs Selling price
1 E 2/3 A, 1/3 B 1.5 €/kg E 4.0 €/kg E
2 F 2/3 A, 1/3 B 0.5 €/kg F 3.3 €/kg F
3 G 1/2 A, 1/6 B, 1/3 C 1.0 €/kg G 3.8 €/kg G
A
B
C
x1
x2
x3
Process 1
Process 2
Process 3
E
x4
F
x5
G
x6
Example #1: Operating profit
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Statement: We want to find the maximum daily profit.The dof are the flowrates of the single components [kg/d]
• Profit from selling the products [€/d]
4x4 + 3.3x5 + 3.8x6• Cost of raw materials [€/d]
1.5x1 + 2.0x2 + 2.5x3• Operating costs [€/d]
1.5x4 + 0.5x5 + 1.0x6• Objective function
f(x) = 4x4 + 3.3x5 + 3.8x6 ‐ 1.5x1 ‐ 2.0x2 ‐ 2.5x3 ‐ 1.5x4 ‐ 0.5x5 +‐ 1.0x6 = 2.5x4 – 2.8 x5 + 2.8 x6 – 1.5 x1 – 2x2 – 2.5x3
• Constraints on material balances
x1 = 2/3 x4 + 2/3 x5 + 1/2 x6x2 = 1/3 x4 + 1/3 x5 + 1/6 x6x3 = 1/3 x6
Example #1: Operating profit
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• Upper & lower limits on the dof
0 x1 40,000
0 x2 30,000
0 x3 25,000
• The problem is LINEAR in the objective function and constraints.
• We use LINEAR PROGRAMMING techniques (e.g., the simplex method) to solve
the optimization problem. Since the objective function is a hyperplane with a
research area bounded by hyper‐lines (i.e. equality and inequality linear
constraints) the optimal solution is on the intersection of constraints and more
specifically of equality constraints.
Example #1: Operating profit
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Statement:
We want to determine the optimal ratio, L/D, for a given cylindrical pressurized vessel with a given volume, V.
Hypotheses:The extremities are closed and flat.Constant wall thickness t.The thickness t does not depend on the pressure.The density of the metal does not depend on the pressure.Manufacturing costs M [€/kg] are equal for both the side walls and the bottoms. There are not any production scraps
Unrolling:
We can write three equivalent objective functions:
DLDDLDStot
242
22
tDLDMf
tDLDf
DLDf
2
2
2
2
3
2
2
2
1
Example #2: Investment costs
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By using the specification on the volume V:
By differentiating we obtain:
Then:
N.B.: by modifying the assumptions and considering the bottoms characterized by
an ellipsoidal shape with higher manufacturing cost, the thickness being also a
function of the diameter D, the pressure and the corrosiveness of the liquid,
we get a different optimal L/D :
DVD
DVDDf 4
24
2
2
2
2
1
042
1 DVD
dDdf 3
4VDopt 3
4VLopt
1
optDL
42
optDL
Example #2: Investment costs
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Statement: we want to determine the optimal thickness s of the insulator for a large diameter pipe and a high internal heat exchange coefficient. We need to find a compromise between the energy savings and the investment cost for the installation of the refractory material.
• Heat exchanged with the environment in presence of the refractory:Q = U A T = A T / (1/he+s/k)
• Cost of installation of the refractory material [€/m2]F0 + F1 s
• The insulator has a five‐year life. The capital for the purchase and installation is borrowed. r is the percentage of the capital + interests to be repaid each year. It follows that r > 0.2
• Ht is the cost of the energy losses [€/kcal]• Y are the working hours in a year [h/y]• Each year we must return to the bank which provided the loan:
(F0 + F1 s ) A r [€/y]
Example #3: CAPEX + OPEX
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• Heat exchanged with the environment without the refractory material:
Q = U A T = he A T
• Annual energy savings due to the refractory:
[he A T ‐ A T / (1/he+s/k)] Ht Y [€/y]
• The objective function in the dof s becomes:
fobj = [he A T ‐ A T / (1/he+s/k)] Ht Y – (F0 + F1 s) A r
• The problem is solved analytically by calculating:
d fobj / ds = 0
• We obtain:
sopt = k [((T Ht Y)/(k F1 r))½ ‐ 1/he]
• Note that sopt depends neither on A nor on F0
Example #3: CAPEX + OPEX
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Solution methods
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• The problem: can always be turned into:by changing the sign of the objective function
• The optimization problem can deal with two distinct approaches:
• Equation oriented: based on an overall approach model that describes the process through a single system of equations (in general differential‐algebraic) that solves the problem by considering it as a set of constraints
• Black‐box or Sequential modular: the process model is called by the optimization routine and returns the data required to evaluate the objective function
• The simulation model can then work in terms of either FEASIBLE PATH or INFEASIBLE PATH, depending on whether the equations related to the recycle streams are solved for each call or if the consistency of the recycles is introduced as a linear constraint in the structure of the optimization problem
)( xfMax )( xfMin
Solution methods
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Equation-oriented formulation
The equality constraints h contain all the equations describing the model of the
device/equipment/process/plant to be optimized.
