Optimal Use of the Compensating Capability of Synchronous Motors at Existing Industrial Enterprises

OPTIMAL USE OF THE COMPENSATING CAPABILITY OF SYNCHRONOUS MOTORS AT OPERATING INDUSTRIAL ENTERPRISES

V. V. ARKHIPENKO, V. A. TROSHIN, N. E. CHERNYI.

(Presented by the scientific seminar of the Department of Electrical Power Stations and Electrical Systems and Networks)

The practice that has developed at industrial enterprises in the operation of synchronous motors (SM)—the use of the full compensating capability of all motors regardless of the specific costs of compensation and of the electrical remoteness of the SM from the supply centers—leads in many cases to unjustified overconsumption of electric power and to a reduction in voltage quality at electrical receivers.

Analysis shows that the industrial enterprises of the Union possess a reserve of electric-power savings of 1 billion kWh/year, which can be realized through optimal use of the compensating capability of SMs. In this connection, the introduction into the practice of SM operation of scientifically substantiated engineering methods for calculating optimal excitation modes appears highly urgent.

The present article gives the theoretical prerequisites and general provisions of engineering methods for solving the optimization problem as applied to operating enterprises.

Fig. 1. Calculation diagram of a typical load node

The optimal excitation mode of synchronous motors at an operating enterprise is selected from the condition of a minimum of the total equivalent losses in the load node (Fig. 1).

\[ p_{\Sigma}^{\mathrm{e}}=p_{\Sigma}^{\mathrm{I}}+p_{\Sigma}^{\mathrm{U}}, \tag{1} \]

where

\(p_{\Sigma}^{\mathrm{I}}\) — losses depending on the reactive current in the supply and distribution networks, in the stators of synchronous machines (motors, generators, compensators), and losses depending on the excitation currents and exciters of synchronous machines, kW;

\(p_{\Sigma}^{\mathrm{U}}\) — losses in networks and electrical receivers depending on voltage, as well as equivalent losses by means of which changes in productivity, equipment, the service life of motor insulation, and other factors [1] are taken into account when the voltage at electrical receivers deviates from optimal values, kW.

It is convenient to represent the components of the total losses in the following form:

\[ p_{\Sigma}^{\mathrm{I}}= \sum_{j=1}^{l} \left[ \frac{c_{\phi j}}{2}Q_{\phi j}^{2} + \frac{c_j}{2}Q_j^{2} + \sum_{i=1}^{m_j} \left( \frac{d_{ij}}{2}Q_{\mathrm{c}ij}^{2} + \beta_{ij}Q_{\mathrm{c}ij} \right)n_{ij} + \frac{a_0}{2}(Q_{\phi 1}+Q_{\mathrm{op}})^2 + \beta_0(Q_{\phi 1}+Q_{\mathrm{op}}) \right], \tag{2} \]

\[ p_{\Sigma}^{\mathrm{U}}= \sum_{j=1}^{l} \left( \frac{e_{\phi j}}{2}k_{\phi j}^{2} + f_{\phi j}k_{\phi j} + \frac{e_j}{2}k_j^{2} + f_jk_j \right). \tag{3} \]

In expressions (2), (3), the designations \(Q\) and \(Q_{\phi}\) with indices correspond to the reactive powers of individual sections of the network, kvar;

\(Q_{\mathrm{c}}\) with indices — to the reactive powers of synchronous motors, kvar;

\(k\), \(k_{\phi}\) with indices — to voltage levels at nodal points, p.u.

\[ k=\frac{U}{U_{\mathrm{n}}}; \]

\(a\), \(b\), \(c\), \(d\) with indices — to specific increments of losses in synchronous motors and networks due to reactive power;

\(e\), \(f\) with indices — to specific increments of losses in the elements of the load node due to the voltage level.

The specific increments are calculated by the following formulas:

\[ a=\frac{2A}{Q_{\mathrm{c.n.}}^{2}},\ \text{kW}/\text{kvar}^{2}; \qquad b=\frac{B}{Q_{\mathrm{c.n.}}},\ \text{kW}/\text{kvar}, \]

\[ c=\sum_{s=1}^{r}\frac{2r_s}{U_s^{2}}\cdot 10^{-3},\ \text{kW}/\text{kvar}^{2}; \qquad d=\sum_{s=1}^{r}\frac{2x_s}{U_s^{2}}\cdot 10^{-3},\ \text{kvar}/\text{kvar}^{2}. \]

\[ e=\frac{2E}{z_{\mathrm{e}}},\ \text{kW}/\text{p.u.}^{2}; \qquad f=\frac{F}{z_{\mathrm{e}}},\ \text{kW}/\text{p.u.}, \]

where

\(A, B\) — constant losses of a synchronous machine, kW [2];

