Optimum Allocation of Reactive Power for Voltage Stability Improvement in ACDC Power Systems

8/3/2019 Optimum Allocation of Reactive Power for Voltage Stability Improvement in ACDC Power Systems

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Optimum allocation of reactive power for voltagestability improvement in AC–DC power systems

D. Thukaram, L. Jenkins and K. Visakha

Abstract: The dependence of the system voltage stability on reactive power distribution forms thebasis for reactive power optimisation. The technique attempts to utilise fully the reactive powersources in the system to improve the voltage stability and profile as well as meeting the reactivepower requirements at the AC–DC terminals to facilitate the smooth operation of DC links. Themethod involves successive solution of steady-state power flows and optimisation of reactive powercontrol variables using linear programming techniques. The proposed method has been applied toa few systems and the results obtained on a real-life equivalent 96-bus AC and a two-terminal DCsystem are presented for illustration.

1 Introduction

In big developing countries like India, HVDC transmissionis becoming an acceptable alternative to AC and isproviding an economic solution for bulk power transferover long distances, and also as a means of interconnectingsystems with problems of frequency and stability. Thegrowing number of schemes in existence and underconsideration demands corresponding methods of model-ling and analysis for planning and day-to-day operation.Considerable work has been reported in the literature inregard to integrated AC–DC system performance evalua-tion procedures, notably for load flow and stability studies[1–5]. There is very limited work in the area of reactivepower control in AC/DC systems. Even though DC

transmission lines carry no reactive power, real power flowinto the converters is accompanied by some reactive powerflow because of the phase control. The considerations inthe operation of a DC transmission system are to satisfy theneed for reactive power at the terminals, maintain goodvoltage profile and voltage stability.

During the peak load condition the entire reactive powerdemand at the DC terminals generally may not be met bythe AC system, and it is not desirable anyway, so, reactivepower compensation is generally provided at the DCterminals. The net reactive power absorption by theconverters can be varied by the converter controls. Duringlight load conditions considerable reactive power isgenerated in the AC system by EHV lines and this can beused to meet the reactive demand at the DC terminalsand locally provided compensation at the DC terminals canbe suitably switched off to maintain a satisfactory voltageprofile.

With the increased loading of existing power transmissionsystems the problem of voltage stability and voltage collapsehas become a major concern in power system planning andoperation. Voltage stability is concerned with the ability of a

power system to maintain acceptable voltages in the system

both under normal conditions and after being subjected to adisturbance. From a system operations view point a heavilyloaded system has to be carefully monitored and adequatecontrol action taken when the operating point approachesthe limit of voltage stability. In day-to-day operation andcontrol of power systems these decisions require very fastcomputations in the energy control centre. The staticaspects of voltage stability can be considered as an effectivetool to assess system security with respect to voltagecollapse. Various voltage stability and voltage collapseprediction methods have been reported in literature [4–12].

On load transformers taps, generator excitations andswitchable VAr compensators are the reactive powercontrol variables in the AC system. These variables are

optimised for the purpose of improved voltage stabilityand voltage profile in the system. In an AC/DC powersystem these control variables have to be optimised in aco-ordinated manner taking account of the reactive powerrequirements at the DC terminals.

This paper is mainly concerned with development of a method for co-ordinated optimum allocation of reactivepower in AC/DC power systems with an objective of enhancement of steady-state voltage stability based on theL-index [6]. An algorithm is proposed for optimisation of reactive power control variables using linear programming.The proposed method has been tested on sample systemsand the results obtained for an equivalent 96-bus AC and atwo-terminal DC power system with a peak and light-loadconditions are presented.

2 Static voltage stability analysis

Static voltage stability is primarily associated with thereactive power support. The real power (MW) loadability of a bus in a system depends on the reactive power supportthat the bus can receive from the system. Several analyticaltools have been presented in the literature for the analysis of the static voltage stability of a system.

In the day-to-day operation of power systems the analysisof voltage stability for a given system state involves anevaluation of how close the system is to voltage instability,

which gives a measure of voltage security and what thecontributing factors are, and the operating strategy to beused to prevent voltage instability. In the past, utilities

The authors are with Department of Electrical Engineering, Indian Institute of Science, Bangalore 560 012, India

E-mail: [email protected], [email protected]

r IEE, 2006

IEE Proceedings online no. 20045210

doi:10.1049/ip-gtd:20045210

Paper first received 14th October 2004 and in final revised form 16th June 2005

IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 2, March 2006 237

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depended on conventional power-flow programs for thestatic analysis of voltage stability by computing V – P andQ – V curves at selected load buses. These methods do notreally provide information useful in gaining insight into thecauses of instability problems [2–8].

