IS IT PERMISSIBLE IN ELECTRIC NETWORK CALCULATIONS TO NEGLECT CHANGES IN LOAD POWER DURING VOLTAGE DEVIATIONS AT CONSUMER SUBSTATIONS?

V. K. Shcherbakov and G. A. Nazarenkova

1. Calculations of electric networks are usually carried out under the assumption that the power consumed by each load of the network remains constant, regardless of changes in the network voltage. Loads are specified by active power and by power factor at the nominal voltage of the network (and, consequently, of the loads). In the course of calculations of various operating conditions of the network, the loads are thus taken into account by currents whose magnitude changes together with the change in the network voltage and is equal to

\[ I_{\text{load}} = I_a - j I_r = \frac{P_{\text{load}} - jQ_{\text{load}}}{\sqrt{3}\,\hat U_{\text{network}}}. \]

In reality, the power of loads supplied from the network changes together with the change in voltage in the network. Fig. 1 gives the widely known typical static characteristics of a composite load of large electric systems, including power losses in transformers and lines, compiled by “Teploelektroproekt.” The characteristics give the change in power of a composite load connected to a 6 kV network. The composite load includes:

large asynchronous motors . . . . . . . . . . 15%,
small asynchronous motors . . . . . . . . . . 35%,
large synchronous motors . . . . . . . . . . . 9%,
furnaces and mercury rectifiers . . . . . . . 11%,
lighting and household load . . . . . . . . . 22%,
losses . . . . . . . . . . . . . . . . . . . . 8%.

The curves of Fig. 1 show that when the voltage in the network deviates from the nominal value toward an increase by 5%, the active power of the load increases by 3%, and the reactive power by 13%.

When the voltage is reduced by 10%, the active power of the load decreases by 6%, and the reactive power by 17%.

The error in calculating voltage losses in high-voltage networks, which is usually made by assuming the load power to be unchanged, can be estimated approximately from the following considerations. In the formula for calculating the voltage loss, equated to the longitudinal component of the voltage drop,

\[ \Delta U = \frac{PR + Q \cdot X}{U}, \]

one may take the ratio

\[ \frac{X}{R} = 3 \]

and the ratio

\[ \frac{Q}{P} = 0.75. \]

Then

Table 1

It follows from Table 1 that the error in voltage calculations in a particularly unfavorable case (\(U_1=110\) kV) may reach almost 8%, and in calculations of voltage loss—33%.

Naturally, the magnitude of the error decreases and may be equal to zero when the calculated voltage approaches the true value.

One more example. Fig. 3 gives a diagram of a 220-kilovolt electric transmission line, which it is considered impermissible to calculate by the Π-shaped equivalent circuit without correction factors for the distributed parameters of the circuit.

However, it turns out that neglecting the correction factors introduces an error in calculating voltages of only 0.76%, whereas

Fig. 3

neglecting the change in load power leads to an error of 3.64%. In this same case the voltage loss will be calculated with an error of 17%.

The current at the beginning or end of the line must be known, specified, which is possible only under the condition of a specified voltage at the corresponding end of the line. If this specified voltage is equal to the nominal voltage, then the current is determined simply as

\[ I=\frac{P-jQ}{\sqrt{3}\,U_{\text{nom}}}, \]

where \(P\) and \(Q\) are the assumed loads under nominal conditions. In other cases it is necessary to take the load power from the static load characteristics.

1. Exact equations

\[ \begin{aligned} \dot U_1 &= A\dot U_2+B\dot I_2,\\ \dot I_1 &= C\dot U_2+D\dot I_2 \end{aligned} \tag{1} \]

used for calculating long lines, relate the currents and voltages at the beginning and end of the line. If the calculation is carried out for a specified voltage at the end of the line, determined by the nominal load voltage and the transformation ratio of the transformers between the line and the load, then unambiguous

the error at reduced voltage will be approximately 15%, and at increased voltage relative to the nominal—9%. In the latter case, that is, at increased voltage, the voltage deviation from nominal was taken as 5%. With a voltage deviation of 10%, the error will probably reach 20%.

The error is especially significant in determining the loss and drop of voltage in a transformer, as an inevitable element of a high-voltage network. The ratio of the reactive and active resistances of transformers—

\[ \frac{X_T}{R_T} \]

for large and medium transformers is approximately equal to 10.

For \(Q = 0.75P\), the error in determining the voltage loss will amount to 18.5%.

Often, considerable inaccuracies in calculating voltage losses in transmission lines are readily tolerated on the grounds that, in the final analysis, this has little effect on the magnitude of the voltage at the calculated points of the network, since an inaccuracy in calculating the voltage loss affects the accuracy of determining the voltage at the ends of a transmission line in the ratio

\[ \frac{\Delta U}{U}. \]

However, in such reasoning it is overlooked that, in planning and calculating various operating conditions of electrical-system networks, as well as in designing electrical systems, one has to deal with an unregulated system with voltage deviations from nominal at individual points of the network of 20 percent and more. Consequently, an inaccuracy in calculating voltage losses even of 20% gives an error in calculating voltages of 4%, which may already completely distort conclusions regarding the necessary nature and magnitude of regulating devices in the network.

Fig. 1

In addition, independent importance should be attached to the accuracy of calculating losses and voltage drops in the network. A correct estimate of the magnitudes of equalizing currents and powers in networks depends entirely on the accuracy of calculating losses and voltage drops (and not voltages).

Fig. 2

2. To confirm this, let us give an example of calculating a transmission line according to the diagram in Fig. 2.

The calculation results are given in Table 1, taking into account

\[ S_{\text{load}} = f(U) \]

the voltage at the beginning of the line can be calculated exactly. However, it is usually necessary to find several values of the voltage at the end of the line, corresponding to many operating conditions of the line, with an unchanged or specified voltage at the beginning of the line. In all cases, practically accurate calculation results can be obtained only on the condition of repeated recalculations, using the static characteristics of the loads, or on the condition of introducing into equations (1) a current dependent on the voltage \(U_2\), i.e., depriving equations (1) of linearity. But one cannot fail to take into account the circumstance that the so-called static load characteristics can be compiled only as averaged dependences of the load power on the network voltage.

It is practically impossible to take into account the full diversity of electric motors and apparatus when compiling static characteristics, and therefore the latter will always be compiled with one or another degree of error relative to reality. Moreover, this error may be considerable when using static characteristics of a “typical load.” Consequently, the desire to apply exact equations in calculations of electrical networks is in many cases inexpedient, since it does not justify the expenditure of the great amount of computational labor that calculations using exact equations usually require.

Conclusions

1. Voltage deviations at consumer substations from the nominal value are associated with a change in load power, especially reactive power, relative to the specified load power at nominal voltage. This change in load power can be very significant and can substantially affect the magnitude of losses and voltage drops in the network.

2. Calculations of high-voltage electrical networks that neglect the change in load power, when there is a significant deviation of voltage from the nominal value at consumer substations, give a noticeable error, which in many cases cannot be allowed.

3. The error in the calculation results can be reduced or brought to zero on the condition of repeated refinements using the static characteristics of the loads.

4. Repeated recalculations to achieve the necessary accuracy of calculations require much time and effort, and therefore it is necessary to find simple methods for approximate allowance for the variability of load power, which at the same time would ensure the required accuracy of calculations of electrical networks.