Model and Discretiza tion Err or Ad aptivity
within St a tionar y Gas Transpor t Optimiza tion
V olker Mehrmann 1 , Mar tin Schmidt 2 , 3 , Jer oen J. Stol wijk 1
Abstra ct.
The minimization of op eration costs for natural gas transp ort
net works is studied. Based on a recen tly developed mo del hierarch y ranging from
detailed mo dels of instationary partial differen tial equations with temp erature
dep endence to highly simplified algebraic equations, mo deling and discretization
error estimates are presen ted to control the o v erall error in an optimization
metho d for stationary and isothermal gas flo ws. The error con trol is realized
b y switching to more detailed models or finer discretizations if necessary to
guaran tee that a prescrib ed mo del and discretization error tolerance is satisfied
in the end. W e prov e con vergence of the adaptiv ely con trolled optimization
metho d and illustrate the new approac h with numerical examples.
De dic ate d to Hans Ge or g Bo ck on the o c c asion of his 70th birthday.
1. Intr oduction
In this pap er w e discuss the minimization of op eration costs for natural gas
transp ort net works based on a model hierarch y , see [
11
,
21
], whic h ranges from
detailed mo dels based on instationary partial differen tial equations with temp erature
dep endence to highly simplified algebraic equations. The detailed mo dels are
necessary to ac hiev e a go o d understanding of the system state, but in many practical
optimization applications only the stationary algebraic equations—or ev en further
simplifications lik e piecewise linearizations as in [
15
,
16
,
32
]—are used in order to
reduce the high computational effort of ev aluating the state of the system with
the more sophisticated mo dels. Ho w ev er, it is then unclear how goo d the true
state is appro ximated b y these simplified mo dels and error b ounds are typically not
a v ailable in this con text; see the chapter [
22
] in [
23
] for a more detailed discussion
of this issue. Recen tly , in [
37
], a detailed error and p erturbation analysis has b een
dev elop ed for several components in the model hierarch y and it has b een shown ho w
the more detailed mo del comp onents can be used to estimate the error obtained in
the simplified mo dels.
Here, w e use these error estimates from the mo del hierarc h y together with classical
error estimate grid adaptation tec hniques for the space discretization within an
optimization metho d to con trol the error adaptively b y switc hing to more detailed
mo dels or finer discretizations if necessary . Moreo v er, our adaptiv e metho d also
allo ws to lo cally switch bac k to coarser mo dels or to coarser discretizations if they
are sufficien tly accurate with resp ect to the lo cal flo w situation. Our new approac h
can, in general, b e used for the en tire mo del hierarc h y by also using space-time grid
adaptation. Ho w ev er, to keep things simple and to illustrate the functionalit y of
the new adaptiv e approac h, w e will use three stationary isothermal mo dels from the
hierarc h y in [ 11 ].
Date : Decem b er 7, 2017.
2010 Mathematics Subje ct Classific ation. 35Q31, 65G99, 65L70, 90C30, 93C40.
Key wor ds and phr ases. Gas transp ort optimization, Isothermal stationary Euler equations,
Mo del hierarc hy , A daptive error con trol, Marking strategy .
1

2 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
Using adaptiv e tec hniques to achiev e a trade-off b etw een computational efficiency
and accuracy b y using adaptiv e discretization metho ds in the con text of optimization
and optimal con trol problems is an imp ortant researc h topic, in particular in the
con text of real-time optimal con trol of constrained dynamical systems, see, e.g.,
[
3
,
9
,
8
,
29
], or in the con text of optimal con trol of problems constrained b y partial
differen tial equations; see, e.g., [
1
,
24
,
25
,
26
]. W e extend these ideas and com bine
adaptiv e grid refinemen t and mo del selection in a mo del hierarc h y in the con text of
nonlinear optimization problems. W e also theoretically analyze the new algorithm.
First promising n umerical results for suc h an approac h w ere presen ted in [ 34 , 35 ].
The pap er is structured as follo ws. The mo dels used in this pap er are describ ed in
Sect. 2 together with a simple first-order Euler metho d for the space discretization.
In Sect. 3 w e in tro duce mo del and discretization error estimators, whic h are used
in Sect. 4 to deriv e an adaptiv e mo del and discretization con trol algorithm for the
nonlinear optimization of gas transp ort net w orks that, in the end, deliv ers solutions
that satisfy prescrib ed error tolerances. Numerical results are presen ted in Sect. 5
and the pap er concludes in Sect. 6 .
2. Pr oblem Description, Modeling Hierar chy, and Discretiza tions
In this section w e in tro duce the problem of op eration cost minimization for
natural gas transp ort net works. W e presen t our ov erall mo del of a gas transp ort
net w ork in v olving contin uous nonlinear mo dels describing a stationary flo w for all
the considered net w ork elemen ts. Since the ma jorit y of the elements are pipes, our
fo cus lies on the precise and ph ysically accurate mo deling of these pip es. The t ypical
mo dels for the pip e flo w are nonlinear instationary partial differen tial equations
(PDEs) on a graph and their appropriate space-time discretizations. T o address
the fact that the b eha vior of the flow and the accuracy of the mo del ma y v ary
significan tly in differen t regions of the net w ork, w e discuss a small part of the
complete mo del hierarc hy of instationary models, see [
11
], where the lo w er lev e l
mo dels in the hierarc hy are simplifications of the higher lev el mo dels. Whic h mo del
is most appropriate to obtain a computationally tractable, adequately accurate, and
finite-dimensional appro ximation dep ends on the task that needs to b e p erformed
with the mo del.
Our mo deling approac h is based on the follo wing ph ysical assumptions. First, we
only consider a stationary gas flo w, i.e., w e neglect all time effects of gas dynamics,
so that w e ha ve ordinary differen tial equations (ODEs) in space instead of systems
of PDEs on a graph. Second, w e assume an isothermal regime, i.e., we neglect all
effects arising from c hanges in the gas temp erature.
These assumptions are tak en carefully suc h that we still obtain ph ysically mean-
ingful solutions and suc h that w e are still able to deriv e and analyze an adaptiv e
mo del and discretization con trol algorithm—without unnecessarily ov erloading the
mo dels with all tec hnical details of the application that may distract us from the
main mathematical ideas.
2.1.
The Net w ork.
W e mo del the gas transp ort net w ork b y a directed and con-
nected graph
G
= (
V , A
) . The no de set is made up of en try no des
V +
, where gas
is supplied, of exit no des
V −
, where gas is disc harged, and of inner no des
V 0
, i.e.,
w e ha ve
V
=
V + ∪ V − ∪ V 0
. The set of arcs in our mo dels comprises pip es
A pi
and
compressor mac hines A cm , i.e., w e hav e A = A pi ∪ A cm .
Real-w orld gas transp ort netw orks con tain many other elemen t t yp es lik e (control)
v alv es, short cuts, or resistors. F or detailed information on mo deling these devices,
see [
14
] in general or [
34
,
35
] for a fo cus on nonlinear programming (NLP) t yp e
mo dels. Ho w ev er, we restrict ourselv es to mo dels with pip es and compressors in
order to streamline the presen tation of our basic ideas and metho ds, and to sho w in

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 3
a protot ypical w a y that our approac h of space discretization and mo del adaptivit y
leads to ma jor accuracy and efficiency improv ements.
As basic quan tities w e in tro duce gas pressure v ariables
p u
at all no des
u ∈ V
and
mass flo w v ariables
q a
at all arcs
a ∈ A
of the net w ork. Both t yp es of v ariables are
b ounded due to tec hnical constraints on the pipes, i.e.,
p u ∈ [ p u , p u ] for all u ∈ V , (1a)
q a ∈ [ q a , q a ] for all a ∈ A. (1b)
All other required quan tities are in tro duced where they are used first.
2.2.
No des.
In stationary gas net w ork mo dels, the no des
u ∈ V
are mo deled b y a
mass balance equation, i.e., w e ha v e the constrain t
X
a ∈ δ in ( u )
q a − X
a ∈ δ out ( u )
q a = q u for all u ∈ V , (2)
where for ingoing arcs w e use the notation
δ in ( u ) : = { a ∈ A : there exists w ∈ V and a = ( w , u ) }
and for outgoing arcs
δ out ( u ) : = { a ∈ A : there exists w ∈ V and a = ( u, w ) } .
Moreo v er,
q u
mo dels the supplied or disc harged mass flow at the corresponding
no de, i.e., w e hav e
q u 




