Optimization of hydraulic modes of single-source tree-like heat networks

. The article deals with the problem of optimizing the hydraulic modes of radial heat supply systems with a single heat source and pumping stations. The features of the problem under consideration are the presence of several target functions and a fixed flow distribution. The topology of the networks under consideration allows using the equivalent branching methods of branches connected in parallel and in series to reduce the design schemes to single equivalent branches. To solve this problem, the original method proposed by the authors earlier is adapted. The proposed method was tested on an aggregated scheme of a real heat supply system. Computational experiments have shown the performance and computational efficiency compared with the methods proposed earlier.


Introduction
Heat supply systems (HSS) have significant reserves of energy saving [1 -3], which can be realized by optimizing their operation modes. The construction of new residential areas, the closure of industrial facilities leads to significant changes in the structures of HSS loads and an increase in the risk of emergencies. The emergence of new types of units raises questions of their effective use. In practice, the task of planning HSS operation modes is solved by multivariate calculations [4], which does not guarantee the optimality of the obtained modes. Automating the solution of these problems is hampered by several factors: the high dimensionality of the flow distribution models [5,6], their nonlinearity, the presence of several objective functions, etc. For these reasons, there are no methods and software packages suitable for practical use. This determines the relevance of the development of methods and software systems for calculating the optimal HSS modes.
The literature discusses the problem of optimizing the HSS modes, but most of the work has drawbacks. A large number of papers consider small-scale HSS, for example, [7,8]. Some works (for example, [9,10] and others) consider approximation of the relationship between the value of the objective function and the mode parameters chosen as the basis, which makes it difficult to account for discrete mode parameters. In the remaining works, the HSS aggregated schemes are used [11,12], which does not guarantee obtaining optimal solutions with the required accuracy. The practice of using ready-made software packages for solving systems of equations and inequalities ( [13,14]) is widespread, which leads to the impossibility of adapting methods to problems and too high computational costs. Even greater computational costs are required by the use of genetic algorithms ( [15]). Private objective functions are often considered. For example, in [16] , the total fuel consumption at heat sources (HS) is minimized, and in [9 , 10] power consumption at pumping stations (PS) is minimized. The operational management is mainly considered, for example, [16 -18], and there are practically no works devoted to the planning tasks of the established HSS operation modes in preparation for the heating season.
The article provides the formalization of the optimization problem for the steady-state hydraulic modes (HM) of a HSS of a radial structure, having one HS and PS for the typical case of parallel operation of pumps of the same type. To solve this problem, the previously proposed loop reducing dynamic programming method (LRDP) [19] is adapted. This method has several advantages: an increase in computational costs linear in the dimension of the problem; guaranteed finding optimal HM; providing the possibility of optimization for several objective functions at the same time, etc.

Problem statement
The task of optimizing HM HSS is to find a HM that meets the requirements of admissibility and optimality according to a given system of objectives. Energy saving requirements is laid down in the economic objective function. Technological objective functions are associated with the desire to reduce the labor intensity of adjustment activities, to reduce leakages and the risks of accidents.
Pipeline sections, PS and consumers are the main elements of the HSS. Denote the sets of branches that model the elements of the first, second and third types as IPL, IPS, IC. Then Let J be the set of all nodes, Jm  . As the hydraulic characteristics of the i-th branch can be taken [20]: Here: ibranch number; hi, xipressure drop and coolant flow to the branches; si, zihydraulic resistance and its relative increase (for example, increased by throttling); i In (1), to simulate a control that is absent or forbidden to change, one can equate the corresponding parameter to a constant. Also (1) can be considered as a generalization of the hydraulic characteristics of the pipeline section.
Part of the parameters of HM, depending on the external environment, we call the boundary conditions. As such, for HSS, pressure in the feed nodes and flows in the remaining nodes are usually indicated. The considered HSS have features: 1) tree in the single-line representation and multi-loop in the two-line one; 2) supply and return pipelines are symmetrical to each other with the exception of PS; 3) supply and return pipelines are connected through HS and consumers with fixed flows. From these features and the form of setting the boundary conditions follows a fixed flow distribution. In this case, the model of controlled flow distribution used in [20], will take the form: ( , , , ) where: A is the incidence matrix of the HSS calculated scheme; P is the nodal pressure vector; y is the vector of pressure drops on the branches; ( , , , ) h x z γκ -vector function with components ( , , , ) The requirements of acceptability and realizability of the mode are usually written as [20] jjj PPP , The variable component of the cost of maintaining the mode consists of the cost of electricity for the PS and the cost of fuel. In the case of HSS with one HS, fuel costs can not be changed due to the redistribution of load between HS. The electric power consumed on a separate PS is equal to