In general h can be a system of differential‐algebraic equations, DAE, in the form:
, , , t h x x p 0
Optimization routine
xUser defined:
f(x)h(x) = 0g(x) ≤ 0, ,f h g
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Black-box formulation
Optimization routine
x User defined:f(x,v)
hreal (x,v) = 0g(x,v) ≤ 0
User defined solver:
hmodel (x,v) = 0
The model of the device/equipment/process/plant
is solved by a user‐defined solver outside of the
real constraints that are directly dealt and satisfied
by the optimization routine.
Also in this case, hmodel can be a DAE system: , , , ,model t h x x v v 0
x
, ,realf h g
v
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Methods for multidimensional unconstrained optimization
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• There is a necessary condition to be fulfilled for the optimal point:that is the gradient of f(x) must be zero (this is not true for cusp points and more in general for discontinuous functions)
• Sufficient condition for the minimum is that: the Hessian matrix of
f(x) must be positive definite.
• There are three distinct classes of methods that differ in the use of the
derivatives of the objective function during the search for the minimum:
• HEURISTIC methods do not use the derivatives of f(x). They are more robust because they are slightly if not affected by the discontinuities of the problem to be solved.
• FIRST ORDER methods work with the first order partial derivatives of f(x) i.e. the GRADIENT of the objective function.
• SECOND ORDER methods use also the second order partial derivatives of f(x) i.e. the HESSIAN of the objective function.
0)( * xf
0)( *2 xf
Multidimensional unconstrained optimization
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• The numerical algorithms are intrinsically iterative and usually perform a series of direction‐searches. At the k‐th iteration we have the k‐th direction sk and the method minimizes f(x) along sk.
• DIRECT or HEURISTIC methods:• Random search (Montecarlo)• Grid search (heavy but exhaustive)• Univariate research we identify n directions (where n is the number of
dof) with respect to which perform iteratively the optimization.
x0
xOtt
Multidimensional unconstrained optimization
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Simplex method (Nelder & Mead, 1965)• The simplex is a geometric figure having n+1 vertices for n dof. We identify the worst
vertex (i.e. having the highest value for f(x)) and we reverse it symmetrically with respect to the center of gravity of the remaining n‐1 vertices. We identify a new simplex respect to which continue the search. The overturning of the simplex may be subject to expansion or contraction according to the actual situation.
x0
xOtt
Multidimensional unconstrained optimization
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Conjugate directions methodConsidering a quadratic approximation of the objective function it is possible to identify its conjugate directions.
Hp.: f(x) is quadratic1. x0 generic2. s generic3. xa minimum on s4. x1 generic5. t parallel to s6. xb minimum on s7. u from joining xa and xb
u is the conjugated direction with respect to s and t and by minimizing it we identify the optimal point xott of f(x) (for that quadratic approximation).
x0
x1
xa
xb
xott
s
t
u
Multidimensional unconstrained optimization
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First order indirect methodsA possible candidate search direction smust decrease the function f(x). It must satisfy the condition:
In fact:
only if:
The gradient method selects the gradient of the objective function (in the opposite direction) as the search direction .The idea of moving in the direction of the maximum slope (i.e. “Steepest Descent”) may be not optimal.
0)( sfT x
)(xf
s
10 8 6
0cos)()( sfsf TT xx
cos 0 90
Multidimensional unconstrained optimization
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Gradient method
Easy search
Difficult searchx0
x0
xott
Multidimensional unconstrained optimization
xott
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Second order indirect methodsThey exploit the second order partial derivatives of the objective function.Implementing the Taylor series truncated at the second term and equating the gradient to zero we get:
Consequently, it must be:
N.B.: the Hessian matrix is not inverted, we solve the resulting linear system via LU factorization.In addition, the Hessian matrix is NOT calculated directly as it would be very expensive in terms of CPU computing time. On the contrary, the BFGS formulas (Broyden, Fletcher, Goldfarb, Shanno) allow starting from an initial estimation of H(often the identity matrix) and through the gradient of f(x) they evaluate iteratively H(x).The solving numerical methods are: Newton, Newton modified: Levemberg‐Marquardt, Gill‐Murray.
0)()( kkkf xxHx
)()(11 kkkk f xxHxx
Multidimensional unconstrained optimization
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Some peculiar objective functions
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Some peculiar objective functions
-10-5
05
10
-10-5
05
100
50
100
150
200
250
Rastrigin
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Some peculiar objective functions
01
23
4
01
23
40
0.5
1
1.5
2
Michalewicz
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Methods for multidimensionalconstrained optimization
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• The objective function and the equality and inequality constraints are all LINEAR. Thus, the objective function is neither concave nor convex. Actually, it is either a plane (2D) or a hyperplane (with n dof).If the region identified by the constraints is consistent we have to solve a problem (“feasible”) that will take us through the constraints and more specifically towards their intersection.
• Simplex Method LP
1210
86
4
xOtt
x0
It is first necessary to identify a starting point that belongs to the “feasible” region.Then we move along the sequence of constraints until we reach the optimal point.The problem may also NOT have a “feasible” region of research.
