Three phase algorithm

For assessing the unbalanced operation of an interconnected HVAC-DC system three phase LF algorithm is necessary

Converter operating from unbalanced ac system

Unbalance in converter control equipment

Asymmetry in converter transformer etc

Analysis of such system requires model of three phase converter during unbalanced operation and its integration into three phase fast decoupled load flow

State variables of combined system is defined as

-vectors of the balanced internal voltages at the generator internal bus bars - vectors of the three-phase voltages at every generator terminal bus bar and every load bus bar -vector of the d.c. variables

Equations describing the ac system is

Where,

(mismatches at converter terminal is shown separately)

DC system equation are given by

3-phase ac-dc load flow problem is formulated as the solution of the below eqn for the set of variables (V,θ,x)

=0

The 3 phase ac-dc load flow equations are based on the Fast Decoupled ac algorithm

and Newton Raphson algorithm for dc eqns.

------1

---------2

---------3

Solution technique used is sequential method with block successive iteration of three

equations. Iteration sequence is -P, Q, DC-

For ac iteration (eqns(1,2)) dc variables x are treated as constants i.e the d.c. system is modelled by real and reactive power injections at the converter terminal bus bar.

Obtained from latest solution of dc eqn

For the d.c iteration, the a.c variables at the terminal bus bars are taken to be constant

The converter operation is strongly related to the magnitude of the terminal voltages and more weakly dependent on their phase angles.

In the three-phase case, final convergence

is comparatively slow because the d.c. system behaviour is dependent on the phase-angle unbalance as much as on the voltage unbalance

The vector R of dc variables x can now be defined.

Consider the HVDC interconnection as in fig

Each bridge in fig. must be modelled independently

due to the difference in the converter transformer connections (operate with a different degree of

unbalance). This is in contrast to the balanced d.c.

model where it is possible to combine bridge in

series and in parallel into an equivalent single bridge.

The dimensions of the three-phase d.c. model will, therefore, be much greater than the balanced d.c. model

Assumptions made are 1. The three a.c phase voltages at the

terminal bus bar are sinusoidal.2. The direct voltage an direct current

are smooth.3. The converter transformer is

lossless and the magnetizing admittance is ignored

ai - off-nominal tap ratios on the primary side U13∟C1, U23∟C2, U21∟C3 - phase-to-phase

source voltages for the converter referred to the transformer secondary.

Ci are the zero crossings for the timing of firing pulses

αi - Firing delay angle measured from the respective zero crossing

Vd -total average d.c. voltage from complete bridge

Id - Average d.c. current. where i = 1,2,3 for the three phases

The phase-to-phase source voltages referred to the transformer secondary depends on the transformer connection and off-nominal turns ratio.

consider the star-star transformer

Gives a total of 6 equations

DC voltage is given by integrating the output waveform

Depending on system topology an eqn for each converter is given by f(Vd,Id)=0 which relates dc voltages and current)

Finally the control strategies are specified,

Such as minimum firing angle control, off nominal taps etc

Remaining are derived from system operating conditions

The mathematical model of the converter includes the formulation of equations

for the individual phase real and reactive power flows on the primary of the converter transformer.

The calculation of the individual phase, real and reactive powers at the terminal bus bar requires the values of both the magnitude and angle of the fundamental components of the individual phase currents flowing into the converter transformer.

-the fundamental component of the voltage wave shape at the transformer secondary bus bar

-the fundamental component of the secondary current wave shapes; where i = 1,3 for the three phases

The calculation of real and reactive power in three phase analysis is influenced by the three phase transformer connection. Such influence is accounted by including the converter transformer in the dc model.

The inclusion of the converter transformer within the d.c. model requires 12 extra variables

The three-phase converter transformer is

represented by its nodal admittance model

Thus a total set of 26 variables is required for each converter in the d.c. system

The 26 variable vector (x) is:

Take common power and voltage bases on both sides of the converter.

MVAb = base power per phase Vb= phase-neutral voltage base. (the

phase-neutral voltage is used as the base)

The current base on the a.c. and d.c. sides are also equal.

The p.u. system does not change the form of any of the converter equations.

The sequential method (P, Q, DC) produces fast and reliable convergence.

The reactive power convergence is slower than for the a.c. system alone.

The overall convergence rate of the a.c.-d.c. algorithms depends on the successful interaction of the two distinct parts.

The a.c. system equations are solved using the

fast-decoupled algorithm, whereas the d.c. system equations are solved using Newton-Raphson approach.

The solution time of the d.c. equations is normally small compared to the solution time of the ac. equations.