MYNTS: Multi-phYsics NeTwork Simulator

Tanja Clees, Kl

¨

are Cassirer, Nils Hornung, Bernhard Klaassen, Igor Nikitin, Lialia Nikitina,

Robin Suter and Inna Torgovitskaia

Fraunhofer Institute for Algorithms and Scientiﬁc Computing, Schloss Birlinghoven, 53754, Sankt Augustin, Germany

Keywords:

Simulation Tools and Platforms, Complex Systems Modeling and Simulation, Optimization of Non-Linear

Systems, Simulation of Transport Networks.

Abstract:

We present a generic approach for the simulation of transport networks, where the steps of physical modeling

and numerical simulation are effectively separated. The model is described by a list of physical equations and

inequalities as problem constraints for non-linear programming (NLP). This list is translated to the language of

expression trees and is made accessible for the numerical solution by standard NLP solvers. Various problem

types can be solved in this way, including stationary and transient network simulation, feasibility analysis and

energy-saving optimization. The simulation is provided for different disciplines, such as gas transport, water

supply and electric power networks. We demonstrate the implementation of this approach in our multiphysics

network simulator.

1 INTRODUCTION

The simulation of transport networks in civil engi-

neering becomes increasingly important for the plan-

ning and stable operation of modern infrastructure.

Many software systems exist for the solution of spe-

cialized problems in the simulation of gas transport

networks (Scheibe and Weimann, 1999; Aymanns

et al., 2008), water supply (Rogalla and Wolters,

1994; Stevanovic et al., 2009), electric power trans-

mission (Milano, 2015; Zimmerman and Murillo-

Sanchez, 2015), etc. Our purpose is to develop a uni-

versal system capable of solving all problems of this

kind, as well as of providing for them an energetically

efﬁcient optimal solution. The system has to be max-

imally open in terms of mathematical modeling and

numerical solver, allowing the user to extend the sys-

tem by new element types and new conditions and to

upgrade it by modern highly efﬁcient algorithms.

The possibility of such a universal solution fol-

lows from the great similarity of the considered prob-

lems. Indeed, different network disciplines represent

the network topology as a graph on which problems

of the following types are formulated:

• stationary problems – systems of non-linear equa-

tions f (x) = 0;

• transient problems – differential-algebraic equa-

tions, DAEs, f (x, ˙x) = 0;

• feasibility problems – ﬁnd (any) solution of the

system of equations and inequalities f (x) = 0,

g(x) ≥ 0;

• constrained optimization – ﬁnd min h(x) such that

f (x) = 0, g(x) ≥ 0.

Note that constrained optimization is the most general

formulation, while the three others represent its par-

ticular cases: stationary problems – in the absence of

the inequalities g(x) ≥ 0 and with a trivial objective

function h(x) one has to solve only non-linear sys-

tems f (x) = 0; the sequential solution of non-linear

systems represents a process of (semi-)implicit inte-

gration of a DAE; feasibility problems also belong to

constrained optimization, where one can select an ar-

bitrary objective function.

In its most general form constrained optimization

is also known as non-linear programming, NLP, for

which, since the 1970s, many efﬁcient algorithms

have been developed (Bazaraa and Shetty, 1979; Bert-

sekas, 1999; Avriel, 2003; Fletcher, 2013) and imple-

mented in a number of software packages (Murtagh

and Saunders, 1978; Gill et al., 2005; W

¨

achter and

Biegler, 2006; Nocedal and Wright, 2006; Belotti

et al., 2013). Nowadays the solution of the above

listed problems is as easy as the solution of a linear

sparse system, one only needs to represent the prob-

lem in a standard form, accepted by the correspond-

ing software package. Continuing this analogy, the

solution of a linear sparse system requires to spec-

ify the system matrix in a triplet, a column or a row

Clees, T., Cassirer, K., Hornung, N., Klaassen, B., Nikitin, I., Nikitina, L., Suter, R. and Torgovitskaia, I.

MYNTS: Multi-phYsics NeTwork Simulator.

DOI: 10.5220/0005961001790186

In Proceedings of the 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2016), pages 179-186

ISBN: 978-989-758-199-1

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

179

Figure 1: Gas transport network simulation in Mynts.

compressed form, while NLP solvers accept systems

of non-linear equations in the form of so-called ex-

pression trees (Gay, 2005). This de facto standard

is supported by many existing NLP solvers, while

newly developed algorithms can be easily equipped

with the necessary interface. In this way the mathe-

matical modeling becomes effectively separated from

the solution algorithms, providing the required ﬂexi-

bility and modularity of the software.