Linear Programming
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Method of the Lagrange multipliers
The inequality constraints, , if violated, are rewritten as equality
constraints by introducing the slack variables: 0)( 2 xg
( ) 0g x
Nonlinear Programming
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Method of the Lagrange multipliers
The objective function is reformulated to contain both equality and inequality
constraints:
There are necessary and sufficient conditions to identify the optimal point that
simultaneously satisfies the imposed constraints.
It is easy to see how the dimensionality of the problem increases.
2
1 1
( , , ) ( ) ( ) ( )NEC NCTOT
i i i i ii i NEC
L f h g
x ω σ x x x
Nonlinear Programming
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Penalty Function method
We change the objective function by summing some penalty terms that quantify
the violation of inequality and equality constraints:
More generally:
2 ( ) ( ) min 0, ( )Min f h g x x x
1 1
22
( ) ( ) ( )
( ) ( ) min(0, )
NEC NCTOT
i ii i NEC
nn
Min f h g
y y y y
x x x
Nonlinear Programming
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SQP method (Successive Quadratic Programming)The objective function f(x) is approximated iteratively through a quadratic function, while the constraints are linearized and added to the objective function:
the search for the optimal point is made through a direction s (identified by the vector x) over which the objective function and constraints have been formulated.Matrix B is an approximation of the Hessian matrix H and is calculated through the BFGS formulas (Broyden, Fletcher, Goldfarb, Shanno).
bAx
Bxxxcx
..21 )(
ts
MinfMin TT
Nonlinear Programming
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Case-study #1
On-line optimization ofcontinuous processes
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Waste to energy plant with DeNOx catalytic section
On-line optimization of continuous processes
D. Manca, M. Rovaglio, G. Pazzaglia, G. Serafini. Comp. & Chem. Eng., 22(12), 1879‐1896, (1998)
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• Requirements of the optimization procedure
• Economic optimization of the process:
• maximize the steam production and then electrical energy.
• Minimize the operating costs
• Respect the process constraints for a correct plant operation
• Respect the law constraints
• Alternatively
• Minimize the production of micropollutants
• Reduce environmental impact
• Optimal mixing of wastes having different nature
On-line optimization of continuous processes
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Objective function to be maximized
Degrees of freedomWaste flowrate Total air flowrate to the furnace Air flow first drumAir flow rate second drum Air flow third drum Air flow fourth drumSecondary air flow to furn. Air flow afterburner CH4 flow rate afterburnerFirst drum speed Second drum speed Third drum speed Fourth drum speed NaOH flow CH4 DeNOx flowNH3 DeNOx flowrate
Law constraints Process constraints% vol. min. O2 afterburner Delta P max on every drumT out min. afterburner T in max. and min. DeNOx reactorHCl max to the stack % max. unburnt in ashesSO2 max to the stack Max. and min. steam producedNOx max to the stack % vol. max. O2 afterburnerNH3 max to the stack Delta max. combustion on the first 3 drums
T out max. and min. primary combustion chamber
Higher and lower constraints on the degrees of freedom
4 4 4 3 3, ,obj rif rif vap vap CH PC CH DeNOx CH NH NHF W c W c W W c W c
On-line optimization of continuous processes
D. Manca, M. Rovaglio, G. Pazzaglia, G. Serafini. Comp. & Chem. Eng., 22(12), 1879‐1896, (1998)
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Problem solution
• We must adopt a nonlinear constrained multivariable optimization routine which is efficient (in terms of CPU time) and robust (able to identify the solution).
• We must implement a detailed model of the process able to simulate the response of the system whenever the optimization routine proposes a new degrees of freedom vector.
• The process optimizer has as its main task to bring the system to operate in the “feasible” region, where the constraints are respected. In some cases, it may happen that the objective function worsens compared to the initial conditions since the process is brought to operate within the feasibility region. Then, within this region, the optimizer maximizes the objective function.
• Note that the explicit computation of the objective function is almost instantaneous. This does not occur for the evaluation of each single term which composes the objective function as they come from the simulation procedure of the process.
