Model and Algorithm of Electricity Consumption Management for Household Consumers in the Republic of Tajikistan

In the process of searching for a solution to the scientific problem posed, we collected data on power consumption in 9 cities of Tajikistan. The study was based on consumers living in 9 cities of Tajikistan Dushanbe (D), Khorog (X), Bokhtar (B), Kulyab (K), Garm (G), Istaravshan (I), Isfara (Is), Khadzhent (Khad.) and Panjakent (P) total 980 consumers (people).

A visual analysis of the data obtained made it possible to put forward an assumption about the dependence of power consumption on a number of parameters. The purpose of further research was to substantiate this assumption and identify these parameters. The method of achieving the goal was mathematical modeling:

$\mathrm{W}=\mathrm{f}(\mathrm{t} ; \mathrm{h} ; \mathrm{c} ; \mathrm{s})$     (1)

where, W- the amount of power consumption given to us in units of measurement, kWh;

t-temperature coefficient in showing the extreme temperature difference during the study period, given to us ℃.

h-is the geographical coefficient associated with the height of the local area of residence of energy consumers above sea level, the reference value given to us in meters.

с-coefficient of constructive and technological readiness of buildings and structures, consumer places of residence, taking into account the number of storeys of the building, percentage of wear and tear and physical characteristics of networks and wiring in the building, network throughput, type of electricity meters, technologies for collecting data on electricity consumption adopted in this group consumers, given in conventional level units (1 - zero readiness, 2 - low readiness, 3 medium readiness, 4 readiness above average, 5 - high readiness).

S-is the welfare coefficient of consumers living compactly, taking into account the population density, the price per kWh of electricity and the average annual income per capita, given to us per day in units / person.

The modeling was carried out in two stages.

At the first stage:

1) statistical processing of empirical data collected in 9 cities of Tajikistan for 36 months (three years);

2) checking consumer groups for homogeneity in terms of electricity consumption;

3) analysis of the statistical significance of each of the parameters in homogeneous groups

4) search for a form of mathematical recording of the dependence of energy consumption on the identified parameters.

The presented electrical consumers were selected after a qualitative analysis of the structural and technological features of buildings and structures: the main development of cities and the choice of methods of power supply and metering of power consumption were formed in these groups before the beginning of the study period, are identical and have undergone minimal transformation during the study period. Thus, the homogeneity of the groups by parameter "c" was revealed through observations, interviews of experts, work with technical documentation. It turned out to be at the average level.

Statistical processing of empirical data was carried out by the method of finding the average seasonal (winter, spring, summer, autumn) and average annual values of electricity consumption and temperatures in order to level them out and find a trend.

$\mathrm{t}_{0}=\left(\mathrm{T}_{\max }-\mathrm{T}_{\min }\right)_{\mathrm{i}} / 36$      (2)

where, T_max and T_min respectively, the extreme maximum and minimum temperature values for the investigated month of the year, i is the month of the year for the three-year study period, t₀- is the average annual temperature coefficient.

$t_{\text {winter}}=\left(T_{\max }-T_{\min }\right)_{i} / 9$     (3)

where, T_max and T_min respectively, the extreme maximum and minimum temperature values for the studied month of the winter period, i-is the month of the winter period for the three-year period of the study, T_winter-is the average seasonal winter temperature coefficient.

$t_{\text {Spring}}=\left(T_{\max }-T_{\min }\right)_{i} / 9$     (4)

where, T_max and T_min respectively, extreme maximum and minimum temperature values for the studied month of the spring period, i-is the month of the spring period for the three-year period of the study, t_spring-is the average seasonal spring temperature coefficient.

$t_{\text {summer}}=\left(T_{\max }-T_{\min }\right)_{i} / 9$     (5)

where, T_max and T_min respectively, the extreme maximum and minimum temperature values for the investigated month of the summer period, i - is the month of the summer period for the three-year period of the study, t_summer-is the average seasonal summer temperature coefficient.

$t_{\text {autumn}}=\left(T_{\max }-T_{\min }\right)_{i} / 9$     (6)

where, T_max and T_min respectively, the extreme maximum and minimum temperature values for the studied month of the winter period, i-is the month of the autumn period for the three-year study period, t_autumn-is the average seasonal autumn temperature coefficient.

As a result of processing, the statistical data took the form (Table 1).

By pairwise comparison of the parameter value in the groups with an average sample value of the parameter, all nine groups were tested for homogeneity in terms of the average annual energy consumption parameter. Pearson's X² statistical goodness test was used for testing.

The statistical null hypothesis Ho was that the two compared groups of consumers are similar in terms of average annual energy consumption for the three years under study.

The X² value for the groups ranged from 6 to 46.23.

К (degree of freedom)=1,

Р (significance level)=0.05,

Table value $\mathrm{X}_{\text {critical}}^{2}$=3.84.

