Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

Contents lists available at ScienceDirect

Journal of the Taiwan Institute of Chemical Engineers

journal homepage: www.elsevier.com/locate/jtice

A two-stage approach of multiplicative dimensional reduction and

polynomial chaos for global sensitivity analysis and uncertainty

quantification with a large number of process uncertainties

Le Quang Minh

a , 1 , Pham Luu Trung Duong

b , 1 , ∗, Jorge Goncalves b , Ezra Kwok

c , Moonyong Lee

a , ∗

a School of Chemical Engineering, Yeungnam University, Gyeongsan, Republic of Korea b Luxembourg Centre for Systems Biomedicine, Systems Control Group, University of Luxembourg, Esch-sur-Alzette, Luxembourg c Chemical & Biological Engineering, University of British Columbia, Vancouver, Canada

a r t i c l e i n f o

Article history:

Received 28 February 2017

Revised 19 May 2017

Accepted 7 June 2017

Available online 30 June 2017

Keywords:

Uncertainty quantification

Variance-based sensitivity analysis

Multiplicative dimensional reduction

method

Polynomial chaos

Gaussian process regression model

a b s t r a c t

Uncertainties associated with estimates of model parameters are inevitable when simulating and mod-

eling chemical processes and significantly affect safety, consistency, and decision making. Quantifying

those uncertainties is essential for emulating the actual system behaviors because they can change the

management recommendations that are drawn from the model. The use of conventional approaches for

uncertainty quantification ( e.g ., Monte-Carlo and standard polynomial chaos methods) is computation-

ally expensive for complex systems with a large/moderate number of uncertainties. This paper develops

a two-stage approach to quantify the uncertainty of complex chemical processes with a moderate/large

number of uncertainties (greater than 5). The first stage applies a multiplicative dimensional reduction

method to approximate the variance-based global sensitivity measures (Sobol’s method), and to simplify

the model for the uncertainty quantification stage. The second stage uses the generalized polynomial

chaos approach to quantify uncertainty of the simplified model from the first stage. A rigorous simula-

tion illustrates the proposed approach using an interface between MATLAB and HYSYS for three com-

plex chemical processes. The proposed method was compared with conventional approaches, such as the

Quasi Monte-Carlo sampling-based method and standard polynomial chaos-based method. The results

revealed the clear advantage of the proposed approach in terms of the computational efforts.

© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

p

a

M

t

(

e

t

a

s

e

t

s

1. Introduction

Uncertainties associated with process variables are inevitable

when modeling and designing chemical processes and can signifi-

cantly affect safety, consistency, and decision making. Conventional

process design, based on a nominal case without considering un-

certainties, can have negative influences on the design accuracy.

The problems of process design under uncertainties have at-

tracted considerable attention recently especially regarding safety,

reliability, and economic decisions [1] . At the design level, the

uncertainties, which depend on several input parameters, are

classified into two common types: epistemic and stochastic uncer-

tainties [2] . Propagation of all these input uncertainties onto the

∗ Corresponding author.

E-mail addresses: pluutrungduong@yahoo.com (P.L.T. Duong), mynlee@yu.ac.kr

(M. Lee). 1 These authors contributed equally to this work.

a

t

r

t

s

e

http://dx.doi.org/10.1016/j.jtice.2017.06.012

1876-1070/© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All r

rocess output of interest is one of the main tasks of uncertainty

nalysis [3,4] .

Probabilistic approaches, such as Monte-Carlo (MC) and Quasi

onte-Carlo (QMC) methods, provide a common framework for

he uncertainty quantification (UQ) and uncertainty propagation

UP) in the model input to its output [5–9] . MC/QMC methods gen-

rate an ensemble of random realizations from its uncertainty dis-

ribution to evaluate the model for each element of a sample set

nd estimate the relevant statistical properties, such as the mean,

tandard deviation, and quantile of output [10] . Furthermore, it can

xamine the different parameter values one by one and combina-

ions using a more comprehensive approach, performing a global

ensitivity analysis [11] . Sensitivity analysis (SA) of a process model

ims to characterize how the process model outputs respond to

he variation in inputs with an emphasis on finding the input pa-

ameters to which the outputs are the most sensitive [12,13] . Note

hat it is suitable for factor interactions and non-linear relation-

hips between factors and the output [14] since it focuses on the

ntire field of potential alterations of the input parameters. Despite

ights reserved.

L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264 255

t

g

M

c

[

t

p

r

r

a

g

[

i

i

p

m

l

[

t

B

t

t

t

p

[

d

a

t

a

l

b

N

g

b

/

t

[

n

t

h

v

a

b

t

n

p

m

t

c

c

w

t

m

t

t

t

i

g

t

f

2

m

m

g

m

w

d

s

a

a

i

t

a

t

i

y

w

o

μ

D

c

y

w

y

w

t

E

E

D

w

D

i

a

i

S

he simplicity in their implementation, however, the mean conver-

ence rate was estimated to be in the order of O (1 / √

M ) , where

is the number of samples, which makes MC-based approaches

omputationally expensive and they are only used as a last resort.

To tackle practical and time-consuming problems, Celse et al.

15] constructed an accurate, efficient-to-evaluate surrogate model

hat can be used in place of expensive simulations. For the pur-

ose of global uncertainty assessment, Li et al. [16] exploited the

andom samples to obtain an efficient high dimensional model

epresentation (RS-HDMR), where the HDMR component functions

re approximated by orthogonal polynomials. Currently, there is

rowing demand for computationally efficient surrogate models

17,18] that can ensure an acceptable degree of accuracy [19] . Sim-

larly, quantifying the dependence on the uncertain parameters us-

ng a surrogate model for generalized polynomial chaos (gPC) ex-

ansion achieved faster convergence rate in various areas, such as

odeling, control, robust optimal design, and fault detection prob-

ems [10] . The gPC method, which was first proposed by Wiener

20] , is a spectral expansion of a random process based on the or-

honormal polynomials in terms of the random variables. Nagy and

raatz [21] considered a polynomial chaos expansion for UQ and

he robust design of a batch crystallization process. They reported

hat the gPC approach can reduce the computational cost for a sys-

em with a relatively small number of design inputs and uncertain

arameters compared to MC/QMC methods. Also, Shen and Braatz

22] developed a new polynomial chaos based algorithm on the

esign of batch and continuous-flow chemical reactors with prob-

bility uncertainties. Duong and Lee [23,24] examined a PID con-

roller design using the gPC method. Du et al. [25] demonstrated

fault detection solution by combining the gPC with a maximum

ikelihood framework. Bavdekar and Mesbah [26] proposed a gPC

ased particle filtering for a state estimation of nonlinear system.

immegeers et al. [27] compared sigma points, linearization and

PC methods for developing a robust multi objective design of

iosystems. Recently, Duong et al. [10] studied the problem of UQ

SA of chemical processes using the standard gPC method for sys-

ems with a small number of random inputs. Xiu and Karniadakis

28] proposed utilizing the Askey scheme for gPC expansion with

on-standard distributions.