The same transport network problem can have var-

ious representations: a human-readable description of

the network (e.g. in NetList format, used in prac-

tice for solution of real-life problems for gas, water

and electric power transport (Baumanns et al., 2012;

Clees, 2012; Clees, 2016; Clees et al., 2016a; Clees

et al., 2016b)), formats understandable only by sim-

ulation kernels (e.g. NL (Gay, 2005)), there is also

an intermediate level of modeling languages (such

as AMPL (Fourer et al., 2002), Modelica (Modelica,

2012)). Well designed simulator should be able to

translate the problem between its different represen-

tations and do it fast for large realistic problems. In

this paper we describe the algorithms suitable for this

purpose.

Related work: in (Clees, 2012) we present gen-

eral concepts common for all types of transport net-

work problems, in (Clees, 2016) we describe the gas

part of our simulator applied for parameter studies in

gas transport networks, in (Clees et al., 2016a; Clees

et al., 2016b) the detailed modeling of water cool-

ing networks is given. Our advancement reported in

the present paper is the development of general algo-

rithms for formulation and solution of transport net-

work problems based on their representation in NLP

form. In the ﬁrst place, we have the task to translate

between different problem encodings, i.e., between a

human-readable description of the transport network

problem and the language understood by NLP solvers.

We tackle these aspects in Section 2. Different ap-

proaches for solving NLPs are discussed in Section 3.

Our system Mynts with its main components and ex-

amples of usage is presented in Section 4.

2 PROBLEM FORMULATION

Network problems can be presented as a list of ele-

ments (nodes, edges), containing a topological deﬁ-

nition (in the form of an incidence matrix, i.e., every

edge should refer to the two nodes it connects to) and

a set of properties used for problem formulation (e.g.,

node coordinates, pipe length, diameter, etc). For in-

stance, a gas transport problem can have the following

form.

SIMULTECH 2016 - 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

180

Network base description:

# Nodes

n, name="n56", H=5.2, X=14400, Y=10800

n, name="n99", H=0, X=18471.821, Y=10670.833

....

# Pipes

p, name="p11", node1="n93", node2="n94",

L=7900.0, D=0.5, k=0.02

p, name="p12", node1="n94", node2="n52",

L=4700.0, D=0.5, k=0.02

....

# Compressors

c, name="CS1|ME1|c0", node1="CS1|ME1|n2",

node2="CS1|ME1|n3"

c, name="CS1|ME2|c0", node1="CS1|ME2|n4",

node2="CS1|ME2|n5"

....

For user convenience, this description can be divided

into sections where the properties for a class of prob-

lems are listed (base) and where problem variations

are set (scenario). They can also be stored in separate

ﬁles.

Network scenario description:

name="n56", PSET=50, gasmix="gas1"

name="n99", PSET=60, gasmix="gas2"

name="n76", QSET=300

name="n91", QSET=1000

name="CS1|ME1|c0", MODE="PO", SPO=80,

model="advanced", cchar="T-SCAI0",

echar="P-SCAI0"

name="CS1|ME2|c0", MODE="M", SM=500,

model="advanced", cchar="T-SCAI1",

echar="P-SCAI1"

A scenario for a gas transport problem typically con-

tains pressure settings in supply nodes (PSET), out-

ﬂow settings in exit nodes (QSET), references to

gas composition (gasmix), compressor mode settings,

such as output pressure (SPO) and output ﬂow (SM),

a model of operation (advanced or free), a reference

to characteristics of compressor (cchar) and its driv-

ing engine (echar), etc. Any properties can be added,

they should only make sense in the context of the con-

sidered discipline.

The discipline description is a list of translation

rules, which allow to compose variables, equations

and inequalities given per element.

Network discipline description:

# Variables

for every node:

var P,rho,T,...

# pressure, density, temperature

for every edge:

var Q... # flow

for every compressor:

var Rev,Perf,Had...

# revolutions per minute, power,

# adiabatic enthalpy difference

.....

# Equations

for every node:

eq gas_law(rho,P,T, gasmix,...)=0

for every pipe:

eq pipe_law(P@node1,P@node2,Q, L,D,...)=0

.....

