Some Issues of the Economic Justification for Switching from One Voltage Class to Another in Distribution Networks
Abstract Generated abstract
This paper examines the economic selection of voltage class and conductor cross section for distribution network lines. It derives reduced cost expressions as functions of transmitted power, nominal voltage, and conductor cross section, then minimizes them to obtain an economic current density and power intervals within which each voltage class is cost preferable. The analysis further incorporates permissible voltage loss as a constraint on transmission distance, using an approximation for line reactance, and formulates the resulting choice of voltage and cross section as a constrained optimization problem.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE NAMED AFTER S. M. KIROV,
HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR
Volume 295 1976
SOME ISSUES OF ECONOMIC JUSTIFICATION FOR TRANSITIONING FROM ONE VOLTAGE CLASS TO ANOTHER IN DISTRIBUTION NETWORKS
N. A. Dulzon, R. P. Alekseev
(Presented by the scientific seminar of the Department of Electrical Systems and Networks)
1. Economically expedient voltage and cross section of power-transmission lines
The reduced calculated costs per unit length of a line depend on the design features of the power transmission, the magnitude of the transmitted power, and the nominal voltage:
\[ Z = pK(a+bF+cU+dUF)+\frac{S^2\tau\beta}{U^2\gamma F}, \tag{1.1} \]
where \(a, b, c, d\) are coefficients constant for the given line design;
\[ p=p_{\mathrm{n}}+p_{\mathrm{a}}+p_{0}+p_{\mathrm{tr}} \]
are deductions from capital expenditures;
\(K\) is the coefficient for converting wholesale prices (according to the price list) to average regional estimated prices;
\(\tau\) is the number of hours of maximum losses;
\(\beta\) is the cost of \(1\) kWh of lost energy;
\(\gamma\) is the specific conductivity of the conductor material;
\(F\) is the conductor cross section;
\(U\) is the line voltage;
\(S\) is the power transmitted over the line.
Minimizing these costs with respect to the cross section makes it possible to find an expression for the economic current density in the form
\[ j_{\mathrm{e}}= \sqrt{\frac{pK\gamma(b+dU)}{3\tau\beta}} = \frac{S}{\sqrt{3}\,UF_{\mathrm{e}}}. \tag{1.2} \]
Substituting the value of \(j_{\mathrm{e}}\) into (1.1), we obtain the minimum of the reduced calculated costs:
\[ Z_{\min}=pK(a+cU)+\frac{S}{U} \left[ \frac{pK}{\sqrt{3}\,j_{\mathrm{e}}}(b+dU) + \frac{\sqrt{3}\tau\beta}{\gamma}j_{\mathrm{e}} \right]. \tag{1.3} \]
The first term in the square brackets of expression (3) can be transformed by substituting \(j_{\mathrm{e}}\) from (1.2):
\[ \frac{pK}{\sqrt{3}\,j_{\mathrm{э}}}(b+dU)=\frac{\sqrt{3}\tau\beta}{\gamma}\,j_{\mathrm{э}}. \]
Taking this into account, expression (1.3) takes the form
\[ Z_{\min}=pK(a+cU)+\frac{2\sqrt{3}\tau\beta S}{\gamma U}\,j_{\mathrm{э}}. \tag{1.3a} \]
As is clear from (1.3a), the reduced calculated costs, in the case where the conductor cross section is selected according to \(j_{\mathrm{э}}\), depend linearly on the transmitted power. At the same time, as the voltage increases, the constant component grows, while the angular coefficient with respect to power decreases. If, for a series of \(U_{\mathrm{n}}\), one constructs the dependence of the reduced calculated costs on power, the following graph is obtained.
From Fig. 1 it is clear that for each nominal voltage there exists a certain range of powers for which the reduced calculated costs per unit length of line are minimal.
Let us determine the boundaries of the interval for voltage \(U_2\). For this it is necessary to equate the costs when transmitting powers \(S_{12}\) and \(S_{23}\), respectively, for voltages \(U_1U_2\), \(U_2U_3\):
\[ pK(a_1+c_1U_1)+\frac{2\sqrt{3}\tau\beta S_{12}}{\gamma U_1}\,j_{\mathrm{э}1} = pK(a_2+ \]
\[ +c_2U_2)+\frac{2\sqrt{3}\tau\beta S_{12}}{\gamma U_2}\,j_{\mathrm{э}2}; \]
Fig. 1.