For engineering purposes, when the output function is smooth,

he gPC expansion with a uniform and Gaussian distribution ex-

ibited rapid convergence; in some cases, even exponential con-

ergence [22–29] . A current limitation of the standard full gPC

pproach, where the coefficients are estimated using the tensor cu-

ature, is that the number of model evaluations grows exponen-

ially and may not be applicable to systems with a moderate/large

umber of uncertainties. This paper aimed to overcome this com-

utational limitation by proposing a two-stage approach using a

ultiplicative dimensional reduction method (M-DRM) [30] and

he standard gPC method [10] . The main target is to reduce the

omputational efforts (simulation time) for UQ significantly over

onventional approaches, such as MC/QMC/standard gPC methods,

hen dealing with moderate/large number of uncertainties. Here,

he M-DRM is first used to detect important inputs by approxi-

ating a complicated function of random variables as a deriva-

ion of the univariate functions. The standard gPC method was

hen applied for UQ by considering only the important inputs de-

ected in the SA step. The proposed two-stage approach is then

llustrated on complex chemical processes, including the propylene

lycol production process, a lean dry gas processing process, and

he synthesis gas production process. A rigorous simulation is per-

ormed by an interface between MATLAB ۛand Aspen HYSYS ۛ.

. Variance based-sensitivity analysis

Sensitivity analysis (SA) examines how the variability of the

odel output is affected by the uncertainty of various inputs. The

ethods for SA can be divided into two categories: local SA and

lobal SA. The local approach uses one-factor-at-a-time experi-

ents with gradients to study the local variability of the model,

hereas global SA deals with the global variations in the output

ue to the given entire range of uncertainties on the inputs. This

tudy focused on the global SA using Sobol’s indices [31] , which

re also known as a variance based method, to determine the vari-

bles most responsible for the uncertainty in the model output.

Probability is a natural framework for modeling an uncertain

nput by assuming that the input is an N -dimensional random vec-

or of mutually intendent components, ξ= ( ξ 1 , ξ 2 ,..., ξN ), with prob-

bility density functions of ρ i ( ξ i ): �i → R + ; A steady-state process with m outputs y = [ y ( ξ, t = 1), ..., y ( ξ,

= m )] T , which is described conceptually in Fig. 1 , is summarized

n the form of nonlinear equations:

= h (ξ , t) (1)

here t ∈ {1, ..., m } is the index variable, y is the vector of process

utputs.

The output mean and variance are defined as

y (t) =

∫ �1

...

∫ �N

y ( ξ1 , ..., ξN , t) N ∏

i =1

ρi ( ξi ) d ξi (2)

y (t) =

∫ �1

...

∫ �N

[(y ( ξ1 , ..., ξN , t)) − μ

y (t) ]2

N ∏

i =1

ρi ( ξi ) d ξi (3)

Global sensitivity analysis is based on a decomposition of the

omputational model in Eq. (1) as follow [33,34] :

(ξ , t) = y 0 (t) +

N ∑

i =1

y i ( ξi , t)

+

N−1 ∑

i =1

N ∑

j>i

y i j ( ξi , ξ j , t) + . . . + y i 1 ... i N ( ξi , . . . , ξN , t) (4)

here y 0 ( t ) = μy ( t ).

The terms in Eq. (4) can be obtained recursively as follows:

y i ( ξi , t) = E [ y (ξ , t) | ξi ] − μy (t)

i j ( ξi , ξ j , t) = E

[y (ξ , t) | ξi , ξ j

]− y i (t) − y j (t) − μy (t)

. . . (5)

here E [ y ( ξ, t )|[]] is the conditional expectation of y ( ξ, t ) when

he corresponding inputs are set. The components defined in

q. (5) can be proven to be orthogonal with each other.

By taking advantage of the independent of the summands in

q. (4) , the output variance can be decomposed as:

y (t) =

N ∑

i =1

D i (t)+

N−1 ∑

i =1

N ∑

j>i

D i j (t)+ ... + D 1 , 2 , ... ,N (t) (6)

here:

D i (t) = var ( E [ y (ξ , t) | ξi ] )

D i j (t) = var (E

[y (ξ , t)

∣∣ξi , ξ j

])− D i (t) − D j (t)

. . .

1 , 2 , ... ,N (t) = D y (t) −N ∑

i =1

D i (t) −. . . −∑

1 ≤i 1 < ···< i n −1 ≤N

D i 1 ... i N−1 (t) (7)

In Eq. (7) , the outer variance is taken over the corresponding

nputs.

The first order Sobol’s sensitivity index (function) quantifies the

mount of the output variance that can be apportioned to the sole

nput variable, ξ i :

i (t) =

D i (t)

D y (t) (8)

256 L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

Fig. 1. Concept of uncertainty analysis in chemical processes.

T

Q

T

n

3

r

d

Z

i

a

g

d

o

s

3

i

ϕ

m

t

ϕ

w

f{

w

i

Similarly, higher order sensitivity functions describe what part

of the total variance is due to the joint effect of inputs, { ξi 1 , ..., ξi s } ,

as

S i 1 ,..., i k (t) =

D i 1 ,..., i k (t)

D y (t) (9)

The Sobol’s total effect functions quantify the total impact of

the factor, ξ i , including all of its interactions with the other inputs:

T i (t) =

E ( var (y (ξ , t) | ξ∼i ))

D y (t) (10)

In other words, if T i (t) is close to zero, the effect of i th input, ξ i ,

on t th output can be neglected.

Furthermore, the Monte Carlo method can be used to estimate

the Sobol indices as follows [13] .

• Generate a A Q×N matrix ( Q is the sample size) from a given

density function of inputs.

A Q×N =

⎡

⎣

ξ (1) 1

... ξ (1) i

... ξ (1) N

... ... ... ... ...

ξ (Q ) 1

... ξ (Q ) i

... ξ (Q ) N

⎤

⎦ (11)

• Generate a B Q×N matrix (independent from A ) from a given

density function of inputs.

B Q×N =

⎡

⎣

ξ (Q+1) 1

... ξ (Q+1) i

... ξ (Q+1) N

... ... ... ... ...

ξ (2 Q ) 1

... ξ (2 Q ) i

... ξ (2 Q ) N

⎤

⎦ (12)

• Form matrices C i from all columns of B except the i th column,

which is taken from A . • Obtain vectors of the model output, y A = M(t, A ) , y B =

M (t, B) , y C i = M (t, C i ) , by computing the output of model in

Eq. (1) for all input values in (sample matrices) A , B, C i . • First order indices are estimated as follows:

S i (t) =

( 1 /Q ) ∑ Q

j=1 y ( j) A (t) y ( j)

C i (t) − M 0 2 (t)

( 1 /Q ) ∑ Q

j=1 ( y ( j)

A ( t)) 2 − M 0

2 ( t) (13)

where M

2 0 = ( ( 1 /Q )

∑ Q i =1

y ( j) A ( t) )

2 is the empirical mean of the

model output.

The total order indices is estimated as follows:

i (t) = 1 − ( 1 /Q ) ∑ Q

j=1 y ( j) B (t) y ( j)

C i (t) − M 0 2

( 1 /Q ) ∑ Q

j=1 (y ( j)

A (t)) 2 (t) − M 0

2 (t) (14)

For a model with N inputs, the total computational cost is

( N + 2), where Q is a large number of samples ( e.g ., 10,0 0 0).

herefore, the computational efforts become expensive when the

umber of process uncertainties increase.