≥ 0 for all u ∈ V − ,
≤ 0 for all u ∈ V + ,
= 0 for all u ∈ V 0 .
2.3.
Pip es.
Isothermal gas flo w through cylindrical pip es is describ ed b y the Euler
equations for compressible fluids,
∂ ρ
∂ t + 1
A
∂ q
∂ x = 0 , (3a)
1
A
∂ q
∂ t + ∂ p
∂ x + 1
A
∂ ( q v )
∂ x = − λ ( q ) | v | v
2 D ρ − g ρh 0 , (3b)
see, e.g., [
13
,
27
] for a detailed discussion. Here and in what follows,
ρ
is the gas
densit y ,
v
is its v elo city ,
λ
=
λ
(
q
) is the friction term,
A
denotes the cross-sectional
area of the pip e,
h 0
is its slop e, and
D
is the diameter of the pip e. F urthermore,
g
is the acceleration due to gra vit y ,
t
is the temp oral co ordinate, and
x ∈
[0
, L
] is the
spatial co ordinate with L b eing the length of the pip e. Equation ( 3a ) is called the
con tin uit y equation and
( 3b )
the momen tum equation. Since we only consider the
stationary case, all partial deriv ativ es with resp ect to time v anish and w e obtain
the simplified stationary mo del
1
A
∂ q
∂ x = 0 , (4a)
∂ p
∂ x + 1
A
∂ ( q v )
∂ x = − λ ( q ) | v | v
2 D ρ − g ρh 0 . (4b)
Th us, the con tin uit y equation in its s tationary v arian t simply states that the mass
flo w along the pip e is constant, i.e., q ( x ) ≡ q = const for all x ∈ [0 , L ] .
T o simplify the stationary momen tum equation
( 4b )
, w e consider t w o more mo del
equations. First, the equation of state
p = ρc 2 with c = p R s T z ,

4 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
where
c
is the sp eed of sound,
R s
is the sp ecific gas constan t, and
z
is the com-
pressibilit y factor. The second mo del is the relation of gas mass flo w, densit y , and
v elo city giv en b y
q = Aρv .
Substituting b oth these mo dels in to ( 4b ), w e obtain
∂ p
∂ x  1 − q 2
A 2
c 2
p 2  = − λc 2
2 A 2 D p | q | q − g h 0
c 2 p, (M 1 )
i.e., the stationary momen tum equation written in dep endence of the gas pressure
p = p ( x ) , x ∈ [0 , L ] , and the mass flo w q .
A simplified v ersion of the latter equation can b e obtained by ignoring the ram
pressure term
1
A
∂ ( q v )
∂ x ,
in
( 4b )
, i.e., the total pressure exerted on the gas b y the pip e wall, or, equiv alently ,
the term
− q 2
A 2
c 2
p 2
∂ p
∂ x (5)
in
( M 1 )
. F or a discussion of this simplification step, see [
38
]. Neglecting the ram
pressure term ( 5 ) yields
∂ p
∂ x = − λc 2
2 A 2 D p | q | q − g h 0
c 2 p. (M 2 )
Finally , one ma y also neglect gra vitational forces, i.e., set the term
g h 0 p/c 2
to 0 and
obtain ∂ p
∂ x = − λc 2
2 A 2 D p | q | q . (M 3 )
Analytical solutions for the mo dels
( M 1 )
–
( M 3 )
are only rarely kno wn; see, e.g.,
[
18
,
19
,
34
]. Th us, in order to obtain finite-dimensional nonlinear optimization
mo dels, we discretize these differen tial equations in space. Applying, e.g., the
implicit Euler metho d w e obtain
p k − p k − 1
h  1 − q 2
A 2
c 2
p 2
k  = − λc 2
2 A 2 D p k
| q | q − g h 0
c 2 p k , k = 1 , . . . , n, (D 1 )
p k − p k − 1
h = − λc 2
2 A 2 D p k
| q | q − g h 0
c 2 p k , k = 1 , . . . , n, (D 2 )
p k − p k − 1
h = − λc 2
2 A 2 D p k
| q | q , k = 1 , . . . , n, (D 3 )
where
p k
=
p
(
x k
) and Γ =
{ x 0 , x 1 , . . . , x n }
is an equidistan t spatial discretization of
the pip e with constan t stepsize
h
=
x k − x k − 1
and
x 0
= 0
, x n
=
L
. Of course, one
could also apply a higher-order Runge–Kutta metho d, whic h w ould allo w a larger
stepsize and w ould th us reduce the computational cost.
These discretizations extend the mo del hierarc hy
( M 1 )
–
( M 3 )
for the Euler equa-
tions b y infinitely man y mo dels that are parameterized b y the discretization stepsize
h
applied in
( D 1 )
–
( D 3 )
. In summary , w e obtain the pip e mo del hierarc hy of stationary
Euler equations depicted in Fig. 1 .
2.4.
Compressors.
Compressor mac hines
a
= (
u, w
)
∈ A cm
increase the inflo w gas
pressure to a higher outflo w pressure, i.e., they can b e describ ed in a simplified w a y
b y
p w = p u + ∆ a , ∆ a ∈ [0 , ¯
∆ a ] for all a ∈ A cm . (6)
Moreo v er, for simplicit y , w e assume that w e are giv en cost co efficien ts
ω a ≥
0 for
ev ery compressor
a ∈ A cm
that con v erts pressure increase to compression cost. Of

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 5
Mo del
( M 1 )
( M 2 )
( M 3 )
h 0 = 0
∂ ( q v )
∂ x ≈ 0
( D 1 )
( D 2 )
( D 3 )
h 0 = 0
∂ ( q v )
∂ x ≈ 0
discr.
discr.
discr.
Figure 1. Pip e mo del hierarch y based on the Euler equations.
The space con tin uous mo dels are p ositioned in the left column and
their space discretized coun terparts are p ositioned in the righ t
column.
course, this is an extremely coarse approximation of a compressor mac hine. An
alternativ e w ould b e to use a simple input-output surrogate mo del obtained from a
realization or system iden tification of an input-output transfer function; see, e.g., [
5
].
Ho w ev er, our fo cus is on an accurate mo deling of the gas flow in pipes and on
deriving an adaptiv e mo del and discretization control algorithm. Mo del ( 6 ) allo ws
for setting up a reasonable ob jective function for our NLPs and is th us appropriate
in this w ork. F or more details, see [ 31 , 34 , 35 ] or [ 14 ].
2.5.
The Optimization Problem.
W e will use the adaptiv e mo del and discretiza-
tion con trol algorithm in the con text of the following nonlinear ODE-constrained
optimization problem
min X
a ∈ A cm
ω a ∆ a (7a)
s.t. v ariable b ounds ( 1 ) , (7b)
mass balance ( 2 ) , (7c)
compressor mo del ( 6 ) for all a ∈ A cm , (7d)
pip e mo del ( M 1 ) for all a ∈ A pi , (7e)
where our ob jectiv e function mo dels the cost for the compressor activit y that is
constrained b y an infinite-dimensional description of the gas flo w in pip es. Prob-
lem
( 7 )
is a classical nonlinear optimal con trol problem. A t ypical approac h to
solv e suc h problems in practice is the first-discretize-then-optimize paradigm; see,
e.g., [
2
]. In this setting, one replaces the ODE constraints b y finite sets of nonlinear
constrain ts that arise, e.g., from implicit Euler discretizations lik e
( D 1 )
for
( M 1 )
.
Moreo v er, practical exp erience suggests that for the ev aluation of the constrain ts,
it is often not required to apply the most accurate mo del lik e
( D 1 )
with a small
stepsize for ev ery pip e in the netw ork. Instead, in man y situations it is sufficien t to
use simplified mo dels lik e
( D 2 )
and
( D 3 )
with a coarse grid, whic h then t ypically
yields fast execution times for the ev aluation of the constrain t functions.
T o this end, w e define discretized problem v arian ts of Problem
( 7 )
b y sp ecifying
the mo del lev el
` a ∈ {
1
,
2
,
3
}
for ev ery arc
a ∈ A pi
(i.e., the discretized mo del
( D 1 )
,

