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**Gas pipelines design **

~~Mathematical models~~

### Modeling and algorithmic considerations:

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Natural gas is considered by many to be the most important energy source=
for the future. The objectives of energy commodities strategic problems ca=
n be mainly related to natural gas and deal with the definition of the *"optimal*" gas pipelines design which i=
ncludes a number of related sub problems such as: Gas stations (compression=
) location and Gas storage locations. Needless to say these problems involv=
e amount of money of the order of magnitude of the tens of billions =E2=82=
=AC and often these problems can be a multi-countries problem. From the eco=
nomic side, the natural gas consumption is expected to continue to grow lin=
early to approximately 153 trillion cubic feet in 2030, which is an average=
growth rate of about 1.6 percent per year. Because of the properties of na=
tural gas, pipelines were the only way to transport it from the production =
sites to the demanding places, before the concept of Liquefied Natural Gas =
(LNG). The transportation of natural gas via pipelines remains still very e=
conomical.

From an optimization standpoint, the gas pipeline design problems can be= divided in the following main sub problems:

- how to setup the pipeline network, i.e. its topology;
- how to determine the optimal diameter of the pipelines;
- how to allocate compressor stations in the pipeline network;

Typically, the mathematical programming formulations of these optimizati= on problems contain a lot of nonlinear/nonconvex and even nonsmooth constra= ints and objective functions because of the underlying physic of the gas fl= ows that needs to be considered. The classic constraints are the so-called = Weymouth panhandle equations, which are a potential-type set of constraints= and relate the pressure and flow rate through an arc (m,n) of the pipeline= .

As in many other situations problems 1-3 are a single problem but a *=
divide et impera* principium is applied. Therefore the problems 1 and 2=
are somehow determined via simulations and normally there are - in the fir=
st but also in the 2 - a lot of economic drivers, and also political driver=
s when many countries are involved. From a technical point of view, instead=
probably the most challenging problem is the number 3, the compression sta=
tions allocation. Because of the high setup cost and high maintenance cost,=
it is desirable to have the best network design with the lowest cost. This=
problem concerns a lot of variables: the number of compressor stations whi=
ch is an integer variable, the pipeline length between two compressor stati=
ons, and the suction and discharge gas pressures at compressor stations. Th=
is problem is computationally very challenging since it includes not only n=
onlinear functions in both objective and constraints but, in addition, also=
integer variables.

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**District Heating Ne=
twork Design**

### Introduction

### Mathematical models

### Optimization methods

~~Data and Software~~

### <=
span class=3D"confluence-anchor-link" id=3D"EnergyCommodities,Strategic-Pla=
nning-DistrReferences">References

=20
In the current Energy market context, district heating has an important =
role as it often leverages on existing significant sources of heat generate=
d by industrial processes, a mix of renewable sources and Combined Heat and=
Power (hereafter CH&P) units, all of them environmentally beneficial b=
ecause of their high energy efficiency when compared to conventional conden=
sing power plants (not to mention that single, large scale plants are signi=
ficantly more efficient and safe than numerous low scale heat-generation un=
its).

From a management standpoint, heat distribution becomes a strategi=
c business issue, related to the design of the district heating network tha=
t require large investments, due to the cost of materials and civil works f=
or the realization of the network.

Proper strategic design of network (i=
.e. definition of the most convenient backbone pipelines to lay down) and t=
actical targeting of most promising potential customers are both aimed at m=
aximizing the Net Present Value (NPV) of the investment.

The problem calls for finding the extension plan for an existing (or eve=
ntually empty) district heating network that maximizes the NPV at a given t=
ime horizon. It is therefore necessary to decide: (i) the set of potential =
new customers that should be reached, (ii) which new pipelines should be in=
stalled, and (iii) their diameter.

Research on representation and simula=
tion in details of the behavior of the thermo-hydraulic network through set=
s of non-linear equations can be found in literature, for example (Bohm, 19=
99) and (Y.S. Park, 2000). In (Aringhieri, 2003), an integer programming mo=
del is proposed for the optimal selection of the type of heat exchangers to=
be installed at the users=E2=80=99 premises in order to optimize the retur=
n temperature at the plant. The authors achieve good system efficiency at a=
reasonable cost. Boldrin et al. (Bordin, 2016) developed a mathematical mo=
del to support district heating system planning by identifying the most adv=
antageous subset of new users that should be connected to an existing netwo=
rk. In (Bettinelli, 2016), an economic and a thermo-hydraulic Mixed Integer=
Linear Programming (MILP) models have to be considered. The economic model=
takes into account:

- Production cost and selling revenues;
- Cost for network link activation, that depends on the diameter of the s= elected pipes;
- Cost for customer connections;
- Amortization;
- Taxes;
- Budget constraints.

Moreover, while the investment on the backbone pipelines is done on the =
first year, new customers are not connected immediately, but following an e=
stimated acquisition curve (e.g., 25% the first year, 15%, the second year,=
=E2=80=A6). Hence, the corresponding costs and revenues have to be scaled a=
ccordingly

The thermos-hydraulic model must ensure the proper operation=
of the extended network. The following constraints are to be imposed:

- Flow conservation at the nodes of the network;
- Minimum and maximum pressures at the nodes;
- Plants operation limit: maximum pressure on the feed line, minimum pres= sure on the return line, minimum and maximum flow rate;
- Pressure drop along the links;
- Maximum water speed and pressure drop per meter.

