Compensation by Sectioning (Dynamic Compensation) 173 with itself and, therefore, also with P (through Equation This

can be written

P = sin 6 ,

where

cos .

can always be determined from Equation because E, and are constants.) gives rise to the concept of a series of sinusoids varying amplitudes as shown in Figure 32 (dotted curves), Each corresponds to a fixed value of the compensating susceptance B,. For example, suppose P i s equal to the steady-state limit of the uncompensat- ed line, = From Equation 136 with = = 2 sin-' Operation is at the point A in Figure 32. From Equa- tion 138, = 1.155 that is, the "current" static power transmission characteristic has a maximum transmissible power of 1.155

If the transmitted power were increased to 1.2 would increase to 2 sin-' (1.212) = and would increase to

= 1.250 Operation is at the point B in Figure 32. The

I

\

- capacitive

limit reached

Steady-State Reactive Power Control i n Electric Transmission Systems

controlling effect of the midpoint compensator constrains the operating point to move along the locus given by Equation 136, passing smoothly from one constant-By _{sinusoid to another as P varies.}

For P

>

6 is greater than 90". The system is stable as long as as long as an increase in transmission angle is accom- panied by an increase in transmitted power; see Section However, the operating point is now on the unstable side of the "current" con- stant-By sinewave. For example, if P = 1.8 6 = 2 sin-' =

and = 2.294 Operation is at point in Figure 32. The system owes its stability to the fact that if the transmission angle 6 increases slightly, the compensator responds by changing immediately in such a way as to keep constant and so to increase to the value which satisfies Equation 138, so that the operating point moves along the stable characteristic, P = sin

Typically, in transmission systems which are compensated by section- ing and which are operating at high power levels, the effective compensat- ing susceptance is capacitive. In practice there is an economic limit to the capacitive reactive power of the compensator. With many types of compensator, when the limit is reached the compensator ceases to main- tain constant voltage at its terminals and behaves instead like a fixed ceptance. The operating point then leaves the dynamically stable charac- teristic P = 2 sin and moves on to the constant-By sinusoid

corresponding to the maximum value of that is, This depar- ture is shown at point D in Figure 32. The higher the value of the steeper is the (negative) slope of the constant-By characteristic at the point of departure. If were smaller, the point of departure could occur on the stable side of one of the dotted sinusoids in Figure 32, such as at point A or B. However, the power transmitted P and the transmis- sion angle would then have to be kept to much lower values. In other words, the capacitive current rating of the compensator is one of the main factors limiting the achievable increase in the maximum transmissible power.

Reactive Power ar From Equations

133, and 134 it can be shown that the reactive power requirement at the sending end is given by

- cos

with E, = = E . By symmetry, - Q,.

For a synchronous condenser or saturated-reactor compensator is the maximum capacitive current divided by

2.6. Compensation by Sectioning (Dynamic Shunt Compensation)

Reactive Power A practical

control system for varying B, would not be based on Equation 134, be- cause there is no information as to the value of 6 available at the inter- mediate compensator station. Instead, a feedback control system would be used to keep In the saturated-reactor compensator an in- herent regulating process achieves the same end.

With constant the compensator reactive power Q, =

is a function of 6 also:

Through the parameter 6, is related to the transmitted power P by Equations 135 and 140. Note that = being the elec- trical length of the line in radians.

Example of Line Compensated by Sectioning?

As an example of compensation by sectioning, or "dynamic shunt com- pensation," consider a 400-mi line with a midpoint constant-voltage com- pensator. As in Section 2.5.3, = 0.8108 and = The power transmission characteristic is given by Equation 132 as

6 6

sin 2.4667 sin

-

0.8108 2 2

with = = = pu. is quite close to the value 2.6084 achieved with 50% series compensation (with shunt reactors connected). The reactive power requirement of the compensator is given by Equa- tion 140 as

and the terminal reactive powers are given by Equation 139 as

126 Steady-State Reactive Power Control in Electric Transmission

At no-load = -0.2027 corresponding to the line-charging reactive power of the 100 mi of line nearest to the sending end. (Note that this differs from the value -0.2055 calculated from Equation 30. This is due to approximation involved in the circuit.) At no- load, = 0.4054 corresponding to the line-charging reactive power of the central 200 mi of the line.

The variation of the main parameters is shown in Table as the transmitted power varies. It can be seen that the reactive power absorbed or supplied by each terminal is just half that of the compensator. The two vary in concert, the terminal synchronous compensating the extreme 100-mi portions of the line, while the compensator compen- sates the central 200-mi portion.