6 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
Γ 1 = { x k } L a /h a
k =0
Γ 2 = { x s } L a / (2 h a )
s =0
Γ 3 = { x r } L a / (4 h a )
r =0
h a
2 h a
4 h a
Figure 2. Ov erview of the three considered discretization grids Γ
1
,
Γ
2
, and Γ
3
with gridp oin ts
x k
,
x s
, and
x r
and stepsizes
h a
, 2
h a
,
and 4
h a
, resp ectiv ely . The v ertical lines represen t the ev aluation
grid Γ 3 for the error estimators in ( 9 ) and ( 10 ).
( D 2 )
, or
( D 3 )
, resp ectiv ely) together with a stepsize
h a
. This yields the family of
finite-dimensional NLPs
min X
a ∈ A cm
ω a ∆ a (8a)
s.t. v ariable b ounds ( 1 ) , (8b)
mass balance ( 2 ) , (8c)
compressor mo del ( 6 ) for all a ∈ A cm , (8d)
pip e mo del (D ` a ) with stepsize h a for all a ∈ A pi . (8e)
Note that the constrain ts
( 7b )
–
( 7d )
in the infinite-dimensional problem are exactly
the same as constrain ts ( 8b )–( 8d ) in the family of discretized problems.
3. Err or Estima tors
In this section w e in tro duce a first-order estimate for the error b etw een the most
detailed infinite-dimensional and an arbitrary space-discretized mo del. This error
estimator is obtained as the sum of a discretization and a mo del error estimator.
Since w e consider the stationary case, mass flo ws in pip es are constan t in the spatial
dimension. This is wh y w e base our error estimators on the differences of the
pressures p ( x ) for differen t mo dels and discretizations.
Supp ose that for a giv en pip e
a ∈ A pi
, the mo del lev el
` a ∈ {
1
,
2
,
3
}
with
discretization stepsize
h a
is curren tly used for the computations. The o v erall
solution of the optimization problem for the en tire net w ork, also including pressure
increases in compressors etc., is denoted b y
y
and con tains the discretized pressure
distributions of the separate pip es
a ∈ A pi
, whic h w e denote b y
p ` a
(
x k
;
h a
) with
discretization grid Γ
1
=
{ x k } L a /h a
k =0
obtained b y using the stepsize
h a
. W e no w
compute an estimate for the error b et ween the solution of the curren tly used
mo del (
D ` a
) and the solution of the reference mo del
( M 1 )
. Let the solution of
mo del ( M 1 ) for pip e a ∈ A pi b e denoted by ˆ p ( x ) with x ∈ [0 , L a ] .
F urthermore, let the solutions of Mo del
( D 1 )
with discretization grids Γ
2
=
{ x s } L a / (2 h a )
s =0
and Γ
3
=
{ x r } L a / (4 h a )
r =0
using stepsizes 2
h a
and 4
h a
, b e denoted by
p 1
(
x s
; 2
h a
) and
p 1
(
x r
; 4
h a
) , resp ectiv ely . Due to the larger stepsize, the computation
of these t w o solutions is in general less exp ensiv e than computing a solution of
Mo del (
D ` a
) on the grid Γ
1
. Since the discretization grid Γ
3
is the coarsest grid
and all computed pressure profiles can b e ev aluated on this grid, Γ
3
is called the
ev aluation grid. This grid is used in the definitions of the follo wing error estimators.
The considered discretization grids and the ev aluation grid are depicted in Fig. 2 .

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 7
Mo del
( M 1 )
Mo del
( D 1 )
with step-
sizes 2
h a
and 4
h a
Mo del (
D ` a
) with stepsize
h a
η d ,a ( y )
η m ,a ( y )
η a ( y )
Figure 3. Ov erview of the mo dels and stepsizes used for the com-
putation of the o v erall error estimator
η a
(
y
) b et ween models
( M 1 )
and (
D ` a
) in
( 11 )
. Here, for a pip e
a
,
η d ,a
(
y
) is the discretization
error estimator and η m ,a ( y ) is the mo del error estimator.
F or a pip e a ∈ A pi , let the discretization error estimator b e defined b y
η d ,a ( y ) : = 
 p 1 ( x r ; 2 h a ) − p 1 ( x r ; 4 h a ) 
 ∞ (9)
and let the mo del error estimator b e defined by
η m ,a ( y ) : = 
 p 1 ( x r ; 2 h a ) − p ` a ( x r ; h a ) 
 ∞ . (10)
Here,
p ` a ( x r ; h a ) = [ p ` a ( x 0 ; h a ) , . . . , p ` a ( x n ; h a )] > , n = L a / (4 h a ) ,
denotes the solution of Mo del (
D ` a
) computed with stepsize
h a
that is ev aluated
at the gridp oin ts
x r
, i.e., on the grid Γ
3
. If
` a
= 1 , i.e., if the considered solution
already corresp onds to the most accurate mo del, then we set the model error to
zero, i.e.,
η m ,a
(
y
)=0 . F urthermore, let the o v erall error estimator
η a
(
y
) for a
pip e
a ∈ A pi
b e defined to b e a first-order upp er b ound for the maximum error
b et ween the solutions of models
( M 1 )
and (
D ` a
) at gridp oin ts
x r
with stepsize 4
h a
.
Th us, w e ha v e

 ˆ p ( x r ) − p ` a ( x r ; h a ) 
 ∞
≤ 
 ˆ p ( x r ) − p 1 ( x r ; 2 h a ) 
 ∞ + 
 p 1 ( x r ; 2 h a ) − p ` a ( x r ; h a ) 
 ∞
.
= η d ,a ( y ) + η m ,a ( y ) = : η a ( y ) ,
(11)
where
.
=
denotes a first-order appro ximation in
h a
, see [
36
, page 420], and w e use
that the implicit Euler metho d has con v ergence order 1. The error estimator
η a
(
y
) is
the absolute coun terpart of the comp onen t wise relativ e error estimator given in [
37
].
An o v erview of the considered mo dels in this section together with the considered
stepsizes is depicted in Fig. 3 .
W e close this section with a remark on the computation of the discretization
error estimator in
( 9 )
. A straigh tforw ard w a y is to solv e Mo del
( D 1 )
once with
stepsize 2
h a
and once again with stepsize 4
h a
for ev ery
a ∈ A pi
. Another p ossibility
w ould b e to use an embedded Runge–Kutta metho d, see, e.g., [
20
], whic h in general
sa v es computational cost due to the reduced n um b er of function ev aluations.
4. The Grid and Model Ad apt a tion Algorithm
In this section w e presen t and analyze an algorithm that adaptiv ely switc hes
b et ween the model levels in the hierarc h y of Fig. 1 and adapts discretization stepsizes
in order to find a con v enien t trade-off b etw een ph ysical accuracy and computational
costs. T o this end, the algorithm iterativ ely solv es NLPs and initial v alue problems

8 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
(IVPs). Solutions of the latter are used to ev aluate the error estimators discussed in
the last section and to decide on the mo del lev els and the discretization stepsizes
for the next NLP .
Consider a single NLP of the sequence of NLPs that are solv ed during the
algorithm and assume that pip e
a ∈ A pi
is mo deled using mo del (
D ` a
) and stepsize
h a
.
Let the solution of this NLP b e denoted b y
y
. A ccording to the last section, the
o v erall mo del and discretization error estimator for this pip e is giv en by
η a
(
y
) as
defined in
( 11 )
. Th us, it is giv en by the error estimator betw een the solutions of the
most accurate mo del ( M 1 ) and the curren t mo del ( D ` a ) .
The o v erall goal of our metho d is to compute a solution of a mem b er of the
family of discretized problems
( 8 )
for whic h it is guaran teed that this solution has
an estimated a v erage error p er pip e with resp ect to the reference mo del
( M 1 )
, that
is less than an a-priorily giv en tolerance
ε >
0 . This leads us to the follo wing
definition:
Definition 1
(
ε
-feasibilit y)
.
Let
ε >
0 b e giv en. W e sa y that a solution
y
of
problem
( 8 )
with discretized mo dels (
D ` a
),
` a ∈ {
1
,
2
,
3
}
, and stepsizes
h a
for the
pip es a ∈ A pi is ε -fe asible with resp ect to the reference problem ( 7 ) if
1
| A pi | X
a ∈ A pi
η a ( y ) ≤ ε.
The remainder of this section is organized as follo ws. Sect. 4.1 introduces rules
ab out ho w the mo del lev els and discretization stepsizes are mo dified. The strategies
for marking pip es for mo del or grid adaptation are explained in Sect. 4.2 . The
adaptiv e mo del and discretization control algorithm is in tro duced in Sect. 4.3 ,
together with a theorem for the finite termination of the algorithm. Finally , some
remarks regarding the adaptiv e con trol algorithm are giv en in Sect. 4.4 .
4.1.
Mo del and Discretization A daptation Rules.
Before w e presen t and dis-
cuss the o v erall adaptiv e mo del control algorithm w e ha v e to
(1)
describ e the mec hanisms of switching up or do wn pip e mo del lev els as w ell
as that of refining and coarsening the discretization grids, and
(2)
discuss our marking strategy that determines the arcs on whic h the mo del
or grid should b e adapted.
W e start with the first issue and follo w the standard PDE grid adaptation tec hnique;
see, e.g., [
6
,
7
,
12
] or [
4
]. The general strategy is as follo ws. W e switch up one
lev el in the mo del hierarch y if this yields an error reduction that is larger than
ε
;
otherwise, w e switch up to the most accurate discretized model
( D 1 )
. Hence, for
pip e a ∈ A pi w e hav e the rule
` new
a = ( ` a − 1 , if η m ,a ( y ; ` a ) − η m ,a ( y ; ` a − 1) > ε,
1 , otherwise , (12)
for switc hing up lev els in the mo del hierarc hy . W e apply this rule b ecause it is
p ossible that the effects of neglecting the ram pressure term (whic h is the difference
b et ween model levels
`
= 1 and
`
= 2 ) and neglecting gra vitational forces for
non-horizon tal pip es (which is the difference betw een mo del levels
`
= 2 and
`
= 3 )
balance eac h other out in the computation of the pressure profile of mo del
( D 3 )
. In
this case, switc hing from mo del (
D 3
) to (
D 2
) w ould increase the mo del error, which
is wh y w e switc h from ( D 3 ) to ( D 1 ) directly .