\(Q_{\mathrm{c.n.}}\) — nominal value of the reactive power of a synchronous machine, kvar;

\(r_s, x_s, U_s\) — active and inductive resistances, ohm, and the average value of the voltage, kV, of the \(s\)-th series ele-

network elements (transformers, reactors, overhead transmission lines of various cross sections and voltages);

\(r\) — number of series-connected network elements;

\(E, F\) — coefficients in a polynomial of the form \(y = Ek^2 + Fk + L\) rubles/hour, by which the voltage-damage function is approximated [1];

\(z_{\mathrm e}\) — cost of electric-energy losses, rubles/kWh.

In expressions (2), (3), the unknown variables (reactive powers of synchronous machines, networks, and voltage levels) are related to one another by equations of the form:

\[ \varphi_{1j}=Q_{aj}-\sum_{i=1}^{m_j} Q_{cij} n_{ij}-Q_j=0. \tag{4} \]

\[ \varphi_{2j}=Q_{\phi j}-Q_{\phi j+1}-Q_j-q_{\phi j+1}-q_j-Q'_j=0, \tag{5} \]

\[ \varphi_{3j}=k_{\phi j-1}-\frac{c_{\phi j}}{2}P_{\phi j} -\frac{c_{\phi j}}{2\varepsilon_{\phi j}}Q_{\phi j} +\delta U_{\phi j}-k_{\phi j}=0, \tag{6} \]

\[ \varphi_{4j}=k_{\phi j}-\frac{c_j}{2}P_j -\frac{c_j}{2\varepsilon_j}Q_j+\delta U_j-k_j=0, \tag{7} \]

where

\(Q_{aj}\) — reactive load of the \(j\)-th substation with SM, kvar;

\(Q_j\) — reactive loads of substations that do not have SM, kvar;

\(P_{\phi}, P\) — with indices, active powers, kW;

\(q_{\phi}, q\) — with indices, reactive-power losses (including no-load losses) in networks, kvar;

\(\delta U_{\phi}, \delta U\) — with indices, voltage increments in transformers with on-load tap changers, the magnitude of which depends on the operating positions of the off-circuit tap switches [3];

\(\xi_{\phi}, \xi\) — with indices, the ratio of the active resistance of the network to the inductive resistance

\[ \xi=\frac{r}{x}=\frac{c}{d}. \]

The variables are subject to constraints of the form:

\[ Q_{cij}^{\min} \leq Q_{cij} \leq Q_{cij}^{\max}, \tag{8} \]

\[ Q_j^{\min} \leq Q_j \leq Q_j^{\max}. \tag{9} \]

\[ Q_{\phi j}^{\min} \leq Q_{\phi j} \leq Q_{\phi j}^{\max} \tag{10} \]

The solution of the optimization problem must satisfy the condition of a minimum of the total equivalent losses (1) when (4)—(10) are satisfied. Using the method of undetermined Lagrange multipliers, the solution of the specified problem can be reduced to finding the minimum of the function

\[ \psi=p_{\Sigma}^{\mathrm e} -\sum_{j=1}^{e} \left( \lambda_j\varphi_{1j} -\lambda_{\phi j}\varphi_{2j} -\gamma_j\varphi_{3j} -\gamma_{\phi j}\varphi_{4j} \right), \tag{11} \]

where

\(\lambda_{\phi j}^{1}, \lambda_j, \gamma_j, \gamma_{\phi j}\) — certain constants corresponding to the optimal distribution of reactive loads among sources, kW/kvar.

