Particular Conditions for Minimum Active Power Losses in Power Transmission Lines
Abstract Generated abstract
This paper examines the formulation of conditions for minimizing active power losses in power transmission line sections when the receiving-end voltage is fixed. It argues that a commonly used partial-derivative condition is incomplete because it treats active and reactive power at the receiving end as independent, although reactive power generally depends on active power and other operating factors. To address cases where analytic determination of optimal reactive power and losses is difficult, the paper proposes an iterative procedure that varies sending-end voltage, load, and receiving-end voltage to identify minimum-loss operating points. The resulting calculations can be approximated as a functional relation between optimal reactive power, active power, and voltage, supporting subsequent selection of voltage values that minimize losses under changing load.
Full Text
Particular Conditions for Minimum Active Power Losses in Power Transmission Lines
A. G. Milyushkin
(Presented by the scientific seminar of the Department of Electrical Systems and Networks)
The solution of optimization problems, in particular, the determination of laws of deep voltage regulation in power transmission lines and of changes in reactive power corresponding to optimal operating modes of electric transmission, must conform to the accepted assumptions.
It appears incompletely formulated to determine particular conditions for a minimum of total losses at a fixed voltage at one of the ends of a section, following from the conditions [1]
\[ \frac{\partial \Delta P_{\Sigma}}{\partial Q}=0. \tag{1} \]
When this equation is solved for the section in Fig. 1, one obtains
Fig. 1.
\[ \Delta P_{\ell} = \operatorname{Re} \left[ \hat{A}\hat{C}U_2^2 + \frac{\hat{B}\hat{D}}{U_2^2}(P_2^2+Q_2^2) + P_2(\hat{A}\hat{D}+\hat{B}\hat{C}-1) + jQ_2(\hat{A}\hat{D}-\hat{B}\hat{C}) \right] = f(U_2,\,P_2,\,Q_2). \tag{2} \]
It further follows that, for a given \(P_2\), i.e., for a compulsory character of \(P_2\) caused by the need to cover the load, the particular condition for a minimum of losses at fixed \(U_2\) is determined from the equation
\[ \frac{\partial \Delta P_{\ell}}{\partial Q_2}=0. \tag{3} \]
Solving this equation, we obtain
\[ \frac{2Q_2}{U_2^2}\operatorname{Re}(\hat B\hat D)-\operatorname{Im}(\hat A\hat D-\hat B\hat C)=0, \tag{4} \]
i.e., it is assumed that \(\dfrac{\partial P_2}{\partial Q_2}=0\).
In fact, \(\dfrac{\partial P_2}{\partial Q_2}\ne 0\) for fixed \(U_2\).
The point is that the condition \(\dfrac{\partial P_2}{\partial Q_2}=0\) is fulfilled only when the quantities \(P_2\) and \(Q_2\) are mutually independent. In reality, the quantity \(Q_2\) depends on \(P_2\) [2] and also on other factors.
Thus, for fixed \(U_2\), the reactive power \(Q_2\) is a function of \(P_2\), including the value \(Q_2=Q_{2\eta}\), which gives the minimum \(\Delta P_\lambda\) for fixed \(U_2\). On the basis of the above, for cases of solving problems in which the determination of the optimal values \(Q_\eta\), \(\Delta P^\eta\) analytically is associated with certain difficulties or is impossible, the following method for determining them is proposed.
In contrast to the previously proposed methods [1], the particular conditions for the minimum active-power losses at a fixed voltage of the receiving-end node are found for specific section lengths by an iterative method:
a) we specify the maximum permissible value of \(U_2\), for example, \(1.05\,U_{\text{nom}}\);
b) we adopt a value of \(P_2\); then, by varying the voltage \(U_1\), we determine the parameters \(Q_2\), \(Q_1\), \(\Delta P_\Sigma\); at some value \(U_1=U_{1\eta}\), the quantity \(\Delta P_\Sigma\) reaches a minimum; we determine the value of \(Q_2\) at \(U_1=U_{1\eta}\) and the adopted \(P_2\) and \(U_2\) from formula (5):
\[ Q_2=\frac{U_2^2}{B^2}\operatorname{Im}(\hat A\hat B)+ \sqrt{\left(\frac{U_1U_2}{B}-\left[P_2+\frac{U_2^2}{B^2}\operatorname{Re}(\hat A\hat B)\right]\right)}; \tag{5} \]
c) changing the values of \(P_2\) and repeating the same operations as in item “b,” we obtain the dependence
\[ Q_{2\eta}=f(P_2)\quad \text{for } U_2=U_{2\max}; \tag{6} \]
d) next, decreasing the value of \(U_2\), leaving it fixed while varying \(P_2\), \(U_1\), the calculations according to items “b,” “c,” “d” are repeated.
The calculation results obtained can, by approximation, be brought to the form of the analytical dependence
\[ Q_{2\eta}=f(P_2,U_2), \tag{7} \]
then, substituting expression (7) into (2), we obtain
\[ \Delta P_\Sigma=f(P_2U_2). \tag{8} \]
Subsequently, when the load changes, for each value \(P_2\) the voltage value corresponding to the minimum active-power losses is found iteratively.
REFERENCES
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V. A. Venikov, I. P. Siuda. Calculations of Operating Conditions of Long-Distance AC Power Transmissions. Moscow, “Vysshaya shkola,” 1966.
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R. I. Borisov, A. G. Milyushkin. Probabilistic analysis of power flows over intersystem ties. Present collection.