. Sensitivity analysis using a multiplicative dimensional

eduction method

This section briefly describes the multiplicative dimensional re-

uction method (M-DRM) for the global sensitivity analysis from

hang and Pandey [30] . The important role of the M-DRM method

s to approximate a complicated function of random variables by

product of one-dimensional functions. Therefore, the simple al-

ebraic expressions can be derived for the first order and total or-

er sensitivity coefficients associated with a significant reduction

f computation efforts. The M-DRM method will then be used to

implify the model for UQ with the polynomial chaos method.

.1. Brief about M-DRM

Consider an overall indexed response function, y ( t ) = h ( ξ, t ). Us-

ng the logarithmic transformation, one can obtain,

(ξ , t) = log [ abs (y, t)] = log { abs [ h (ξ , t)] } (15)

Following the univariate conventional dimensional reduction

ethod (C-DRM) in the literature [32] , an approximation of ϕ( ξ,

) can be written as

(ξ , t) ≈N ∑

i =1

ϕ( ξi , c −i , t) − (N − 1) ϕ 0 (t) (16)

here the functions are referred to those in the original space as

ollows:

ϕ 0 (t) = log { abs [ h ( c 1 , c 2 , ..., c N )] } ϕ( ξi , c −i , t) = log { abs [ h ( c 1 , ..., c i −1 , ξi , c i +1 , c N , t)] } (17)

here c = ( c 1 , c 2 ,..., c N ) is a cut point. Occasionally, the mean vector

s referred as the cut point c [33] .

L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264 257

t

e

t

h

t

E

r

o

t

o

t

m

3

t{

m

μ

f

α

w

t

E

E

a

D

m

S

c

v

o

E

a

r

o

v

E

E

s

T

w

a

k

To invert the transformation, the original function can be writ-

en as

xp [ ϕ(ξ , t)] ≈ exp

[

N ∑

i =1

ϕ( c 1 , ..., c i−1 , ξi , c i +1 , c N , t) −(N−1) ϕ 0 (t)

]

= exp [(1 −N) ϕ 0 (t)]

× exp

[

N ∑

i =1

ϕ( c 1 , ..., c i −1 , ξi , c i +1 , c N , t)

]

(18)

Substituting for the expressions from Eq. (17) into Eq. (18) leads

o a multiplicative approximate of the response function:

(ξ , t) ≈ [ h (c, t) 1 −N ] .

N ∏

i =1

h ( c 1 , ..., c i −1 , ξi , c i +1 , c N , t)

= h 0 (t) 1 −N N ∏

i =1

h i ( ξi , c −i , t) (19)

This approximate model of the original input-output rela-

ion is known as the univariate M-DRM. The log transform in

q. (15) aims to utilize the property of the log function: the loga-

ithm of product of a univariate function is equivalent to the sum

f the logarithm of the univariate function. Thus, a standard addi-

ive model reduction method can be applied to the log transform

f the function and the inverse transform can be taken to obtain

he M-DRM. Furthermore, the M-DRM leads to a simple approxi-

ation of SA indices as will be discussed in the next section.

.2. Variance based sensitivity analysis by the M-DRM

Denote the mean and mean square of the k dimensional func-

ion of t th output as

ρk (t) = E [ h k ( ξk , c −k , t) ]

θk (t) = E

[h k ( ξk , c −k , t)

2 ] (20)

Utilizing the M-DRM approximation in Eq. (19) , the mean and

ean square of the output can be approximated as

�

y (t) ≈ h o 1 −N

(t) N ∏

k =1

ρk (t) ; E

[y 2 (ξ , t)

]≈ h 0

2 −2 N (t)

N ∏

k =1

θk (t) ;

D Y (t) = E

[y 2 (ξ , t)

]− (

�

μy (t)) 2 = μy 2 (t)

[

N ∏

k =1

θk (t) /ρ2 k (t) − 1

]

(21)

Under the M-DRM approximation, an i th conditional expectation

unction of y ( t , ξ) can be evaluated as

i ( ξi , t) = E [ y (ξ , t) | ξi ] ≈ E

[

h

1 −N 0 (t) .

N ∏

k =1

h k ( ξk , c −k , t)

]

= h

1 −N 0 (t) . h i ( ξi , c −i , t) .

N ∏

k =1 ,k = i ρk (t) (22)

here the last equality is available because the expectation is

aken with respect to all other inputs except the input ξ i .

The expectation (with respect to ξ i ) of the squared of

q. (22) is

( αi 2 ( ξi , t)) = E

⎡

⎣

[

h

1 −N 0 (t) . h i ( ξi , c −i , t) .

N ∏

k =1 ,k = i ρk (t)

] 2 ⎤

⎦

= h

2 −2 N 0 (t) θi (t)

N ∏

k =1 ,k = i ρ2

k (t) =

�

μy

2 (t) θi (t) / ρ2

i (t) (23)

By substituting Eqs. (22) and (23) into Eq. (7) , the primary vari-

nce (computed with respect to ξ i ) can be approximated as

i (t) = var ( E [ y (ξ ) | ξi ] ) = var ( αi ( ξi , t))

= E ( αi 2 ( ξi , t)) − E ( αi ( ξi , t))

2

=

�

μy

2 (t)( θi (t) / ρi

2 (t) − 1) (24)

Using Eqs. (21) and (24) , the first order index can be approxi-

ated as

i (t) =

D i (t)

D Y (t) ≈ θi (t) /ρ2

i (t) − 1 (∏ N

k =1 θk ( t) /ρ2 k ( t) )

− 1

(25)

To evaluate the total sensitivity index, T i (t) , it is essential to cal-

ulate the following conditional variance:

ar (y (ξ , t) | ξ∼i ) = E ( ( y ( ξ , t) | ξ∼i ) 2 ) − E ((y (ξ , t) | ξ∼i ))

2 (26)

Owing to the M-DRM, the expectations in the right-hand side

f Eq. (26) can be approximated as

[(y (ξ , t) | ξ∼i )

2 ]

≈ E

⎛

⎝

[

h 0 (t) 1 −N

N ∏

i =1

h i ( ξi , c −i , t)

] 2

| ξ∼i

⎞

⎠

= E

[

h

2 −2 N 0 (t) .

(

N ∏

k =1 ,k = i [ h k ( ξk , c −k , t)]

2

)

. h i 2 ( ξi , c −i , t) | ξ∼i

]

= h

2 −2 N 0 (t) .

(

N ∏

k =1 ,k = i [ h k ( ξk , c −k , t)]

2

)

. θi (t) (27)

nd

( E [ (y (ξ , t) | ξ∼i ) ] ) 2 ≈ h

2 −2 N 0 (t) .

(

N ∏

k =1 ,k = i [ h k ( ξk , c −k , t)]

2

)

.ρ2 i (t)

(28)

Note that the expectations in Eqs. (27) and (28) are taken with

espect to the input ξ i

The (inner) conditional variance in the numerator of Eq. (10) is

btained as

ar (y (ξ , t) | ξ∼i ) ≈h

2−2 N 0 (t) .

(

N ∏

k=1 ,k = i [ h k ( ξk , c −k , t) ]

2

)

. ( θi (t) −ρ2 i (t))

(29)

The expectation (with respect to all other inputs except ξ i ) of

q. (29) is obtained as

( var (y (ξ , t) | ξ∼i )) ≈ h

2 −2 N 0 (t) .

