VOLTAGE REGULATION IN LOAD SUPPLY CENTERS

R. I. BORISOV, V. V. LITVAK

(Presented by the Department of Electrical Systems and Networks)

The conditions for economical operation of electrical receivers with respect to voltage require the selection of a voltage regulation law in supply centers. In [1] a sufficiently simple method is proposed for determining an expedient regulation law on the basis of minimizing the annual national-economic damage due to voltage. In this approach, however, the economic indicators of electrical networks, which depend substantially on the value of the operating voltage, are not taken into account.

It is proposed to refine the method for selecting transformer ratios and establishing the desired law of voltage regulation, taking into account power and energy losses in electrical networks.

The active-power losses in the longitudinal elements incident to node \(q\) of the system are equal to

\[ \Delta P_q = 3 \sum_{j=1}^{m} I_{jq}^{2} r_{jq}, \tag{1} \]

where \(j, m\) are the number and the total number of branches connected to node \(q\) of the network; \(I_j, r_{jq}\) are the current and resistance of the \(j\)-th branch.

The annual calculated costs in the power system caused by losses of active power and electrical energy in the longitudinal resistances of the network elements are equal to

\[ Z = 3 \sum^{T} (\alpha_M p_2 + \Delta t p_3) \sum_{j=1}^{m} I_{jq}^{2} r_{jq}, \tag{2} \]

\[ \alpha_M = \begin{cases} 1, & \text{when } \displaystyle \sum_{j=1}^{m} I_{jq}^{2} r_{jq} = \max,\\ 0, & \text{in all other cases.} \end{cases} \]

Here the losses of electrical energy are determined from the curves of variation of the active and reactive load currents. The operator \(\alpha_M\) is introduced to account for the cost of additionally installed capacity at the power station, necessary to cover the maximum active-power losses in the networks.

For the entire network of the system (the aggregate of \(n\) nodes and \(m\) circuits connected to each node), the calculated costs are equal to

\[ Z_1 = \frac{3}{2} \sum_{q=0}^{n} \sum^{T} (\alpha_M p_2 + \Delta t p_3) \sum_{j=1}^{m} I_{jq}^{2} r_{jq}. \tag{3} \]

Losses in the magnetizing circuit of a power transformer are proportional to the square of the applied voltage [2]. Significant overexcitation of it may lead to technically impermissible operating conditions. Therefore the rules for technical operation [3] regulate the excess of the transformer primary voltage by an amount of no more than 5% above the voltage corresponding to the given tap.

Thus, the active-power losses in the magnetizing circuit of a power transformer can be expressed as follows:

\[ \Delta P_{\mathrm{st}}=\frac{\Delta P_{\mathrm{xx}q}}{U_{\mathrm{otv}}^{2}}U_q^2 . \tag{4} \]

It is expedient to express the tap voltage \(U_{\mathrm{otv}}\) through the sought values of the transformation ratios \(k_{\mathrm{t}q}\) and the desired voltages on the secondary side of the transformers \(U_{\mathrm{zh}q}\)

\[ \Delta P_{\mathrm{st}}=k_{\mathrm{t}q}^{2}\frac{\Delta P_{\mathrm{xx}q}}{U_{\mathrm{zh}q}^{2}}U_q^2 . \tag{4'} \]

Then the calculated costs in the power system, taking into account the cost of additionally installed capacity at the power station, development of the fuel base, and the cost of the electric energy lost in the transformer steel, with allowance for expression (1) [1], are equal to:

\[ Z=\sum_{q=0}^{n}\sum_{T}\Delta P_{\mathrm{xx}q}k_{\mathrm{t}q}^{2} \frac{\beta_{\mathrm{m}}p_2+\Delta t p_3}{U_{\mathrm{zh}q}^{2}} \left( U_0-0.5\sqrt{3}\sum_{p=1}^{n} I_{\mathrm{np}} z_{pq} \right)^2 . \tag{5} \]

