LIMITING OPERATING MODES OF LONG-DISTANCE AC POWER TRANSMISSION LINES WITH INTERMEDIATE SYSTEMS UNDER STATIC STABILITY CONDITIONS

R. I. BORISOV, V. I. GOTMAN

(Presented by the Department of Electrical Systems and Networks)

The influence of intermediate systems (IS) on the operating modes, transmission capacity, and stability of long-distance power transmission lines continues to remain an insufficiently studied question, but one that is very important from the standpoint of using the internal reserves of schemes to improve the techno-economic indicators of the aforementioned power transmission lines.

In the present work, the limiting values of active-power transits over a long-distance power transmission line are determined, taking into account the influence of the location, parameters, and operating mode of the IS, as well as the limiting values of relative angles between stations corresponding to the stability limit.

Fig. 1. Schematic diagram of a long-distance power transmission line with an intermediate system

The investigations are carried out as applied to the scheme in Fig. 1, where a transmitting station, through a long-distance power transmission line with one IS, operates into a receiving system.

Initial Provisions and Adopted Assumptions

It is assumed that all electrical and mechanical regulators of the system are adjusted so that self-swinging is excluded; by virtue of this, all generators are replaced by equivalent electromotive forces \(k_E E_q\), applied behind the resistance \(\Delta x = k_x x_d\), where \(k_E, k_x\) are coefficients depending on the type of generator, the type of regulator, and its setting.

With sufficient accuracy, for a generator–transformer unit one takes [3] \(\Delta x = k_x x_d = x'_d + x_{\mathrm{T}}\) and \(E'\) for proportional-type regulators, and \(\Delta x = x_{\mathrm{T}}\) and \(U_{\mathrm{G}}\) for fast-acting regulators.

The electrical system, each of whose stations is represented by one equivalent generator, is considered as a positional one [4]. The equivalent circuit of the system, in which the loads are represented by constant resistances, is characterized by complex values of self and mutual admittances.

The equivalent circuits of synchronous machines, in accordance with what was stated above, are adopted taking into account the automatic voltage regulators (AVR) installed on them.

Under these conditions, with the emf of synchronous machines constant, the powers of each station are functions of the relative angles between their rotors [2].

\[ P_i = P_{ii} + \sum_1^{n-1} P_{ij}\sin(\delta_{ij}-\alpha_{ij}), \tag{1} \]

where

\[ P_{ii}=U_i^2 y_{ii}\sin\alpha_{ii}, \qquad P_{ij}=U_i U_j y_{ij}. \]

The self and mutual admittances in equations (1) are expressed through the generalized constants of the four-terminal networks of the corresponding parts of the system.

Of the three mutual angles, only two are independent variables and are related by

\[ \delta_{12}+\delta_{23}+\delta_{31}=0. \tag{2} \]

The limiting operating condition of the system is found by successive tightening of the normal initial condition by redistributing the active powers of the generating stations; in this process the active powers of \((n-2)\) stations remain unchanged, one is loaded, and the other is the balancing station. The limiting condition of the system corresponds to the maximum of the power characteristic of the station being loaded, or \(dP/d\delta=0\).

For the first (most distant) generator, the maximum value of power that can be transmitted from it to the rest of the system, as follows from (1), will occur at \(\delta_{12}=90^\circ+\alpha_{12}\), \(\delta_{13}=90^\circ+\alpha_{13}\) and will be

\[ P_1 = P_{11}+P_{12}+P_{13}. \]

To determine \(P_{1\mathrm{pr}}\) in the general case, it is necessary to establish the relationship between \(P_1\) and the two indicated independent mutual angles. The required dependence can be obtained from the equation

\[ \frac{dP_1}{d\delta_{12}}=0, \]

which, after transformations, has the form

\[ P_{12}P_{13}\cos(\vartheta_{12}-\alpha_{12})\cos(\vartheta_{13}+\alpha_{13}) +P_{12}P_{23}\cos(\vartheta_{12}-\alpha_{12})\cos(\vartheta_{23}+\alpha_{23})+ \]
\[ +P_{13}P_{23}\cos(\vartheta_{13}-\alpha_{13})\cos(\vartheta_{23}+\alpha_{23})=0. \tag{3} \]

Equation (3), taking relation (2) into account, makes it possible to obtain the relationship between the mutual angles of three stations at which the maximum power of the first station \(P_{1\text{pr}}\) occurs.