The optimal values of the variables corresponding to the minimum of (11) are determined from the solution of a system of equations of the form:

\[ \frac{\partial \psi}{\partial Q_{\phi_1}} = a_0 Q_{\phi_1}+\theta_0+a_0(q_{\phi_1}+Q'_1)-\lambda_{\phi0}=0; \tag{12} \]

\[ \frac{\partial \psi}{\partial Q_{c_{ij}}} = a_{ij}Q_{c_{ij}}+\theta_{ij}-\lambda_j=0; \tag{13} \]

\[ \frac{\partial \psi}{\partial Q_{\phi_j}} = c_{\phi_j}Q_{\phi_j}+\lambda_{\phi_{j-1}}-\lambda_{\phi_j} -\frac{c_{\phi_j}}{2\xi_{\phi_j}}\gamma_{\phi_j}=0; \tag{14} \]

\[ \frac{\partial \psi}{\partial Q_j} = c_jQ_j+\lambda_{\phi_j}-\lambda_j -\frac{c}{2\xi_j}\gamma_j=0; \tag{15} \]

\[ \frac{\partial \psi}{\partial k_{\phi_j}} = e_{\phi_j}k_{\phi_j}+f_{\phi_j}-\gamma_{\phi_j}=0; \tag{16} \]

\[ \frac{\partial \psi}{\partial k_j} = e_jk_j+f_j-\gamma_j=0; \tag{17} \]

\[ j=1,2,\ldots,l \]

and equations (4)—(7), taking into account (8)—(10).

As is evident from (12)—(17), the determination of the optimal excitation modes of the SM is based on the principle of equality of the specific increments of active losses that depend on reactive power and voltage levels.

The general form of the optimal-control law follows directly from (13)

\[ \alpha_{ij}=\frac{\lambda_j-\theta_{ij}}{\theta'_{ij}}, \tag{18} \]

where

\[ \alpha_{ij}=\frac{Q_{c_{ij}}}{Q_{c.\mathrm{n.}ij}} \]

is the relative value of the reactive power of the SM of the \(i\)-th group of the \(j\)-th substation, p.u.;

\[ \theta'_{ij}=a_{ij}Q_{c.\mathrm{n.}ij},\ \mathrm{kW/kvar}. \]

The specific increment \(\lambda_j\) entering into (18) is determined from the solution of the system of equations (13)—(17) and (4—7), and can be represented as the sum of two terms

\[ \lambda_j=\lambda'_j+\Delta_j,\ \mathrm{kW/kvar}. \tag{19} \]

The first term is equal to

\[ \lambda'_j=\lambda_{\phi0}+2\xi'_{\phi_1}k_{\phi0} -\left(2\xi_j+\frac{c_j}{2\xi_j}e_j\right)k_j,\ \mathrm{kW/kvar}, \tag{20} \]

where

\[ \lambda_{\phi0}=\theta_0+a_0(q_{\phi_1}+Q'_1), \tag{21} \]

\[ \xi'_{\phi_1}=\xi_{\phi_1}+\frac{a_0}{d_{\phi_1}}, \tag{22} \]

and the second is

\[ \Delta_j=\Delta_{1j}-\Delta_{2j}+\Delta_{3j}-\Delta_{4j}-\Delta_{5j},\ \mathrm{kW/kvar}, \tag{23} \]

where

\[ \Delta_{1j}=2\left[\xi_j\frac{c_j}{2}P_j+\sum_{\nu=1}^{j}\xi_{\phi_\nu}\frac{c_{\phi_\nu}}{2}P_{\phi_\nu}\right], \tag{24} \]

\[ \Delta_{2j}=2\left[(\xi_j-\xi_{\phi_j})k_{\phi_j}+\sum_{\nu=1}^{j-1}(\xi_{\phi_{\nu+1}}-\xi_{\phi_\nu})k_{\phi_\nu}\right], \tag{25} \]

\[ \Delta_{3j}=2\left[\xi_j\cdot\delta U_j+\sum_{\nu=1}^{j}\xi_{\phi_\nu}\cdot\delta U_{\phi_\nu}\right], \tag{26} \]

\[ \Delta_{4j}=\frac{c_j}{2\xi_j}f_j+\sum_{\nu=1}^{\nu=j}\frac{c_{\phi_\nu}}{2\xi_{\phi_\nu}}f_{\phi_\nu} +\sum_{\nu=1}^{\nu=j}\frac{c_{\phi_\nu}}{2\xi_{\phi_\nu}}e_{\phi_\nu}k_{\phi_\nu}, \tag{27} \]

\[ \Delta_{5j}=c_jq_j+\sum_{\nu=1}^{j}(q_\nu+Q_{a_\nu})c_\nu . \tag{28} \]

The second term \(\Delta_j\) may be regarded as an addition to \(\lambda'_j\), which is due to:

a) active loads (component \(\Delta_{1j}\));

b) network nonuniformity (component \(\Delta_{2j}\));

c) voltage increments in transformers with tap changers, the magnitude of which depends on the operating positions of the tap switches (component \(\Delta_{3j}\));

d) the economic disadvantage of voltage deviations at electrical receivers from the optimal values (component \(\Delta_{4j}\));

e) reactive-power losses in the networks and the reactive loads of substations that do not have synchronous motors (component \(\Delta_{5j}\)).