On-line optimization of continuous processes
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1
2
2.1
2.2
2.3
2.4
2.5
3.1 3.2 3.3 3.4
4
Homogeneous combustion section
Heterogeneous
combustion section
Process Model – Primary kiln
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i,RIFOUT
i,RIFIN
i,RIFi,RIF RFF
dtdM
Solid phase
Gaseous phase
0 OUTi,kk
rif
i,RIFOUTi,k Wx
PMR
W
NGiNCk ,1,1
OUTi,RIFFIN
i,RIFF
OUTi,kW
INi,kW
i‐th drum
NGi ,1
Material balances on each drum
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OHkmNxy
NOxpSOkHClnCONClOSHC iiykqpmn
iO222 22
,2
yx i,NOi
1
315025001
2
2
100
outi,OG
%i,NOG
i,NO
i,NO
W)T/exp(TW
Combustion reactions in the solid phase
(Bowman, 1975)
2 21CO O CO2
2 21 1N O NO2 2
Combustion reactions in the homogeneous phase
Chemical reactions
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),,,( ,,,, iinertiirifiipisc MMdfA
)exp(11 ,
iCG
i
Nisciisc AA ,
*,
rifisciO
iOixiRIF PMA
xkR
*
,,
,,,
2
2
Corrective factors:(adaptive parameters)
),,,,( ,,,, TxWdfk ikiariaiipix Granular solid bed
Kinetic determining step: O2 diffusion
,
Primary kiln – combustion kinetics
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
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Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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<
Manca D., M. Rovaglio, Ind. Eng. Chem. Res. 2005, 44, 3159‐3177
Process model – the equations
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Results:
On-line optimization of continuous processes
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Results:
On-line optimization of continuous processes
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On-line optimization of continuous processes
Rovaglio, M., Manca, D., Rusconi, F. Waste Management 18, 525‐538, (1998)© Davide Manca – Process Systems Engineering – Master Degree in ChemEng – Politecnico di Milano 67
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On-line optimization of continuous processes
Rovaglio, M., Manca, D., Rusconi, F. Waste Management 18, 525‐538, (1998)
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Case-study #2
On-line optimization ofdiscontinuous processes
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Slop‐cutP1
P2
Two-components batch distillation
1. Total reflux
2. Collection of P1 in the dedicated tank
3. Out-of-spec collection in the “slop-cut”
4. Collection of P2 in the still pot
On-line optimization of discontinuous processes
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0 0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
0.55 0.6
0.65 0.7
0.75 0.8
0.85 0.9
0.961
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
Fraz
ione
mol
are
Tempo (h)
In the distillate
-0.10
0.10.20.30.40.50.60.70.80.9
1 1.1
0 0.5 1 1.5 2 2.5 3 3.29 4
Fraz
ione
nel
ser
bato
io P
1
Tempo (h)
In tank P1
0.40.45
0.50.55
0.60.65
0.70.75
0.80.85
0.9
0.961
0 0.5 1 1.5 2 2.68 3.29 4
Fraz
ione
mol
are
Tempo (h)
In the distillate
0 0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
0.55 0.6
0.65 0.7
0.75 0.8
0.85 0.9
0.96 1
0 0.5 1 1.5 2 2.5 3 3.51 4
Fraz
ione
mol
are
Tempo (h)
In the reboiler
3.29
Light component Heavy component
Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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• There are three distinct approaches to the selection of the objective functionwhen distillation batch processes are involved:
1. Maximizing the product quantity
Converse and Gross (1963) were the first researchers to face the optimizationproblem for a batch distillation column. Logsdon et al. (1990) solved the NLP.
2. Minimization of the distillation time
The reflux profile is divided into a number of intervals with the target ofreducing the total distillation time (Coward, 1967). Mujtaba and Macchietto(1988) solved the problem adopting the SQP algorithm.
3. Profit maximization
The method is based on a profit function, for instance the capacity factor, thattakes into account both the quantity/quality of the product and the totaldistillation time. Kerkhof and Vissers (1978), Logsdon et al. (1990), Diwekar(1992).
Optimal trajectory
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• There are three distinct approaches to the selection of the degrees of freedomwhen distillation batch processes are involved:
1. Constant reflux distillation
The degrees of freedom are: pressure, vapor flowrate inside the column,distillate flowrate.
2. Constant composition
The degrees of freedom are: pressure, vapor flowrate and the purity of the keycomponent that remains constant throughout the batch.
3. Variable reflux profile
The degrees of freedom are: pressure and reflux ratio (or equivalently thedistillate flowrate) at every time of the batch Optimal trajectory.
Optimal trajectory
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t (h)
D (mol/h)
Δti Di
t (h)
D (mol/h) Variable reflux profile
Optimal trajectory
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1 2
, 0.5i it D tot
P PtMax
Capacity FactorLuyben (1978)
, , , , 0g x x u t spec
Pi Dix x
max0 iD D
max0 it t
. .s tThe DAE system comprises
(NS+2)(NC‐1)+1 ODE (mass + energy balances)
(NS+1) NC AE (thermodynamic equilibria)
(NS+2) AE (stoichiometric equations)
Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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Objective function + constraints Variables
Initial values
OPTIMIZER
Optimal solution
MODEL
SQP
Simplex
Robust method searching the solution within the entire domain of the dof
OPTIMIZATION METHODS
Optimal trajectory
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CAP
D1
3039
4857
6675
t1
0.2
2.44
0.76
1.32
1.88
Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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Second group
First group
Third group
30 39 48 75
D1
3
2.44
1.88
1.32
0.76
0.2
t1
Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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10 20 30 40 50 60 70 80
0 0.5 1 1.5 2 2.5 3 3.5
CA
P
t1
0
10
20
30
40
50
60
70
80
0 0.5 1 1.5 2 2.5 3 3.5
CA
P
t3
20
30
40
50
60
70
80
10 20 30 40 50 60 70 80 90 100
CA
P
D2
25 30 35 40 45 50 55 60 65 70 75 80
0 0.5 1 1.5 2 2.5 3 3.5
CA
P
t2
20
25
30
35
40
45
50
55
60
65
30 40 50 60 70 80 90 100
CA
P
D1
25
30
35
40
45
50
55
60
65
70
75
80
30 40 50 60 70 80 90 100
CA
P
D3
The objective function, CAP, as a function of the six dof respect to the absolute optimum value.
