EXPERIMENTAL AND STATISTICAL METHODS OF RESEARCH AND OPTIMIZATION OF ELECTRIC POWER SYSTEMS MODES UNDER UNCERTAINTY CONDITIONS

ABSTRACT

The mathematical modeling issues of the electric power systems modes in conditions of inhomogeneities such as time drift of the main indicators are considered. In incomplete knowledge conditions of the mechanism of phenomena, mathematical modeling was carried out by experimental and statistical methods, namely: experiment planning was used. Also, optimization problems are considered, in particular gradient problems, including the "steep ascent" method, which is preceded by a local description of the response surface using a full or fractional factorial experiment.

АННОТАЦИЯ

Рассмотрены вопросы математического моделирования режимов электроэнергетических систем в условиях неоднородностей типа временного дрейфа основных показателей. В условиях неполного знания механизма явлений, математическое моделирование производилось экспериментально-статистическими методами, а именно: использовалось планирование эксперимента. Также рассмотрены оптимизационные задачи, в частности градиентные задачи, в том числе метод «крутого восхождения», которому предшествует локальное описание поверхности отклика с помощью полного или дробного факторного эксперимента.

Keywords: time drift, electric power systems modes, experiment planning, mathematical models, electrical circuits.

Ключевые слова: временной дрейф, режимы электроэнергетических систем, планирование эксперимента, математические модели, электрические цепи.

One of the urgent tasks is to determine the optimal modes of power systems in the process of production, transmission and distribution of electric energy. At the same time, special attention is paid to the problems of optimal planning of short-term regimes in power systems under conditions of uncertainty, taking into account the probabilistic nature of the initial information.

With optimal planning of power system modes, the calculated parameters of electrical networks and loads of power consumption nodes are used [1,3].

To achieve the optimal mode, calculations should be carried out taking into account the probabilistic nature and partial uncertainty of the initial data on the loads of the elements of electrical circuits.

Traditional methods are associated with an experiment that requires a lot of effort and money, since it is time-consuming and based on alternating variation of independent variables in conditions when other parameters tend to remain unchanged.

In this case, there is no real possibility of a comprehensive study of the high quality of the elements of power systems and as a result, many decisions are made on the basis of incoming information, which is random.

Currently, two approaches for solving of modeling problems have been defined: deterministic and experimental – statistical. In the case of a deterministic approach, the solution of problems is preceded by a comprehensive study of the process and, as usual, it is given in the form of some system of differential equations.

Most of the real processes in the electric power industry are complex and are influenced by a large number of interrelated factors, and in this regard, the use of deterministic methods is largely difficult. On the other hand, theoretical consideration is not able to take into account the whole variety of really acting factors, and therefore the theoretical mathematical model loses its power to a greater extent when moving to real conditions.

Under conditions of incomplete knowledge of the mechanism of phenomena, the problems of mathematical modeling can be solved by experimental and statistical methods.

In this regard, mathematical methods of experiment planning are becoming increasingly widespread in solving problems related to the modes of operation of electric power systems under conditions of uncertainty such as time drift of the main operating parameters of power systems.

The main mathematical apparatus for processing the results of observations using experimental planning methods is regression analysis, which consists in estimating the regression coefficients using the least squares method followed by statistical analysis of the resulting polynomial model [2,4,5]:

When planning an experiment, when studying the modes of power systems, one has to deal with inhomogeneities sources of a continuous type. These sources cause a continuous change in energy regimes – the drift of their output indicators over time and are expressed in the form of additive drift, that is, the displacement of the response surface without its deformation, and the drift function itself has a fairly smooth character and can be represented by a polynomial of a low degree.

                                                    (2)

During the planning an experiment under inhomogeneities conditions, in the general case, the results of observations Y (power balance, load, conditional fuel consumption, etc.) are an additive mixture of changes in the output f(x), caused by a vector of variable factors x of continuous or discrete drift Y(ω) caused by a vector of uncontrolled factors ω and some error ε with normal distribution:

                                                 (3)

The task is reduced to choosing a plan that is orthogonal to the identified drift and provides the best estimates of the effects of the studied factors, regardless of the influence of drift.