Here the section of variables extends the list of prop-

erties per given element by new properties whose

values will be computed in the network simulation.

Equation prototypes combine the properties with ex-

pressions representing the underlying physical laws.

Gas transport simulation can use several types of laws

for gas compressibility, pressure drop in pipes and

compressor working conditions, in more detail con-

sidered in Section 4. They can also be extended by

the user, by specifying a formula per element or a

class of elements. Let us demonstrate this by deﬁn-

ing a quadratic pipe law (Mischner et al., 2011) by

the equation:

pow(P@node1,2)-pow(P@node2,2)-R*Q*abs(Q)=0

A translation utility represents this equation in the

form of an expression tree:

then, using the correspondence

P@node1 P@node2 Q R 2

v1 v2 v3 n0.1 n2

pow abs * -

o5 o15 o2 o1

the tree is traversed by the left-hand rule and encoded

in Polish preﬁx notation (PPN):

o1 o1 o5 v1 n2 o5 v2 n2 o2 o2 v3 o15 v3 n0.1

In the obtained representation of the expression tree

the leaves are variables (v...) and constants (n...),

while the branches are operators (o...). A represen-

tation in the form of an expression tree makes equa-

tions suitable for automatic differentiation using the

chain rule. A detailed description of the encoding

procedure can be found in (Gay, 2005). The interface

MYNTS: Multi-phYsics NeTwork Simulator

181

supports more than 70 basis operators and a possibil-

ity to extend them by arbitrary user deﬁned functions.

PPN encoded expression trees are the language which

many NLP solvers understand.

In terms of general theory (Harrison, 1978) all

these representations belong to formal languages, be-

ing sets of strings of symbols, constrained by speciﬁc

rules. The network description and the lists of equa-

tions in human-readable or NLP format deﬁne differ-

ent formal languages, while the description of the dis-

cipline provides a means for translation, in a way:

discipline ∗ network description = NLP.

Algorithmically, this can be done as follows.

Algorithm 1: Problem translation.

read network graph (base and scenario);

read vars and eqs prototypes (discipline);

translate eqs prototypes to NLP format

with placeholders for vars and consts;

loop over element:

create vars per element according

to the discipline description;

place them into the graph as sequentially

numbered ids (v1...vn);

loop over element:

create eqs per element according to

the discipline description;

substitute actual vars and consts

to the placeholders;

output the eqs.

Since the main part of the algorithm is a straightfor-

ward substitution (stamping), it requires O(N + E)

operations, where N and E are the numbers of nodes

and edges in the network graph, respectively.

3 PROBLEM SOLUTION

In the ﬁeld of applied optimization the problems are

often subdivided into two parts: simulation and opti-

mization. Simulation solves the part of the problem

deﬁned by equations f (x, y) = 0 with respect to some

part of the variables y, while the remaining variables

x are considered as constant parameters. Then opti-

mization is applied to the objective function h(x, y),

with respect to the variables x, while y is permanently

updated for the given x using the simulation. Algo-

rithmically this can be expressed as follows.

Algorithm 2: Optimization on top of the simulation.

define y(x):

solve f(x,y) w.r.t. y;

return y;

optimize h(x,y(x)) w.r.t. x.

An alternative is to use NLP algorithms which solve

the optimization and simulation parts simultaneously.

Algorithm 3: NLP.

optimize h(x,y) w.r.t. (x,y) s.t. f(x,y)=0.

The ﬁrst variant is often used if the simulation and/or

the optimization are considered as blackbox algo-

rithms, e.g., if simulation y(x) is encapsulated in a

proprietary software tool. In this case one is re-

duced to applying the ﬁrst algorithm indeed. The

second variant composes a uniﬁed system of equa-

tions f (x, y) = 0 and optimality conditions for h(x, y),

which are solved simultaneously by means of super-

convergent methods (usually a stabilized Newton’s

method). The ﬁrst algorithm ﬁnds, for every value

of x, an exact solution of the system f (x, y) = 0. This

is completely superﬂuous for intermediate optimiza-

tion steps when x is only an approximation to the

optimum. For the second algorithm the precision of

the solution for the equations and the precision of the

optimality conditions are improved simultaneously in

the solution process, so that a precise solution of the

simulation problem is obtained only at the last step,

being also the optimal one. For the ﬁrst algorithm

the required computational effort can be estimated as

O(Nsim Nopt), where Nsim is the number of itera-

tions per simulation and Nopt is the number of op-

timization steps. The second algorithm solves a uni-

ﬁed system, which is almost entirely composed of the

equations from the simulation part, so that the over-

head related to the optimality conditions is usually

small and the overall effort becomes O(Nsim). Thus,

NLP algorithms deﬁnitely win in performance and,

therefore, they are our preferred choice.