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 9
A discretization grid refinemen t or coarsening with a factor
γ >
1 is defined b y
taking the new stepsize as
h new
a : = ( h a /γ , for a grid refinement ,
γ h a , for a grid coarsening . (13)
F or a discretization sc heme of order
β
it is w ell-kno wn that a first-order approxima-
tion for the discretization error in
x ∈
[0
, L a
] is giv en b y
e d ,a
(
x
)
.
= c
(
x
)
h β
a
, where
c
(
x
)
is indep enden t of
h a
; see, e.g., [
36
]. F rom this, it follows that the new discretization
error after a grid refinemen t or coarsening can b e written as
e new
d ,a ( x ) .
= ( h new
a /h a ) β e d ,a ( x ) .
Since the implicit Euler metho d has con v ergence order
β
= 1 , with
h new
a
in
( 13 )
and
γ
= 2 , for the new discretization error estimator after a grid refinemen t or
coarsening, it holds that
η new
d ,a ( y ) .
= ( η d ,a ( y ) / 2 , for a grid refinemen t ,
2 η d ,a ( y ) , for a grid coarsening . (14)
4.2.
Marking Strategies.
W e no w describ e our marking strategies, i.e., ho w w e
c ho ose which pipes should b e switc hed up or do wn in their mo del lev el and which
pip es should get a refined or coarsened grid. Giv en marking strategy parame-
ters Θ
d ,
Θ
m ∈
[0
,
1] , w e compute subsets
R , U ⊆ A pi
suc h that they are the minimal
subsets of arcs that satisfy
Θ d X
a ∈ A pi
η d ,a ( y ) ≤ X
a ∈R
η d ,a ( y ) (15)
and
Θ m X
a ∈ A >ε
pi
( η m ,a ( y ; ` a ) − η m ,a ( y ; ` new
a )) ≤ X
a ∈U
( η m ,a ( y ; ` a ) − η m ,a ( y ; ` new
a )) (16)
with
A >ε
pi : = { a ∈ A pi : η m ,a ( y ; ` a ) − η m ,a ( y ; ` new
a ) > ε } ,
where
` new
a
is giv en in
( 12 )
. Analogously , given marking strategy parameters
Φ
d ,
Φ
m ∈
[0
,
1] and
τ ≥
1 , w e compute
C , D ⊆ A pi
suc h that they are the maximal
subsets of arcs that satisfy
Φ d X
a ∈ A pi
η d ,a ( y ) ≥ X
a ∈C
η d ,a ( y ) (17)
and
Φ m X
a ∈ A <ε
pi ( τ )
( η m ,a ( y ; ` new
a ) − η m ,a ( y ; ` a )) ≥ X
a ∈D
( η m ,a ( y ; ` new
a ) − η m ,a ( y ; ` a )) (18)
with
A <ε
pi ( τ ) : = { a ∈ A pi : η m ,a ( ` new
a ) − η m ,a ( ` a ) ≤ τ ε } .
In
( 18 )
,
` new
a
is alw a ys set to
min { ` a
+ 1
,
3
}
. F or ev ery arc
a ∈ R
(
a ∈ C
) w e
refine (coarsen) the discretization grid b y halving (doubling) the stepsize, i.e., w e
set
γ
= 2 in
( 13 )
. W e note that these marking strategies are v ery similar to the
greedy strategies on a net w ork describ ed in [
10
], where those pip es are mark ed for a
spatial, temp oral, or mo del refinemen t whic h yield the largest error reduction.

10 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
Algorithm 1: Adaptiv e Mo del and Discretization Control
Input:
A full sp ecification of the gas net w ork
G
= (
V , A
) , a tolerance
ε >
0 ,
initial marking strategy parameters Θ 0
d , Θ 0
m , Φ 0
d , Φ 0
m ∈ [0 , 1] , τ 0 ≥ 1 ,
and an initial safeguard parameter µ 0 ∈ N .
Output: An ε -feasible solution of the reference problem ( 7 ).
1 Cho ose an initial mo del lev el ` 0
a and a stepsize h 0
a for ev ery a ∈ A pi .
2 Solv e Problem ( 8 ) and let y 0 denote the optimal solution.
3 Compute η a ( y 0 ) for ev ery a ∈ A pi .
4 if y 0 is ε -fe asible then
5 return y 0 .
6 Set k = 1 and Θ k
d = Θ 0
d , Θ k
m = Θ 0
m , Φ k
d = Φ 0
d , Φ k
m = Φ 0
m , µ k = µ 0 , τ k = τ 0 .
7 for k = 1 , 2 , . . . do
8 for j = 1 , . . . , µ k do
9 Compute the sets U k ,j , R k ,j ⊆ A pi according to ( 15 ) and ( 16 ).
10 Switc h up the mo del level for ev ery pip e a ∈ U k,j .
11 Refine the discretization grid for ev ery pip e a ∈ R k ,j .
12 Solv e Problem ( 8 ) and let y k ,j denote the solution.
13 Compute η a ( y k ,j ) for ev ery a ∈ A pi .
14 if y k ,j is ε -fe asible then
15 return y k ,j .
16 Compute the sets D k , C k ⊆ A pi according to ( 17 ) and ( 18 ).
17 Switc h do wn the mo del lev el for every pipe a ∈ D k .
18 Coarsen the discretization grid for ev ery pip e a ∈ C k .
19 Increase k ← k + 1 and up date parameters Θ k
d , Θ k
m , Φ k
d , Φ k
m , µ k , τ k .
4.3.
The Algorithm.
With these preliminaries w e can no w state the o v erall adap-
tiv e mo del and discretization control algorithm for finding an
ε
-feasible solution of
the reference problem ( 7 ). The formal listing is giv en in Alg. 1 .
The algorithm mak es use of the safeguard parameter
µ ∈ N
. This parameter
ensures that the algorithm p erforms grid coarsenings and switc hes do wn the mo del
lev el only after applying
µ
rounds of grid refinemen ts and switc hing up mo del lev els.
It prev en ts an alternating switc hing up and do wn mo del lev els or an alternating
refining and coarsening of the discretization grid. W e note that this tec hnique
is similar to the use of h ysteresis parameters; see, e.g., [
28
]. By employing this
safeguard, w e can pro v e that Alg. 1 terminates after a finite n um b er of iterations
with an ε -feasible p oin t of the reference mo del ( M 1 ).
T o impro v e readabilit y , w e split the pro of of our main theorem in to tw o parts.
The first lemma states finite termination at an
ε
-feasible p oin t if only discretization
grid refinemen ts and coarsenings are applied, whereas the second lemma considers
the case of switc hing levels in the model hierarch y only , i.e., with a fixed stepsize
for ev ery pip e.
Lemma 1.
Supp ose that the mo del level
` a ∈ {
1
,
2
,
3
}
is fixe d for every pip e
a ∈ A pi
.
L et the r esulting set of mo del levels b e denote d by
M
. Supp ose further that
η a
(
y
) =
η d ,a
(
y
) holds in
( 11 )
and that every NLP is solve d to lo c al optimality. Consider
A lg. 1 without applying the mo del switching steps in Lines 10 and 17 . Then, the
algorithm terminates after a finite numb er of r efinements in Line 11 and c o arsenings
in Line 18 with an
ε
-fe asible solution with r esp e ct to mo del level set
M
if ther e exists