Optimal trajectory
D. Manca. Chemical Product and Process Modeling, 2,12 (2007)
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Case-study #3
Model based control ofindustrial processes
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• Industrial processes are characterized by the following control problems:
1. The process is almost usually multivariate
• several controlled variables: y1, y2,…, yn
• several manipulated variables: u1, u2,…, um
• several disturbance variables: d1, d2,…, dk
2. Complex dynamic behaviour:
• time delays due to the inertia of the system (either material or energetic),
mass flow in the pipes, long measuring times
• Inverse response
• possible instability at open‐loop
Introduction
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3. Intrinsic nonlinearity of the system
4. Operative constraints that are quite dissimilar and complex
• constraints on the input and output variables
• constraints on the changing rate of the input variables
• Constraints on the optimal value of the input variables (e.g., economic
value)
• process and law constraints
• soft and hard constraints
Introduction
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An ideal control system should be:
• Multivariable and capable of managing:
• time delays,
• inverse response,
• process and law constraints,
• measurable and non‐measurable disturbances
• Minimize the control effort
• Able of inferring the unmeasured/unmeasurable variables from the measured ones
• Robust respect to the modeling errors/simplifications and the noise of the measured variables
• Able to manage both the startups and shutdowns (either programmed or emergency) as well as the steady‐state conditions
Model based control
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• The availability of dynamic numerical models of:
• chemical/industrial processes,
• unit operations
• process units,
• plant subsections
• …
allows forecasting the response of the simulated plant/process to possible disturbances and manipulated variables.
• The availability of such dynamic numerical models paves the way to the so‐called: model based control.
• The model of the process can be used to forecast the system response to a set of control actions originated by modifying a suitable set of manipulated variables.
Model based control
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• We are going to answer the following question:
Whatistheresponseofthesystemtoamodificationofthemanipulatedvariables?
• More specifically, we can imagine to deploy an optimizing procedure that looks for the best response of the system subject to the manipulation of the process variables.
• According to the most simplified approach, we have:
• the control specifications, i.e. the setpoint
• the objective function that measures the distance of the controlled variable from the setpoint
• the dynamic model of the system usuallydescribed by a DAE system, which playsthe role of the equality constraints
• the manipulated variables that are thedegrees of freedom of optimizationproblem
Model based control
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ysp = y set point (setpoint trajectory)y = y model responseyr = y real, measured responseu = manipulated variable
y(k‐1)
y(k)y(k+1)
u(k)
FuturePast
k k+1 k+2 k+3 k+4 k+5 …
Time horizon
yr(k‐1)
yr(k)
ysp(k‐1)
ysp(k)ysp(k+1)
Model predictive control
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L6—
• The system follows a specified trajectory optimal setpoint trajectory, ysp
• The model is called to produce a prediction, y, of the real response of the system.
• We have:
• response in the future: y(k+1), y(k+2), y(k+3), …
• respect to past real inputs: u(k), u(k1), u(k2), …
• respect to future manipulated inputs: u(k+1), u(k), …
• The numerical model of the process to be controlled is used to evaluate a sequence
of control actions that optimize an objective function to:
• Minimize the system response y respect to the optimal set‐point trajectory, ysp
• Minimize the control effort
MPC features
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L6—
• Since the model is a simplified representation of the real system, it is intrinsically not
perfect. This means that there is a discrepancy between the real system and the
modeled one.
• The present error k between the real system and the model is:
• This error is kept constant and it is used for future forecasts.
k ry k y k
MPC features
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L6—© Davide Manca – Process Systems Engineering – Master Degree in ChemEng – Politecnico di Milano 90
1 12 2 2
( ), ( 1), , ( 1) 1
2
min
, ,
p p p
c
k h k h k h
y y y u u T Tk k k h j k i k l k
y spy y r real model
sp
MAXy
MAX
j j i i l
j k jj k k k k k
j
j jj Max Min
u u uω e PF ω Δu PF ω δu
y δ ye δ y y y y
y
y y yPF 0 0
y
2
2 2
1
1
, ,
, , ,. .
, , , ,
MIN
MIN
MIN MAX
MAX MINu
MAX MIN
T T
i ii i
i
i ii Max Min
l l l
ts t
t
yy
u uΔu Δu Δu Δu
u
u u u uPF 0 0
u u
δu u u
h y u d 0
f y y u d 0
MPC mathematical formulation
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L6—
( ), ( 1), , ( 1)
min ...ck k k h u u u
MPC mathematical formulation
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L6—
2
( ), ( 1), , ( 1) 1
2 2
min ...
, ,
p
c
k h
y y yk k k h j k
y spy y r real model
sp
MAX MINy
MAX MIN
j j
j k jj k k k k k
j
j jj Max Min
u u uω e PF
y δ ye δ y y y y
y
y y y yPF 0 0
y y
MPC mathematical formulation
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L6—
12
( ), ( 1), , ( 1)
2 2
min ... ...