Representing the IS of a certain power system by an equivalent four-terminal network is not the only possible representation and is not always the most convenient.

In a number of cases, when the IS contains several generating stations, as well as sources of reactive power, the influence of the IS on the operating mode of the long-distance power transmission line and on the limiting values of power flows can be taken into account by representing the IS by generalized static characteristics of response with respect to voltage at the point of their connection [2, 5]

\[ P=P(U),\qquad Q=Q(U). \]

Representing the IS by static characteristics of active and reactive power substantially simplifies the solution of the problem posed, since the study of the static characteristics of local systems and the limiting operating modes of the long-distance power transmission line can be carried out separately, which facilitates the rapid determination of the factors defining the limiting power of the long-distance power transmission line under aperiodic stability conditions.

The regulating properties of each static characteristic are determined by the slope coefficients \((\kappa_{qu},\ \kappa_{pu})\) and by the range of variation of its variables.

The slope coefficient \(\kappa_{pu}\) is determined by the magnitude of the active power of the load nodes, their regulating effects, and the electrical distance of the loads from the IS connection node.

The coefficient \(\kappa_{qu}\) of the static characteristics of reactive power depends to a considerable degree on the coupling impedances of the IS connection node with sources of active and reactive power and on their voltage gain coefficients.

Equation (4) was obtained for the transmitted power at the end of the first section of the long-distance power transmission line (Fig. 1) as a function of the voltage of the IS connection point and of its static characteristics

\[ P_1=(0.5dc-a_1U^2)+a\sqrt{\,b-0.25d^2\,}, \tag{4} \]

where

\[ a=[U^2(r_1+r_2)+Q(U)];\qquad r_1=-\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2};\qquad r_2=-\frac{\operatorname{Im}(\dot D_2\hat B_2)}{B_2^2}; \]

\[ c=[U^2(\alpha_1+\alpha_2)+P(U)];\qquad \alpha_1=\frac{\operatorname{Re}(\dot A_1\hat B_1)}{B_1^2};\qquad \alpha_2=\frac{\operatorname{Re}(\dot D_2\hat B_2)}{B_2^2}; \]

\[ b=\frac{U_1^2}{B_1^2}\cdot\frac{U^2}{(a^2+c^2)};\qquad d=1+\left[\frac{U_1^2}{B_1^2}-\frac{U_2^2}{B_2^2}\right]\frac{U^2}{(a^2+c^2)}. \]

The studies carried out show that, for a given initial operating mode, the slope of the static characteristics is stable and, with a sufficient degree of accuracy, within the operating range of voltage variation the static characteristics can be linearized.

\[ P(U)=\pm P_0+(U-U_0)\kappa_{pu}, \tag{5} \]

\[ Q(U)=\pm Q_0+(U-U_0)\kappa_{qu}, \]

where \(U_0\) and \(U\) are, respectively, the voltage in the initial operating mode and the current voltage at the IS connection point;

\(P_0\) and \(Q_0\) are the power exchanges in the initial mode; the sign \((+)\) corresponds to withdrawal in the IS.

The limiting mode of remote station No. 1 can be found from (4) by successively reducing the voltage \(U\) at the point of connection of the IS, to which \(dP_1/dU=0\) will correspond.

The static-stability margin of the power transmission under consideration should be characterized by the margin factor for each section

\[ \kappa_{3i}=\frac{P_i-P_{i0}}{P_{i0}}, \]

where \(P_1=P_{1\mathrm{lim}}\) is the limiting value of the power of the first section, determined taking into account the influence of the IS;

\(P_2\) is the power of the second section corresponding to \(P_{1\mathrm{lim}}\);

\(P_{i0}\) is the power of the sections in the initial mode.