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 11
a c onstant C > 0 such that 1
2 Θ k
d µ k > Φ k
d + C (19)
holds for al l k .
Pr o of. W e consider the total discretization error
η d ( y ) = X
a ∈ A pi
η d ,a ( y )
and sho w that for ev ery iteration
k
the difference b et ween the decrease obtained
in the inner for-lo op and the increase obtained due to the coarsenings applied in
Line 18 is p ositiv e and uniformly b ounded a w ay from zero. In what follo ws, w e only
consider a single iteration and drop its index k for b etter readabilit y .
First, w e consider one refinemen t step in Line 11 . Let
η j − 1
d ,a
denote the discretiza-
tion error b efore the
j
th inner iteration and let
η j
d ,a
denote the discretization error
after the j th inner iteration. With this, w e ha v e
X
a ∈ A pi
η j − 1
d ,a − X
a ∈ A pi
η j
d ,a
= X
a ∈R j
η j − 1
d ,a + X
a ∈ A pi \R j
η j − 1
d ,a − X
a ∈R j
η j
d ,a − X
a ∈ A pi \R j
η j
d ,a
= X
a ∈R j
η j − 1
d ,a − X
a ∈R j
η j
d ,a
= X
a ∈R j
1
2 η j − 1
d ,a
for ev ery
j
= 1
, . . . , µ
. F or the last equalit y w e hav e used that the implicit Euler
metho d has con vergence order 1, whic h (for small stepsizes
h a
) implies
η j
d ,a
=
1
2 η j − 1
d ,a
when w e tak e the new stepsize as half the curren t stepsize; see
( 14 )
. Summing
up o v er all
µ
inner iterations w e obtain a telescopic sum and finally get an error
decrease of
µ
X
j =1 X
a ∈ A pi
η j − 1
d ,a − X
a ∈ A pi
η j
d ,a ! = X
a ∈ A pi
η 0
d ,a − X
a ∈ A pi
η µ
d ,a = 1
2
µ
X
j =1 X
a ∈R j
η j − 1
d ,a .
W e no w consider the coarsening step. F or this, let
η µ
d ,a
denote the discretization
error b efore and
η µ +1
d ,a
the discretization error after the coarsening step in Line 18 .
Using similar ideas lik e ab ov e we obtain
X
a ∈ A pi
η µ +1
d ,a − X
a ∈ A pi
η µ
d ,a
= X
a ∈ A pi \C
η µ +1
d ,a + X
a ∈C
η µ +1
d ,a − X
a ∈ A pi \C
η µ
d ,a − X
a ∈C
η µ
d ,a
= X
a ∈C
η µ +1
d ,a − X
a ∈C
η µ
d ,a
= 2 X
a ∈C
η µ
d ,a − X
a ∈C
η µ
d ,a
= X
a ∈C
η µ
d ,a .

12 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
Th us, w e are finished if w e pro v e that
1
2
µ
X
j =1 X
a ∈R j
η j − 1
d ,a − X
a ∈C
η µ
d ,a
is p ositiv e and uniformly b ounded a w ay from zero. Using
η j − 1
d ,a ≥ η µ
d ,a , for all j = 1 , . . . , µ,
( 15 ), ( 17 ), and ( 19 ), w e obtain
1
2
µ
X
j =1 X
a ∈R j
η j − 1
d ,a ≥ 1
2 Θ d
µ
X
j =1 X
a ∈ A pi
η j − 1
d ,a ≥ 1
2 Θ d
µ
X
j =1 X
a ∈ A pi
η µ
d ,a
= 1
2 Θ d µ X
a ∈ A pi
η µ
d ,a > (Φ d + C ) X
a ∈ A pi
η µ
d ,a > X
a ∈C
η µ
d ,a + C | A pi | ε,
whic h completes the pro of. 
Next, w e pro ve an analogous lemma for the case that we fix the stepsize of every
arc a ∈ A pi and only allo w for mo del switching.
Lemma 2.
Supp ose that the discr etization stepsize
h a
is fixe d for every pip e
a ∈ A pi
.
Supp ose further that
η a
(
y
) =
η m ,a
(
y
) holds in
( 11 )
and that every NLP is solve d to
lo c al optimality. Consider A lg. 1 without applying the discr etizat ion r efinements in
Line 11 and the c o arsenings in Line 18 . Then, A lgorithm 1 terminates after a finite
numb er of mo del switches in Lines 10 and 17 with an
ε
-fe asible solution with r esp e ct
to the stepsizes h a , a ∈ A pi , if ther e exists a c onstant C > 0 such that
Θ k
m µ k > τ k Φ k
m | A pi | + C (20)
holds for al l k .
Pr o of. W e consider the total mo del error
η m ( y ) = X
a ∈ A pi
η m ,a ( y )
and sho w that the difference b etw een the decrease obtained in the inner lo op and
the increase obtained due to switc hing mo del levels do wn in Line 17 is p ositiv e and
uniformly b ounded a wa y from zero for ev ery iteration
k
. W e again consider only a
single iteration and drop the corresp onding index.
First, w e consider a single step of switc hing up the mo del lev el in Line 10 . Let
η j − 1
m ,a
denote the mo del error b efore the
j
th inner iteration and
η j
m ,a
the mo del error after
the j th inner iteration. W e then ha ve
X
a ∈ A pi
η j − 1
m ,a − X
a ∈ A pi
η j
m ,a
= X
a ∈U j
η j − 1
m ,a + X
a ∈ A pi \U j
η j − 1
m ,a − X
a ∈U j
η j
m ,a − X
a ∈ A pi \U j
η j
m ,a
= X
a ∈U j
η j − 1
m ,a − X
a ∈U j
η j
m ,a
for ev ery
j
= 1
, . . . , µ
. Summing up o v er all
j
yields the o v erall mo del error decrease
after µ for-lo op iterations of
µ
X
j =1 X
a ∈ A pi
η j − 1
m ,a − X
a ∈ A pi
η j
m ,a ! = X
a ∈ A pi
η 0
m ,a − X
a ∈ A pi
η µ
m ,a =
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) .

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 13
W e no w consider the step of switc hing down the model level in Line 17 . Let
η µ
m ,a
denote the mo del error b efore and
η µ +1
m ,a
the mo del error after this step. It holds
that
X
a ∈ A pi
η µ +1
m ,a − X
a ∈ A pi
η µ
m ,a
= X
a ∈D
η µ +1
m ,a + X
a ∈ A pi \D
η µ +1
m ,a − X
a ∈D
η µ
m ,a − X
a ∈ A pi \D
η µ
m ,a
= X
a ∈D
( η µ +1
m ,a − η µ
m ,a ) .
Th us, the pro of is finished if we sho w that
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) − X
a ∈D
( η µ +1
m ,a − η µ
m ,a )
is p ositiv e and uniformly b ounded a w ay from zero. With similar ideas as in the
pro of of Lemma 1 and using ( 16 ), ( 18 ), and ( 20 ), w e obtain
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) ≥ Θ m
µ
X
j =1 X
a ∈ A >ε
pi
( η j − 1
m ,a − η j
m ,a ) > Θ m
µ
X
j =1 X
a ∈ A >ε
pi
ε
= Θ m µ   A >ε
pi   ε ≥ Θ m µε > τ Φ m | A pi | ε + C ε ≥ Φ m X
a ∈ A <ε
pi ( τ )
ετ + C ε
≥ Φ m X
a ∈ A <ε
pi ( τ )
( η µ +1
m ,a − η µ
m ,a ) + C ε ≥ X
a ∈D
( η µ +1
m ,a − η µ
m ,a ) + C ε,
where w e used that | A >ε
pi | ≥ 1 . This completes the pro of. 
Let
η new
m ,a
(
y
) denote the new mo del error estimator after a grid refinemen t or
coarsening. In order to pro ve our main theorem w e need to assume that, for ev ery pip e
a ∈ A pi
, the change in the model error estimator after a grid refinement or coarsening
can b e neglected as compared to
η m ,a
(
y
) , i.e.,
| η m ,a
(
y
)
− η new
m ,a
(
y
)
|  η m ,a
(
y
) , suc h
that w e ma y write
η new
m ,a
(
y
) =
η m ,a
(
y
) . A sufficient condition for this assumption to
hold is giv en b y
η d ,a
(
y
)
 η m ,a
(
y
) for ev ery
a ∈ A pi
. This condition also implies
that
η m ,a
(
y
) is a first-order appro ximation of the exact mo del error
e m ,a
(
y
) and is
th us reliable for small stepsizes h a .
Lemma 3.
L et the discr etization and mo del err or estimator
η d ,a
(
y
) and
η m ,a
(
y
)
as define d in
( 9 )
and
( 10 )
b e given for every
a ∈ A pi
. L et further
e m ,a
(
y
) b e the
exact err or b etwe en mo dels (M
1
) and (M
` a
) and let
η new
m ,a
(
y
) b e the new mo del err or
estimator after a grid r efinement or c o arsening. Then, the implic ations
(1) η d ,a ( y )  η m ,a ( y ) = ⇒ η m ,a ( y ) .
= e m ,a ( y ) ,
(2) η d ,a ( y )  η m ,a ( y ) = ⇒ η new
m ,a ( y ) = η m ,a ( y )
hold for every a ∈ A pi .
Pr o of.
Let pip e
a ∈ A pi
b e arbitrary . T o impro ve readabilit y , in the following w e
drop the dep endencies of the exact errors and the error estimators on
a
and
y
.
Without loss of generalit y , w e consider only one arbitrary spatial gridp oint x k .
Let us first in tro duce some notation. The exact mo del error is giv en b y
e m
(
x k
) =
ˆ p
(
x k
)
− p M ` a
(
x k
) for the curren t mo del level
` a
, the exact discretization error for
mo del (D
1
) is giv en b y
e 1
d
(
x k
) =
ˆ p
(
x k
)
− p 1
(
x k
; 2
h a
) and the exact discretization
error for mo del (
D ` a
) is denoted b y
e ` a
d
(
x k
) =
p M ` a
(
x k
)
− p ` a
(
x k
; 2
h a
) . F urthermore,
the mo del error estimator is giv en by
η m
(
x k
) =
p 1
(
x k
; 2
h a
)
− p ` a
(
x k
; 2
h a
) , see
( 10 )
,
and w e define the discretization error estimators
η 1
d
(
x k
)
: = p 1
(
x k
; 2
h a
)
− p 1
(
x k
; 4
h a
)

