Selection of Power Source Placement in a Distribution Network under Technical Specification Constraints
Abstract Generated abstract
The paper formulates the problem of locating a power source in radial distribution networks as a technical and economic optimization subject to voltage deviation constraints at consumer substations. It replaces the conventional load center approach with minimization of reduced calculated costs, expresses costs as functions of distances from the source to substations, and determines admissible placement regions from limiting line lengths imposed by voltage losses. A procedure using Newton’s method followed by a feasibility check and boundary adjustment is illustrated on a three substation example, where the constrained optimum differs from the load center and yields lower calculated costs while satisfying required bus voltage levels.
Full Text
SELECTION OF POWER SOURCE PLACEMENT IN A DISTRIBUTION NETWORK WITH TECHNICAL CONSTRAINTS
N. A. Dulzon, R. I. Zakirov
(Presented by the Department of Electrical Systems and Networks)
Under the new conditions of development in the electric-power industry, the development of new, more modern calculation methods for selecting elements and power-supply systems is an important link in solving the problem of increasing production efficiency. Among such issues is the problem of designing a distribution network.
In designing power-supply systems, one of the factors affecting technical-and-economic indicators is the selection of the location of the power source.
The choice of the location of the power source in distribution networks must be made while simultaneously taking into account economic and technical indicators. This article poses the problem of finding the optimal placement of the power source in radial distribution networks under technical constraints on voltage deviations.
Radial schemes of electrical networks are widely used in power-supply systems for cities and industrial enterprises; therefore, the question of selecting the location of the power source in such schemes is a pressing one.
The existing method for finding the location of the power source, based on applying the concept of the “center of gravity of loads,” is not optimal in the sense of minimizing the reduced calculated costs.
The criterion of optimality for selecting the location of the power source should be taken as the minimum of reduced calculated costs, determined in accordance with the methodology of technical-and-economic calculations in power engineering [1] and with consideration of technical constraints on voltage deviations.
For the adopted optimality criterion, we obtain the following mathematical formulation of the problem:
$$ Z = \sum_{\kappa=1}^{m} (pK_{\kappa} + C_{\kappa}) = \min \tag{1} $$
subject to the constraints
$$ (U_{\text{ip}} - \Delta U_{\kappa} - \Delta U_{\text{t}\kappa}) \frac{1}{K_{\text{t}\kappa}} \geq U_{\text{ж}\kappa}, \tag{2} $$
where \(p\) is the total coefficient of annual deductions;
\(K_k\) — capital costs of the \(k\)-th consumer;
\(C_k\) — annual operating expenses of the \(k\)-th consumer;
\(U_{\mathrm{ip}}\) — voltage on the buses of the power-supply source, kV;
\(\Delta U_k\) — voltage losses in the lines from the power-supply source to the consumer substation, kV;
\(\Delta U_{\mathrm{tk}}\) — voltage losses in the transformers, kV;
\(K_{\mathrm{tk}}\) — transformation ratio of the transformers;
\(U_{\mathrm{zhk}}\) — desired voltage on the buses of the consumer substations, kV.
Capital costs and annual operating expenses are functions depending on the length of the lines, while the line length can be expressed through the coordinates of the consumer substation \(x_k, y_k\) and of the power-supply source \(x, y\). Then the reduced calculated costs for the construction and operation of the network will be represented in the form of an explicit function of the sought coordinates of the power-supply source:
\[ Z=\sum_{k=1}^{m} z_k \sqrt{(x-x_k)^2+(y-y_k)^2}, \tag{3} \]
where \(z_k\) are the specific reduced calculated costs.
To solve the stated problem in the presence of constraints in the form of inequalities, the following procedure is proposed.
From (2), after substituting the known quantities, it is not difficult to determine the limiting values of the line lengths \(l_k\), above which the voltage on the buses of the consumer substations will be less than \(U_{\mathrm{zhk}}\), according to the expression
\[ l_k \le \frac{ n U_{1k}\left\{ n'\left[U_{\mathrm{ip}}U'_{1k}-(U'_{1k})^2\right]-(P_{\mathrm{nk}}r_{\mathrm{tk}}+Q_{\mathrm{nk}}x_{\mathrm{tk}})\right\} }{ n' U'_{1k}(P''_{\mathrm{nk}}r_{0k}+Q''_{\mathrm{nk}}x_{0k}) }, \tag{4} \]
where \(U_{1k}\) is the voltage on the high-voltage side of the transformers of the consumer substations, which is equal to
\[ U_{1k}=U'_{1k}+\frac{P_{\mathrm{nk}}r_{\mathrm{tk}}+Q_{\mathrm{nk}}x_{\mathrm{tk}}}{nU'_{1k}}, \tag{5} \]
\(U'_{1k}\) is the voltage on the low-voltage side of the transformers of the consumer substations, referred to the high-voltage side, which is equal to
\[ U'_{1k}=U_{\mathrm{zhk}}K_{\mathrm{tk}}; \tag{6} \]
\[
S_{\mathrm{nk}}=P_{\mathrm{nk}}-jQ_{\mathrm{nk}}
\]
— line load;
\[
S''_{\mathrm{nk}}=P''_{\mathrm{nk}}-jQ''_{\mathrm{nk}}
= S_{\mathrm{nk}}+\Delta S_{\mathrm{mtk}}+\Delta S_{\mathrm{st,tk}}
\]
— line load taking into account power losses in the transformers;
\(r_{0k}, x_{0k}\) — active and reactive resistance of 1 km of line;
\(r_{\mathrm{tk}}, x_{\mathrm{tk}}\) — active and reactive resistance of the transformer;
\(n\) — number of circuits of the distribution line;
\(n'\) — number of transformers at the substation.
