Surrogate Modelling for PSA Design forCarbon Capture
Joakim Beck and Eric S. Fraga
Centre for Process Systems EngineeringDepartment of Chemical EngineeringUniversity College of London (UCL)
Workshop on Mathematical Modelling and Simulation of PowerPlants and CO2 Capture, March 20-21, 2012
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 1 / 18
Outline
1. Design of Pressure Swing Adsorption
2. Simulation and optimisation for PSA
3. Surrogate modelling
4. A case study for CO2 capture
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 2 / 18
Pressure swing adsorption
� A cyclic process forgas separation.
� A promising carboncapture technique.
Figure: PSA plant in China
Blowdown
Desorption
Desorption Pressurisation Adsorption
Pressurisation Adsorption Blowdown
productproduct
purge
purge
exhaustexhaustfeedfeed
feedfeedexhaustexhaust
Figure: 2-bed/4-step Skarstrom:classical PSA cycle
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 3 / 18
PSA design
Design variables cycle time, feed, bed pressure profile,bed dimension, cycle schedule, number ofbeds, and bed flow interconnections.
Design objectives higher product purity and recovery,and lower energy consumption, economicalcosts, and waste, etc.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 4 / 18
Optimisation problem
maxx∈D y(q(t, x))such that F (q(t, x)) = 0
W (q(t, x)) ≤ 0cylic steady state condition
(1)
When the design domain D is large, extensive searchwith conventional optimisers is precluded due to thecomputational work required to solve the modelequations F , which often include coupledparabolic/hyperbolic PDAEs.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 5 / 18
Advances in Simulation and Optimisation
Simulation
Unibed approximation [Kumar et al., 1994]Cyclic steady state acceleration
Quasi-Newton approach [Smith and Westerberg,1992; Ding and LeVan, 2001]Newton methods [Croft and Levan, 1994; Jiang et al.2003, 2005]Optimisation-based approach [Latifi et al. 2008,2011]
Parallel implementation [Jiang et al. 2005]
Reduced-order modelling [Agarwal et. al. 2009]
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 6 / 18
Optimisation
Complete discretisation [Nilchan and Pantelides,1998]
Dimension reduction techniques [Cruz et al.,2003]
Black-box optimisation with simpler model[Smith and Westerberg, 1991; Lewandowski et al.1998; Sundaram 1999; Agarwal et. al. 2009]
Simultaneous tailored optimisation [Ding andLeVan 2001, Jiang et al. 2003]
Super-structure framework [Agarwal et al. 2010]
Surrogate-based optimisation [Faruque Hasan etal. 2011]
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 7 / 18
Surrogate modelling
Problem setting
Consider black-box optimisation
maxx∈D y(q(t, x)),
where the constraints are implicitly embedded in y .
For PSA optimisation, y is a computational expensivefunction, which often take minutes or hours to evaluate.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 8 / 18
Surrogate modelling
Replace y with a computationally less expensive y - asurrogate model fitted to the responses of y at a set ofdesign points.
Examples
Quadratic response surface, Lagrange interpolation,Radial basis function interpolation, Artificial neuralnetwork, Kriging, etc.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 9 / 18
Kriging interpolation
Kriging predictor
y(x) =∑n
j=1 βj fj(x) + ε(x)
where the first term is a regression model, and thesecond term is a Gaussian process with zero mean andwith covariance
cov(x1, x2) = σ2e−∑N
`=1 θ|x1,`−x2,`|2.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 10 / 18
Example
Figure: A kriging example. Here the dashed line is a test problem ywith known points ◦, the solid lines are the kriging predictor y andits corresponding standard deviation σ.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 11 / 18
Surrogate-based optimisation strategy
Optimisation problem
x = arg maxx∈D y(q(t, x))
Surrogate optimisation problem
x = arg maxx∈D y(q(t, x))
A sequential optimisation strategy
(1) Apply optimiser to surrogate model y to predict theoptimal design x .
(2) Evaluate y(q(t, x)) through simulation.
(3) Update y with the new knowledge of y .Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 12 / 18
Choice of optimisers
Genetic Algorithm - A stochastic global optimiseroften with slow convergence.
SQP - A Newton-based local optimiser withquadratic convergence given good initial guess andif some regularity assumptions are satisifed.
EGO - A global optimiser that tries to find designpoint x with highest expected improvement to thedata known about y .
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 13 / 18
Efficient Global Optimization (EGO)
Kriging assumes the error residual to be normallydistributed, so the optimiser attempts to find design xthat gives the highest expected improvement to the bestdesign point we know so far for y , lets denote it by ymax.
x = arg maxx∈D EY∈N(y(x),σ2(x))[max{Y − ymax, 0}]
= arg maxx∈D∫∞Y=−∞max{Y − ymax, 0} 1√
2πσ2(x)e−(
y(x)−Y√2σ(x)
)2
dY
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 14 / 18
Numerical test: dual piston PSA
The case study is the separation of a binary gas mixtureof 80% N2 and 20% CO2. We consider a closed dualpiston PSA system with Zeolite 13X.
Figure: Schematic of the Dual Piston PSA system
Offset angles of piston 1 and 2, φ1 and φ2 ∈ [0, 2π], volumes of
piston chamber 1 and 2, V1 and V2 ∈ [0.5, 15.0]Vc (m3), and
temperature T ∈ [15, 70] (◦C ). Cycle time tc ∈ [1, 20] (s).Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 15 / 18
Numerical results
0
20
40
60
80
100
0 20 40 60 80 100 120 140 160
Pur
ity (
%)
Number of full evaluations
GA
SbGA
SbSQP
EGO
Figure: CO2 purity obtained using 5 different initial datasets foreach method, where the range between the best and worseperformance curves is filled.
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 16 / 18
0
0.2
0.4
0.6
0.8
1
tc φ1 φ2 V1 V2 T Purity (%)
Var
iabl
e va
lues
(no
rmal
ised
)
Design variables and objective
Figure: High dimensional visualisation of best designs found withSbSQP.
Beck, J., Friedrich, D., Brandani S., Guillas S., and Fraga E. S. Surrogate based
optimisation for the design of pressure swing adsorption systems. In Proceedings
to 22nd European Symposium on Computer Aided Process Engineering. In
press, 2012.Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 17 / 18
Conclusions
The use of surrogate-based optimisation reducescomputational costs without loss of accuracy
Kriging interpolation can be used efficiently withGA, multi-start SQP, and EGO
Guidelines to the design of dual-piston PSA for CO2
capture were proposed
Joakim Beck and Eric S. Fraga (UCL) Surrogate Modelling for PSA March 20, 2012 18 / 18