The possibility to represent gas network simu-

lations in NLP form has been earlier discussed in

(Schmidt et al., 2015a), where the mathematical mod-

eling of gas transport networks has been considered in

great detail. However, the implementation (Schmidt

et al., 2015b) used the simplest possible NLP ver-

sion, a penalty method, taking a cumulative norm of

the residuals of the physical equations as single ob-

jective function. This approach admits a violation of

the physical constraints (gas laws, etc). Although the

global minimum of this objective coincides with the

solution for feasible problems, there is danger that the

solution process gets stuck in a local minimum and

the physical solution will be lost. Also, one cannot

simply add a real objective there, like the total power

of all compressors in energy optimization, since the

improvement of the objective would be achieved at

the cost of the violation of the physical equations.

Fortunately, this is easy to correct and in our imple-

mentation all physical equations are set as equality

SIMULTECH 2016 - 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

182

constraints f (x) = 0 and, at solutions, are satisﬁed ex-

actly.

For solving NLPs many efﬁcient algorithms have

been developed (Bazaraa and Shetty, 1979; Bertsekas,

1999; Avriel, 2003; Fletcher, 2013), e.g., those imple-

mented in the optimization package Ipopt (W

¨

achter

and Biegler, 2006):

• an interior-point algorithm (also known as primal-

dual barrier method) for the solution of generic

NLPs with equations and inequalities;

• a line-search ﬁlter with second order corrections,

serving as a stabilizer for Newton’s method;

• an inertia correction method, driving the iterations

towards a critical point of the desired type (e.g.,

minimum, not maximum, not saddle point);

• a feasibility restoration method, applying an

emergency restart in the case that the convergence

becomes too slow;

• an automatic initialization, appropriate problem

scaling, interfaces to many sparse linear solvers,

etc.

Our approach, using the translation of network

problems to the standard NLP format, also allows to

employ for their solution any other package, such as

Snopt (Gill et al., 2005), Minos (Murtagh and Saun-

ders, 1978), Knitro (Nocedal and Wright, 2006), etc.

For the solution of mixed integer NLPs one can use

Couenne (Belotti et al., 2013), while multiobjective

optimization can be done with the approaches de-

scribed in (Clees et al., 2015).

4 NETWORK SIMULATION IN

MYNTS

The backbone of the system is formed by linear

Kirchhoff equations and non-linear element equa-

tions, depending on the ﬂow through an element and

on its two adjacent nodal variables:

∑

e

I

ne

Q

e

= Q

(s)

n

, f

e

(P

in

, P

out

, Q

e

) = 0, (1)

where indices n = 1...N denote the nodes and e =

1...E the edges of the network graph, I

ne

is the in-

cidence matrix of the graph, Q

e

are ﬂows through the

edges, Q

(s)

n

are source/sink contributions, localized in

the supply/exit nodes, P

n

are nodal variables (pres-

sure, voltage, etc – depending on the discipline). For

example, in gas transport networks the simplest pipe

law has a quadratic form (Mischner et al., 2011):

f

e

= P

2

in

− P

2

out

− RQ

e

|Q

e

|, (2)

where R is a resistance coefﬁcient, depending on pipe

length L, diameter D, friction factor λ, temperature

T , compression factor z and on pressure, temperature

and density at normal conditions (P

norm

, T

norm

, ρ

norm

):

R =

16 λ L T z P

norm

ρ

norm

π

2

D

5

T

norm

. (3)

System (1) describes a stationary network problem,

while for dynamical (transient) problems additional

terms with time derivatives are included. Speciﬁc for

every medium, new equations can be introduced, re-

lating additional nodal and element variables.