14 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
and
η ` a
d
(
x k
)
: = p ` a
(
x k
; 2
h a
)
− p ` a
(
x k
; 4
h a
) as in
( 9 )
. Then, we ha v e
η 1
d
(
x k
)
.
= e 1
d
(
x k
)
and η ` a
d ( x k ) .
= e ` a
d ( x k ) ; see [ 36 , page 420]. F urther, it holds that
  η 1
d ( x k )    | η m ( x k ) | ⇐ ⇒ | η ` a
d ( x k ) || η m ( x k ) | , (21)
b ecause
η 1
d
(
x k
) and
η ` a
d
(
x k
) use the same stepsizes 2
h a
and 4
h a
to compute the
discrete pressure distributions.
W e no w pro ve implication
( 1 )
. Using the previously defined notation it holds
that
e m ( x k ) = ˆ p ( x k ) − p M ` a ( x k )
= e 1
d ( x k ) + p 1 ( x k ; 2 h a ) − e ` a
d ( x k ) − p ` a ( x k ; 2 h a )
.
= η 1
d ( x k ) + p 1 ( x k ; 2 h a ) − η ` a
d ( x k ) − p ` a ( x k ; 2 h a )
= η 1
d ( x k ) − η ` a
d ( x k ) + η m ( x k ) .
Th us, if
| η 1
d
(
x k
)
|
and
| η ` a
d
(
x k
)
|
ma y b e neglected as compared to
| η m
(
x k
)
|
, then w e
ha v e e m ( x k ) .
= η m ( x k ) , i.e.,
  η 1
d ( x k )    | η m ( x k ) |∧| η ` a
d ( x k ) || η m ( x k ) | = ⇒ e m ( x k ) .
= η m ( x k ) .
Considering also the equiv alence relation ( 21 ) it follo ws that
  η 1
d ( x k )    | η m ( x k ) | = ⇒ e m ( x k ) .
= η m ( x k ) ,
from whic h implication ( 1 ) follo ws directly .
Finally , w e prov e implication
( 2 )
. W e sho w that this implication holds for the
case that
η new
m
(
x k
) is the new mo del error estimator after a grid coarsening. The
case for a grid refinemen t can b e shown analogously . It holds that
η new
m ( x k ) = p 1 ( x k ; 4 h a ) − p ` a ( x k ; 4 h a )
= − η 1
d ( x k ) + p 1 ( x k ; 2 h a ) + η ` a
d ( x k ) − p ` a ( x k ; 2 h a )
= − η 1
d ( x k ) + η ` a
d ( x k ) + η m ( x k ) .
This yields
  η 1
d ( x k )    | η m ( x k ) |∧| η ` a
d ( x k ) || η m ( x k ) | = ⇒ η new
m ( x k ) = η m ( x k ) . (22)
Again, considering ( 21 ) and ( 22 ) results in
  η 1
d ( x k )    | η m ( x k ) | = ⇒ η new
m ( x k ) = η m ( x k ) ,
from whic h implication ( 2 ) follo ws immediately . 
With the three preceding lemmas at hand, w e are no w ready to state and prov e
our main theorem ab out finite termination of Alg. 1 .
Theorem 1
(Finite termination)
.
Supp ose that
η d ,a  η m ,a
for every
a ∈ A pi
and
that every NLP is solve d to lo c al optimality. Then, A lgorithm 1 terminates after a
finite numb er of r efinements, c o arsenings and mo del switches in Lines 10 , 11 , 17 ,
and 18 with an ε -fe asible solution with r esp e ct to the r efer enc e pr oblem ( 7 ) if ther e
exist c onstants C 1 , C 2 > 0 such that
1
2 Θ k
d µ k > Φ k
d + C 1 , Θ k
m µ k > τ k Φ k
m | A pi | + C 2
hold for al l k .
Pr o of.
W e consider the total error
P a ∈ A pi η a
and sho w that the difference b etw een
the decrease obtained in the inner lo op and the increase obtained due to switc hing
do wn the mo del level and coarsening the grid is p ositiv e and uniformly b ounded
a w ay from zero for ev ery iteration
k
. Again, w e consider only a single iteration and

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 15
drop the corresp onding index. W e first consider Lines 10 and 11 for fixed
j
. It holds
that
X
a ∈ A pi
η j − 1
a − X
a ∈ A pi
η j
a
= X
a ∈ A pi
η j − 1
m ,a + X
a ∈ A pi
η j − 1
d ,a − X
a ∈ A pi
η j
m ,a − X
a ∈ A pi
η j
d ,a
= X
a ∈ A pi \ ( U j ∪R j )
η j − 1
m ,a − X
a ∈ A pi \ ( U j ∪R j )
η j
m ,a + X
a ∈U j \R j
η j − 1
m ,a − X
a ∈U j \R j
η j
m ,a
+ X
a ∈R j \U j
η j − 1
m ,a − X
a ∈R j \U j
η j
m ,a + X
a ∈R j ∩ U j
η j − 1
m ,a − X
a ∈R j ∩ U j
η j
m ,a
+ X
a ∈ A pi \ ( U j ∪R j )
η j − 1
d ,a − X
a ∈ A pi \ ( U j ∪R j )
η j
d ,a + X
a ∈U j \R j
η j − 1
d ,a − X
a ∈U j \R j
η j
d ,a
+ X
a ∈R j \U j
η j − 1
d ,a − X
a ∈R j \U j
η j
d ,a + X
a ∈R j ∩ U j
η j − 1
d ,a − X
a ∈R j ∩ U j
η j
d ,a
= X
a ∈U j
η j − 1
m ,a − X
a ∈U j
η j
m ,a + X
a ∈R j
η j − 1
d ,a − X
a ∈R j
η j
d ,a
= X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) + 1
2 X
a ∈R j
η j − 1
d ,a ,
where w e use that
η j
m ,a
=
η j − 1
m ,a
for ev ery
a ∈ R j \ U j
since
η j − 1
d ,a  η j − 1
m ,a
for ev ery
a ∈ A pi
; see Lemma 3 . Moreo v er, the discretization error estimator
η d ,a
do es not
c hange after a switc hing up the mo del lev el.
Again, summing up o v er all
j
= 1
, . . . , µ
yields the o v erall error decrease after
µ
for-lo op iterations of
µ
X
j =1 X
a ∈ A pi
η j − 1
a − X
a ∈ A pi
η j
a ! = X
a ∈ A pi
η 0
a − X
a ∈ A pi
η µ
a
=
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) + 1
2 X
a ∈R j
η j − 1
d ,a ! .
With similar argumen ts as b efore for Lines 10 and 11 w e consider Lines 17 and 18
and obtain
X
a ∈ A pi
η µ +1
a − X
a ∈ A pi
η µ
a
= X
a ∈ A pi
η µ +1
d ,a + X
a ∈ A pi
η µ +1
m ,a − X
a ∈ A pi
η µ
d ,a − X
a ∈ A pi
η µ
m ,a
= X
a ∈C
η µ +1
d ,a − X
a ∈C
η µ
d ,a + X
a ∈D
η µ +1
m ,a − X
a ∈D
η µ
m ,a
= X
a ∈C
η µ
d ,a + X
a ∈D
( η µ +1
m ,a − η µ
m ,a ) .
Finally , it remains to pro v e that
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) + 1
2 X
a ∈R j
η j − 1
d ,a ! − X
a ∈C
η µ
d ,a − X
a ∈D
( η µ +1
m ,a − η µ
m ,a )