The regulating effect of the active power of the IS somewhat reduces the limiting value of the power of the first section, while increasing that of the second. In this case, the maxima of the angular characteristics of the sections occur at different angles \(\delta_{12}\) \((\delta_{12}/P_{1\mathrm{lim}}<\delta_{12}/P_{2\mathrm{lim}})\).

Figure 2 gives the results of calculations of limiting modes according to (4), (5) for a long-distance lossless EHV transmission line 1000 km long as a function of the location and the parameters of the IS mode.*

In the initial mode, the voltage at the point of connection of the IS was taken as \(U_0=1\).

The IS, having a natural positive regulating effect, increases the limiting values of power along the EHV line.

If the IS is balanced in active power, then the greatest effect occurs when it is located at the middle of the line (in the case \(U_1=U_2\)). For the example considered (Fig. 2, 1), the increase is 22.5%. In the presence of a voltage difference \((U_1\ne U_2)\), the location of the maximum can be determined from expression (4).

\[ \frac{dP_1}{dl_1}=0,\qquad l_1=\frac{1}{a_0}\arctg\frac{U_1\sin\lambda}{U_2+U_1\cos\lambda}\;(\mathrm{km}), \]

where \(a_0=0.06\ \mathrm{grad}/\mathrm{km}\), \(\lambda\) is the wave length of the line, which for \(U_1=U\) gives \(l_1=l/2\).

Withdrawal of active power in the IS shifts the maximum of the limiting values of transmitted power toward the sending end, and the more so the greater the withdrawal for an unchanged value of \(k_{qu}\).

In Fig. 3, using equations (1)--(3), the region of limiting modes of station No. 1 is constructed as a function of the initial mode of the IS \((P_3')\), connected at the middle of an EHV line 1000 km long.

It follows from the above that the limiting value of the power transmitted from station No. 1 (or part of the system) depends substantially on the point of connection, the values of the tie impedances, and the magnitude and direction of the flows of the intermediate system, i.e., on the parameters of the initial mode.

Determination of the limiting values of the relative angles under static-stability conditions reduces to analysis of the roots of the characteristic equation compiled for a three-machine scheme [2]

\[ p^4+A_2p^2+A_4=0, \tag{6} \]

where

\[ A_2=a_{21}^{12}+a_{31}^{13};\qquad A_4=a_{21}^{12}a_{31}^{13}-a_{21}^{13}a_{31}^{12}; \]

\[ \text{* Calculations were performed in relative units at } U=500\ \mathrm{kV},\ S=P_{\mathrm{nat}}\simeq 900\ \mathrm{MW} \]

\[ a_{21}^{12}=\frac{1}{T_{J1}}\frac{\partial P_1}{\partial \delta_{12}}-\frac{1}{T_{J2}}\frac{\partial P_2}{\partial \delta_{12}};\qquad a_{21}^{13}=\frac{1}{T_{J1}}\frac{\partial P_1}{\partial \delta_{13}}-\frac{1}{T_{J2}}\frac{\partial P_2}{\partial \delta_{13}}; \]

\[ a_{31}^{12}=\frac{1}{T_{J1}}\frac{\partial P_1}{\partial \delta_{12}}-\frac{1}{T_{J3}}\frac{\partial P_3}{\partial \delta_{12}};\qquad a_{31}^{13}=\frac{1}{T_{J1}}\frac{\partial P_1}{\partial \delta_{13}}-\frac{1}{T_{J3}}\frac{\partial P_3}{\partial \delta_{13}}. \]