Thus, the variable part of the operating costs, depending on the voltage supplied to the network, in the power-supply system consists of the voltage damage (2) [1] and the costs of electric-energy losses in the longitudinal and transverse elements:

\[ \begin{aligned} Z&=Z_1+Z_2+Y =\sum_{q=0}^{n}\sum_{T}\Bigg[ 1.5(\alpha_{\mathrm{m}}p_2+\Delta t p_3)\sum_{j=1}^{m} I_{jq}^{2} r_{jq} \\ &\quad +\Delta P_{\mathrm{xx}q}k_{\mathrm{t}q} \frac{\beta_{\mathrm{m}}p_2+\Delta t p_3}{U_{\mathrm{zh}q}^{2}} \left( U_0-0.866\sum_{p=1}^{n} I_{\mathrm{np}} z_{pq} \right)^2 \\ &\quad +k_{\mathrm{u}q}\frac{\Delta t_*}{U_{\mathrm{zh}q}^{2}} \left( U_{\mathrm{zh}q}-k_{\mathrm{t}q}U_0 -0.866k_{\mathrm{t}q}\sum_{p=1}^{n} I_{\mathrm{np}}z_{pq} \right)^2 \Bigg]. \end{aligned} \tag{6} \]

Minimization of the calculated-cost functional makes it possible to establish the optimal conditions for voltage regulation in distribution networks. The method of solution is described in more detail in [1]. The calculation expressions presented below for determining the voltage at the supply center \(U_0\), the regulated \(k_{\mathrm{tr}}\) and unregulated \(k_{\mathrm{tn}}\) transformation ratios of distribution transformers ensure a minimum of calculated costs:

\[ U_0= \frac{ \displaystyle \sum_{q=0}^{n} \left[ \frac{k_{\mathrm{t}q}}{U_{\mathrm{zh}q}}\,k_{\mathrm{u}q}\Delta t_* +0.866\,\frac{k_{\mathrm{t}q}^{2}}{U_{\mathrm{zh}q}^{2}}\times \right] }{ \displaystyle \sum_{q=0}^{n} \frac{k_{\mathrm{t}q}^{2}}{U_{\mathrm{zh}q}^{2}} \left[ k_{\mathrm{u}q}\Delta t_*+\Delta P_{\mathrm{xx}q}(\beta_{\mathrm{m}}p_2+\Delta t p_3) \right] } \]

\[ \frac{ \times \left[ k_{yq}\Delta t_*+\Delta P_{xxq}(\beta_M p_2+\Delta t p_3)\right]\sum_{p=1}^{n} I_{nr}z_{pq} }{ \sum_{q=0}^{n}\frac{k_{Tq}^{2}}{U_{zh q}^{2}}\left[k_{yq}\Delta t_*+\Delta P_{xxq}(\beta_M p_2+\Delta t p_3)\right] }; \tag{7} \]

\[ k_{\mathrm{tr}}= \frac{ K_y\Delta t_* U_{zh} }{ \left(U_0-0.866\sum_{p=1}^{n}I_{nr}z_{pq}\right) \left[k_{yq}\Delta t_*+\Delta P_{xxq}(\beta_M p_2+\Delta t p_3)\right] }; \tag{8} \]

\[ k_{\mathrm{tn}}= \frac{ k_{yq}\sum^{T}\frac{\Delta t_*}{U_{zh q}} \left(U_0-0.866\sum_{p=1}^{n}I_{nr}z_{pq}\right) }{ \sum^{T}\frac{1}{U_{zh q}^{2}} \left(U_0-0.866\sum_{p=1}^{n}I_{nr}z_{pq}\right)^2 \left[k_{yq}\Delta t_*+\Delta P_{xxq}(\beta_M p_2+\Delta t p_3)\right] }. \tag{9} \]

Voltage regulation at the supply center according to form (7) proves to be impossible with regulators of the ordinary type because of the need to process a large volume of diverse information. The resulting regulation law can be implemented by means of a special program device using time as its argument. At the same time, the use of such a rather complex and expensive device should be additionally justified.

In this connection, it seems advisable to investigate the conditions under which the use of approximate, easily realizable regulation laws remains sufficiently effective.

The investigation of the conditions under which one or another practical regulation law is most effectively used was carried out for the power-supply scheme shown in Fig. 1. The nodal substation feeds two groups of consumers whose load schedules are known. Centralized voltage regulation is carried out by means of the on-load tap changer of the step-down transformer. The optimal regulation law for the adopted conditions is determined by expression (7).