Continuous variables model pressures at nodes and flow rate on the links=
, and binary variables model decisions on the connection of new customers, =
on the installation of new links, on the diameter choice and on flow direct=
ion on the links. The last ones are necessary since district-heating networ=
ks contains cycles: the potential network usually corresponds to the street=
network. Thus, it is not possible to know a priori the flow direction on t=
he links (at least not for all of them) and such decision must be included =
into the model.

The pressure drop along a pipe is a non-linear function=
that depends on flow rate, and on the diameter of the pipe. This can be ap=
proximated using a piecewise linear function, that translates into a set of=
linear constraints. Solving systems of non-linear equations is difficult a=
nd computationally expensive. For this reason, aggregation techniques of th=
e network elements are often used to model large district heating networks,=
at the expense of some accuracy (Zhao, 1995), (H. Zhao, 1998), (Larsen H. =
V., 2002), (Loewen A. a., 2001), (Loewen A. a., 2001), (Larsen H. V., 2004)=
. The higher the number of segments in the linear function, the smaller wil=
l be the approximation error. At the same time, the number of constraints g=
rows (there is one piecewise-linear function for each combination of pipe a=
nd diameter) and the solving time increases. To keep the number of segments=
small, while obtaining a good accuracy, breakpoints of the piecewise-linea=
r function can be concentrated in the most probable range of flow rate.

District heating networks can be quite large (hundreds of existing and p= otential users, thousands of links) making it difficult to solve the proble= m directly with the full MILP. Solution methods developed in (Bettinelli, 2= 016) approach the problem in three steps.

- solve the linear relaxation of the MILP model and use it to select wate= r direction in all the pipes. Then, solve to integrality the MILP model, wi= th the directions fixed, obtaining a first heuristic solution.
- In the solution found at step 1, detect the conflict points, which are = the nodes of the network where different water direction meet. The flow dir= ection is released for the nodes close to conflict points, and the MILP mod= el is solved again, obtaining a second heuristic solution
- The full MILP, initialized with the best solution found in the previous= steps, is solved, until either optimality or the time limit are reached.

Optit srl has developed a decision support system, in collaboration with= the University of Bologna, based on the modelling mentioned above, that ha= s been successfully used in two of largest multi-utility companies operatin= g in the Italian District Heating market. The application leverages on open= source Geographical Information System (GIS) to allow a simple user interf= ace and a number of plug-in tools to manage the specific optimization issue= .

1. Aringhieri, R., Malucelli, F.: Optimal operations management and netw=
ork planning of a district heating system with a combined heat and power pl=
ant. Annals of Operations Research120(1-4), 173=E2=80=93199 (2003)

2. Be=
ttinelli, A. a. (2016): Decision support systems for energy production opti=
mization and network design in district heating applications. Technical Rep=
ort OR-16-5, DEI =E2=80=93University of Bologna, under review by Integrated=
Series in Information Management.

2. B=C3=B8hm, B., Palsson, H., Larsen=
, H.V., Ravn, H.F.: Equivalent models for district heating systems. In: Pro=
ceeings of the 7th International Symposium on District Heating and Cooling.=
Nordic Energy Research Programme (1999)

3. Bordin, C., Gordini A., Vigo=
D., An optimization approach for district heating strategic network design=
, European Journal of Operational Research, Volume 252, Issue 1, 1 July 201=
6, Pages 296-307, ISSN 0377-2217,

4. Larsen, H.V., B=C3=B8hm, B., Wigbel=
s, M.: A comparison of aggregated models for simulation and operational opt=
imisation of district heating networks. Energy conversion and management45(=
7), 1119=E2=80=931139 (2004)

5. Larsen, H.V., Palsson, H., B=C3=B8hm, B.=
, Ravn, H.F.: Aggregated dynamic simulation model ofdistrict heating networ=
ks. Energy conversion and management43(8), 995=E2=80=931019 (2002)

6. Lo=
ewen, A., Wigbels, M., Althaus, W., Augusiak, A., Renski, A.: Structural si=
mplification of complex dh-networks-part 1. EUROHEAT AND POWER FERNWARME IN=
TERNA-TIONAL30(5), 42=E2=80=9344 (2001)

7. Loewen, A., Wigbels, M., Alth=
aus, W., Augusiak, A., Renski, A.: Structural simplification of complex dh-=
networks-part 2. Euroheat and Power-Fernwarme International30(6), 46=E2=80=
=9351(2001)

8. Park, Y.: State of the art report of Denmark, Germany and=
Finland. Simple models for operational optimization. Department of Mechani=
cal Engineering, Technical University of Denmark(2000)

9. Zhao, H.: Anal=
ysis, Modelling and Operational Optimazation of District Heating Systems. C=
entre for District Heating Technology, Technical University of Denmark (199=
5)

10. Zhao, H., Holst, J.: Study on a network aggregation model in dh s=
ystems. Euro heat & power27(4-5), 38=E2=80=9344 (1998)

**Contributor**:

Robert Schwarz, ZIB