According to the assumptions adopted, only an aperiodic loss of stability can occur in the system. The condition determining the stability boundary is the equality to zero of the free term of the characteristic equation, which for the case under consideration has the form

\[ \begin{aligned} A_1={}&P_{12}P_{23}\biggl[ \frac{1}{T_{J1}T_{J2}}\cos(\delta_{12}-\alpha_{12})\cos(\delta_{23}-\alpha_{23})+\\ &\quad+\frac{1}{T_{J1}T_{J3}}\cos(\delta_{12}-\alpha_{12})\cos(\delta_{23}+\alpha_{23})+\\ &\quad+\frac{1}{T_{J2}T_{J3}}\cos(\delta_{12}+\alpha_{12})\cos(\delta_{23}+\alpha_{23}) \biggr]+\\ &+P_{12}P_{13}\biggl[ \frac{1}{T_{J1}T_{J2}}\cos(\delta_{12}+\alpha_{12})\cos(\delta_{13}-\alpha_{13})+\\ &\quad+\frac{1}{T_{J1}T_{J3}}\cos(\delta_{12}-\alpha_{12})\cos(\delta_{13}+\alpha_{13})+\\ &\quad+\frac{1}{T_{J2}T_{J3}}\cos(\delta_{12}+\alpha_{12})\cos(\delta_{13}+\alpha_{13}) \biggr]+\\ &+P_{13}P_{23}\biggl[ \frac{1}{T_{J1}T_{J2}}\cos(\delta_{13}-\alpha_{13})\cos(\delta_{23}-\alpha_{23})+\\ &\quad+\frac{1}{T_{J1}T_{J3}}\cos(\delta_{13}-\alpha_{13})\cos(\delta_{23}+\alpha_{23})+\\ &\quad+\frac{1}{T_{J2}T_{J3}}\cos(\delta_{13}+\alpha_{13})\cos(\delta_{23}-\alpha_{23}) \biggr]. \end{aligned} \tag{7} \]

As is evident from (7), the stability regions are determined only by the ratio of the amplitudes of the mutual powers and by the inertia constants of the generators.

Suppose that the remote station with the intermediate system operates into a powerful concentrated system with constant frequency, which makes it possible to take \(T_{J2},T_{J1}\gg T_{J}^{\xi}\). This reduces the degree of freedom of the system; expression (7) is simplified and takes the form of equation (3), i.e., the region of limiting operating modes in this case coincides with the region of the stable state of the system.

In Fig. 4, in the coordinates of two independent angles \(\delta_{12}\) and \(\delta_{13}\), the region corresponding to the limiting operating modes of system (3) and the stability limit (7) is constructed. The calculation was carried out for the same case as in Fig. 3.

Fig. 2. Dependence of the power limit of a long-distance AC transmission line on the location of the intermediate system:
\(\alpha_{pU} = 0.2;\ \alpha_{qU} = 4;\) in the initial operating condition: \(1\) — \(P_0 = 0,\ Q_0 = 0;\ 2,\ 3\) — \(P_0 = 0.5,\ Q_0 = 0;\ 4,\ 5\) — \(P_0 = -0.5;\ Q_0 = 0\)

Fig. 3

The dependences obtained indicate that stable operation of the system can take place at mutual angles considerably exceeding \(90^\circ\).

Fig. 4. 1 — region of limiting modes; 2 — stability region for \(T_{j1} = T_{j2} = 2T_{j3}\)

Conclusions

The intermediate system stabilizes the voltage at the connection point to the long-distance AC transmission line and increases the limiting transmitted power under the conditions of static stability (by up to 20% for the calculation conditions considered).

References

1. I. M. Markovich. Operating Conditions of Power Systems. Energiya, Moscow, 1969.
2. P. S. Zhdanov. Stability of Electrical Systems. Gosenergoizdat, Moscow, 1948.
3. V. A. Venikov, N. D. Anisimova et al. Investigation, by means of a digital computer, of the static stability of complex automatically controlled electrical systems. Elektrichestvo, 11, 1967.
4. V. A. Venikov. Transient Electromechanical Processes in Electrical Systems. Energiya, Moscow, 1964.
5. M. L. Levinshtein, O. V. Shcherbachev. A method for calculating the static stability of complex electrical systems using equivalent regulating effects of stations and loads. Izv. vuzov SSSR, Energetika, 8, 1962.