Determination of the Optimal Location of a Power Supply Source in Distribution Networks Using an Analog Device
Abstract Generated abstract
This paper describes the use of an analog device to determine the economically optimal location of a power supply source in distribution networks whose radial line sections run parallel to rectangular coordinate axes. The authors show that the cost function for such networks can be minimized by applying the device separately along each coordinate axis, extending a method previously used for radial-beam networks and allowing analogous treatment in three-dimensional space. An example from a residential district block compares the computed optimum with a load center method and an existing installation: the optimal and load-center placements yield nearly identical reduced costs, while the existing location gives costs about 22 percent higher.
Full Text
Determining the Optimal Location of a Power Source in Distribution Networks Using an Analog Device
N. A. Dulzon, R. I. Zakirov
(Presented by the scientific seminar of the Departments of Electric Power Stations and Electrical Systems and Networks)
At the Department of Electrical Systems and Networks of TPI, an analog device has been developed that makes it possible to determine the location of a power source (PS) in radial-ray networks [1]. It appears possible to use the same device to find the location of the PS in cases where sections of radial lines are laid parallel to the axes of rectangular coordinates (for two- and three-dimensional space). In this case, the function of the reduced calculated costs for the network (Fig. 1) will be
\[ Z=\sum_{k=1}^{n} z_k l_k =\sum_{k=1}^{n} z_k\left(|x-x_k|+|y-y_k|\right)= \tag{1} \]
\[ =\sum_{k=1}^{n} z_k |x-x_k|+\sum_{k=1}^{n} z_k |y-y_k|, \]
where
\(z_k\) — reduced calculated costs per unit length of line;
\(l_k\) — length of the line to the \(k\)-th load;
\(x, y\) — coordinates of the PS;
\(x_k, y_k\) — coordinates of the \(k\)-th load.
Fig. 1. \(1, 2, \ldots, k, \ldots, n\) — numbers of electric-power receivers; PS — power source.
The analog device [1] makes it possible to find the minimum of the function:
\[ З=\sum_{k=1}^{n} з_k \sqrt{\rho^2+\rho_k^2-2\rho\rho_k\cos(\varphi-\varphi_k)}, \tag{2} \]
where
\[ \begin{aligned} \rho,\ \varphi &\text{— the radius vector of the PS and the corresponding polar angle;}\\ \rho_k,\ \varphi_k &\text{— the corresponding quantities for each load.} \end{aligned} \]
For \(\varphi=\varphi_k\), this expression takes the form:
\[ З=\sum_{k=1}^{n} з_k(\rho-\rho_k), \]
which is analogous to the components of function (1). Thus, with the aid of the analog device it appears possible to find the minimum of function (1), if the minimization is carried out separately along the \(x\)-axis and the \(y\)-axis. To obtain the total reduced calculated costs for the entire network, it is necessary to sum the costs along the axes. One must proceed similarly when finding the location of the PS in three-dimensional space.
As an example, let us consider the electric power supply of an actual block of a residential district (Fig. 2), for which we shall determine the optimal location of the PS and compare it, in terms of total reduced calculated costs, with the location of the PS found by the “center of gravity of loads” [2] method and with the actually existing location of the PS.
Fig. 2. Plan of a block of a residential district
The initial data for solving the problem of determining the optimal location of the PS on the analog device are the coordinates of the consumers in a rectangular coordinate system and the specific reduced calculated costs.
As a result of the calculation on the analog device, the optimal location of the PS was obtained with coordinates \(x=115\ \text{m}\); \(y=122\ \text{m}\) (point \(O_1\)) and with total reduced calculated costs of 1.077 thousand rubles. But it is not possible to install the PS at this point (the corner of a residential building), and therefore the PS must be shifted to point \(O_2\), with coordinates \(x=110\ \text{m}\), \(y=111.5\ \text{m}\); in this case the total reduced calculated costs became equal to 1.080 thousand rubles.
Calculation by the “center of gravity of loads” showed that the PS should be placed at point $O_3$ with coordinates $x = 133\ \text{m}$, $y = 105\ \text{m}$; in this case the total reduced calculated costs became equal to 1.085 thousand rubles, i.e., in practice there proved to be no difference in the reduced calculated costs between the optimal placement of the PS and placement of the PS according to the “center of gravity of loads” in the given case.
For the actually existing location of the PS, the total reduced calculated costs proved to be equal to 1.317 thousand rubles, which is 22% greater than the costs when the PS is located at the optimal point.
Conclusion. The given analog device makes it possible to find the optimal location of a PS both in radial-beam networks and in those cases when sections of radial lines are laid parallel to the axes of rectangular coordinates. In addition, it is quite simple to estimate the increase in costs when the PS is located at any other point, which is especially important when carrying out practical calculations for PS placement.
REFERENCES
- N. A. Dulzon, R. I. Zakirov. Electrification of metallurgical enterprises of Siberia (based on the materials of a scientific-technical conference), Novokuznetsk, 1971.
- A. A. Fedorov. Fundamentals of power supply for industrial enterprises. “Energiya,” Moscow, 1967.