The regularities governing the formation of the components \(\Delta_{1j}\div\Delta_{5j}\) are determined by formulas (24)—(28). These regularities, as well as the regularities governing the formation of \(\lambda_j\), determined by formula (20), as is not difficult to show, are universal, i.e., valid for a load node of arbitrarily complex branching.

To implement law (18), a very complex automatic-control system is required, using logic units common to the entire load node and measuring the magnitude of the additions \(\Delta_{1j}\div\Delta_{5j}\).

Prerequisites for simplifying the optimization problem

The mathematical model of optimal control proposed above describes the real process with a probable error of no less than \(\pm 10\%\). This circumstance makes it possible to simplify the solution of the optimization problem. The basic requirement in simplifying any problem is the preservation of the principal relationships that determine the essence of the physical or techno-economic phenomenon.

For the problem under consideration, the principal relationships can be found as a result of analyzing the initial information on the modes of electricity consumption by characteristic load nodes with synchronous motors. As the analysis shows, load nodes with a substantial fleet of synchronous motors (concentrating mills, clay-

cement and tire plants, oxygen and compressor stations), where, above all, there are reserves of electric-power savings through optimization of operating conditions, are characterized by uniform schedules of reactive loads and load diagrams of synchronous motors, rhythmic operation, and the relative proximity of supply sources.

As a result, the mode of electric-power consumption of such load nodes can be characterized quite fully by average indices, such as the average active and reactive loads of substations with synchronous motors in the characteristic modes determined by the production technology.

In most cases, exhaustive information is provided by the principal technological mode, which makes it possible to reduce the solution of the optimization problem to setting, for the synchronous motors, certain constant base excitation currents selected from the average indices of electric-power consumption in the principal technological operating mode of the enterprise.

The calculations of base modes reduce to solving the system of equations (12)—(17) and (4)—(7), taking into account the constraints (8)—(10).

Fig. 2. For calculations of base modes. \(P_{\text{net}}\)—losses in networks,
\(P_{\text{gen}}\)—losses in generators, \(P_{\text{SM}}\)—losses in synchronous motors

Figure 2 gives the results of calculations of “base” modes for a characteristic load node having a substantial fleet of synchronous motors. As can be seen from the figure, through the optimal distribution of reactive loads among the sources (synchronous motors and generators), it is possible to reduce substantially the value of the total losses \(p_{\Sigma}^{\mathrm{e}}\) in the principal technological operating mode of the enterprise. At the same time, any limitation of reactive power on the head section \(Q_{\varphi_1}\) reduces the amount of savings in total losses.

As calculations on an electronic digital computer for characteristic load nodes with synchronous motors [4] show, more than 90% of the savings in electric-power losses is provided by setting

base currents, and only the remaining 10% can be implemented by means of automatic regulation of the excitation of SMs according to law (18).

The introduction of base modes without automation means at a number of enterprises may prove impossible because of the conditions for stable operation of SMs and load nodes. In addition, as a result of setting base excitation currents—usually reduced relative to the rated ones—a reserve of reactive power is created in the load node. Without automation means, effective use of this reserve is practically impossible, if one takes into account that up to 30 or more SMs are concentrated in load nodes, while the maintenance personnel capable of manually introducing the reserve are few in number [5].

A techno-economic analysis [4] shows that, under the conditions of operating industrial enterprises with a large fleet of SMs, the use of the simplest automation means is economically justified, their cost not exceeding 40–80 rubles per motor.

In this connection, the development of regulators is of considerable interest which, on the one hand, would be inexpensive and, on the other, would realize the laws of optimal regulation with a known approximation.

Simplified Laws of Optimal Regulation

With accuracy sufficient for practical purposes, it may be assumed that:

1) deviation of the magnitude of the increment \(\Delta_j\) in (19) from the value corresponding to the base mode does not cause a change in the optimal value of the reactive power \(\alpha_{ji}\) in (18);

2) deviations of the voltage level \(k_{\phi 0}\) and of the specific increment at the supply center \(\lambda_{\phi 0}\) from a certain average value are absent.