(

N ∏

k =1 ,k = i θk (t)

)

. ( θi (t) − ρ2 i (t))

(30)

Finally, by substituting Eqs. (30) and (21) into Eq. (10) , the total

ensitivity index of ξ i can be approximated as

i (t) ≈ 1 − ρ2 i (t ) / θi (t )

1 −(∏ N

k =1 ρ2 k (t ) / θk (t )

) (31)

Let us obtain the Gaussian quadrature with q nodes { ξ k ( l ) ,

kl } l = 1, ..., q for k th input. The functional relationship is denoted

s h ( c 1 ,..., c k −1 , ξ k ( l ) , c k + 1 , ..., c N , t ) = h kl ( t ), whereas the following

th dimensional functions for all outputs can be computed as

258 L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

w

(

h

e

P

s

t

t

a

μ

t

D

c

v

t

s

n

m

s

s

m

f

T

t

g

e

s

1

5

m

i

s

t

5

u

a

⎡

⎢ ⎢ ⎣

ρk (1) ...

ρk (t) ...

ρk (m )

⎤

⎥ ⎥ ⎦

=

⎡

⎢ ⎢ ⎣

h k 1 (1) h k 2 (1) ... h kq (1) ... ... ... ...

h k 1 (t) h k 2 (t) ... h kq (t) ... ... ... ...

h k 1 (m ) h k 2 (m ) ... h kq (m )

⎤

⎥ ⎥ ⎦

⎡

⎢ ⎢ ⎣

w k 1

w k 2

...

...

w kq

⎤

⎥ ⎥ ⎦

⎡

⎢ ⎢ ⎣

θk (1) ...

θk (t) ...

θk (m )

⎤

⎥ ⎥ ⎦

=

⎡

⎢ ⎢ ⎣

( h k 1 (1)) 2 ( h k 2 (1)) 2 ... ( h kq (1)) 2

... ... ... ...

( h k 1 (t)) 2 ( h k 2 (t)) 2 ... ( h kq (t)) 2

... ... ... ...

( h k 1 (m )) 2 ( h k 2 (m )) 2 ... ( h kq (m )) 2

⎤

⎥ ⎥ ⎦

⎡

⎢ ⎢ ⎣

w k 1

w k 2

...

...

w kq

⎤

⎥ ⎥ ⎦

(32)

Therefore, through utilizing the indexed variable t and matrix

multiplication formation as in Eq. (32) , one can easily obtain the t -

indexed k th dimensional functions. By substituting these functions

into Eqs. (25) and (31) , approximations of the sensitivity analy-

sis indices can be obtained simultaneously for all system outputs.

Note that owing to the special capability on handling the matrix

computation of MATLAB ۛ, the computation time for all m outputs

will not be increased significantly in comparison with that for sin-

gle output only.

Regarding the standard of the M-DRM approximation, only Nq

total number of functional evaluations are required for complete

sensitivity analysis. Note that only a small number of nodes q (5–

10) is normally used in Gaussian quadrature. On the other hand,

the MC method indices require ( N + 2) Q to estimate the sensitivity

[34–35] .

4. Uncertainty quantification with polynomial chaos

Assume that n t < N important inputs ξn t = ( ξk 1 , ..., ξn t ) are de-

tected via the SA indices in the previous step for the t th output.

The following polynomial chaos based method [10] can be used re-

peatedly for UQ of the model output. The unimportant inputs are

fixed to their nominal value. Therefore, the t th output is expanded

into a series of n t variate P th order polynomials:

y ( ξn t , t) =

M t ∑

i =1

�

f i (t) i ( ξn t ) (33)

M t + 1 =

(n + P n

)=

(n + P )!

n ! P ! (34)

where �

f m

(t) is the coefficient of gPC expansion that satisfies

�

f i (t) = E [ i (t ) f (y, t )] =

∫ �

f (y, t) i ( ξn t ) ρ( ξn t ) d ξn t . (35)

In Eq. (34) , these coefficients of the gPC expansion can be cal-

culated numerically through a discrete projection according to the

procedure reported by Xiu [36] :

• Choose a n t -dimensional integration rule with q 1 × ... ×q n t cu-

bature nodes/weights

q 1 ×.., ×q n t [ g] =

q 1 ∑

j 1 =1

...

q n t ∑

j N =1

g(ξ ( j 1 ) 1

, ..., ξ ( j n ) n t ) ( w 1

( j 1 ) ... w 1 ( j n ) )

∼=

∫ �

g( ξn t ) ρ( ξn t ) d ξn t (36)

where q 1 ×... ×q n t [ ·] refers to the cubature numerical integration.

• Using the numerical integration rule in Eq. (36) , the gPC coeffi-

cients can be approximated as:

˜ �

f j (t) = q 1 ×... ×q n t [ f (y, ξn t , t)( ξn t ) ρ] =

q 1 ×... ×q n t ∑

m =1

f ( ξn t (m )

, t)

× j ( ξn t (m )

, t) w

(m ) for j = 1 , ..., M (37)

here �

f j (t) is approximated numerically by ˜ �

f j (t) and f ( ξn t , t)

j ( ξn t ) plays a role of g( ξn t ) in Eq. (36) . The number of nodes

simulation) in the cubature rule rises exponentially. On the other

and, because only the important inputs (2–5 inputs) are consid-

red, the number of simulations is still acceptable. An n -variate

th order gPC approximation of the response function can be con-

tructed in the form,

˜ f P N (t) =

M ∑

j=1

˜ �

f j (t) j ( ξn t ) (38)

The truncated gPC expansion in Eq. (38) contains all informa-

ion about the statistical properties of the random output. Owing

o the orthonormality of the gPC polynomials, the mean and vari-

nce of output may be calculated directly from the gPC coefficients.

The mean value is the first expansion coefficient:

y (t) =

∫ �

˜ f P N (t) ρ( ξn t ) d ξn t =

∫ �

[

M ∑

j=1

˜ �

f j (ξ , t) j ( ξn t )

]

ρ( ξn t ) d ξn t

=

˜ �

f 1 (t) (39)

The variance of the output y ( ξ) can be calculated as the sum of

he square of the remaining coefficients:

y = σ 2 y = E [ (y − μy )

2 ] =

M ∑

j=2

˜ �

f

2

j (t) (40)

The output density function can be obtained by sampling the

heap-to-evaluate truncated gPC model in Eq. (38) .

Different output may have different key inputs. For this obser-

ation, we denote j ( ξn t ) and ρ( ξn t ) as a polynomial basis of the

th outputs to emphasize that they have different basis function

ets. Let us denote Q (t) = q 1 × ... × q n t as the number of cubature

odes needed for the constructed gPC model for the t th output. For

outputs, therefore, m different surrogates have to be constructed

eparately and require ∑

i = m

Q (i ) model simulations.

Remarks : Note that this study considers the reduced dimen-

ional surrogate which is different from the case where a full di-

ensional surrogate was considered [27,28,37] . In the case of the

ull dimensional surrogates, they are constructed with all inputs.

hus, one can totally use Q simulations with the same experimen-

al design set { ξ 1 ( i ) ,..., ξN

( i ) }, i = 1, ..., Q and solve m different re-

ressions to obtain m surrogate models . In this study, under differ-

nt outputs, it is necessary to simulate the model with

∑ m

t=1 Q (t)

imulations for different experimental sets { ξk 1

(i ) , ..., ξ

n t

(i ) } , i = , ..., Q (t) . Table 1 summarizes the steps for the proposed method.

