Estimation of the Value of the Stress Damage Coefficient

R. I. BORISOV, V. V. LITVAK

(Presented by the Department of Electrical Networks and Systems)

The tasks of organizing the operation of electrical networks and of designing newly constructed and expanded power-supply schemes consist in the continuous improvement of the quality indices of power supply. A very important index of this kind is the magnitude of voltage nonuniformity, which characterizes voltage damage. As was shown earlier [1], power-supply schemes of industrial enterprises and the main networks of the system must be designed with allowance for the economic characteristics of equipment in terms of quality parameters. However, considerable organizational and methodological difficulties arise in the practical determination and calculation of these characteristics [2]. Voltage deviations are accompanied by simultaneous changes in other variables (composition of raw material, labor intensity, etc.), which affect the cost of producing a unit of output. Separating these variables proves to be very difficult or impossible. Therefore, along with direct methods for determining economic characteristics, in our opinion, indirect methods also deserve attention, making it possible to estimate the value of the damage coefficient from other initial data.

Voltage deviations at consumers under different variants of the supply and distribution network turn out to be different, which leads to different costs of producing a unit of output. Each variant of network construction has corresponding calculated costs; therefore the cost of damage from voltage deviations should be related to its costs, since the cost of production depends on the quality of electric power under the given variant.

\[ З = p_{\mathrm{н}}K + C + y; \tag{1} \]

\[ y = k_y \sum_{\tau} \left(\frac{U_{\mathrm{н}} - U_i}{U_{\mathrm{н}}}\,100\right)^2 \frac{\Delta t_i}{T} = k_y \cdot N_t,\ , \tag{2} \]

where
\(N_t\) — voltage nonuniformity,
\(k_y\) — coefficient of voltage damage,
\(U_{\mathrm{н}}, U_i\) — nominal and actual voltage at the consumer,
\(\Delta t_i, T\) — duration of the \(i\)-th step and the considered interval of the load curve.

Additional expenditures on means of voltage regulation and maintenance reduce these deviations and decrease the damage. If

Fig. 1. Network diagram.

the reduced calculated costs for voltage-regulating equipment prove to be greater than the amount of damage from voltage deviations over the calculation period, this means that, for consumers of this kind, such damage is permissible. With the reverse ratio of the components, the expediency of additional expenditures for eliminating voltage deviations should be recognized. The amount of damage proves to be commensurate with the expenditures on additional regulating devices. Namely: equality of the damage and the reduced expenditures on additional means for voltage regulation determines the possible value of the damage coefficient.

Let a compensator be installed at the substation (Fig. 1), whose capacity is sufficient to maintain the desired voltage on the secondary-voltage buses under any operating load condition. Then it may be assumed that the voltage deviations in each condition are minimal, and the total damage is equal to zero. In the absence of voltage-regulating equipment, the damage will be greatest. Equating the costs of the two variants, we obtain the relation for determining the damage coefficient:

\[ Z_1 - Z_2 = 0, \]

\[ (\Delta P_{\mathrm{M}1}+\Delta P_{\mathrm{l}1}) \left[\rho_{\mathrm{n}} k_1 K_{\mathrm{st}}+ \tau_1(\rho_{\mathrm{n}} k_{\mathrm{TT}} b_{\mathrm{st}}+\beta_{\mathrm{st}})\right] +(\rho_{\mathrm{n}}+p_{\mathrm{a}})Q_{\mathrm{k}}k_{\mathrm{k}}+ \]

\[ +\Delta P_{\mathrm{k}}\left[\rho_{\mathrm{n}}k_1K_{\mathrm{st}}+ t(\rho_{\mathrm{n}}k_{\mathrm{TT}}b_{\mathrm{st}}+\beta_{\mathrm{st}})\right] -k_{\mathrm{y}}N_t- \]

\[ -(\Delta P_{\mathrm{M}2}+\Delta P_{\mathrm{l}2}) \left[\rho_{\mathrm{n}}k_1K_{\mathrm{st}}+ \tau_2(\rho_{\mathrm{n}}k_{\mathrm{TT}}b_{\mathrm{st}}+\beta_{\mathrm{st}})\right]=0. \]

If the active losses in the compensating device and the increments of losses in the line and transformer are neglected, we obtain

\[ k_{\mathrm{y}}N_t=(\rho_{\mathrm{n}}+p_{\mathrm{a}})Q_{\mathrm{k}}k_{\mathrm{k}}, \tag{3} \]

where \(Q_{\mathrm{k}}\) is the capacity of the compensating device,

\[ k_{\mathrm{k}} \]

is the unit cost of the compensating device.