Modeling of gas transport networks in Mynts includes

the following building blocks, described in detail in

(Schmidt et al., 2015a; Mischner et al., 2011; CES,

2010):

• main elements: supplies, exits, pipes, valves,

compressors, drives, regulators, resistors, short-

cuts, ﬂaptraps, heaters, coolers;

• gas laws: ideal, AGA, Papay, AGA8-DC92 (ISO

standard);

• pressure drop in pipes: besides the quadratic law,

more accurate Hofer and Nikuradze formulae are

used;

• various models for compressors (turbo, piston)

and their drives (gas turbine, steam turbine, gas

motor, electro motor), with characteristic dia-

grams calibrated on real compressors;

• control logic of compressors and regulators, im-

plemented in the form of control equations or in-

equalities, e.g., a compressor/regulator can have a

control goal to keep ﬁxed output pressure (SPO),

input pressure (SPI) or ﬂow value (SM);

• gas temperature: propagation over the network,

including non-linear heat capacity, heat exchange

with the soil, Joule-Thomson effect (temperature

drop in gas due to free expansion through a valve,

regulator, etc);

• gas composition: propagation and mixing (includ-

ing detailed molar components and effective gas

properties like critical temperature and pressure,

caloriﬁc value, molar mass, etc).

Modeling of water networks strongly resembles the

modeling of gas transport networks, except that a set

of important simpliﬁcations can be applied. In Mynts

it is assumed, e.g., that transient effects and the com-

pressibility of the ﬂuid do not play a role. Hence,

the constitutive assumption, which relates the pres-

sure to the density of a compressible ﬂuid, is omitted.

Moreover, only the temperature is considered as inter-

nal energy and only one phase of liquid water is be-

ing modeled. Mynts includes the following building

MYNTS: Multi-phYsics NeTwork Simulator

183

Figure 2: Gas transport network simulation in Mynts.

Closeup to a compressor station.

Figure 3: Gas transport network simulation in Mynts. Con-

ﬁguration of the solver.

blocks, which are described in detail in (Clees et al.,

2016a; Clees et al., 2016b; Colburn, 1933; Colebrook,

1939; VDI/VDE, 1962; KSB AG, 2005; Brki

´

c, 2011):

• main elements: supplies, exits, pipes, resistors,

two-way and three-way valves, pumps and heat

exchangers;

• pressure drop in pipes: Bernoulli’s equation with

friction including a variety of implementations for

Colebrook’s friction factor (more complex arma-

tures are considered as simple resistors via their

k

vs

-values);

• models for regulated and unregulated pumps (in-

cluding user deﬁned characteristic curves), heat

exchangers and two-way or three-way valves with

linear and equal percentage characteristics;

• simple control via goals deﬁned either for pump

speed or valve lift, or – at arbitrary elements of the

network – for pressure, pressure drop, ﬂow rate or

temperature;

• water temperature: propagation and mixing over

the network, power supply and exit at dedicated

elements, heat exchange with the environment

and with other network elements.

Modeling of electric power networks (Herman, 2011;

Milano, 2010):

• main elements: buses (generators, consumers,

prosumers, internal) and branches (transmission

lines, transformers and phase shifters, shortcuts);

• voltage drop in lines due to resistances (direct-

current, DC) or impedances (alternating-current,

AC): analogously to Ohm’s law for DC circuits,

the voltage drop in an AC circuit is the product

of the current and the impedance of the circuit;

electrical impedance is the vector sum of electri-

cal resistance, capacitive reactance and inductive

reactance;

• Norton scheme (admittance formulation): all con-

stant impedance elements of the model are in-

corporated into a complex bus admittance matrix

that relates the complex nodal current injections to

the complex node voltages; similarly, the system

branch admittance matrices relate the bus voltages

to the vectors of branch currents at the from and

to ends of all branches, respectively;

• various types of transformers, their parameters

(e.g. tap ratio, phase shift) and connections

(Delta, Wye, Wye-G).

Fig. 1 shows an example of a gas transport simula-

tion in Mynts. The network, used here for illustration,

contains 100 nodes, 111 pipes and other connecting

elements. We note that real life problems are much

larger. In cooperation with our partners we solve sta-

tionary and transient problems for gas networks with

thousands of elements.

The color in Fig. 1 shows the pressure distribution

in the network, which starts from 50bar and 60bar,

respectively, at the two supply nodes, is increased to

80bar by two compressor stations and then gradually

falls to 55bar at the exit nodes.

