A new approach to the mathematical model of thermal energy balance

Formation of balance of thermal energy and the heat carrier for a thermal network is carried out for the purpose of increase of efficiency of work of the heat supplying organization. The standard approach proposes to write off the difference between the released and realized heat energy for heat losses. The article proposes a new approach to the formation of the balance of thermal energy and coolant. It includes statistical analysis of telemetry data relating to the released thermal energy. Heat consumption was estimated by contractual load for heating, ventilation and hot water supply. In the mathematical model of the thermal balance for each term weight coefficients were introduced. To obtain the numerical values of these coefficients, the method of least squares was used. The results of checking the adequacy of mathematical models that take into account or neglect thermal losses are presented. The obtained coefficients for the mathematical model of heat balance were used for the mathematical model of mass balance. The results can be used to predict the cost of production and transmission of heat energy and coolant, to assess the efficiency of the heat network and the formation of tariff applications for the future.


Introduction
The heat pipelines as others engineering systems are a part of every human settlement. If the city turns into a Smart city, then engineering systems must be satisfied to the status Smart Grid. There are no opponents of the necessity of development and introduction of ideas, which increase "the intellectuality" of heat pipelines. The process of engineering systems turning into the other intellectual level can be rather long. collection and transfer. There is practically no discussion of issues related to the processing and analysis of the collected information, the Range of tasks that can be solved with new approaches to the automation of information collection from all objects of the network will expand. One of the aim is a aim of preparation of heat and material balance of the network. One of these tasks is the task of drawing up the thermal and material balance of the network. To solve this problem, it is necessary to collect synchronized information from the sources supplying the network and from the consumers connected to the network. The full amount of data will allow to estimate the heat losses and leaks that occur during the transport of the coolant. The time period for which the balance is drawn is determined by the frequency of obtaining the initial information.

Methods
The mathematical model of heat balance is based on the law of heat energy conservation: Where Q, Gcal/dayis the amount of thermal energy released into the network; Where c, Gcal/(t °С) -the heat capacity of water; G s i , t/day, -coolant flow rate in the i supply pipeline; t s i °С, -coolant temperature at the inlet of the i supply pipeline; G r i , t/day, -coolant flow rate in the I-th return pipeline; t r i °С, -coolant temperature at the outlet of the i return pipeline.
The number of terms in the formula (2) is determined by the number of working pipes k. The energy consumed by heating, ventilation and hot water systems can be estimated from the data of consumer metering units. This requires not only the presence of these nodes, but also the synchronous transmission of information. If the subscriber uses the meter readings to pay for the consumed thermal energy and does so when it is convenient, even partially use the actual load of the systems is not possible. Another way to estimate the energy consumed is to recalculate the calculated loads on the current conditions by the formula: Where 1 1 k c Q HV j j   , Gcal/h, -total calculated load of heating and ventilation; c t o , °С -calculated outdoor temperature taken according to [12] equal to -39 °C; t o , °Сcurrent outdoor temperature; t i , °С -calculated internal air temperature taken as 20 °C. Assuming that the thermal load on the hot water systems is constant throughout the year, the thermal energy consumed is determined by the formula: , . . .
q as и . q ar , kcal/(m h) -specific hourly heat losses, defined according to the norms of heat losses in accordance with the design standards of thermal insulation for piping and equipment for each diameter pipeline with an annual average conditions of operation of the heat network, for underground strip total for supply and return piping and separately for ground installation; L, m, -length of pipelines on a site of a thermal network with diameter d in two-pipe calculation at under-ground laying and on the giving (return) line at overground laying; β -coefficient of local heat losses, taking into account the thermal losses of valves, compensators, supports; adopted for underground channel and aboveground gaskets equal to 1.2 with diameters of pipelines up to 150 mm and 1.15 with diameters of 150 mm and more, as well as with all diameters of the channel-free gaskets. Analysis of the calculated load raises doubts about their absolute reliability. The calculated load on the DHW corresponds to the maximum load, rarely achievable in reality. Therefore, to use the contractual loads as calculated to estimate the actual consumption for heating, ventilation and hot water, the coefficients m1, m2, m3 were introduced into equation (1), after which the equation took the form: To obtain the coefficients m1, m2, m3 it is necessary to use the least squares method [14]. The initial data are: -telemetry data -average monthly values of ambient air temperature and the coolant supply and return lines, the values of the costs in the supply and return lines; -contractual (settlement) load for heating and ventilation, calculated on the actual average conditions; -heat losses converted to actual average monthly conditions. The adequacy of the mathematical model to the real process was checked for all days included in the sample. The deviation of the predicted values from the actual values was estimated by the formula: %, 100 where Q , Gcal/day, -predicted value of thermal energy consumption; Q, Gcal/day, -actual heat release.   (14); m 1 , m 2 -linear regression coefficients (11). The adequacy of the model to the actual data can be checked only for the heating period, since the coolant consumption for heating and ventilation is 0 in the summer. The criterion is the deviation of the amount of coolant costs for heating, ventilation and hot water from the amount of measured costs in the supply lines:

Research object
The heat and mass balance were made for the real heat network. The way of regulation is qualitative. Wiring diagram for domestic hot wateropen. The source has 3 mains, in the summer period only 2 of them more often work. The periodicity of the data collection is a day. The data of telemetry include period of 629 days. Information about actual consumption of thermal energy is practically absent. It did not make sense to judge the consumption of thermal energy by payments for it, since the analysis of payments showed the absence of the system both in the payment periods and in the methods of calculation. Only heating network was connected to 899 subscribers.
The system of pipelines consisted of 1295 sections of different diameters, methods and years of laying.

Results
When compiling the heat balance, it is important to use daily telemetry data at the source. The outdoor temperature was presented both actual and predicted. The samples of each of them did not fully cover all 629 days, while there was a time period when both temperatures were present. The frequency distribution shows that the difference between the predicted and actual temperatures is described by the normal distribution with the parameters m=0.0, =1.4 (Fig.1), and in the absence of the actual temperature can use the predicted. The accumulation of data makes it possible to assess climate change (Fig.2). Temperature fluctuations during the month make it difficult to assess the nature of the changes (Fig.2), averaging of daily temperatures during the month allows to compare them with reference data [12]. The average monthly average actual data is the same as the reference data. For example, cold February 2014 is compensated by warm March and April. Analysis of the temperature in the supply and return pipeline at the source (Fig.4) makes it possible to estimate deviations from the accepted temperature schedule. The temperature of the mains water is too high in summer (Fig.5), in the heating period at low outside temperatures, the coolant temperature is lower than required by the schedule To analyze the data, it is not enough to visualize them on the charts, the use of statistical methods allows to quantify the results. For example, the statistical analysis presented in the table. 1, showed that it is possible to accept the hypothesis of the constancy of the daily total flow rate in all supply lines only for heating periods 2013-2014 (sample 1) and 2014-2015 (sample 3).  The results of the regression analysis are presented in the table. 2. Model 1 includes the 3 components of the balance sheet (11); model 2 -2 terms -heat loss of the piping to be equal to 0; model 3 -3 components, the coefficient of m 3 is initially equal to 1. In all three models, the deterministic coefficient r 2 is close to 1, indicating no difference between actual and estimated values. The actual and predicted values of heat output to the network are shown in Fig. 6.
Repeating the numbers of the months is the month of transition from the heating mode to summer and back Values calculated according to the formula (12), is the same for models 2 and 3. The greatest error have days, which coincide with the period of transition from the heating period to the summer and back. During the summer period, periods of 8 days are allocated, which are characterized by significant deviations of the predicted values from the actual ones. This is due to the change in the mode of operation of the heat network (preventive maintenance and disconnection of consumers). Average  values are -3.8% for model 2 and -5.5% for model 3. The ratio of the coolant flow G HW calculated by the formula (14) to the actual makeup consumption G was analyzed separately for each season. Table 3 shows the average ratio, standard deviation, and confidence intervals. The coefficients m1 and m2 obtained for the approximating dependence (11) were used to predict the total flow rate in the supply lines according to the formula (15). The prediction s -0.7% for sample 1 and 1.4% for sample 3.