The minimum capacity of the compensating device not used in the underexcitation mode is determined [3]:

\[ Q_{\mathrm{k}}=\frac{1}{x} \left[P_1 r+Q_1 x-k_{\mathrm{t}}U_{\mathrm{zh}1} \left(U_0-k_{\mathrm{t}}U_{\mathrm{zh}1}\right)\right], \tag{4} \]

\(P_1, Q_1, U_{\mathrm{zh}1}\) — active and reactive powers of the load, and the desired voltage on the secondary side of the substation in the maximum-load condition,

\(U_0\) — voltage of the supply source.

The transformation coefficient \(k_{\mathrm{t}}\) for a transformer without on-load tap changing is determined from the condition of maintaining the desired voltage in the minimum-load condition:

\[ k_{\mathrm{t}}= \frac{U_0 U_{\mathrm{n}}-P_2 r-Q_2 x} {U_{\mathrm{zh}2}U_{\mathrm{n}}}. \]

The desired voltage on the low-voltage buses of the substation, in accordance with the law of counter-regulation, may be determined as follows:

Fig. 2. Dependence of the voltage-damage coefficient on the length of the line ($a$), the hours of utilization of maximum power ($b$), the load power factor ($v$), and the consumer power ($g$)

\[ U_{\mathrm{zh}}=\left(1+0.05\frac{P}{P_1}\right)U_{\mathrm{n}}. \]

Calculations for determining the voltage damage $k_y$, thousand rub./$(\%)^2$, for the scheme in Fig. 1 were carried out on an M-20 computer, and the results are shown in Fig. 2$a$, $b$, $v$, $g$.

Dependences 1 and 2 were constructed for values of the power factor 0.9 and 0.6, respectively; 3, 4, 5 (Fig. 2$v$) were constructed for values of the time of utilization of maximum load $T_M$ equal to 0.333, 0.7, and 0.85, respectively.

It may be considered that the damage coefficient does not depend on $\cos\varphi$, $T_M$, and the length of the transmission line, since the changes in $k_y$ occur within the limits of $2 \div 6$ thousand rub./$(\%)^2$ when $\cos\varphi=0.6 \div 1$, $T_M=0.333 \div 0.95$, $l=10 \div 150$ km.

The dependence of the damage coefficient on the magnitude of the transmitted power is shown in Fig. 2$g$.

Conclusions

1. Estimation of the damage coefficient from the network initial data divides all consumers into two groups. For consumers of the first group, the damage coefficient is insignificant; its value is less than 0.1 thousand rub./$(\%)^2$, and the necessity of taking it into account when designing power-supply schemes requires justification.

Consumers of the second group have values of damage coefficients of 0.1 thousand rub./$(\%)^2$ and higher; they require means of additional voltage regulation, and the obtained values of the coefficients must be taken into account when designing power-supply schemes.

1. Assignment of a type of consumer to one or the other group according to the magnitude of the damage coefficient must be justified.

2. The weak dependence of the damage coefficient on $\cos\varphi$, the length of the feed-

curves and the number of hours of utilization of the maximum makes it possible to outline the regions of its possible values in the working range of the specified parameters.

1. The voltage damage coefficient is nonlinearly related to the consumer’s maximum power.

REFERENCES

1. R. I. Borisov, V. V. Litvak. On the need to take account of damage from voltage deviations in the design of distribution networks. Izv. TPI, vol. 159, 1968.

2. B. A. Konstantinov. On voltage quality at industrial enterprises. Elektrichestvo, No. 5, 1963.

3. A. A. Glazunov, A. A. Glazunov. Electric networks and systems, GEI, 1960.