These assumptions make it possible, while preserving the basic relationships in the object under study, to express the law of optimal regulation in the form of linear dependences:

\[ \alpha_{ij}=\alpha_{0.U}-\alpha_U k_j, \tag{29} \]

\[ \alpha_{1j}=\alpha_{0.q}+\alpha_q Q_j, \tag{30} \]

where

\[ \alpha_{0.U}=\frac{1}{b'_{ij}}\left[ b_0+\alpha_0(q_{\phi 1}+Q_1)+2\left(\xi_{\phi 1}+\frac{a_0}{d_{\phi 1}}\right)k_{\phi 0}+\Delta_j^{\delta}-b_{ij} \right], \tag{31} \]

\[ \alpha_U=2\left(\xi_j+\frac{c_j}{2\xi_j}e_j\right)\frac{1}{b'_{ij}}, \tag{32} \]

\[ \alpha_{0.q}=\alpha_{0.U}-\alpha_U\left(k_{\phi i}-\frac{c_j}{2}P_j+\gamma U_j\right)^{\delta}, \tag{33} \]

\[ \alpha_q=-\frac{c_j}{2\xi_j}\alpha_U \tag{34} \]

are constant coefficients.

As is known, the reactive power of an SM, \(\alpha_{ij}\), is in a very complex nonlinear dependence on the following operating-mode parameters: excitation current, active load, and voltage at the stator terminals. In solving the optimization problem, a number of serious constraints are imposed on these parameters: the excitation current is limited to 60–100% of rated, according to the conditions of stable operation and overheating of the equip-

motor torque, the limits of voltage variation do not exceed 90–110%, and the active load is 50–100% of the rated values.

Under these specific conditions the nonlinearities manifest themselves very weakly, which makes it possible to approximate the complex dependences of reactive power on the operating parameters by linear characteristics of the form [6]

\[ \alpha = h_i i - h_\beta \beta - h_k k - h_0, \tag{35} \]

where

\[ \alpha,\ i,\ \beta,\ k \]

are the relative values of reactive power, excitation current, active load, and stator voltage, respectively;

\[ h_i,\ h_\beta,\ h_k,\ h_0 \]

are certain constant coefficients.

For electric drives characterized by a small change in torque on the motor shaft, it is quite permissible to assume that the deviation of the active load \(\beta\) from the mean value does not cause a noticeable change in the excitation current.

These assumptions make it possible to express laws (29) and (30) through the excitation current of the SM:

\[ i_{ij} = i_{0.U} - i_U k_j, \tag{36} \]

\[ i_{ij} = i_{0.q} + i_q Q_j, \tag{37} \]

where

\[ i_{0.U} = \frac{\alpha_{0.U} + h_0 + h_\beta \beta}{h_i}, \tag{38} \]

\[ i_U = \frac{\alpha_U - h_k}{h_i}, \tag{39} \]

\[ i_{0.q} = i_{0.U} - i_U \left(k_{\varphi j} - \frac{c_j}{2} P_j + \delta U_j\right)^\delta, \tag{40} \]

\[ i_q = \frac{c_j}{2 \xi_j} i_U \]

are constant coefficients. \(\tag{41}\)

Figure labels: ordinate \(\alpha\); abscissa \(k\); “optimal characteristic”; “regulating \(\alpha\)”; “self-regulating characteristic, \(\alpha_c\)”; \(\alpha^\delta\); \(k^\delta\).

Fig. 3. Simplified linear control laws

The simplified laws of optimal control (29), (30), and (36), (37) presuppose an increase in the generation of reactive power by the synchronous motor when the stator voltage is reduced (Fig. 3), which in the general case can be accomplished by the corresponding regulation of the excitation current.

A distinctive feature of synchronous motors with SCF \(>1\) is their ability to increase reactive-power generation when the voltage is reduced without an artificial increase in the excitation current—the so-called self-regulation effect.

With some approximation, the law of self-regulation can be represented in the form

\[ \alpha_{cij}=\alpha_{o.U}-(\alpha_U-h_k)-h_k k . \tag{42} \]

For the same slope of the linear characteristics (29) and (42), i.e., when

\[ \alpha_U=h_k \tag{43} \]

the implementation of the optimal law will not require automatic regulation. In all other cases, automatic regulation of the excitation current is required, enhancing the self-regulation effect.

By substituting into (43) the value of \(\alpha_U\) from (32), one can obtain a calculation formula for determining the critical value of the ratio \(\dfrac{r_j}{x_j}=\xi_j\), at which automatic regulation should not be applied,

\[ \xi_{j\mathrm{cr}}=\frac{h_k b'_{ij}}{4}\pm\frac{1}{2}\sqrt{(h_k b'_{ij})^2-c_j e_j}. \tag{44} \]

In the case \(\xi_j \geqslant \xi_{j\mathrm{cr}}\), the self-regulation law of the synchronous motor satisfies the optimal conditions.