On the bottom right a characteristic diagram of

one compressor is shown, where the horizontal axis

refers to the volume ﬂow through the compressor, the

vertical axis to the adiabatic enthalpy difference, the

blue parabolic curves correspond to ﬁxed revolution

numbers, the magenta and dark cyan lines to ﬁxed adi-

abatic efﬁciency, the green line to maximal efﬁciency

and the red line to the minimal possible throughput

ﬂow for the given pressure increase (the so-called

surge line). The curves for the maximal and the min-

imal revolution numbers and the surge line deﬁne a

working region of the compressor, where the working

SIMULTECH 2016 - 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

184

point must be located (shown by a blue cross, here lo-

cated on the surge line). In addition, there is an upper

limit for the power, provided by the compressor drive,

and there are lower and upper limits on input and out-

put pressure and ﬂow, which together with other con-

trol equations and inequalities contribute to the whole

NLP system.

The solution of the NLP system can either be per-

formed as a “pure simulation” (solve>simulate) or as

a simulation with optimization (solve>optimize). For

the optimization the following objectives can be se-

lected:

• energy: minimize the total power, provided by all

drives;

• efﬁciency: maximize the average adiabatic efﬁ-

ciency for all compressors;

• SPSM: minimize the deviation from the pre-

scribed pressure and ﬂow conditions in compres-

sors and regulators (SPO, SPI, SM);

• QP-sets: minimize the deviation from the pre-

scribed ﬂow settings (QSET) for supply nodes

with the given pressure settings (PSET).

The energy optimization allows to achieve the best en-

ergy savings of the gas transport for a given network

scenario. The efﬁciency optimization tries to set all

compressors on their central green lines. The opti-

mization of SPSM/QP-sets allows to control the dis-

tribution of load between compressors/suppliers.

Fig. 2 shows a closeup of a compressor station.

It consists of two compressors, which by means of

a special pattern of open and closed valves are con-

nected parallel to each other. Each compressor has in-

put and output resistors and is equipped with a cooler,

compensating the temperature increase in the com-

pressor, and a bypass regulator. This regulator is au-

tomatically opened if the working point reaches the

surge line and provides an internal gas circulation to

keep a constant output pressure for small outﬂows.

There is also a bypass valve, which opens only if the

compressor is shut down.

Fig. 3 shows a menu for detailed solver conﬁgu-

ration. In particular, it allows to set various physi-

cal parameters, switch between the physical models,

add transient terms, choose a solution method (e.g.,

the whole problem can be solved as a coupled sys-

tem at once or in blocks using an iterative relaxation

scheme), set the maximal number of iterations, the

solver tolerance, select deep options to the solver, etc.

The optimization part allows to select the above de-

scribed objectives or their combinations.

The viewer visualizes the network hierarchically,

switching automatically the level of detail suitable for

the current zoom factor. The user can select a property

shown on the graph by color or by the variable width

of the edges and nodes, change the size and shape of

visual elements, the detailization of the displayed in-

formation, etc.

Graphical user interface of Mynts is not only a net-

work viewer, but also a scalable network editor. It al-

lows to create and modify the network topology (new

nodes/edges, connect, merge, split, etc) and also to

edit the properties of the elements. For example, the

left bottom menu in Fig. 1 shows the details of a com-

pressor element, where the user can inspect different

properties and modify the settings.

The detailed numbers on Mynts performance are

given in our satellite paper (Clees et al., 2016c). In

particular, solution of a realistic gas transport network

scenario with 4466 nodes and 5362 edges requires

1.9sec on a 3 GHz Intel i7 CPU.

5 CONCLUSIONS

We have presented a generic approach for the simula-

tion of transport networks. The main steps of physi-

cal modeling and numerical simulation are effectively

separated. Input data for the modeling come in the

form of lists representing the topology of the network,

physical properties of the elements and a prototypical

description of variables and constraints deﬁning the

discipline. A rapid algorithm translates the problem

statement into the language of PPN encoded expres-

sion trees, making it accessible for the numerical so-

lution by standard NLP solvers. The applicability of

this approach has been demonstrated on various prob-

lem types, including stationary and transient network

simulation, feasibility analysis and energy-saving op-

timization. The approach has been implemented in

our multiphysics network simulator Mynts, support-

ing various disciplines, such as gas transport, water

supply and electric power networks.

Our further plans include the support of other dis-

ciplines (steam engines, mechanical transmissions,

pipelines in the chemical industry), the solution of

cross-disciplinary problems and interfacing to other

simulation kernels and modeling systems (Couenne,

SCIP, Modelica).

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