AC-DC LOADFLOW

INTRODUCTION

• HVDC is accepted as an economical alternative to ac transmission.

• Used in Underground, submarine, long distant transmissions• Also for interconnecting networks under different

frequencies.• This extensive application demands modelling techiques for

planning purposes.• The basic ac load flow is substantially modified to incorporate

the DC variables.• Thus there are single phase and three phase algorithms.

SOLUTIONS

• Done in Fast Decoupled Method• Sequential solution techniques.

AC and DC eqns are solved separately.For AC solution, converter stations are

represented as real & reactive Power sources.For DC soln, AC systems are represented as a

constant voltage source.These are cross substituted until the convergence

is satisfied.

AC-DC LOAD FLOW

The operating state of the combined AC-HVDC power system is defined by the vector,

where• v is a vector of the voltage magnitudes at all a.c.

system bus bars,• θ is a vector of the angles at all a.c. system bus bars

(except the reference bus which is assigned θ = 0) • x is a vector of d.c. variables

AC system eqns are derived from the ac operating conditions with modification required to real & reactive power mismatches at converter terminal bus bar.

where

• Equations derived from ac system conditions

(mismatches at converter terminal bus bars are indicated separately)

• Eqns from dc system conditions

• General AC-DC load flow problem is as

‘term’ refers to converter ac terminal bus bar

Formulation of ac-dc load flow equation based on NR algorithm

Δθ Δθterm

= [ J ] ΔV ΔVterm

Δx

J is the matrix of first order partial derivatives ΔPterm =

ΔQterm=

Modeling of HVDC systems

The representation of the dc systems requires consideration of the following:

• Converter model• DC transmission line/network model• Interface between ac and dc systems• DC system controls model

DC SYSTEM MODEL

Assumptions before selection of and formulation of eqns:• The three a.c. voltages at the terminal bus bar are balanced

and sinusoidal.• The converter operation is perfectly balanced (small voltage

drop is neglected).• The direct current and voltage are smooth.• The converter transformer is lossless and the magnetising

admittance is ignored.

Selection of Converter Variables

• Similar bridges connected to same AC terminal busbars operate identically.

Single phase equivalent circuit

DC variables• Min no. of variables is the no. of independent variables by

which any other system parameter (eg: Pdc & Qdc) can be expressed.

• The set of independent variables which is required to define the operation of the converter (i.e to model the DC system) and to define the control specification of the HVDC converter are given by

[ x ]=[Vd, Id, a, cosα, ø]T

dc per system Use common power and voltage base parameters on both

sides of the converter, i.e. the a.c. and d.c. sides for simplicity.

To preserve consistency of power in per unit, the direct current base, obtained from (MVAB)/VB has to be √3 times larger than the a.c. current base.

In pu the eqn is,

K=0.995

Equations describing the DC system are given as , 1.The fundamental current magnitude on the converter

side is related to the direct current by

2.The fundamental current magnitudes on both sides of the lossless transformer are related by the off-nominal tap,

3. The d.c. voltage may be expressed in terms of the bus bar voltage on the system side of the converter transformer, Vterm

4. The d.c. current and voltage are related by the d.c. system configuration

5. Real power equation relates dc power to transformer primary real power and secondary real power (lossless transformer)

Eqn 1,2&5 can be combined to give

The equations for Pdc and Qdc may now be written as

i.eReal power injection into the ac bus is P=Vd Id=Pdc

Reactive power injection into the ac bus is Q = P tanø

• DC system model can be summarized as follows k - no of converters present

R is the vector of DC system equations and x is the

vector of independent variables for dc system

Incorporating control Equations

At each converter, the angle ‘α’ and the transformer tap ‘a’ can be controlled with in limits to achieve

current control, dc voltage control, real & reactive power control.

Limits on the control variables must be considered when specifying Id, Vd, Pd , Qd

Each additional converter in the d.c. system contributes two further constraint equations which must be derived from the control strategy of the system to define the operating state.

Inversion operation

• All eqns for rectifier are also applicable to inverter.• Due to unidirectional nature of converter valves power

reversal requires direct voltage polarity reversal –achieved by delay angle control i.e α>900 (average inverter voltage will be negative)

• Inverter voltage in terms of extinction angle is

AC-DC LOAD FLOW

The operating state of the combined AC-HVDC power system is defined by the vector,

where• v is a vector of the voltage magnitudes at all a.c.

system bus bars,• θ is a vector of the angles at all a.c. system bus bars

(except the reference bus which is assigned θ = 0) • x is a vector of d.c. variables

AC system eqns are derived from the ac operating conditions with modification required to real & reactive power mismatches at converter terminal bus bar.

where

• Equations derived from ac system conditions

(mismatches at converter terminal bus bars are indicated separately)

• Eqns from dc system conditions

Formulation of ac-dc load flow equation based on NR algorithm

Δθ Δθterm

= [ J ] ΔV ΔVterm

Δx

J is the matrix of first order partial derivatives ΔPterm =

ΔQterm=

• Applying the ac fast decoupled assumptions to all jacobian elements related to ac system eqns gives

Solution of AC-DC power flow

The solution methodology for AC-DC power flow can be classified as

1. Simultaneous or unified2. Sequential or alternating

Simultaneous method• In the first approach the AC & DC eqns are solved

simultaneously.

• The matrices [B’] & [B’’] are usual single phase FD jacobians and are constant in value.

• The jacobian element related to dc variables are non constant and must be reevaluated at each iteration.

Sequential Solution• In the second approach, the AC and DC system equations are

solved separately and sequentially.

• The AC system is solved to some degree of convergence using a simple model for the DC system based on its last solution.

• The DC system is then solved using a simplified representation of the AC system.

Flowchart

Convergence criterion

The normal criterion for convergence in ac load flow is that the bus bar power mismatches should be small,

i.e ΔPi < ε for all PV &PQ bus bars

ΔQi < ε for all PQ bus bars

where ε=1MW/MVAR =0.1 MW/MVAR

DC convergence tolerance

• General eqn is

• An acceptable p.u tolerance for the d.c residuals (d.c. powers, voltages and currents) is typically 0.001 p.u on a 100MVA base, i.e. the same as that normally adopted for the ac system.

Multiterminal d.c system

• Each additional converter adds a further 5 variables and a set of 5 equations to the basic algorithm developed for single converter.

• Jacobian elements in the a.c system corresponding to the converter terminal bus bar gets modified in the solution

consider the multiterminal dc system shown in fig

Real power - dc jacobian in the unified method has the structure shown

ΔP/V Δθ ΔPterm1/Vterm1 Δθterm1 . . . .

ΔPterm6/Vterm6 = [J] Δ θterm6 ΔR1 ΔX1 . ΔR6 ΔX6

Reactive power - dc jacobian in the unified method has the structure shown

ΔQ/V ΔV ΔQterm1/Vterm1 ΔVterm1 . . . .

ΔQterm6/Vterm6 = [J] Δ Vterm6 ΔR1 ΔX1 . ΔR6 ΔX6