On the Need to Account for Damage from Voltage Deviations in the Design of Distribution Networks
Abstract Generated abstract
The paper addresses the design of 10 to 0.4 kV distribution networks, where conductor cross sections are commonly chosen by permissible voltage loss criteria without explicitly considering energy losses or economic damage from voltage deviations at load nodes. It proposes an expanded annual cost function that combines capital costs, power and energy losses, and a damage term assumed proportional to the square of voltage deviation, expressed through line currents, lengths, load curves, and conductor cross sections. Minimizing this function yields economically optimal line cross sections, and a numerical example for a 10 kV cable network indicates that including voltage deviation damage can substantially affect the selected conductor sizes.
Full Text
On the Need to Account for Damage from Voltage Deviations in the Design of Distribution Networks
R. I. Borisov, V. V. Litvak
(Presented by the Department of Electrical Systems and Networks)
Experience in the design and operation of distribution networks shows that it is precisely in 10–0.4 kV distribution networks that the greatest relative magnitude of losses of electrical energy and voltage occurs [1]. In designing such networks, it is customary to determine the cross sections of line conductors from the conditions of permissible voltage losses to the most remote consumer. The magnitude of the permissible voltage losses is then compared with the actual losses under the maximum-load regime. Power and electrical-energy losses are not taken into account in this process.
This paper proposes to refine the existing methods for designing electrical networks by taking into account the damage caused by voltage reduction at the nodes where loads are connected.
The calculated costs for an electrical network, reduced to one year, as a function of the cross sections of the network lines can be written in the following form:
\[ З = (p_{\text{н}} + p_a)\left(A \sum^m l_i + B \sum^m l_i S_i\right) + \tag{1} \]
\[ + 3\rho(c_1 k_m p_{\text{н}} + c_2 \tau)\sum^m \frac{l_i I^2_{i\max}}{S_i}, \]
where \(A\) and \(B\) are the coefficients of the function of the specific capital costs for a line as a function of cross section,
\[ K = A + BS. \]
The calculated costs in the form (1) take into account electrical-energy losses over the year and the cost of additional generating capacity in the power system to cover power losses.
Voltage deviations from nominal may cause changes in the rate of technological processes, changes in the service life of electrical devices, an increase in capital expenditures for voltage-regulation equipment, etc. Voltage deviations at consumers are assessed by the magnitude of the damage to the production of a unit of output; moreover, evidently, both a decrease and an increase in voltage lead to positive values of damage in the sense that the costs of producing a unit of output increase.
The magnitude of the damage from voltage deviation at consumers should be taken into account in the formula for the annual calculated costs for the construction and operation of the electric network; here it is assumed that the damage is proportional to the square of the voltage deviation [2]. Then the formula for the annual calculated costs has the form:
\[ З=(p_{\mathrm{н}}+p_{a})\left(A\sum^{m} l_i+B\sum^{m} l_i S_i\right)+ \]
\[ +3\rho\,(c_1\kappa_{\mathrm{м}}p_{\mathrm{н}}+c_2\tau)\sum^{m}\frac{l_i I^2_{i\max}}{S_i} +\sum^{n} D_q\,(U_{\mathrm{н}}-U_q)^2 . \]
If each load is an \(n\)-dimensional vector of currents in accordance with the load curve, then, expressing the damage formula through the voltage losses up to each \(q\)-th node of load connection, the calculated costs can be expressed through the sought cross sections of the designed network:
\[ З=(p_{\mathrm{н}}+p_{a})\left(A\sum^{m} l_i+B\sum^{m} l_i S_i\right)+ \]
\[ +3\rho\,(c_1\kappa_{\mathrm{м}}p_{\mathrm{н}}+c_2\tau)\sum^{m}\frac{l_i I^2_{i\max}}{S_i}+E+ \]
\[ +D\rho^2\sum^{m}\frac{l_i^2}{S_i^2}\sum^{n} I_{ik}^2\Delta t_k+ \tag{2} \]
\[ +2D\rho^2\sum^{m}\frac{l_i l_j}{S_i S_j}\sum^{n} I_{ik}I_{jk}\Delta t . \]
The value \(E\) depends on the permissible voltage losses, the currents in the network sections, the load curve, the reactances, and the lengths of the network lines. At the beginning of the calculation one may consider the value of the line reactance to be independent of the cross section, and then \(E\) becomes a free term.
Minimization of the calculated-cost function (2) gives the optimal cross sections of the network lines.
Thus, the possibility has been shown of accounting for damage from voltage deviations at load nodes already when designing electric networks. Accounting for damage can have a substantial influence on the magnitude of the cross sections of the network lines. Let us show this by an example.
Fig. 1. Diagram of a 10 kV distribution network.
Select the cross sections of 10 kV cable lines for supplying loads in accordance with the diagram in Fig. 1.
\[ I_{a}=I_{\б}=I_{\в}=70\ \text{A};\quad l_1=l_2=l_3=2\ \text{km};\quad \cos\varphi_{a}=\cos\varphi_{\б}=\cos\varphi_{\в}=0.8 . \]
The load curves are assumed identical and have \(T_{\max}=4200\) h. For installation in the ground, an ASB-10 kV cable is assumed \((K=2.8+0.0205\,S)\) [3].
The results of the calculations are presented in Figs. 2, 3. The calculated costs are expressed in relative units. Curves 1, 3 are plotted without accounting for damage from voltage deviations, and curves 2, 4 in accordance with formula (2).
Fig. 2. Dependence of the annual calculated costs on the cross-section of line 1 for \(S_2 = 70\ \text{mm}^2\) and \(S_3 = 50\ \text{mm}^2\).
Fig. 3. Dependence of the calculated costs on the cross-section of line 2 for \(S_1 = 70\ \text{mm}^2\) and \(S_3 = 50\ \text{mm}^2\).
Accepted notation:
$p_{\text{n}} = 0.125$ — standard efficiency coefficient;
$p_{\text{a}} = 0.1$ — allocations for depreciation, repair, and maintenance;
$l$ — line length;
$i, j$ — line number;
$m$ — number of lines;
$S$ — required line cross-section;
$\rho = 31.5 \dfrac{\Omega \,\mathrm{mm}^{2}}{\mathrm{km}}$ — resistivity;
$c_{1} = 154 \dfrac{\text{thousand rubles}}{\mathrm{MW}}$ — cost of an installed megawatt of generating capacity at a power plant;
$k_{\text{m}} = 0.95$ — coefficient of participation in the maximum;
$c_{2} = 72 \dfrac{\text{thousand rubles}}{\mathrm{MW}\cdot\text{year}}$ — cost of lost energy;
$\tau = 0.292$ — loss time;
$I_{i}, I_{j}$ — currents in sections of the network;
$D = 5 \dfrac{\text{thousand rubles}}{\mathrm{kV}^{2}\cdot\text{year}}$ — cost of the lost squared kilovolt;
$q$ — load-node number;
$k$ — number of the stage of the load curve;
$\Delta t_{k}$ — duration of the $k$-th stage of the load curve.
REFERENCES
- A. A. Glazunov, A. A. Glazunov, Electric Networks and Systems. GEI, 1960.
- B. A. Konstantinov, “On voltage quality at industrial enterprises.” Elektrichestvo, No. 5, 1963.
- Electrical Engineering Handbook. GEI, 1962.