16 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
is p ositiv e and uniformly b ounded a w ay from zero. Using the pro ofs of Lemmas 1
and 2 w e ha ve
µ
X
j =1 X
a ∈U j
( η j − 1
m ,a − η j
m ,a ) + 1
2
µ
X
j =1 X
a ∈R j
η j − 1
d ,a
> Θ m µε + 1
2 µ Θ d X
a ∈ A pi
η µ
d ,a
> τ Φ m | A pi | ε + C 2 ε + (Φ d + C 1 ) X
a ∈ A pi
η µ
d ,a
> X
a ∈D
( η µ +1
m ,a − η µ
m ,a ) + C 2 ε + X
a ∈C
η µ
d ,a + C 1 | A pi | ε,
whic h completes the pro of. 
4.4.
Remarks.
Before w e close this section w e dis cuss some details and extensions
regarding Alg. 1 . First, w e giv e an o v erview of the main computations that are
p erformed in the algorithm. In Lines 2 and 12 , the NLP
( 8 )
is solv ed using the
curren t mo del level
` a
and the curren t stepsize
h a
for ev ery pip e
a ∈ A pi
. Most
t yp es of NLP algorithms are iterative methods. That is, the c omputational costs
of the algorithms dep end on the n um b er of iterations required to con verge to a
(lo cal) optimal solution and the costs p er iteration. The latter mainly consist of the
solution of a linear system (e.g., suitable forms of the KKT system for in terior-p oint
or activ e-set metho ds) for computing the search direction. The size of this linear
system t ypically is
O
(
n
+
m
) , where
n
is the n um b er of v ariables and
m
is the
n um b er of constrain ts of the NLP . Both
n
and
m
are directly con trolled b y the
stepsizes
h a
that w e use in our NLP mo dels. The mo del lev el
` a
mainly determines
the sparsit y/densit y of the system matrices of the linear systems and the o verall
nonlinearit y of the NLP , whic h typically influences the n um b er of required iterations.
In Lines 3 and 13 , the o v erall error estimator
η a
(
y
) is computed for ev ery pip e
a ∈ A pi
. Th us, for all pip es, the solution of mo del (
D 1
) is computed with stepsize
b oth 2
h a
and 4
h a
and the solution of mo del (
D ` a
) is computed with stepsize
h a
.
These solutions are obtained b y solving the initial v alue problems consisting of the
ordinary differen tial equations (
M 1
) and (
M ` a
) together with the initial v alue
p
(
x 0
) ,
whic h is con tained in the optimal solution
y
of Problem
( 8 )
. Con tin uing with the
example of the implicit Euler metho d that w e use as numerical in tegration sc heme
throughout this pap er, the initial v alue problems can b e solv ed (i) b y considering
the implicit equations in (
D 1
) and (
D ` a
) and using, e.g., the Newton metho d to
solv e for
p k
in ev ery space in tegration s tep or (ii) b y using an existing soft ware code
and setting the order of the n umerical in tegration sc heme to one.
The subset
R
in Line 9 can b e determined efficien tly , since
η d ,a
(
y
) has already
b een computed in Line 3 or 13 for ev ery
a ∈ A pi
. F or subset
U
in Line 9 and in
( 16 )
the error estimator
η m ,a
(
y
) has also already b een computed in Line 3 or 13 for ev ery
a ∈ A pi
. Moreo v er,
` new
a
in
( 12 )
has to b e computed in order to determine
U
. F or
this, w e compute
η m ,a
(
y
;
` a −
1) if and only if
` a
= 3 . In the case
` a
= 2 w e hav e
η m ,a
(
y
;
` a −
1) = 0 and for
` a
= 1 w e hav e
η m ,a
(
y
;
` new
a
) =
η m ,a
(
y
;
` a
) = 0 . Subset
C
in Line 16 can also b e computed efficien tly , since
η d ,a
(
y
) has already b een computed
in Line 3 or 13 for ev ery
a ∈ A pi
. F or subset
D
in Line 16 and in
( 18 )
the error
estimator
η m ,a
(
y
) has b een computed already in Line 3 or 13 for ev ery
a ∈ A pi
. If
` a ∈ {
1
,
2
}
, then
η m ,a
(
y
;
` a
+ 1) has to b e computed for ev ery
a ∈ A pi
in order to
determine D .
W e note that the optimal solution
y
of Problem
( 8 )
con tains, among others,
the mo del lev el
` a
, stepsize
h a
, and pressure
p ` a
(
x 0
) at the b eginning of the pip e,

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 17
for ev ery
a ∈ A pi
. Using
` a
,
h a
, and
p ` a
(
x 0
) , the discretization and mo del error
estimator for pip e a ∈ A pi can b e computed without information from other pip es.
Hence, the error estimators, e.g., in Line 13 , can b e computed in parallel.
Up to no w, w e ha ve discussed t wo t yp es of errors: mo deling and discretization
errors. Both are handled b y Alg. 1 and w e hav e shown that the algorithm terminates
with a com bined mo del and discretization error that satisfies a user-sp ecified error
tolerance
ε >
0 . What w e ha v e ignored so far is that the NLPs are also solv ed b y
a n umerical metho d that introduces numerical errors as w ell. Ho wev er, it is easy
to in tegrate the con trol of this additional error source into Alg. 1 . Let
ε opt >
0 b e
the optimalit y tolerance that w e hand o v er to the optimization solver and suppose
that the solv er alw a ys satisfies this tolerance. F urthermore, let the tolerance
ε
considered so far no w b e denoted by
ε dm
. Using the triangle inequalit y we easily see
that the upp er b ound of the total error (that is aggregated mo deling, discretization,
and optimization error) is
ε opt
+
ε dm
. Hence, in order to satisfy an o v erall error
tolerance
ε >
0 , w e ha ve to ensure that
ε opt
+
ε dm ≤ ε
holds, whic h can b e formally
in tro duced in Alg. 1 by replacing ε with ε opt + ε dm .
Finally , note that this additional error source directly suggests itself for adaptiv e
treatmen t as w ell. In the early iterations of Alg. 1 it is not imp ortan t that
ε opt
is
small. That is, the optimization is allo w ed to pro duce coarser approximate local
solutions. Ho w ever, in the course of the algorithm, one can observ e the ac hieved
mo deling and discretization error and can adaptiv ely tigh ten the optimization
tolerance. Since this strategy allo ws the optimization metho d to pro duce coarse
appro ximate solutions in the b eginning, it can b e exp ected that this leads to a
sp eed-up in the o verall running times of Alg. 1 .
The c hoice of the error tolerance
ε
that has to b e pro vided in Alg. 1 will dep end
on the user requiremen ts, ho wev er, one should b e a w are that due to the round-off
errors committed during ev ery single step of the pro cedure, and due to p ossible
ill-conditioning of the linear systems solv ed b y the NLP solv er, none of the three
errors, the discretization error, the mo deling error, and the NLP error can b e c hosen
extremely small. Since the bac kward error and the asso ciated condition n um b er of
the linear systems can b e estimated during the pro cedure, see [
17
], and since the
error estimates for the discretization metho d are at hand, it is just the mo deling
error whic h is not kno wn a priori. T o estimate this latter error (of the finest m o del)
usually requires a comparison with exp erimen tal data. If these are a v ailable during
a real-w orld pro cess, then it is p ossible to adjust the required tolerances
ε
in a
feedbac k lo op using a standard PI con troller, see, e.g., [
30
], i.e., if measured data
are a v ailable that sho w that the finest mo del has a giv en accuracy , then
ε
should
not b e c hosen smaller than this.
Finally , w e w ant to stress that the described adaptiv e error con trol algorithm can
b e used with an y num b er of mo del lev els in the hierarc h y , with any higher order
discretization sc heme, and with an y n um b er of grid refinemen t levels.
5. Comput a tional Resul ts
In this section w e presen t numerical results obtained b y the adaptiv e error con trol
algorithm. T o this end, w e compare the efficiency of the metho d with an approac h
that directly solv es an NLP that satisfies the same error tolerance and that is
obtained without using adaptivit y . Before w e discuss the results in detail w e brief ly
men tion the computational setup and the gas transp ort net w ork instances that we
solv e.

