Heating networks are classified as topological connected objects; therefore performance of networks is largely determined by their structural features. City heating networks were developed with a growth of city and that is why their structure is random.
Structural analysis allows determining special properties, weak and strong point of these networks. Proposed method of structural analysis was carried out for heating networks of Kazakhstan cities as of 1980-1990.
Created graph models of city heating networks were reflected at the first stage of the analysis by network graph vertex-branches incidence matrixes (A) and independent loops matrix (B).
When performing systems structural analysis it is often necessary to have method allowing determining some structural features of systems and giving them quantitative estimation. Expediency of determining such features is in the fact that the necessity in evaluation of system structure and its elements quality from the position of overall system approach appears already at early design stage. Consider some of them.
Structure connectivity. This quantitative characteristic allows detecting presence of breaks, hanging vertexes and ect. in the structure. Complete quantitative determination of directed graph elements connectivity is given by connectivity matrix С = ∣ ∣ cij ∣ ∣
The right side of the inequation determines minimum required number of connections in the structure of undirected graph containing n vertexes.
Parameter ε2 characterizes capacity slackness of the set structure with m edges and n vertexes in achievement of maximal connectivity. This parameter in relative terms is used for comparison of various automated control systems structures.
Structural compactness. Parameter reflecting elements proximity is entered for quantitative estimation of structural compactness. Proximity of two elements i and j will be determined through minimal path length for directed graph (circuits — for undirected) dij. Then the value Q = ∑p=1 ∑'' 1 dij (i ≠ j) reflects total structural proximity of elements in the system. Relative parameter
Qoth = ^^— 1 is very often used for quantitative estimation of structural compactness
where Qmuh =n (n — 1) — minimal value of compactness for system structure of —complete graph” type.
Structural compactness can be also characterized by other characteristic – structure diameter: d = max. Taking into consideration prevailing information character of communications in technological networks it can be stated that value Qoth as well as d give integral estimation of inertance of information processes in system, and at equal values of ε2 and R their increase reflects
increase of separating communications number thus characterizing reduction of general reliability.
Element rank is used when representing system structure in the form of directed graph. This characteristic allows distributing of system elements in the order of their magnitude. Element magnitude is defined here only by number of connections of this element with other ones. Certainly, element rank in such definition doesn't give сотрìеţе сĥагасţегǐśţǐс of element İтрогţапсе in system as in this case accuracy, information and other functional characteristics of element are not considered. However, having characterized element by rank the following plausible assumption can be made: the higher element rank the stronger its connection with other system elements and therefore the more severe effect of its performance quality change. Strict definition of element rank is integrated with certain computing difficulties therefore at this stage of structural analysis approximate way is quite enough. For practical tasks this way gives almost true values of element relative ranks and doesn't require big calculations. Values of elements ranks are quite useful information for distribution of temporary, cost and technical resources for achievement of tasks set at technological networks design stage. Quantitative characteristics entered above may be used when performing comparative evaluation of systems structures topological properties.
Model of city heating networks is presented in table 1. At that due to volume representation of network graph (e.g. Almaty (Fig.1)) column 2 of table 1 contains only generalized model of network graph.
- for disconnected structures R < 0; for structures without redundancy (consequential, radial, tree shaped) R = 0; for structures with connections redundancy (ring, «complete graph» type) — R > 0;
- structures (consequential, radial, tree shaped) with R = 0 are distinguished by the parameter ε2 ; radial structure has the greatest nonuniformity of connections;
- tlιe strudure of ‘Complete ĝгарĥ” type ĥаś tlιe grentest elements proximity (parameter Q); the least - consequential; radial and ring structures undistinguishable with regard to parameter d have different Q values;
- radial and tree shaped structures having equal or near to equal R, Q, d values are significantly different as per ε2 and δ parameters, that corresponds to
physical content as displacement from full centralization in structure results in greater uniformity of elements connections distribution.
Structure processing shows:
- HN graph has high centralization degree in Almaty, Leninogorsk, Karaganda. In Uralsk, Arkalyk - low.
- HN graph in Almaty (part 1) Almaty (part 2) has greater structure compactness ratio Q (27.6), the lowest in Uralsk (10).
- HN graph in Almaty has grater structural redundancy (R=6.23 part 1, R= 4.98 part 2), Karaganda (2.7), the rest cities have structure poorly connected among themselves, that is failure of elements with significant ranks will result in —collapse” of city heating network, that is in potential break-down. In general this parameter of HN reflects proximity and interaction of elements.
Reviewed structural characteristics were received only based on the information about composition of elements and their connections. Further development of structural parameters construction methodology for solving structural analysis problems can be based on non-structural information by entering numerical functions onto graph. It allows considering other relevant sides of interaction (temporary, reliability, cost and ect.) along with elements composition and interaction directedness when solving structural analysis problems.
Table 2 shows:
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