12th International Symposium on District Heating and Cooling

1600The <strong>12th</strong> <strong>International</strong> <strong>Symposium</strong> on District Heating and Cooling,September 5 th to September 7 th , 2010, Tallinn, Estonia1. In the conceptual design planning phase for thecost estimate and the principle decision for thedistrict heating (yes or no).2. In the detailed planning phase the localization ofthe pipeline route occurs for the approvalplanning.3. In the execution planning phase the finaldetermination of the dimension occurs, but nomechanical calculation (stress-strain analysis) isdone by the program. Still the required proofsaccording to e.g. EN 13941 have to be done.In addition, the program can be used for the hydrauliccalculation of existing district heating networks.ModelThe hydraulic calculations establish the technical basiswhich performs constraints of the optimization model.In district heating systems a distinctive turbulent flowcan be presumed. In this case a good approximationwith the surface roughnesscoefficient of friction is applied (4).of the pipe and theIn the mentioned planning phases the coefficients ofdrag are included blanket into the pressure lossaccording to (5):(4)(5)If the edge is not used for the site development ( ),then holds.As variables are required beside the diameter furthervariable than auxiliary variables to the formulation ofthe constraints: vector of the mass flows of the edges vector of the pressures of the verticesbinary variable to the capture of the jump atThus the constraints can be formulated. These are theequation (6), local and technical limitations as well asequations. (8) and (9).First Kirchhoff‘s law: Point rule. The sum of all massflows in a vertex is equal zero. ( - vertex matrix)Second Kirchhoff‘s law: Mesh rule. The sum of thepressure losses along a mesh is equal zero ( - meshmatrix):This mathematical model is simple to describe, butdifficult to solve (already for medium-sized graphs). Ifthe diameters are eliminated by the equation (6) as avariable, the variables and whose impact on theobjective function is discussed in detail in [5] remain.The principal dependency of the objective function onthe vector of the mass flows is displayed in Fig 6schematically.(8)(9)whereas a extra charge of length.For a pipe of the length and the diameterfor the pressure loss of a plain pipe.(6) arises(6)Thus the following mathematical optimization modelarises:The investment costs of every new route come into theobjective function (investment costs, annual costs ornet present value). They are included in the form of (7)in the model.The bracket of the first summand contains theinvestment costs of the route per meter as a total lumpsumprice and must be multiplied according to by thelength of pipeline. In the second summand"obstacles" can be included as direct costs dependentfrom the diameter. The exponent is set =1 in thepresent program version for linear dependence. Theparameters and are input data.(7)K konkavK BaumFig. 6 Schematic dependence of the objective function onthe mass flowOn the abscissa the circulatory mass flowof a meshis displayed. The ordinate shows the non-convexobjective function which shows jumps by the binaryvariables with the rhombuses (the filled rhombus is thefunction value).322