$\mathrm{X}_{\text {critical}}^{2}<\mathrm{X}^{2} \min =3.84<6$

Since the critical value is less than the calculated one, then the null hypothesis at this level of significance can be accepted, we consider the groups to be homogeneous in the average annual consumption.

As a result of testing at a significance level of 0.05, the hypothesis was confirmed for all experimental groups, but a large scatter of X² values required an additional study for homogeneity in some groups. As a result of increasing the power of the criterion, that is, at a significance level of 0.01, a block of groups heterogeneous with the rest of the groups according to the proposed parameter was revealed.

Table 1. Statistical data on temperatures and electricity consumption for the cities under consideration

So the value of X² in the groups fluctuated equal to 6 and 6.2, respectively.

When К (degree of freedom)=1,

Р (significance level)=0.01,

Table value $\mathrm{X}_{\text {critical}}^{2}$=6.64.

$\mathrm{X}_{\text {critical}}^{2}>\mathrm{X}^{2} \max =6.64>6.2$

Therefore, the hypothesis of their homogeneity should be rejected (Table 2).

Table 2. Blocks of homogeneous and heterogeneous groups in terms of height above sea level

Consequently, in the groups Had., Kh., D., G., I. there is a factor statistically significantly influencing, other things being equal, on electricity consumption.

A detailed qualitative analysis of the differences between these groups confirmed the assumption and statistical conclusions: Electricity consumption is a function of the consumer's location above sea level, that is, altitude above sea level is a statistically significant factor affecting electricity consumption.

Further analysis of the empirical data revealed that experimental groups homogeneous in terms of average annual electricity consumption have significant statistical distinction when comparing seasonal energy consumption.

The null hypothesis Ho was as follows: the experimental groups compared in pairs are statistically indistinguishable in terms of energy consumption. Statistical method for assessing agreement - Pearson X² test.

As a result of statistical calculations, groups were revealed that are homogeneous and heterogeneous in terms of seasonal power consumption.

Within each season, the value of X² in the studied experimental groups Had., D., K., B., H., G., P., Is., fluctuated:

in the range [2.48; 3.5] in groups, Hud., D., K., B.

in the range [6, 6; 9.2] in groups X., G., P., Is.

Insofar as К=1,

at р=0.05, 3.84>3.5 i.e. $\mathrm{X}_{\text {critical}}^{2}>\mathrm{X}^{2} \max$

Hence there are grounds for the given groups Had., D., K., B. to accept the hypothesis of their homogeneity, but since $\mathrm{X}_{\text {critical}}^{2}<\mathrm{X}^{2} \min =3.84<6.6$, then for the groups X., G., P., Is. it makes sense to reject the hypothesis.

The results are shown in Table 3.

Table 3. Blocks of homogeneous and heterogeneous groups by the parameter of well-being

Revealed dependences of the influence of factors on seasonal power consumption (see Tables 1-3), we present the comparative dependences for several cities with different factor parameters, Figures 1-2. When comparing, we will take data from winter and summer periods.

1.png

Figure 1. Comparison of the distribution of power consumption values with different temperatures and altitudes above sea level

2.png

Figure 2. Comparison of the distribution of energy consumption values with the same temperatures and well-being but at different altitudes above sea level

The groups were subjected to a qualitative analysis of their characteristics, in the course of which it was found that they were distinguishable by the criterion of consumer welfare. So at low temperatures, the amount of electricity consumption increases statistically in those cities where consumers with high per capita income live, since they turn on additional power consumption devices.

Thus, as a result of the first stage of mathematical modeling, statistically significant factors of influence on energy consumption were identified:

t-temperature coefficient,

h-is the height of the local area of residence of energy consumers above sea level,

с-coefficient of structural and technological features of buildings and structures.

S-is the consumer welfare coefficient.

As parameters, these factors were statistically checked for significance, mathematically justified the feasibility of their inclusion and accounting when constructing a mathematical predictive model of power consumption.

Based on the identified factors of influence on power consumption in the conditions of the Republic of Tajikistan, a model for forecasting power consumption is proposed.

$W_{\text {forecast.}}=W_{\text {temporary norms}} \cdot\left(1-\alpha_{i}\right)$     (7)

The coefficient taking into account the factors affecting power consumption is presented in the following form:

$\alpha_{i}=\left(\left(t_{i} \cdot h+c\right) \cdot s\right) /(s+k)$     (8)

where, (ti·h+c)-coefficient that takes into account temperature, geographic (altitude) and design and technological coefficients, an analogue of the constant coefficient in the Tornquist function.

s-consumer welfare, an analogue of income in the Tornquist function.

k-coefficient that takes into account the dynamics and seasonal fluctuations in consumer welfare.

i–month of study.