Experimental studies have established that the optimization parameter Y – the consumption of conventional fuel undergoes a time drift with properties close to linear. In this regard, planning orthogonal to drift can be obtained using a complete factorial experiment (CFE) at two levels.

For our case, the matrix of the complete factorial is given in Table 1.

Table 1.

The matrix of the complete factorial

For represent N values of the linear drift,  is necessary for the first N columns of such matrix:

                                        (4)

where

   ;                                                       (5)

                                                                (6)

Equality (6) follows from the fact that the first Chebyshev polynomial is , used to represent a linear combination of K first column vectors of the matrix   [2,6,7].

Any of the remaining columns can be considered as a vector of the desired planning and the rule for obtaining planning orthogonally to linear drift can be formulated as follows: for N observations, make a CFE matrix and discard the first  columns in it. The remaining part of the matrix is the desired planning for determining l≤N–L-1 coefficients of influence and mutual influence on the output of controlled factors.

By calculating the expansion coefficients, we can check the drift linearity. If (6) is satisfied well enough, then the drift linearity is performed.

Significance evaluation of the mathematical coefficients model is carried out by the usual method according to the Fisher criterion [3].

When constructing a mathematical model of the power systems operating parameters using the planned experiment methods, the issue of identifying the main parameters and their controllability becomes important. The research of the power systems regime processes, as well as the analysis of a priori information contained in the practical experience of technologists and operators, allowed us to identify the main parameters that have a significant impact on the course of the process and, thereby, determine the qualitative indicators [4,5]:

 – fuel outgoings in thermal power plants;

 – power balance in the power system;

 – powers at stations;

 – power flow in power transmission lines;

Y is the consumption of conventional fuel at thermal power plants.

The interrelation of the process parameters can be represented as:

                                         (7)

As a result of the lack of a priori information about the degree of the polynomial model of the process by parameter Y, a linear mathematical model was developed at the first stage of statistical processing of experimental data accumulated according to type  СFE [6,7]. By checking according to the F – Fisher criterion, its inadequacy was established. The inclusion of interaction effects also did not give positive results. At the next stage, the implementation of the orthogonal plan, followed by the calculation and statistical analysis of the results obtained, allowed us to develop a mathematical model of conventional fuel consumption at a 5% significance level:

                           (8)

For  the calculated value of the Fisher criterion F_r=2.5/1.3=1.92 is determined. With =10 and , we find . Thus, equation (8) can be considered correct with a 95% confidence probability [8]:

()

The power system with thermal power stations is characterized by the presence of such uncontrollable factors as fuel costs in thermal power stations, power balance in the power system, power at power stations, power flows in power transmission lines, etc.

All this leads to the fact that the output parameter Y changes indefinitely over time. There is a time drift of the power system characteristics.

Taking into account the time drift, the total time of the experiment was reduced to a minimum. In order to determine under these conditions k=4 linear effects of the influence of controlled factors according to factor planning, a minimum of N= 8 experiments is required. With the time of one experiment ∆t=5-6 hours, the total time required for the entire experiment was Т=∆t∙N=48 hours [9].

It was assumed that time drift in this interval could not differ much from the linear one. The absence of interactions of controlled factors was also postulated (the variation steps were chosen as little in order to neglect the interactions).

For evaluating the desired linear effects, it was decided to use the scheme of the experiment type  (Table 1), putting in it ; ; ; .

As a result of the experiment, eight values of optimization parameters were found, each of which had three repetitions.

The values of the linear model coefficients were calculated using the usual factor planning formulas:

The values of the linear model coefficients calculated by the results of the experiment turned out to be equal:

The coefficients  in the drift decomposition ( calculated by formulas (10) are equal to: .

Thus, the equation of the desired dependence for the encoded variables , free from time linear drift, has the form:

A variance analysis of this mathematical model is carried out. The total sums of squares of  , are calculated, and the residual sums of squares of   are determined.