Method
To use LRDP, the following conditions are necessary and sufficient: 1) additivity of the objective functions; 2) the ability to reduce the design scheme of the network to one branch by the equivalent of serial and parallel branches; 3) fixed flow distribution. The topology of the considered HSS differs from the topology of the distribution heat networks for which LRDP was developed in that the hanging nodes are connected by a branch corresponding to HS. Obviously, the other conditions are also satisfied. Thus, in order to apply the LRDP to solve problem (4), it is necessary and sufficient to ensure correct consideration of the work of the PS.
The intervals of allowable change in nodal pressure are divided into subintervals of  (pockets). Renumber the pockets so that their numbers represent discrete pressure readings with a certain multiplier. For definiteness, assume that all branches are downstream. The initial node of the i-th branch is denoted as fi, lithe final one. Each branch (i) is associated with a set of segments of possible piezometric graphs    . The forward stroke of LRDP consists in the «convolution» of the HSS design scheme with discarding of unacceptable and non-optimal variants of piezometric graphs due to the methods of equivalence of serial and parallel branches to one branch, on which the optimal piezometric graph will remain.
To equivalent parallel branches, the following is done. If there are a pair of piezometric graph segments ( 1 pairs of segments are found that connect the same pockets, the pair that has the worst value of the increment of the objective function is discarded. Each of the pairs of segments found on the original HSS fragment turns into a piezometric plot of the equivalent branch with the corresponding initial and final pockets. The increments of the objective function are summed up.
To restore the optimal piezometric graph, it is necessary, for each equivalent branch, to memorize which fragment it is equivalent to, and for each equivalent segment of the piezometric graph, to memorize which segments it is equivalent to.

Method testing
The efficiency of the presented method was tested using the example of solving the problem (4) for the aggregated HSS of the city of Baikalsk (Fig. 1) in two cases: 1) changing  is prohibited on all PS; 2) allowed. On the PS-1, the pumps are installed on the supply pipe, on the rest of the PS -on the return. Since all PSs receive electricity at the same cost, the total power consumption is minimized. Considered two options loads. The first corresponds to the winter mode of operation HSS. In the second mode (lightweight), all loads were reduced by half. PS of this network have a different number of pumps. Table  1 shows their number per PS, type, power and hydraulic factors.  The following approach was used as a reference method. First, the values of all Boolean variables were set equal to one and fixed, then the HSS HM with the economic objective function was optimized using the method presented in [20]. After that, to fix the values of FC, the values of i  and i  , iI  were fixed and optimization was performed for the technological objective functions using the method described in [21]. The results of the calculation of LRDP coincided with the results of the calculation by the reference method. Controls on the passive branches are not required. Table 2 shows the number of pumps in operation. In modes marked with an asterisk, changing  is allowed.
The LRDP speed exceeds the reference speed by more than 2 orders of magnitude.

Conclusions
1. The article summarizes the original method of multi-objective discrete-continuous optimization of hydraulic modes of distribution thermal networks in the case of radial thermal networks with a single heat source and pumping stations.
2. The new method is implemented in the form of a research program and tested on an aggregated scheme of a real HSS. Calculational experiments confirmed its performance and high computational efficiency.
The study was carried out within the framework of the SB RAS Basic Research Program (AAAA-A17-117030310437-4, project III. 17.4.3) and with the financial support of the RFBR and the Government of the Irkutsk Region .