. Case studies

In this study, the UQ for complex chemical processes was esti-

ated based on the two-stage gPC-based approach . The emphasis

s to explain the efficient, practical framework, and the principal

teps involved SA and UQ, while further analysis can lead to a bet-

er understanding of those process models.

.1. Example 1: Propylene glycol production process with six uniform

ncertainties

Propylene glycol (PG) has been used widely in industries, such

s a food grade coolant in food industry [38] , solvent in cosmetics

L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264 259

Table 1

Summary of the proposed algorithm.

• Obtain one dimensional quadrature rule with q points. Calculate the k th dimensional functions for the t th output as in Eq. (32) . It requires computing (2 ×k )

matrix vector multiplications. The size of matrices in Eq. (32) is m ×q . Note that for a single objective the vector matrix multiplications reduce to a dot product

between two vectors. • Utilizing the k th dimensional functions for the t th output, the sensitivity function of first order and total order can be computed as in Eqs. (25) and (31) . From

these SA functions, one can detect the important inputs for the t th output. • Perform Q (t) simulations with a Q (t) nodes of cubature for the t th output • Compute the gPC coefficients for all t th outputs by Eq. (37)

Fig. 2. Propylene glycol (PG) production process (Example 1).

[

m

d

w

(

l

l

i

9

s

p

c

p

r

t

v

{

p

t

t

o

t

a

f

t

t

t

r

t

c

m

b

w

b

Fig. 3. Density distributions of reboiler duty in PG production process using the

QMC/gPC methods with six and two uniform random inputs (Example 1).

e

m

g

t

g

e

o

m

c

Q

[

t

r

Q

t

s

39] , and de-icing fluid in aviation [40] . Referring to the conceptual

odel from HYSYS ۛ, Fig. 2 presents a flow diagram of a PG pro-

uction process. In this process, propylene oxide (PO) is combined

ith water to produce PG in a continuously-stirred-tank reactor

CSTR). Owing to the exothermic reaction, a coolant stream circu-

ates within the reactor jacket to remove any extra heat. The out-

et stream is then introduced to a distillation column, in which PG

s essentially recovered from the bottom stream with a purity of

9.5 wt. %. The distillation column operating at atmospheric pres-

ure has 10 stages with a full reflux condenser and reboiler.

Adapted from a practical point of view, the variations in tem-

erature, feed flowrate, pressure, and other properties should be

onsidered. In this study, the flowrates of PO and water, the tem-

erature and pressure of the mixed stream, the temperature of the

eactor effluent, and the reflux ratio of the column were assumed

o be independently uncertain and distributed uniformly in inter-

als of

[61 . 2 , 74 . 8 kmol / h ] ; [249 . 3 , 304 . 7 kmol / h ] ; [21 . 5 , 26 . 3

◦ C] ;[1 . 28 , 1 . 56 bars ] ; [54 , 66

◦ C] ; [0 . 9 , 1 . 1] } . Other parameters, such as the reactor vessel volume, column

ressure, and number of theoretical stages were assumed to be de-

erministic. According to the M-DRM, a set of 60 Gaussian quadra-

ure nodes, which was generated using the MATLAB ۛcodes from the

rthogonal polynomial toolbox [41] , was passed to HYSYS ۛ, where

he PG process in Fig. 2 was modeled rigorously. The reboiler duty

nd PO conversion obtained from HYSYS ۛwere collected and used

or SA in Eq. (31) . The Sobol’s sensitivity indices were used to iden-

ify the inputs of the model which have the largest influence on

he prediction. Table 4 lists the sensitivity indices obtained from

he M-DRM for two outputs. As a result, for the heat duty, the two

andom variables, such as the flowrate of water and the reflux ra-

io, were detected as being important. Other random variables, in-

luding the PO feed flow rate, the temperature and pressure of the

ixed stream, and the outlet temperature of the reactor effluent

ecome non-influential factors that can be omitted in UQ. Mean-

hile, the uncertainty of the PO conversion is mostly influenced

y the variation of the reactor effluent temperature. Owing to the

ffective detection of the non-influential input of the M-DRM, the

odel can be largely simplified. A 10th order with two variates

PC, which requires only 10 × 10 simulations, was constructed for

he UQ of the reboiler duty, while a 10th order univariate-based

PC was exploited for UQ of the PO conversion. Because precise

stimates of the process output cannot be accessible, the results

f the proposed method were compared with those of the QMC

ethod with a sufficiently large number of simulations. For an ac-

urate estimation of the probability, the number of samples for the

MC method was selected from Chernoff bound in Tempo et al.

42] . Fig. 3 compares the density functions of the reboiler duty ob-

ained using the proposed and QMC methods with two influential

andom inputs (water flow rate and reflux ratio) and that by the

MC method with all six random inputs (using the 10,0 0 0 simula-

ions from Halton sequence). Similarly, as seen in Fig. 4 , the den-

ity functions of the PO conversion obtained by the proposed and

260 L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

Fig. 4. Density distributions of PO conversion in PG production process using the

QMC/gPC methods with six and two uniform random inputs (Example 1).

Fig. 5. Density distributions of reboiler duty in PG production process using the

QMC/GPR methods with six uniform random inputs (Example 1).

Fig. 6. Density distributions of PO conversion in PG production process using the

QMC/GPR methods with six uniform random inputs (Example 1).

a

c

m

h

l

g

s

b

i

f

QMC methods with the influential random input of the reactor ef-

fluent temperature was compared with that by the QMC method

with all six random inputs. Table 2 lists the statistical properties

of the column reboiler duty and PO conversion achieved from the

proposed and QMC methods. The computational time for the pro-

posed method includes the computational time for 60 M-DRM sim-

ulations for SA and constructed two gPC models with (100 + 10)

quadrature nodes for UQ. The QMC/MC methods were computa-

tionally expensive requiring a large number of simulations to esti-

mate the expected values, variances, and density accurately.

The surrogate model by the proposed method was also com-

pared with that by the Gaussian process regression (GPR) [43] .

Two experimental sets with 160 and 20 0 0 data points from QMC

sequences were considered for training the Gaussian processes.

Figs. 5 and 6 compare the density functions of the reboiler duty

and PO conversion predicted by the Gaussian process with 160

/20 0 0 training data points and those by the reference (QMC)

method with six random inputs. The MATLAB ۛfunction fitrgp was

used to train the GPR model with a quadratic trend and an

automatic relevance detection (ARD) squared exponential kernel.

Table 2 lists the computational time for training the GPR model for

each set of training data. The computational time for the GPR in-

cludes the simulation time for 160/20 0 0 simulations and training

the two GPR models. The kernel scale of the trained GPR model

for the PO conversion was {18, 11,727, 28, 7059, 0.1 , 10,826}, which

indicates that the GPR correctly identifies the reactor effluent tem-

perature as an important input. However, the kernel scale obtained

for the reboiler duty was {2.898, 0.023 , 2.038, 2.652, 0.347, 2.374},

which implies that the random inputs of water flow rate and tem-

perature of the reactor effluent are relevant (important) for the re-

boiler duty. The GPR incorrectly detected that the reactor effluent

temperature is relevant for the reboiler duty. It was observed from

Figs. 3, 4, 5 , and 6 that the proposed (M-DRM + gPC) method could

Table 2

Simulation parameters and computational time profiles fo

MDRM-gPC / GPR / QMC methods for Example 1 (with six ran

Method Example 1

No. of simulations Runt

QMC 10,0 0 0 5308

GPR 160 85.6

GPR 20 0 0 1098

Proposed (M-DRM + gPC) 170 90.7

chieve acceptable results with only a small computational cost in

omparison with other two methods. Poor performance of the GPR

odel is possibly affected by the result of optimization with seven

yper parameters (six kernel scales and one signal variance) in a

ocal solution.