The implementation of simplified laws of optimal control can be carried out by linear regulators (LR) of reactive power (laws 29, 30) or by substantially simpler LR of excitation current (laws 36, 37).

The control parameters may be the voltage at the stator terminals, if there are no transformers with OLTC in the load node (laws 29, 36), or the reactive power of the feeder supplying the substation, if there are transformers with OLTC in the load node (laws 30, 37). For synchronous motors with SCF \(>1\), in a number of cases the optimal conditions are satisfied by the self-regulation law (43).

On practical methods for solving the optimization problem

Because of the specific features set forth above, it is advisable to divide the solution of the problem of optimal use of the compensating capability of synchronous motors at operating industrial enterprises into two stages.

The first stage is associated with calculations and setting the base excitation currents on the synchronous motors. The second involves the introduction on the synchronous motors of the simplest linear regulators that implement the simplified laws of optimal control.

In practical calculations of the base currents in computing regulator settings, it is quite justified to neglect, in the function of total losses (1), the component \(p_{\Sigma}^{U}\) [7].

A practical method for calculating the base currents is set forth in [8]. The regulator settings are calculated successively by the formulas

(31)—(34) and (38), (41) after the base currents have been calculated and the increments \(\Delta_j^{\delta}\) have been determined.

The advisability of introducing automatic regulation can be checked by the simple formula

\[ \xi_j \leq \xi_{j\mathrm{cr}}=\frac{h_k \delta'}{2}, \]

which is obtained from (44) if \(e=0\) is assumed.

The results of industrial experiments on introducing base currents and pilot tests of linear regulators at operating industrial enterprises confirm the technical and economic premises underlying the calculations of optimal modes.

Statistical processing of the results of introducing optimal modes at nonferrous-metallurgy enterprises made it possible to establish that the average value of the savings in electric-power losses due to optimal use of the compensating capability of synchronous motors is estimated at 0.5% of the total electric-power consumption by the load node. On the basis of the studies carried out, “Methodological Guidelines for Optimizing Modes” [9] were developed.

The “Guidelines” were approved by Gosenergonadzor and, by a circular letter of the Ministry of Nonferrous Metallurgy, were recommended to power engineers at nonferrous-metallurgy enterprises as a manual for the optimal use of the compensating capability of synchronous motors under operating conditions. The “Guidelines” can also be used successfully at other facilities equipped with a substantial fleet of synchronous motors: ore-dressing plants of ferrous metallurgy, tire and other plants of the chemical industry, oxygen and compressor stations.

REFERENCES

1. L. A. Soldatkina. On a quantitative assessment of voltage quality in distribution networks. Elektricheskie stantsii, 1963, No. 9.

2. I. A. Syromyatnikov. Operating modes of asynchronous and synchronous electric motors. Gosenergoizdat, 1963.

3. N. A. Melnikov, L. A. Soldatkina. Voltage regulation in electric networks. Energiya Publishing House, Moscow, 1968.

4. V. V. Arkhipenko, V. A. Troshin. On optimal regulation of the excitation mode of synchronous motors at ore-dressing enterprises. Reports to the Sixth Scientific and Technical Conference on Problems of Production Automation. Tomsk, 1969.

5. R. A. Svirskii. Problems of improving operating modes of electrical equipment of an ore-dressing combine. Promyshlennaya energetika, 1968, No. 11.

6. V. V. Arkhipenko. Determination of the coefficients of a linear approximation of the dependence of the reactive power of a synchronous motor on mode parameters. Collection of materials for the All-Union Conference. Tsvetmetinformatsiya, 1970.

7. Yu. M. Tokhanov, V. V. Arkhipenko. Determination of the region of economic stability of the function of calculated costs depending on the voltage level at ore-dressing enterprises. Collection: Voltage Regulation in Electric Networks. Energiya, 1968.

8. V. V. Arkhipenko, V. A. Troshin. A practical method for calculating excitation modes of synchronous motors. Elektrichestvo, 1969, No. 10.

9. Methodological guidelines for optimizing excitation modes of synchronous motors at operating enterprises of nonferrous metallurgy. Ministry of Nonferrous Metallurgy of the USSR, Moscow, 1969.