1
1
, ,
p
c
k h
u uk k k h i k
MIN MAX
MAX MINu
MAX MIN
i i
i ii i
i
i ii Max Min
u u uω Δu PF
u uΔu Δu Δu Δu
u
u u u uPF 0 0
u u
In general, is negativeMINΔu
MPC mathematical formulation
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L6—
12
( ), ( 1), , ( 1)min ... ...
p
c
k h
T Tk k k h l k
T T
l
l l l
u u uω δu
δu u u
T = target.
It can be the same optimal value of the steady state conditions for the
manipulated variables (e.g., nominal operating conditions).
MPC mathematical formulation
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L6—
1 12 2 2
( ), ( 1), , ( 1) 1
2
min
, ,
p p p
c
k h k h k h
y y y u u T Tk k k h j k i k l k
y spy y r real model
sp
MAXy
MAX
j j i i l
j k jj k k k k k
j
j jj Max Min
u u uω e PF ω Δu PF ω δu
y δ ye δ y y y y
y
y y yPF 0 0
y
2
2 2
, ,
1
,. .
, , , ,
1
, ,
MIN
MIN
MIN MAX
MAX MINu
MAX MIN
T T
i ii i
i
i ii Max Min
l l l
ts t
t
yy
u uΔu Δu Δu Δu
u
u u u uPF 0 0
u u
δu u
h y u d 0
f y y u d 0
u
MPC mathematical formulation
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L6—
c
p
y
u
T
y
model
T
hh
k
y yu
u
Critical elements of the MPC
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L6—
stack fan
inlet water
outlet water
auxiliary burners
vapor inlet
vapor outlet
vapor inlet
vapor outlet
primary burners
inlet water
inlet stream
MPC of the steam reforming process
CH4 + H2O CO + 3H2 H = +206 kJ/mol CH4 + H2O CO + 3H2 H = +206 kJ/mol
CO + H2O CO2 + H2 H = ‐41 kJ/mol CO + H2O CO2 + H2 H = ‐41 kJ/mol
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L6—
natural gas
Furnace
TTTT
Control System
FT
setpoint
fuelsteam
smokes
syn‐gas
FT
Control layout of the steam reforming section
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L6—
We start from the steady‐state model of Singh and Saraf (1979)
Main assumptions:
• pseudo‐homogeneous
• monodimensional model: neither axial nor radial dispersions
• discretization into a series of CSTR
• subdivision into layers of the pipe thickness
Process model
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L6—
dWdt
F F R Vi ki k i k i j j k
j
NR,
, , , , 1
1
c W c V dTdt
T T c Fp mix k tot k p cat catk
k k p mix k tot k, , , , , , , 1 1 1
D h T T dz R H Vin in k p in k k j k j kj
NR
, , , , ,1
Material balance
Energy balance on the gas-catalyst system
V
k-th reactor
Tk-1 Pk-1
Fk-1,i
Tk Pk
Fk,i
k NCSTR i NC 1 1,..... , ......
Process model: inside the reforming pipes
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L6—
met met p meti met
i ii i
i
i
V c dTdt
kx T T dz D D
DD
, /
/
/ln2
1 21 2
1 2
kx T T dz D D
DD
meti i
i i
i
i
2
1 21 2
1 2
//
/
ln
Balance on the i-th discretized layer
Flux continuity
h T Tk T T
D DD
in p inmet p in
inin
,,
ln
2 1
1
( )
ln
, ,GS T TA N
k T T
D DD
t R gas p out
t t
met p out N
outout
N
4 4 2
Internal surface External surface
Process model: pipes
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L6—
Q Q Qf t g t
Q GS E Eg t t R g t Q N A Ef t B f f g t f 1
in in outfuel gas smokes dispH H H Q H
adiabaticT
Flames Smokes
Process model: firebox (radiative chamber)
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L6—
700
750
800
850
900
950
1000
1050
1100
1150
1200
0 2 4 6 8 10 12
reactor length ‐m
Temperature ‐ K
process gas
internal wall
external wall
experimental dataliterature datasimulation
Validation of the numerical model
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L6—
1036
1038
1040
1042
1044
1046
1048
1050
1052
1054
1056
10 15 20 25 30 35
time ‐min
Outlet temperature ‐ K
inlet temperature step disturbance
fuel flowrate step disturbance
initial time of the step
System dynamics at open loop
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L6—
2.46e‐03
2.48e‐03
2.50e‐03
2.52e‐03
2.54e‐03
2.56e‐03
2.58e‐032.60e‐03
Fuel flowrate step
time ‐min10 15 20 25 30
Hydrogen flowrate ‐ kmol/s
Dry flowrate step
time ‐min
2.48e‐03
2.52e‐03
2.56e‐03
2.60e‐03
2.64e‐03
2.68e‐03
10 15 20 25 30
Over‐response: slow dynamics + fast dynamics
System dynamics at open loop
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L6—
Input
Output
Controlled variables:
• outlet gas temperature• hydrogen flowrate
Process
Manipulated variables:• fuel flowrate• steam flowrate
Measurable disturbances:• dry flowrate• inlet gas temperature
Definition of the main variables
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L6—
Real plant
ProcessModel
Optimizer+
+
+
+ +‐ ‐
‐
yset u
u
u
Constraints
d1 d2+
d1 : measurable disturbances
d2 : unmeasurable disturbances
y
y
y
y
minu set
i
h
y i y ip
2
1min ( ( )) ( ( ))
( ).... ( )u k u k h y yj k
k h
ui k
k h
c
p p
e j u i
1
2
1
21
ProcessModel
Model Predictive Control scheme
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L6—
hp: prediction horizon
High values of hpproduce:
• increased predictive capability
• less vigorous control actions
• higher distances from setpoints
1053.5
1054.0
1054.5
1055.0
1055.5
1056.0
1056.5
1057.0
0 5 10 15 20 25 30 35
time ‐min
Outlet temperature ‐ K
Step time
hpsetpoint
hp = 1hp = 3hp = 5hp = 7
Control horizon and prediction horizon
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L6—
hc: control horizon
High values of hcproduce:
• better controllability
• more vigorous control actions
• higher number of dof
dof c cn h n N.B.