However, the result of this case study does not imply that the

PC can work better than the GPR in general. This study used a

tandard tool of MATLAB where the hyper parameters were trained

y the optimization of the log-likelihood. There are several ways to

nfer these hyper parameters [43] and those will be compared in

uture work. It should be interpreted only that the (M-DRM + gPC)

r obtaining the statistical characteristics using the

dom inputs).

ime (s) Mean μ (Q) Mean μ (PO conversion)

.6 5710.1 94.0

5759.8 94.2

.3 5773.1 94.0

5729.1 94.1

L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264 261

Fig. 7. Lean dry gas processing plant (Example 2).

Table 3

Simulation parameters and computational time profiles for obtaining the statistical characteristics using the M-DRM + gPC / QMC methods for

Examples 2 (with six random inputs) and 3 (with seven random inputs).

Method Example 2 Example 3

No. of simulations Runtime (s) Mean μ (BTU/SCF) No. of simulations Runtime (s) Mean μ (HNr)

QMC 10,0 0 0 5754.9 1092.0 10,0 0 0 5590.19 3.1181

Proposed (M-DRM + gPC) 160 85.0 1091.2 170 90.8 3.0616

Table 4

Sobol’s sensitivity indices from the surrogated model by the M-DRM method for Example 1.

Sobol’s sensitivity indices ( S i , T i )

Example 1 (reboiler duty) S1 S2 S3 S4 S5 S6 T1 T2

0.00107 0.7505 0.0829 1.52e −11 0.045 0.1099 0.0108 0.7514

T3 T4 T5 T6

0.0832 1.53e −11 0.0452 0.1104

Example 1 (PO conversion) S1 S2 S3 S4 S5 S6 T1 T2

0.0284 0.0371 0.0033 8.96e-11 0.8756 0.0557 0.0286 0.0372

T3 T4 T5 T6

0.0034 8.98e −11 0.8757 0.0558

m

w

5

t

a

p

p

5

u

d

a

c

E

b

m

d

a

s

a

g

a

c

t

t

h

r

c

i

p

c

t

t

M

i

p

fl

s

t

f

a

i

F

∈

Q

u

o

ethod in this case study can provide a good approximation even

ith the low computational cost.

.2. Other examples

Regarding the previous case study, the proposed method illus-

rated interesting results with the low computational cost. In this

nalysis, the proposed method is highlighted with other complex

rocesses such as lean dry gas production plant and synthesis gas

roduction.

.2.1. Example 2: Lean dry gas processing plant with six uniform

ncertainties

Fig. 7 shows a process schematic diagram of a lean dry gas pro-

uction plant [44] . Two natural gas streams containing N 2 , CO 2 ,

nd C 1 - n -C 4 hydrocarbons were mixed together and then pro-

essed in a refrigerated cycle to eliminate the heavier components.

nsuring the safe transport and processing of natural gas is crucial

ecause hydrocarbon liquid dropout in gas transmission lines nor-

ally leads to a number of problems, including increased pressure

rops, reduced pipeline capacity, and equipment problems, such

s compressor damage [1,45,46] . In this case study, a mixed feed

tream from two different sources is introduced into an inlet sep-

rator to remove the free hydrocarbon liquids. The overhead outlet

as from the separator is then entered into a low-temperature sep-

rator through two heat exchangers (gas/gas heat exchanger and

hiller), in which the required cooling is accomplished. In the low-

emperature separator, heavy hydrocarbons are removed so that

he eventual sales gas meets the required properties, such as the

eating values and dew point. The lean dry gas from the sepa-

ator is fed to the gas/gas heat exchanger and then sent to the

onsumers through a pipeline network. The liquid streams com-

ng from both separators are mixed and introduced to the de-

ropanizer column to recover the propane and remaining heavy

omponents.

This case study focuses mostly on determining the effects of

he decision factors according to the operability limits for each of

he uncertain variables. An interface under the code developed in

ATLAB

® was used to make a random generation of the process

nputs and quantify the uncertainty performance of the process in-

uts. First, Sobol’s indices via SA were derived to detect the in-

uential factors on the output variations. In this case study, the

ale gas heating value was controlled under uncertainties while

he flow rates ( F 1 and F 2), temperature ( Tn ), pressure ( Pn ) of two

eed natural gas streams, outlet temperature of the cold gas ( Tc ),

nd reflux ratio of the column ( R ) were assumed to be uncertain

ndependently by the uniform distribution in the range, such as

1 ∈ [1.90; 2.32 kg/s], F 2 ∈ [1.25; 1.52 kg/s], Tn ∈ [14.0; 17.1 °C], Pn

[37.2; 45.5 bars], Tc ∈ [ −16.8; −13.8 °C], and R ∈ [0.9; 1.1]. The

MC method was also examined to determine the impact of these

ncertainties on the process. Table 3 lists the statistical properties

btained from the proposed (M-DRM + gPC) and QMC methods for

262 L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

Table 5

Sobol’s sensitivity indices from the surrogated model by the M-DRM method for Examples 2 and 3.

Sobol’s sensitivity indices ( S i , T i )

Example 2 S1 S2 S3 S4 S5 S6 T1 T2

4.58e −04 4.57e −04 0.0038 0.4982 0.4561 0.0418 4.58e −04 4.57e −04

T3 T4 T5 T6

0.0038 0.4983 0.4561 0.0418

Example 3 S1 S2 S3 S4 S5 S6 S7 T1

2.55E −14 1.60E −04 0.5217 8.80E −10 0.0419 0.0 0 02 0.4337 2.57E −14

T2 T3 T4 T5 T6 T7

1.62E −04 0.5240 8.88E −10 0.0423 0.0 0 02 0.4359

Fig. 8. Density distributions of heating value of sale gas in lean dry gas produc-

tion process using the QMC/gPC methods with six and two uniform random inputs

(Example 2).

w

g

o

p

T

d

u

l

r

5

u

s

c

l

w

p

r

t

a

a

i

i

t

T

u

h

a

d

the heating value of the lean gas. In addition, Table 5 provides the

Sobol’s indices obtained by the M-DRM approximation. This indi-

cates that the pressure of the feed natural gas and the outlet tem-

perature of the cold gas are two influential factors for uncertainty

propagation, while the other parameters can be fixed as their nom-

inal values. At the second stage, the standard gPC approach can be

used for UQ with only two random inputs. Fig. 8 compares the per-

formance for the density functions of the lean gas heating value

Fig. 9. Synthesis gas product

ith two decision random inputs (the pressure of the feed natural

as and the output temperature of the cold gas) using two meth-

ds: the red line denotes the standard gPC method (with 100 sam-

les) and the blue one is the QMC method (with 10,0 0 0 samples).

he results from the gPC and QMC methods using the two ran-

om inputs showed good agreement with that of the QMC method

sing all six random parameters with 10,0 0 0 simulations (a green

ine). The proposed M-DRM approach detected the influential pa-

ameter inputs correctly through the derived SA indices.