18 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
T able 1. Statistics for the instances
Net w ork # no des # pip es # compressor stations total pip e length (km)
GasLib-40 40 39 6 1112
GasLib-135 135 141 29 6935
W e implemen ted the adaptiv e error con trol algorithm 1 in Python 2.7.13 and
used the scip y 0.14.0 mo dule for solving the initial v alue problems. All nonlinear op-
timization mo dels ha ve been implemented using the C++ framew ork LaMaTTO++
1
for mo deling and solving mixed-in teger nonlinear optimization problems on net-
w orks. The computations ha v e b een done on a six-core AMD Opteron
TM
Pro cessor
2435 with
2 . 2 GHz
and
64 GB
of main memory . The NLPs ha v e b een solv ed using
Ip opt 3.12; see [ 39 , 40 ].
F or our computational study , w e c ho ose publicly av ailable GasLib instances;
see [
33
]. This has the adv antage that, if desired, all numerical results can be
repro duced on the same data. In what follo ws, w e consider the netw orks GasLib-40
and GasLib-135 , since these are the largest net w orks in the GasLib that only contain
pip es and compressor stations as arc t yp es. Detailed statistics are giv en in T able 1 .
Next, w e describ e the parameterization of Alg. 1 . W e initialize ev ery pip e
a ∈ A pi
with the coarsest mo del lev el
` a
= 3 and with the coarsest p ossible discretization
grid. In order to yield a w ell-defined algorithm, the n umber of discretization grid
in terv als has to b e a multiple of 4 ; see Fig. 2 . Th us, w e initially set
h a
=
L a /
4
and ensure in Step 18 of Alg. 1 that w e never obtain a coarser grid size than
the initial one. The ov erall tolerance is set to
ε
= 10
− 4 bar
. Moreov er, we set
Θ
d
= Θ
m
= 0
.
7 , Φ
d
= Φ
m
= 0
.
3 ,
τ
= 1
.
1 , and
µ
= 4 . Here, w e refrain from
up dating these parameters from iteration to iteration, whic h is p ossible in general.
Note that our parameter c hoice violates the second inequalit y of Theorem 1 . This
could b e fixed b y simply increasing the hysteresis parameter
µ
. How ev er, w e refrain
from using a larger
µ
in order to giv e the adaptiv e algorithm more c hances to also
switc h do wn in the mo del hierarc h y or to coarsen discretization grids. Our numerical
exp erimen ts show that the violation of the second inequalit y of Theorem 1 do es not
harm con v ergence in practice but leads to sligh tly faster computations.
The same rationale holds for the relation b et ween model and discretization error
as assumed in Theorem 1 ; see also Lemma 3 . T o b e fully compliant with the
theory , the initial discretization grids need to b e m uc h finer. Again, coarser initial
discretization grids do not harm con v ergence in our numerical experiments but yield
m uc h faster computations.
W e no w turn to the discussion of the n umerical results. Both instances are solv ed
using 8 iterations. Th us, together with the initially solv ed NLP , w e ha v e to solv e
9 NLPs for solving b oth instances.
Using the adaptiv e con trol algorithm, it tak es
3 . 82 s
to solv e the GasLib-40 instance
and
7 . 50 s
to solv e the GasLib-135 instance. F or the GasLib-40 netw ork, the final
NLP con tains
2026
v ariables and
1988
constrain ts, whereas for the GasLib-135 the
final NLP con tains 3405 v ariables and 3271 constrain ts.
Most in teresting is the sp eed-up that we obtain b y using the adaptiv e con trol
algorithm. Th us, w e compare the ab o v e given solution times with the solution times
for an NLP that satisfies the same error tolerances but that is obtained without using
mo del lev el and discretization grid adaptivity . This NLP contains
40 034
v ariables
and
39 996
constrain ts for the GasLib-40 instance and
144 757
v ariables as w ell as
144 623
constrain ts for the GasLib-135 instance. Compared to the final NLPs that
1 http://www.mso.math.fau.de/edom/projects/lamatto.html

MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 19
02468
0
2
4
6
8
10
Iteration
|R|
|U |
02468
0
5
10
15
20
25
Iteration
|R|
|U |
Figure 4. Num b er of pip es with refined grid (
y
-axis;
|R|
) and
n um b er of pip es where the mo del is switched up in the model
hierarc h y (
y
-axis;
|U |
) o v er the course of the iterations (
x
-axis).
Left: GasLib-40 , righ t: GasLib-135 .
ha v e to b e solv ed within the adaptiv e algorithm, the NLPs obtained without using
adaptivit y are quite large scale. This directly translates to solution times. The
GasLib-40 instance requires
53 . 11 s
and the GasLib-135 instance requires
122 . 42 s
.
Th us, w e get a sp eed-up factor of 13.89 and 16.33, resp ectiv ely .
Figure 4 illustrates the adaptivit y of the algorithm b y plotting how man y pip e
grids are refined (
|R|
) and ho w man y pip e mo dels are switc hed up in the hierarc h y
(
|U |
). It can b e clearly seen that increasing the accuracy is only needed for a small
fraction of the pip es. F or the GasLib-40 net w ork, we nev er refine grids for more than
9 pip es, whereas w e nev er refine grids for more than 21 pip es for the GasLib-135
net w ork. Th us, for the larger netw ork, we nev er refine grids for more than 15 % of
all pip es.
F or b oth net works, the Lines 17 and 18 are only reac hed once. F or the smaller
net w ork, only 1 pip e grid is coarsened, whereas 3 pip e grids are coarsened for the
larger net w ork. M oreo v er, the algorithm nev er switches do wn in the mo del hie rarc h y .
Consequen tly , the NLPs get larger from iteration to iteration. This then yields
increased running times for the NLP solv er as depicted in Fig. 5 . It can b e seen
that the subsequen t NLPs can b e solved quite fast. There are t w o main reasons
for this phenomenon. First, the NLP’s size only increases mo derately due to the
adaptiv e con trol strategy . Second, the o v erall algorithm allows for w arm-starting:
When solving a single NLP w e alw a ys use the last NLP’s solution to set up the
initial iterate.
Lastly , w e consider the decrease in the resp ective errors. In Fig. 6 , the discretiza-
tion, mo del, and total errors are plotted ov er the course of the iterations. Both
profiles sho w the exp ected decrease in the errors.
6. Conclusion
W e ha v e considered the problem of op eration cost minimization for gas transp ort
net w orks. In this con text, w e ha v e fo cused on stationary and isothermal mo dels
and dev elop ed an adaptive model and discretization error control algorithm for
nonlinear optimization that uses a hierarc h y of con tin uous and finite-dimensional
mo dels. Out of this hierarc h y , the new metho d adaptiv ely c ho oses different models
in order to finally ac hiev e an optimal solution that satisfies a prescrib ed com bined

20 V. MEHRMANN, M. SCHMIDT, J. J. STOL WIJK
02468
10 − 1
10 0
10 1
Iteration
CPU time [s]
NLPs
IVPs
02468
10 − 1
10 0
10 1
Iteration
NLPs
IVPs
Figure 5. Aggregated run times (
y
-axis; in s) required for solving
the nonlinear optimization problems (NLPs) and the initial v alue
problems (IVPs) for the computation of the error estimates. Left:
GasLib-40 , righ t: GasLib-135 .
02468
10 − 5
10 − 4
10 − 3
10 − 2
Iteration
Error estimate
η d
η m
η
02468
10 − 5
10 − 4
10 − 3
10 − 2
Iteration
η d
η m
η
Figure 6. Discretization, mo del and total error estimates (
y
-axis)
o v er the course of the iterations (
x
-axis). Left: GasLib-40 , righ t:
GasLib-135 .
mo del and discretization error tolerance. The algorithm is sho wn to b e con vergen t
and its p erformance is illustrated b y several n umerical results.
The results pa v e the w a y for future work in the con text of mo del switc hing and
discretization grid adaptation for nonlinear optimal con trol. On the one hand, it
should b e extended to non-isothermal and instationary mo dels of gas transp ort,
in particular, in a p ort-Hamiltonian form ulation. On the other hand, it would be
in teresting to extend the new tec hnique to mixed-integer nonlinear optimal con trol.
A ckno wledgements
This researc h has b een p e rformed as part of the Energie Campus Nürn b erg and is
supp orted b y funding of the Bav arian State Gov ernmen t. The authors ackno wledge
funding through the DF G T ransregio TRR 154, subpro jects B03 and B08.
References
[1]
R. Bec ker, H. Kapp, and R. Rannac her. A daptive finite elemen t metho ds for optimal con trol
of partial differen tial equations: Basic concept. SIAM Journal on Contr ol and Optimization ,
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MODEL AND DISCRETIZA TION ERR OR AD APTIVITY WITHIN GAS OPTIMIZA TION 21
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1
Institut für Ma thema tik, MA 4-5, TU Berlin, Straße des 17. Juni 136, 10623
Berlin, Germany ({mehrmann,stol wijk}@ma th.tu-berlin.de);
2
Friedrich-Alexander-
Universit ä t Erlangen-Nürnberg, Discrete Optimiza tion, Ca uerstr. 11, 91058 Erlangen,
Germany; 3 Ener gie Campus Nürnberg, Für ther Str. 250, 90429 Nürnber g, Germany

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