Calculating the ratio of the effects variance to the residual variance and comparing them with the tabular value of the F-ratio, the significance of the coefficients a, as well as the presence of time drift were checked [10].

The analysis of the optimization method of power systems modes was carried out, which showed that they differ from each other in convergence, accuracy, and limitations. Gradient methods are the most accessible from the point of view of practical applicability. Gradient methods of finding optimal solutions include the method of "steep ascent". The peculiarity of this method is that in this case, the movement along the gradient is preceded by a local description of the response surface using a full or fractional factorial experiment. The search for the optimum is carried out when moving from the starting point by simultaneously changing all the factors, taking into account the product  for each of them. Taking into account the magnitude of the step ε when choosing a step is necessary due to the fact that the magnitude of the regression coefficients changes when the magnitude of the variation intervals changes.

The "steep ascent" options were planned for the optimization parameters , , , .

The local optimization results showed that simultaneous solution of local optimization problems is impossible, since the criterion of one problem, for example,  conflicts with others, for example, , that is, the optimal solution according to one of the efficiency criteria, turns out to be unsatisfactory according to other criteria [11].

Analysis of local criteria has shown that it is necessary to establish certain requirements for optimal maintenance of the regime in electric power systems.

Based on these requirements, a vector optimality criterion is proposed.

under restrictions  

where  are the upper and lower limits allowed by the technological regulations.

The  function minimization can be implemented by a scanning method characterized by simplicity of calculations. To reduce the amount of calculations, we can use an algorithm with a variable scan step.

References:

1. T.S. Gayibov, S.S. Latipov. To the optimal planning of power system modes in conditions of partial uncertainty of the initial information. Bulletin of TashSTU 2/2019. – pp. 88-94 (05.00 №16).
2. S Amirov, D Rustamov, N Yuldashev, U Mamadaliev, M Kurbanova. Study on the Electromagnetic current sensor for traction electro supply devices control systems// ICECAE 2021 IOP Conf. Series: Earth and Environmental Science 939 (2021) 012009 р
3. T.S. Gayibov, Sh.Sh. Latipov "Optimization of electric power modes in conditions of interval uncertainty of initial information". Problems of energy and resource conservation. Tashkent, No.3-4, 2019, pp. 203-209. (05.00.00, No.2)
4. Kotelnikov A.V. Electrification of railways. Global trends and prospects. Moscow. Intencst, 2002 – 104 p.
5. Mirjalil Yakubov, Kamolbek Turdibekov, Absaid Sulliev, Islom Karimov, Saydiaziz Saydivaliyev, and Sarvar Xalikov. E3S Web of Conferences 304, 02014 (2021) ICECAE 2021. 02014 pp.
6. Allaev K.R. Energy of the world and Uzbekistan. Tashkent. Publishing House “Moliya”. 2007 – 388 p .
7. Vasilyansky A.M., Mamoshin R.R., Yakimov G.B. Improvement of traction power supply of railways electrified with alternating current of 27.5 kV, 50 Hz. Railways of the world. 2002, No. 8 – pp. 40-46.
8. High-speed railway transport. General course; textbook in 2 volumes. I.P.Kiselyov et al. Moscow, 2014. Vol.1(308 p.) – Vol.2 (372s.).
9. Dommel h.w., Tinney w.f. Optimal power flow solution.-IEEE Trans. Power Apparatus and Systems, vol.PAS-87, 1968,p.1866-1876.
10. Hajdu L.P., Peschon w.i., Piercy D.S., Tinney w.f. Optimal load-scheduling policy for power systems. ieee Trans. Power Apparatus and Systems, vol. PAS-3, 1968, p.784-795.
11. Happ H.H., Wirgau K.A. Optimal dispatch and voltage scheduling in power systems. Proc. Int. Symp. Circuits and Systems, 1981, p.759-763.
12. Sasson A.M., Merrill H.H. Some applications of optimization techniques to power systems problem. Proc.IEEE, vol.62, 1974, p.959-972.