.2.2. Example 3: Synthesis gas production with seven uniform

ncertainties

Synthesis gas (or syngas) used as feedstock for the ammonia

ynthesis can be produced by using five reactors, including two

onversion reactors (a reformer and a combustor) and three equi-

ibrium reactors in consequence [44] . Fig. 9 shows the PFD diagram

ith the uncertainty propagation of the synthesis gas production

lant.

The natural gas is mainly reformed in a conversion reactor (or

eformer), in which most of the methane is reacted with steam

o produce hydrogen, carbon monoxide, and carbon dioxide. Air is

dded to the second reactor for combustion. The oxygen from the

ir is consumed in an exothermic combustion reaction while the

nert nitrogen passes through the system. If the content of oxygen

n the air is raised, the effect is the increasing of hydrogen con-

ent in the synthesis gas and the decreasing the nitrogen content.

hese effects are desirable because the hydrogen is the final prod-

ct and combined with nitrogen for the ammonia production. The

ydrogen to nitrogen molar ratio (HNr) in the syngas is expected

t 3:1. The addition of steam in the second reactor serves to the

ual purpose of maintaining the reactor temperature and either to

ion plant (Example 3).

L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264 263

Fig. 10. Density distributions of HNr ratio using the QMC/gPC methods with seven

and two uniform random inputs (Example 3).

e

c

p

o

t

a

u

p

t

s

t⎧⎪⎪⎨⎪⎪⎩r

u

t

t

t

T

p

T

t

i

o

p

m

b

i

s

t

g

a

f

a

a

m

b

Q

t

6

a

o

o

c

I

a

w

p

D

w

t

T

c

s

t

p

p

p

l

p

g

s

p

w

u

c

t

b

w

A

g

f

a

t

t

a

8

R

nsure that the excess of methane from the natural gas stream is

onsumed. In the last three reactors, the equilibrium reaction takes

lace and consecutively with the water–gas shift, the temperature

f the reactor effluent stream is therefore lowered. The set opera-

ions are used to specify the particular pressure of the steam and

ir streams. The Peng–Robinson equation of state, which is widely

sed in the hydrocarbon system, was selected in this simulation to

redict the vapor–liquid equilibrium behavior.

In this example, the NG feed temperature, pressure, flow rate,

he reforming steam flow rate and temperature, the combustion

team flow rate, and the air flow rate were assumed to be uncer-

ain and uniformly distributed independently with a range of

[ (334 . 0 , 408 . 2) ◦

C] ; [(31 . 02 , 37 . 92) bars ] ;[(81 . 65 , 99 . 80) kmol / h] ;[ (221 . 5 , 270 . 7)

◦C] ; [(116 . 73 , 142 . 67) kmol / h] ;

[(212 . 28 , 259 . 46) kmol / h] . [(112 . 70 , 137 . 75) kmol / h]

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

,

espectively. All other process parameters (such as the vessel vol-

me, heat duty, and pressure) were assumed to be determinis-

ic. The impact of the uncertainties on the HNr ratio was con-

rolled through an interface between MATLAB ۛand HYSYS ۛ, where

he syngas production process in Fig. 9 was modeled rigorously.

able 3 lists the statistical properties of HNr obtained from the

roposed method and the conventional MC/QMC methods. Also,

able 5 presents the results of Sobol’s indices, which indicate that

he feed natural gas flow rate and the air flow rate are the most

nfluential factors on the “HNr ratio” output. It is noted that the

ther parameters can be fixed as their nominal values. Fig. 10 com-

ares the density functions of HNr obtained from the gPC/MC/QMC

ethods for two important factors. The sensitivity indices derived

y the proposed M-DRM method correctly identify the influential

nputs. At the second stage, it is worth noting that the number of

imulations required by the gPC method was significantly lower

han those by the other two methods [10] . The results from the

PC method closely matched with those from the traditional MC

nd QMC methods.

Remarks: Owing to the exponential increase in simulation ef-

orts, the standard gPC method was not considered in three ex-

mples with six or seven random inputs, and only the M-DRM

nd QMC methods for UQ were considered. In addition, the QMC

ethod was not used for the SA of the three chemical processes

ecause of the large computational effort required for SA using the

MC method (This will be approximately eight times higher than

he effort for UQ).

. Conclusions

The purposes of UQ and SA in process design and modeling

re important for testing model robustness against uncertainties,

btaining a better understanding of the effects of inputs on the

utput, determining the key inputs that mainly influence the un-

ertainty of model output, and achieving a model simplification.

n this study, to overcome the computational limitation in the UQ

nd SA of complex chemical processes, a two-stage gPC approach

as presented based on M-DRM and gPC for UQ and SA. In the

roposed method, the SA indices were first estimated by the M-

RM in a computationally efficient manner and the gPC method

as then applied to the UQ based on the simplified model from

he M-DRM. HYSYS ۛwas used to obtain a rigorous simulation result.

he M-DRM approximated the complex uncertain problem suc-

essfully in the form of the product of univariate functions using a

mall number of sampling points. Through the M-DRM approxima-

ion, the Sobol’s sensitivity indices for detecting non-influential in-

uts could be obtained with little computational burden. The pro-

osed two-stage approach is superior to the popular QMC/gPC ap-

roaches, primarily in terms of the computational efforts when a

arge number of random inputs is considered. The proposed ap-

roach was rigorously applied to the UQ and SA of the propylene

lycol production process, lean dry gas processing plant, and the

ynthesis gas production plant, respectively. The results showed

recise agreement with those of the conventional QMC method, of

hich the computational cost is in the order of 10 4 model eval-

ations, and not compatible with the UQ and SA of most chemi-

al processes with a moderate/large number of uncertainties. This

wo-stage approach is expected to offer an effective way to handle

oth SA and UQ for other complex chemical processes associated

ith a large number of uncertainties.

cknowledgments

This study was supported by the Basic Science Research Pro-

ram through the National Research Foundation of Korea (NRF)

unded by the Ministry of Education ( 2015R1D1A3A01015621 )

nd by the Priority Research Centers Program through

he National Research Foundation of Korea (NRF) funded by

he Ministry of Education ( 2014R1A6A1031189 ). Pham L.T. Duong

nd Jorge Goncalves were supported by FNR CORE project ref

231540.

eferences

[1] Abubakar U , Sriramula S , Renton NC . Reliability of complex chemical engineer-ing processes. Comput Chem Eng 2015;74:1–14 .

[2] Lucay F , Cisternas LA , Gálvez ED . Global sensitivity analysis for identifying crit-ical process design decisions. Chem Eng Res Des 2015;103:74–83 .

[3] O’Hagan A . Bayesian analysis of computer code outputs: a tutorial. Reliab EngSyst Saf 2006;91:1290–300 .

[4] Saltelli A , Ratto M , Tarantola S , Campolongo F . Update 1 of: sensitivity analysis

for chemical models. Chem Rev 2012;112:1–21 . [5] Binder K . Monte Carlo methods: a powerful tool of statistical physics. In: Pro-

ceedings of a conference at the University of Salzburg, Austria. Springer; 1998.p. 19–39 .

