Beck Workshop on Modelling and Simulation of Coal-fired Power Generation and CCS Process

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


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


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.


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.


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]


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]


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.


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.


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.


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.


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 .


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


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.


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


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