1053.5
1054.0
1054.5
1055.0
1055.5
1056.0
1056.5
1057.0
0 5 10 15 20 25 30 35
time ‐min
Outlet temperature ‐ K
hc = 1 hc = 3 hc = 5
setpoint
Step time
Control horizon and prediction horizon
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L6—
u
High values of u produce:
• small variations of the
manipulated variables
• more sluggish controllers
• more stable controllers
1053.5
1054.0
1054.5
1055.0
1055.5
1056.0
1056.5
1057.0
0 5 10 15 20 25 30 35 40
time ‐min
Outlet temperature ‐ K
u u
Step time
The parameter u
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L6—
0.490
0.495
0.500
0.505
0.510
0.515
0.520
0 5 10 15 20 25 30 35
Hydrogen flowrate ‐ kmol/s
0.090
0.091
0.092
0.093
0.094
0.095
0.096
0.097
0 5 10 15 20 25 30 35
time ‐min
Fuel flowrate ‐ kmol/s
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0 5 10 15 20 25 30 35
time ‐min
Steam flowrate ‐ kmol/s
1051.0
1051.5
1052.0
1052.5
1053.0
1053.5
1054.0
1054.5
1055.0
0 5 10 15 20 25 30 35
Outlet temperature ‐ K
Step time
MPC: closed loop response
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L6—
1054105610581060106210641066106810701072
0 5 10 15 20 25 30 35
Outlet temperature ‐ K
0.494
0.496
0.498
0.500
0.502
0.504
0.506
0.508
0 5 10 15 20 25 30 35
Hydrogen flowrate ‐ kmol/s
Fuel flowrate ‐ kmol/s
0.090
0.095
0.100
0.105
0.110
0.115
0 5 10 15 20 25 30 35
time ‐min
Steam flowrate ‐ kmol/s
0.840
0.850
0.860
0.870
0.880
0.890
0.900
0.910
0.920
0 5 10 15 20 25 30 35
time ‐min
MPC: closed loop response
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L6—
1050
1055
1060
1065
1070
1075
1080
1085
1090
5 10 15 20 25 30 35
time ‐min
Outlet temperature ‐ K
MPC
PI
0.480
0.485
0.490
0.495
0.500
0.505
5 10 15 20 25 30 35
time ‐min
Hydrogen flowrate ‐ kmol/s
MPC
PI
Comparison between MPC and PI control
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L6—© Davide Manca – Process Systems Engineering – Master Degree in ChemEng – Politecnico di Milano 114
• Edgar T. F., D. M. Himmelblau, “Optimization of chemical processes”, McGraw Hill, Singapore, (1989)
• Peters M.S., K.D. Timmerhaus, “Plant Design and Economics for Chemical Engineers, McGraw Hill, New York,
(1993)
• Reklaitis G. V., A. Ravindran, K.M. Ragsell, “Engineering Optimization”, John Wiley, New York, (1983)
• Fletcher R., “Practical methods of optimization”, John Wiley, New York, (1981)
• Gill P.E., W. Murray W., M.H. Wright, “Practical Optimization”, Academic press, London, (1981)
• Manca D. , Rovaglio M., Pazzaglia G., Serafini G. , “Inverse Response Compensation and Control Optimization
of Incineration Plants with Energy Production”, Comp. & Chem. Eng., Vol. 22, 12, 1879‐1896, (1998)
• Manca D., Rovaglio M., “Numerical Modeling of a Discontinuous Incineration Process with On Line Validation”,
Ind. Eng. Chem. Res., 44, 3159‐3177 (2005)
• Manca D., “Optimization of the Variable Reflux Ratio in a Batch Distillation Column Through a Heuristic
Method”, Chemical Product and Process Modeling: Vol. 2, 12, (2007)
• Rovaglio M., D. Manca, F. Rusconi, “Supervisory control of a selective catalytic reactor for NOx removal in
incineration plants”, J. Waste Manag., 18, 525‐538, (1998)
References
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L6—
• Biegler L T., R.R. Hughes, “Infeasible Path Optimization with Sequential Modular Simulators”, AIChE Journal,28, 994, (1982)
• Biegler L.T., R.R. Hughes, “Feasible Path Optimization with Sequential Modular Simulators”, Comp. Chem.Eng., 9, 379, (1985)
• Biegler L.T., “Improved Infeasible Path Optimization for Sequential Modular Simulators: The Interface”,Comp. Chem. Eng., 9, 245, (1985)
• Biegler L.T., J.E. Cuthrell, “Improved Infeasible Path Optimization for Sequential Modular Simulators: TheOptimization Algorithm”, Comp. Chem. Eng., 9, 257, (1985)