Fig. 1. Network diagram

As approximate regulation laws, the following relationships were investigated:

1) stabilization of the voltage at the supply center

\[ U_0=\mathrm{const}; \tag{10} \]

2) voltage regulation with correction by total active power

\[ U_0=U_{0x}+k_cP_{\Sigma}; \tag{11} \]

The values of the coefficients of the linear regulation characteristic \(U_{0x}\), \(k_c\) for the aggregate of \(n\) operating conditions are determined by the least-squares method;

3) voltage regulation with correction by total current or total power

\[ U_0=U_{0x}+k_cS_{\Sigma}; \tag{12} \]

4) voltage regulation at the supply center is carried out in order to stabilize the voltage beyond certain equivalent impedances (compensation of the voltage drop in the equivalent impedance)

\[ U_0=U_{0x}+k_c\Delta U_e. \tag{13} \]

As the parameter of the equivalent voltage drop \(\Delta U_{\mathrm{e}}\), the quantity \(\Delta U_{\mathrm{av}}\) is used—the average voltage drop from the buses of the LSC to consumers 1 and 2.

Since, instead of the optimum law (7), some approximate law of voltage regulation at the LSC is implemented, certain calculated costs arise in the power-supply system; by their magnitude one can judge the effectiveness of a given regulation law and justifiably accept or reject it. The expediency of the named laws of regulation of \(U_0\) was investigated under conditions in which the load schedules of the consumers are close to homogeneous. The degree of nonuniformity is expressed by the parameters of the mutual correlation relationship of the consumed load powers [4] and is characterized by the linear correlation coefficient \(r_{12}\) [5]:

\[ r_{12}=\frac{K_{12}}{\sigma_1\cdot\sigma_2}, \tag{14} \]

Fig. 2

where

\(K_{12}\) is the correlation moment,

\(\sigma_1,\ \sigma_2\) are the standard deviations of the random variables.

Fig. 3. Dependence of costs as a function of the consumer load ratio \(S_1/S_2\) at an average correlation coefficient equal to 0.95

Fig. 4. Steepness of the opposing-regulation characteristic as a function of the ratio of consumer powers (a) and the load correlation coefficient (b)

The effectiveness of practical voltage-regulation laws at the LSC was investigated under the following conditions. The load schedules of the first and second consumers correspond to the actual load schedules of two shops of a machine-building plant that differ in the nature of the production process \((r_{12}=0.6\text{–}1.0)\). The total maximum power of the consumers is \(10\) MVA \((S_1=5\text{–}8,\ S_2=5\text{–}1.6)\). The results of the calculations performed are presented in Figs. 2, 3, and 4. Figure 2 shows the dependences of the calculated costs (6) on the load correlation coefficient for different laws of voltage regulation at the LSC. The dependences shown are constructed for a ratio of the powers of the first and second consumers equal to 0.666.

Conclusions

1. Owing to the difficulties of implementing an exact law of voltage regulation in load centers, studies have been carried out to establish the economical intervals in which it is expedient to apply the regulation laws considered. It is shown that voltage stabilization is economically expedient when supplying nonuniform consumers, when the correlation coefficient is less than 0.7.

2. In a system of uniform consumers (correlation coefficient greater than 0.85), for any ratio of consumer capacities, voltage regulation should be performed by varying the voltage with correlation according to the total active power.

3. The steepness of the characteristic of inverse regulation depends on the ratio of consumer capacities and on the degree of their uniformity, and lies within the limits: 0.15–0.3 for regulation according to active power, and 0.3–0.6 according to total apparent power.

References

1. R. I. Borisov, V. V. Litvak, V. D. Kozyrev. Joint selection of transformation ratios and conditions of centralized voltage regulation in distribution networks. Izv. vuzov. “Energetika”, 12, 1968.

2. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines. “Energiya,” Moscow, 1964.

3. Rules for the Technical Operation of Electric Power Stations and Networks. “Energiya,” Moscow, 1965.

4. N. I. Bilik. Voltage nonuniformity and statistical numerical characteristics of loads of electrical networks. Elektrichestvo, No. 8, 1964.