[6] Coulibaly I , Lécot C . Monte Carlo and quasi-Monte Carlo algorithms for a linearintegro-differential equation. In: Proceedings of a conference at the University

of Salzburg, Austria. Springer; 1998. p. 176–88 . [7] Ferrari A , Gutiérrez S , Sin G . Modeling a production scale milk drying pro-

cess: parameter estimation, uncertainty and sensitivity analysis. Chem Eng Sci

2016;152:301–10 . [8] Kroese DP , Taimre T , Botev ZI . Markov Chain Monte Carlo. Handbook of Monte

Carlo methods. John Wiley & Sons, Inc; 2011. p. 225–80 . [9] Liu JS . Sequential Monte Carlo in Action, Monte Carlo strategies in scientific

computing. New York: Springer; 2004. p. 79–104 .

264 L.Q. Minh et al. / Journal of the Taiwan Institute of Chemical Engineers 78 (2017) 254–264

[

[10] Duong PLT , Ali W , Kwok E , Lee M . Uncertainty quantification and global sen-sitivity analysis of complex chemical process using a generalized polynomial

chaos approach. Comput Chem Eng 2016;90:23–30 . [11] Baroni G , Tarantola S . A general probabilistic framework for uncertainty and

global sensitivity analysis of deterministic models: a hydrological case study.Environ Model Softw 2014;51:26–34 .

[12] Saltelli A , Chan K , Scott EM . Sensitivity analysis. New York: Wiley; 20 0 0 . [13] Saltelli A , Tarantola S , Campolongo F , Ratto M . Global sensitivity analysis for

importance assessment. Sensitivity analysis in practice. John Wiley & Sons;

2004. p. 31–61 . [14] Saltelli A , Ratto M , Andres T , Campolongo F , Cariboni J , Gatelli D , et al. Global

sensitivity analysis: the primer. John Wiley & Sons; 2008 . [15] Celse B , Costa V , Wahl F , Verstraete JJ . Dealing with uncertainties: sensitivity

analysis of vacuum gas oil hydrotreatment. Chem Eng J 2015;278:469–78 . [16] Li G , Wang S-W , Rabitz H , Wang S , Jaffé P . Global uncertainty assess-

ments by high dimensional model representations (HDMR). Chem Eng Sci

2002;57:4 4 45–60 . [17] Papalambros PY , Wilde DJ . Principles of optimal design. Cambridge University

Press; 20 0 0 . [18] Kajero OT , Chen T , Yao Y , Chuang Y-C , Wong DSH . Meta-modelling in chemical

process system engineering. J Taiwan Inst Chem Eng 2017;73:135–45 . [19] Chung PS , Jhon MS , Biegler LT . Chapter 2 - the holistic strategy in multi-scale

modeling. In: Guy BM, editor. Advances in chemical engineering. Academic

Press; 2011. p. 59–118 . [20] Wiener N . The homogeneous chaos. Am J Math 1938;60:897–936 .

[21] Nagy ZK , Braatz RD . Distributional uncertainty analysis using power series andpolynomial chaos expansions. J Process Control 2007;17:229–40 .

[22] Shen DE , Braatz RD . Polynomial chaos-based robust design of systems withprobabilistic uncertainties. AIChE J 2016;62:3310–18 .

[23] Duong PLT , Lee M . Robust PID controller design for processes with stochastic

parametric uncertainties. J Process Control 2012;22:1559–66 . [24] Duong PLT , Lee M . Probabilistic analysis and control of systems with uncertain

parameters over non-hypercube domain. J Process Control 2014;24:358–67 . [25] Du YC , Duever TA , Budman H . Fault detection and diagnosis with para-

metric uncertainty using generalized polynomial chaos. Comput Chem Eng2015;76:63–75 .

[26] Bavdekar VA , Mesbah A . A polynomial chaos-based nonlinear Bayesian ap-

proach for estimating state and parameter probability distribution func-tions. In: Proceeding of the 2016 American Control Conference (ACC); 2016.

p. 2047–52 . [27] Nimmegeers P , Telen D , Logist F , Impe JV . Dynamic optimization of biological

networks under parametric uncertainty. BMC Syst Biol 2016;10(1):86 . [28] Xiu D , Karniadakis GE . The Wiener–Askey polynomial chaos for stochastic dif-

ferential equations. SIAM J Sci Comput 2002;24:619–44 .

[29] Ghanem RG , Spanos PD . Stochastic finite elements: a spectral approach.Courier Corporation; 2003 .

[30] Zhang X , Pandey MD . An effective approximation for variance-based globalsensitivity analysis. Reliab Eng Syst Saf 2014;121:164–74 .

[31] Sobol ′ IM . Global sensitivity indices for nonlinear mathematical models andtheir Monte Carlo estimates. Math Comput Simul 2001;55:271–80 .

[32] Xu H , Rahman S . A generalized dimension-reduction method for multidi-mensional integration in stochastic mechanics. Int J Numer Method Eng

2004;61:1992–2019 .

[33] Sobol IM . Theorems and examples on high dimensional model representation.Reliab Eng Syst Saf 2003;79(2):187–93 .

[34] Saltelli A , Tarantola S , Campolongo F , Ratto M . Methods based on decomposingthe variance of the output. Sensitivity analysis in practice. John Wiley & Sons;

2004. p. 109–49 . [35] Saltelli A , Tarantola S , Campolongo F , Ratto M . Sensitivity analysis in diagnostic

modelling: Monte Carlo filtering and regionalised sensitivity analysis, Bayesian

uncertainty estimation and global sensitivity analysis. Sensitivity analysis inpractice. John Wiley & Sons; 2004. p. 151–92 .

[36] Xiu D . Numerical methods for stochastic computations: a spectral method ap-proach. Princeton University Press; 2010 .

[37] Duong PLT , Minh LQ , Pham TN , Goncalves J , Kwok E , Lee M . Uncertainty quan-tification and global sensitivity analysis of complex chemical processes with a

large number of input parameters using compressive polynomial chaos. Chem

Eng Res Des 2016;115:204–13 Part A . [38] Smolinske SC . Handbook of food, drug, and cosmetic excipients. CRC Press;

1999 . [39] Sullivan CJ . Propanediols, ullmann’s encyclopedia of industrial chemistry. Wi-

ley-VCH Verlag; 20 0 0 . [40] Bausmith DS , Neufeld RD . Soil biodegradation of propylene glycol based air-

craft deicing fluids. Water Environ Res 1999;71:459–64 .

[41] Gautschi W . Orthogonal polynomials: computation and approximation. Claren-don Press; 2004 .

[42] Tempo R , Calafiore G , Dabbene F . Randomized algorithms for analysis and con-trol of uncertain systems: with applications. Springer; 2012 .

[43] Rasmussen CE , Williams CKI . Gaussian processes for machine learning. Adap-tive computation and machine learning. Cambridge, MA: MIT Press; 2006 .

44] AspenTech, Aspen physical property system. Aspen technology, http://www.

aspentech.com/products/engineering/aspen-properties/ ; [accessed 04.12.16]. [45] Mørch Ø, Nasrifar K , Bolland O , Solbraa E , Fredheim AO , Gjertsen LH . Measure-

ment and modeling of hydrocarbon dew points for five synthetic natural gasmixtures. Fluid Phase Equilib 2006;239:138–45 .

[46] Skylogianni E , Novak N , Louli V , Pappa G , Boukouvalas C , Skouras S , et al. Mea-surement and prediction of dew points of six natural gases. Fluid Phase Equilib

2016;424:8–15 .