hence
\[ S_{12}= \frac{pK\gamma(a_2-a_1+c_2U_2-c_1U_1)} {2\sqrt{3}\tau\beta\left(\dfrac{j_{\mathrm{э}1}}{U_1}-\dfrac{j_{\mathrm{э}2}}{U_2}\right)}. \tag{1.4} \]
For lines of the same type, \(a_1=a_2\), \(c_1=c_2\); in this case
\[ S_{12}= \frac{pK\gamma c(U_2-U_1)} {2\sqrt{3}\tau\beta\left(\dfrac{j_{\mathrm{э}1}}{U_1}-\dfrac{j_{\mathrm{э}2}}{U_2}\right)}. \tag{1.4a} \]
In particular cases equality \(j_{\mathrm{э}1}=j_{\mathrm{э}2}=j_{\mathrm{э}}\) is possible; then
\[ S_{12}= \frac{pK\gamma c\,U_1U_2} {2\sqrt{3}\tau\beta\,j_{\mathrm{э}}}. \tag{1.4b} \]
Correspondingly, the second limiting power may be obtained:
\[ S_{23}= \frac{pK\gamma(a_3-a_2+c_3U_3-c_2U_2)} {2\sqrt{3}\tau\beta\left(\dfrac{j_{\mathrm{э}2}}{U_2}-\dfrac{j_{\mathrm{э}3}}{U_3}\right)}; \tag{1.5} \]
\[ S_{23}=\frac{pK\gamma c\left(U_3-U_2\right)} {2\sqrt{3}\,\tau\beta\left(\dfrac{j_{\mathrm{э}2}}{U_2}-\dfrac{j_{\mathrm{э}3}}{U_3}\right)}; \tag{1.5a} \]
\[ S_{23}=\frac{pK\gamma c\,U_2U_3}{2\sqrt{3}\,\beta j_{\mathrm{э}}}. \tag{1.5б} \]
Thus, the economic interval of powers transmitted at voltage \(U_2\) lies within the limits
\[ S_{12}\le S_{\mathrm{э}}\le S_{23}. \]
Analogously, economic intervals can be obtained for the use of other voltages. It should be noted that the economic power intervals obtained do not depend on the distance over which electric power is transmitted. From this point of view, this is a somewhat idealized solution, on which it is necessary to impose a constraint on the distance of electric-power transmission.
2. The Economically Expedient Distance for Electric-Power Transmission
The conductor cross section selected according to the economic current density does not depend on the length of the line. However, restrictions on line length may arise if the maximum permissible voltage loss is taken into account. For lines up to 110 kV, the voltage loss may be taken as equal to the longitudinal component of the voltage drop:
\[ \Delta U=\frac{PR+QX}{U}. \tag{2.1} \]
The voltage loss in the reactive resistance due to reactive power can be determined in advance if the line’s specific reactive resistance is specified:
\[ \Delta U=\frac{QX_0l}{U}. \tag{2.2} \]
For different types of lines, the conductor cross sections used are limited; taking these limitations into account, more accurate results can be obtained if the specific inductive reactances are approximated by an expression of the form
\[ X_0=\beta+\frac{\alpha}{F}. \tag{2.3} \]
Thus, for example, for a 10 kV overhead line \((D_{\mathrm{av}}=1\ \mathrm{m})\), the specific inductive reactance may be determined by the formula
\[ X_0=0.319+\frac{1.5}{F}, \]
where the error does not exceed \(1.7\%\).
Taking (2.3) into account, the voltage loss in the reactive resistance due to reactive power will take the form
\[ \Delta U_p=\frac{Ql}{U}\left(\alpha+\frac{\beta}{F}\right). \tag{2.4} \]
Then the permissible voltage loss in active resistance due to active power will be equal to
\[ \Delta U_{\mathrm{per},a}=\Delta U_{\mathrm{per}}-\frac{Ql}{U}\left(\alpha+\frac{\beta}{F}\right). \tag{2.5} \]
Hence the actual voltage loss in active resistance is determined from the expression
\[ \Delta U_a=\frac{Pl}{U\gamma F}\leq \Delta U_{\mathrm{per}}-\frac{Ql}{U}\left(\alpha+\frac{\beta}{F}\right). \tag{2.6} \]
Let us find the condition imposed on the line length by the permissible voltage loss:
\[ \frac{Pl}{U\gamma F}+\frac{Ql}{U}\left(\alpha+\frac{\beta}{F}\right)\leq \Delta U_{\mathrm{per}}; \]
\[ l_{\mathrm{e}}\leq U\Delta U_{\mathrm{per}}\left(\frac{1}{\alpha Q}+\frac{F}{\beta Q}+\frac{\gamma F}{P}\right). \tag{2.7} \]
The quantity \(l_{\mathrm{e}}\) determines the limiting length over which it is expedient to transmit electric power without violating the constraint on the permissible voltage loss, while simultaneously ensuring the attainable minimum of the reduced calculated costs.
The following procedure for determination appears expedient. For a given load power (Fig. 1), determine the economically expedient line voltage; then, by formula (1.2), determine the economic current density for this line. After this, \(X_0\) is determined. Finally, with the load \(\cos\varphi\) known, formula (2.8) is used to ascertain whether the actual length of the line satisfies the constraints on the permissible voltage loss. If this condition is satisfied, then the problem of selecting the line voltage and cross-section has been solved. Otherwise, it is necessary to determine which is preferable: to adopt a higher voltage or to increase the line cross-section.
This corresponds to the following mathematical programming problem:
minimize
\[ Z=\left[pK(a+bF+cU+dUF)+\frac{S^2\tau\beta}{U^2\gamma F}\right]l \tag{2.8} \]
subject to the constraint
\[ l\leq U\Delta U_{\mathrm{per}}\left(\frac{1}{\alpha Q}+\frac{F}{\beta Q}+\frac{\gamma F}{P}\right), \]
where the variables are \(U\) and \(F\).
The solution of this problem without constraints was given above; it corresponds to the selection of conductor cross-sections according to the economic current density and to the selection of voltage according to economic intervals of power.
References
-
A. A. Glazunov, A. A. Glazunov. Electrical Networks and Systems. 4th ed., Moscow, Gosenergoizdat, 1960.
-
N. A. Dulzon. Selection of conductor cross-sections by the minimum of reduced costs in a branched open network, taking various constraints into account. Scientific report, registration number 7100083, 1973.