Although the overall transmission angle exceeds transmission remains stable up to as long as the midpoint voltage remains con- stant, It can be seen that the power transmission up to 1.25 does not require excessive capacitive reactive power from the compensator.

T A B L E of a 4 0 0 - M i l e T r a n s m i s s i o n L i n e C o m p e n s a t e d by a C o n s t a n t - V o l t a g e Device a t i t s M i d p o i n ta W i t h o u t

Q

C o m p e n s a t i o n 0 10.440 21.249 32.932 46.456 64.966 u n s t a b l e u n s t a b l e u n s t a b l e u n s t a b l e u n s t a b l e Note the conventions for and Q, and are both negatrve for absorption, the compensator then Q, and are both for generation; the conipensator then

References 127

Whether the system has transient stability for major faults at this level of transmission is another question, which will be dealt with in the next chapter.

REFERENCES

1. _{A. Boyajian, "The Physics of Long Transmrssron Lines," Rev. 15-22, July} 1949.

2. and E. "Comparative of Series and Shunt Compensation Schemes for AC Transmission Systems." IEEE Trans. Appar. Syst. 1819-1830 (1977).

3. G . Baum. "Voltage Regulation and Insulation for Large Power Long Distance Transmrssion Systems." AIEE 40, 1017-1032

L. Fortescue and C. Wagner, "Some Theoretical Considerations of Power Transmission," AIEE 43. 106-1 13 (1924).

5. W Kimbark. "How to Improve System Stability Without Riskrng Subsynchronous Resonance." IEEE Trans. Appar. Syst. 96 1608-1613 (1977).

6. G. L. Wilson and P. Zarakas, "Anatomy of a Blackout," IEEE Spectrum. 38-46 (February 1978).

7. B. Crary, Power New York, 1945, 1947 8. E. Kimbark, Power Stability, Wiley, New York, 1948.

9. R. T. and E. Kimbark, Stability of Large Power IEEE Press, 1974.

A. Gross, Poiver Wiley, New York, 1979.

G . Jancke, N. and 0 . Nerf, "Series Capacitors in Power Systems." IEEE Appar. 94, (1975).

A. Miske, "A New Technology for Series Capacitor Protection," Electr.

18-20 (1979).

G. D. Breuer. H. M. Rustebakke, R. A. and H. Simmons Jr., Use of Series Capacitors to Obtain Maximum EHV Transmissron Capability," IEEE Power Appar. 83. 1090-1 102

Barthold et al.. "Static Shunt Devices for Reactive Power Control," CIGRE Paper 31-08, (1974).

E. Friedlander and K. M. "Saturated Reactors for Long Distance Bulk Power Lines," Electr. Rev .. 940-943 (June

Elsliger et "Optimization of

Compensated System Static Compensators on a Large Scale," IEEE PES Power Paper A78 New York, 1978.

D. A. and M. Z. Tarnawecky. "Compensatron of Long Distance Transmission Lines by Shunt Connected Reactance Controllers," IEEE Trans., Power Appar, 94, 655-664

E. Friedlander. "Transient Reactance Effects Static Shunt Reactrve Compensators for Long Lines," IEEE Power Appar, 95. 1669-1680

J. D. Ainsworth et "Long Distance A C Transmissron Using Static Voltage Stabiliz- ers and Switched Linear Reactors," CIGRE Paper 31-01,

J . Ainsworth et al., "Recent Developments towards Long Distance AC sion Using Saturated Reactors," IEE 107, 242-247

M. Boidin and G. Drouin, "Performance Dynamiques des Compensateurs Statrques Thyristors et d e Regulation." Rev. Electr.. 58-73 (1979). (In French)

Chapter 3

3.1.1. The Dynamics of an Electric

electric power system is never in equilibrium for very long. Frequent s disturb the equilibrium so that the system is almost always in on between equilibrium or steady-state conditions. The theory in apter 2 is valid in the steady-state and when changes are taking place This chapter deals with the dynamic behavior of the electric power

during the transition from one equilibrium condition to another. ure 1 depicts the full range of dynamic phenomena that characterize er system performance. The steady-state conditions treated in pter 2 are shown at the far right. Here we are more concerned with namic transitions which occupy a time span near the center of Figure 1.

this regime there is a need to control voltage excursions and

tions of momentum among the synchronous machines which result m faults, switching operations, and load changes. The control must be id and accurate; otherwise, the stability of the system may be lost,

locally or throughout the system.

emphasis in this chapter is on the controlling and stabilizing uence of compensating devices such as shunt and series capacitors, unt reactors, synchronous condensers, and especially static reactive

3.2. Four Characteristic Time Periods

In document [T.J.E.miller] Reactive Power Control in Electric Systems (Page 76-80)