• Biegler L.T., I.E. Grossmann, A.W. Westerberg, “A Note on Approximation Techniques Used for ProcessOptimization”, Comp. Chem. Eng., 9, 201, (1985)
• Coward I., “The Time Optimal Problems in Binary Batch Distillation”, Chem. Eng. Sci., 22, 503, (1967)
• Coward I., “The Time Optimal Problems in Binary Batch Distillation a further note”, Chem. Eng. Sci., 22, 1881,(1967)
• De Deken J. C., E. F. Devos, G. F. Froment, “Steam reforming of natural gas: intrinsic kinetics, diffusionalinfluences, and reactor design”, Chemical Reaction Engineering‐Boston, ACS Symposium series, (1982)
• Diwekar, U. M., R. K. Malik, K. P. Madhavan, “Optimal Reflux Rate Policy Determination for MulticomponentBatch Distillation Columns”, Comp. Chem. Eng., 11, 629, (1987)
References
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L6—
• Diwekar U. M., “Unified Approach to Solving Optimal Design Control Problems in Batch Distillation”, AIChE J.,38, 1551, (1992)
• Diwekar, U. M., “Batch Distillation. Simulation, Optimal Design and Control”, Taylor and Francis, WashingtonDC, (1994)
• Douglas, J. M., “Conceptual Design of Chemical Processes”, McGraw Hill, New York, (1988)
• Eaton J. W., J. B. Rawlings, “Model‐predictive control of chemical processes”, Chem. Eng. Sci., 47, (1992)
• Farhat S., S. Domenech, “Optimization of multiple fraction batch distillation by nonlinear programming”,AIChE J., 36(9), 1349, (1990)
• Gallier P.W., T.P. Kisala, “Process Optimization by Simulation”, Chem. Eng. Progress, 60, (1987)
• Grossmann I.E., K.P. Halemane, R.E. Swaney, “Optimization Strategies for Flexible Chemical Processes”,Comp. Chem. Eng., 7, 439, (1983)
• Hutchinson H.P., D.J. Jackson, W. Morton, “The Development of an Equation‐Oriented Flowsheet Simulationand Optimization Package. The Quasilin Program”, Comp. Chem. Eng., 10, 19, (1986)
• Kerkhof L.H.J., H.J.M Vissers, “On the profit of optimum control in batch distillation”, Chem. Eng. Sci., 33,961, (1978)
• Kilkas A.C., H.P. Hutchinson, “Process Optimization Using Linear Models”, Comp. Chem. Eng., 4, 39, (1980)
References
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L6—
• Kisala T.P., “Successive Quadratic Programming in Sequential Modular Process Flowsheet Simulation andOptimization”, PhD Thesis, (1985)
• Kisala T.P., R.A. Trevino Lozano, J.F. Boston, H.I. Britt, L.B. Evans, “Sequential Modular Approach andSimultaneous Modular Strategies for Process Flowsheets Optimization”, Comp. Chem. Eng., 11, 567, (1987)
• Lang. P., L.T. Biegler, “A Unified Algorithm for Flowsheet Optimization”, Comp. Chem. Eng., 11, 143, (1987)
• Locke M.H., R.H. Edahl, A.W. Westerberg, “An Improved Successive Quadratic Programming OptimizationAlgorithm for Engineering Design Problems”, AIChE J., 29, 871, (1983)
• Logsdon, J. S., U. M. Diwekar, L. T. Biegler, “On the Simultaneous Optimal Design and Operation of BatchDistillation Columns”, Trans. of IChemE, 68, 434, (1990)
• Luenberger D.G., “Linear and Nonlinear programming”, Addison Wesley, Amsterdam, (1984)
• Luyben W. L., “Batch distillation. Binary Systems”, Ind. Eng. Chem. Res., 10, 54, (1971)
• Luyben W. L., “Multicomponent Batch Distillation. Ternary Systems with Slop Recycle”, Ind. Eng. Chem. Res,27, 642, (1988)
• Mujtaba I. M. S. Macchietto, “Optimal Operation of Multicomponent Batch Distillation”, Comp. Chem. Eng.,17, 1191, (1993)
• Mujtaba I. M. S. Macchietto, “Optimal Recycles Policies in Batch Distillation‐Binary Mixtures”, Proc. SIMO’88, 6, 191, (1988)
References
© Davide Manca – Process Systems Engineering – Master Degree in ChemEng – Politecnico di Milano 117
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References
© Davide Manca – Process Systems Engineering – Master Degree in ChemEng – Politecnico di Milano 118