The direction of power flow is determined by the current settings, the rectifier end always having the larger setting. The difference between the settings is the current margin Id,,, and is given by

(11.5.4)

Many d.c. transmission schemes are bidirectional, i.e. each converter operates sometimes as a rectifier and sometimes as an inverter. Moreover, during d.c. line faults, both converters are forced into the inverter mode in order to de-energise the line faster. In such cases each converter is provided with a combined characteristic as shown in Fig. 11.13 which includes natural rectification, constant current control and constant extinction angle control. With the characteristics shown by solid lines (i.e. operating at point A), power is transmitted from converter I to converter 11. Both stations are given the same current command but the current margin setting is subtracted at the inverter end. When power reversal is to be implemented the current settings are reversed and the broken line characteristics apply. This results in operating point B, with direct voltage reversed and no change in direct current.

A common used operating mode is constant power (c.P.) control. As with constant current control either converter can control power. The power setting at the rectifier terminal P d . , must be larger than that at the invertor terminal P d S r by a suitable power margin P d m , that is

= pds,

(11.5.5)

The c.p. controller adjusts the C.C. control setting lipto maintain a specified power flow P i p through the link, which is usually more practical than C.C. control from a system operation point of view. The voltage/current loci now become nonlinear, as shown in Fig. 11.14. Several limits are added to the cap.characteristics as shown in Fig. 11.15. These are:

a maximum current limit with the purpose of preventing thermal damage to the converter valves; normally between 1 and 1.2 times the nominal current a minimum current limit (about 10 % of the nominal value) in order to avoid possible current discontinuities which can cause overvoltages voltage-dependent current limit (line OA in the figure) in order to reduce the power loss and reactive power demand.

In cases where the power rating of the d.c. link is comparable with the rating of either the sending or receiving a.c. system interconnected by the link, the frequency of the smaller a.c. system is often controlled to a 1arge.extent by the d.c. link. With power frequency (p.f.) control if the frequency goes out of pre-specified limits, the output power is made proportional to the deviation of frequency from its nominal value. Frequency control is analogous to the current control described earlier, i.e. the converter with lower voltage determines the direct voltage of the line and the one with higher voltage determines the frequency. Again, current limits have to be imposed, which override the frequency error signal. The c.p.1e.a. and c.c.1e.a. controls were evolved principally for bulk point-to-point power transmission over long distances or submarine crossings and are still the main control modes in present use. Multiterminal d.c. schemes are also being considered, based on the basic controls already described. Two alternatives are possible, i.e. constant voltage parallel and constant current series schemes.

The steady-state behaviour of a multiconductor line at a discrete frequency is described by the equations

(111.1) (111.2)

where [Z] and [Y] are the series impedance and shunt admittance matrices per unit distance and [VI and [I] are the vectors of voltage and current phasors in the various conductors. Differentiating equations (111.1) and (111.2) again with respect to x gives

(111.3) (111.4)

It should be noted that in this case the matrix products [Z].[Y] and [Y'].[Z'] are not equal, except in special cases. These equations are still difficult to solve because all phases are coupled. However, just as three-phase equations with balanced matrices can be transformed into decoupled single-phase equations using symmetrical components, it is possible to transform equations (111.3) and (111.4) into decoupled equations as well. By transforming phase voltages to 'modal' voltages,