With radii equal to \(l_k\), and with centers corresponding to the locations of the consumer substations, arcs are drawn, i.e., the region of admissible values of \(U_{\mathrm{zh}}\) is identified.
After this, the system of equations (1) is solved by approximate methods (in this case the Newton method [4] is used).
Then a check is made of whether the constraint conditions are satisfied, i.e., whether the power source lies in the region of permissible values of \(U_{\text{ж}}\). If the power source is in this region, the problem is solved; if not, the power source must be shifted to the nearest boundary arc in order to satisfy the constraint conditions.
This method is applicable to networks whose cross-section is determined before the start of the calculation and does not depend on the location of the power source; the number of power sources is also determined.
Fig. 1
Example. For the scheme shown in the figure, it is required to select the optimal location of the power source with constraints on voltage deviations. The loads at the consumer substations are specified by schedules, whose characteristics are given in the table.
Table
| No. of step \((i)\) | Duration of step \((t_i)\), h | Load magnitude \((P_i)\), MW | Load magnitude \((P_i)\), MW | Load magnitude \((P_i)\), MW |
|---|---|---|---|---|
| No. of step \((i)\) | Duration of step \((t_i)\), h | 1st substation \((k=1)\) | 2nd substation \((k=2)\) | 3rd substation \((k=3)\) |
| I | 2 | 1.8 | 0.45 | 0.15 |
| II | 6 | 1.35 | 0.45 | 0.15 |
| III | 4 | 3 | 2.7 | 1.2 |
| IV | 2 | 1.35 | 0.9 | 0.45 |
| V | 2 | 3 | 2.4 | 0.3 |
| VI | 4 | 2.4 | 1.05 | 0.3 |
| VII | 4 | 2.1 | 0.45 | 0.15 |
Taking the costs of overhead lines according to data from [2] and the total coefficient of annual charges according to [3], the values of the specific reduced calculated costs in the case where the loads are specified by a schedule will be determined as
\[ z_k=\sum_{i=1}^{n} z_{ik}t_{ik}, \tag{7} \]
where \(z_{ik}\) are the specific reduced calculated costs of the step of the load schedule of the \(k\)-th consumer;
\(t_{ik}\) is the duration of the \(i\)-th step of the load schedule of the \(k\)-th consumer,
and they will be equal to \(z_1 = 1.6584\) thousand rubles/km; \(z_2 = 1.1764\) thousand rubles/km; \(z_3 = 0.6809\) thousand rubles/km.
By Newton’s method, the optimal location of the power source is determined as: \(x = 3.75\) km; \(y = 2.87\) km.
Next, fulfillment of the constraint conditions is checked using expression (4) for the case when \(U_{zhk} = 0.4\) kV,
\[ k_{tk} = \frac{U_{\mathrm{otv\,min}} k}{U_{\mathrm{nn}}}, \]
i.e., the minimum possible value, and the load is maximum. As a result of the calculation, one obtains: \(l_1 \le 17.5\) km, \(l_2 \le 14.85\) km, \(l_3 \le 15.75\) km.
The region of admissible values of \(U_{zh}\) is shown in the figure.
It turns out that the optimal location found for the power source is not in the region of admissible values of \(U_{zh}\); therefore it must be shifted to the boundary arc drawn with radius \(l_2 = 14.85\) km. The new coordinates of the power source, taking into account the voltage-loss constraints, will be: \(x' = 4\) km; \(y' = 3\) km.
The coordinates found for the power source differ from the coordinates of the center of gravity of the electrical loads, which are \(x_0 = 7.4\) km; \(y_0 = 5.8\) km.
The total reduced calculated costs when the power source is located as determined by the proposed method amount to 28.15 thousand rubles, whereas when it is placed at the center of gravity they amount to 28.9 thousand rubles.
Thus, in the first case the costs are reduced by 750 rubles, and at the same time the voltage on the buses of the consumer substations is not less than the desired voltage; i.e., the voltage-deviation constraints are satisfied, which affects the optimal location of the power source in distribution networks.
References
-
Methodology of Technical-Economic Calculations in Power Engineering. State Committee of the Council of Ministers of the USSR for Science and Technology, Academy of Sciences, Moscow, 1966.
-
Electrical Engineering Handbook, vol. II. Energia Publishing House, Moscow, 1964.
-
F. F. Karpov. Calculation of Urban Distribution Networks. Energia Publishing House, Moscow, 1968.
-
B. P. Demidovich and I. A. Maron. Fundamentals of Computational Mathematics. Gosizdat, Moscow, 1963.
-
A. P. Chmutov. Optimal placement of a source of electric-power supply. Elektrichestvo, no. 12, 1969.