In the current Energy market context, District Heating (DH) has an impor= tant role, especially in counties with cold climate, as it often leverages = on Combined Heat and Power (CHP) units, capable to reduce the consumption o= f primary energy to fulfill a given electric and thermal request, as well a= s on existing significant sources of heat generated by industrial processes= or waste-to-energy heat generation. On the top of that heating networks wi= ll need to increase its flexibility in operation due to an increasing mix o= f renewable sources, both heat sources or green electricity utilized by hea= t pumps, distributed generation and smart consumers as well as DH operation= al temperature reduction and heat storage integration [1, 2].

From a management standpoint, the design of the district heating network= is a strategic business issue, since it requires large investments, due to= the cost of materials and civil works for the realization of the network. = Proper strategic design of network (i.e. definition of the most convenient = backbone pipelines to lay down) and tactical targeting of most promising po= tential customers both aims at maximizing the Net Present Value (NPV) of th= e investment.

Finding the extension plan for an existing (or eventually empty) distric= t heating network that maximizes the NPV at a given time horizon is a chall= enging optimization problem that can be stated as follows.

Given:

- A time horizon (e.g., fifteen years)
- A set of power plants, with specific operational limitations (maximum p= ressure, maximum flow rate, =E2=80=A6);
- An existing distribution network, with information on the physical prop= erties of the pipes (length, diameter, =E2=80=A6);
- A set of customers already connected to the network with known heat dem= and;
- A set of potential new pipes that can be laid down;
- A set of potential new customers that can be reached;

find:

I. The subset of potential new customers that should b= e reached;

II. Which new pipelines should be installed;

III. The diameter of the new pipes

that maximize the NPV.

Research on modelling approaches for representing the behavior of the th= ermo-hydraulic network through sets of non-linear equations can be found in= the literature (see for example [3] and [4]). Solving systems of non-linea= r equations is difficult and computationally expensive. For this reason, ag= gregation techniques of the network elements are often used to model large = district heating networks, at the expense of some accuracy [5] [6] [7] [8] = [9] [10].

In [11], an integer-programming model is proposed for the optimal select= ion of the type of heat exchangers to be installed at the users=E2=80=99 pr= emises in order to optimize the return temperature at the plant. The author= s achieve good system efficiency at a reasonable cost.

[12] present a mathematical model to support district heating system pla= nning by identifying the most advantageous subset of new users that should = be connected to an existing network, while satisfying steady state conditio= ns of the thermo-hydraulic system. [13] extend the model proposed by [12] w= ith the selection of the diameter for the new pipes and a richer economic m= odel that takes into account

- Production cost and selling revenues;
- Cost for installing and activating new network links;
- Cost for connecting new customers to the network;
- 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 exte= nded 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 ow 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, o= n the installation of new links, on the diameter choice and on ow direction= on the links. The last ones are necessary since district-heating networks = contains cycles: the potential network usually corresponds to the street ne= twork. Thus, it is not possible to know a priori the flow direction on the = links (at least not for all of them) and such decision must be included int= o the model. The pressure drop along a pipe is a non-linear function that d= epends on ow rate, and on the diameter of the pipe. This can be approximate= d using a piecewise linear function, that translates into a set of linear c= onstraints. The higher the number of segments in the piecewise linear funct= ion, the smaller will be the approximation error. At the same time, the num= ber of constraints grows (there is one piecewise-linear function for each c= ombination of pipe and diameter) and the solving time increases. To keep th= e number of segments small, while obtaining a good accuracy, breakpoints of= the piecewise linear function can be concentrated in the most probable ran= ge 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 full M= ILP directly. Solution methods developed in [13] approach the problem in th= ree 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, the conflict points, which are the nod= es of the network where different water direction meet, are detected. The f= low direction is released for the nodes close to conflict points, and the M= ILP model 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 S.r.l. has developed a decision support system, in collaboration w= ith the University of Bologna, based on the modelling mentioned above, that= has been successfully used in two of largest multi-utility companies opera= ting in the Italian District Heating market. The application leverages on o= pen source Geographical Information System (GIS) to allow a simple user int= erface and a number of plug-in tools to manage the specific optimization is= sue.

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= .

**=
References**

[1] H. Lund, S. Werner, R. Wiltshire, S. Svendsen, J. E. Thorsen, F. Hve= lplund and B. Vad Mathiesen, "4th Generation District Heating (4GDH) Integr= ating smart thermal grids into future sustainable energy systems," Energy, = vol. 68, pp. 1-11, 2014.

[2] V. Verda, M. Caccin and A. Kona, "Thermoeconomic cost assessment in = future district heating networks," Energy, vol. 117, pp. 485-491, 2016.

[3] B. B=C3=B8hm, H. P=C3=A1lsson, H. V. Larsen and H. F. Ravn, "Equival= ent models for district heating systems," in 7th International Symposium on= District Heating and Cooling, 1999.

[4] Y. Park, W. T. Kim and B. K. Kim, "State of the art report of Denmar= k, Germany and Finland," Simple models for operational optimization. Depart= ment of Mechanical Engineering, Technical University of Denmark, 2000.

[5] H. V. Larsen, H. P=C3=A1lsson, B. B=C3=B8hm and H. F. Ravn, "Aggrega= ted dynamic simulation model of district heating networks," Energy conversi= on and management, vol. 43, pp. 995-1019, 2002.

[6] H. V. Larsen, B. B=C3=B8hm and M. Wigbels, "A comparison of aggregat= ed models for simulation and operational optimisation of district heating n= etworks," Energy Conversion and Management, vol. 45, pp. 1119-1139, 2004.

[7] A. Loewen, M. Wigbels, W. Althaus, A. Augusiak and A. Renski, "Struc= tural simplification of complex dh-networks-part 2," Euroheat and Power-Fer= nwarme International, vol. 30, pp. 46-51, 2001.

[8] H. Zhao, Analysis, Modelling and operational optimazation of distric= t heating systems, Centre for District Heating Technology, Technical Univer= sity of Denmark, 1995.

[9] H. Zhao and J. Holst, "Study on a network aggregation model in DH sy= stems," Euroheat \& power, vol. 27, pp. 38-44, 1998.

[10] A. Loewen, W. Wigbels, A. Augusiak, and A. Renski, "Structural simp= lification of complex DH networks - part 1," Euroheat and Power, vol. 30, n= o. 5, pp. 42-44, 2001.

[11] R. Aringhieri and F. Malucelli, "Optimal operations management and = network planning of a district heating system with a combined heat and powe= r plant," Annals of Operations Research, vol. 120, pp. 173-199, 2003.

[12] C. Bordin, A. Gordini and D. Vigo, "An optimization approach for di= strict heating strategic network design," European Journal of Operational R= esearch, vol. 252, pp. 296-307, 2016.

[13] A. Bettinelli, A. Gordini, A. Laghi, T. Parriani, M. Pozzi and D. V= igo, "Decision Support Systems for Energy Production Optimization and Netwo= rk Design in District Heating Applications," in Real-World Decision Support= Systems, Springer, 2016, pp. 71-87.

**Contributor**:

Dr. Robert Schwarz, ZIB

Dr Andrea Bettinelli,Optit

Prof Daniele Vigo, University of Bologna / Optit=