Several researchers have proposed the minimum singularvalue (MSV) of the load-flow jacobian [4–12] as a measureof voltage stability. The singularity of the power-flow

jacobian matrix as an indicator of steady-state stability isused, where the sign of the determinant of the jacobianmatrix determines whether or not the studied operatingpoint is stable. Singularity of the power-flow jacobianmatrix corresponds to that jacobian matrix for which theinverse does not exist and thus there is an infinite sensitivityin the solution to small perturbations in parameter values.The point where this occurs is called a state bifurcationpoint of the system. Several branches of equilibria maycome together and the studied system would experiencea qualitative change in the structure of the solutions due toa small change in the parameter values. At the point of voltage collapse, no physically meaningful load-flow solu-tion is possible as the load-flow jacobian becomes singular.At this point, the MSV becomes zero. Hence the distance of the MSV from zero at an operating point is a measure of

proximity to voltage collapse. The continuation power-flowanalysis overcomes this problem by reformulating thepower-flow equations so that they remain well-conditionedat all possible loading conditions. This allows the solutionof the power-flow problem for a stable as well as unstableequilibrium point [9]. The modal analysis approach [11] hasalso been applied to the voltage stability analysis of practical systems. It may be difficult to describe the MSVin terms of other physical reactive power control variablesand so also the formulation of optimisation problemsuitable for day-to-day operation. However, from theliterature it is seen that this index is helpful in planningthe VAr resources and also identifying the weak spots forreactive power compensation.

The voltage stability index Z ii=Z ij j is proposed in [10].With the aid of Thevenin’s theorem, a general conclusion isdrawn about the condition for maximum power transfer toa node in a system. The maximum power transfer to a bustakes place when the load impedance becomes equal to thedriving point impedance as seen from the load bus underconsideration. At load bus i with load impedance Z i ,for permissible power transfer to the load at bus i wehave Z ii=Z ij j 1. The voltage collapse proximity indicator(VCPI) for all the load nodes is computed asVCPI i ¼ Z ii=Z ij j. The stability margin in this case isobtained as the distance of VCPI from a unit value. Thebus having the maximum value of VCPI is the weakest bus

in the system. If the load at a bus (switching station) is lowthen the VCPI is close to zero indicating a high voltagestability margin at the bus. However, it may not be the caseas the neighbouring nodes may have bulk loads and thusthe switching station bus VCPI also has to be close to theVCPI of neighbouring buses. However, this index may notgive consistent results. Also this index cannot be easilyexpressed in terms of other physical reactive power controlvariables for the purpose of formulation of the optimisationproblem.

2.1 Voltage stability index LConsider a system where n is the total number of buses with1, 2yg generator buses (g), and g+1,g+2yn the load

buses. A load-flow result is obtained for a given systemoperating condition, which is otherwise available from theoutput of an online state estimator. The load-flow algorithm

incorporates load characteristics and generator controlcharacteristics [13]. Using the load-flow results, the L-index[6] is computed as

L j ¼ 1 ÀXgi¼ 1

F jiV i

V j

ð1Þ

where j ¼ g+1yn. The terms within the sigma of (1) arecomplex quantities; V i , V j are the complex voltages of generator buses and load buses, respectively. The values of F ji are obtained from the network Y -bus matrix as follows:

I G

I L

!¼ Y GG Y GL

Y LG Y LL

!V G

V L

!ð2Þ

where I G , I L and V G , V L represent complex current andvoltage vectors at the generator nodes and load nodes;Y GG ½ ; Y GL½ , Y LL½ and Y LG ½ are corresponding parti-

tioned portions of network Y -bus matrix. Rearranging theequation we get

V L

I G

!¼ Z LL F LG

K GL Y GG

!I L

V G

!ð3Þ

where F LG ½ ¼ À Y LL½ À1Y LG ½ are the required values. The

L-indices for a given load condition are computed for all

load buses.For stability, the index L j must not be violated

(maximum limit¼ 1) for any of the nodes j . Hence theglobal indicator L describing the stability of the completesubsystem is given by L ¼ maximum of L j for all j (loadbuses). The indicator L is a quantitative measure for theestimation of the distance of the actual state of the system tothe stability limit. The local indicators L j permit thedetermination of those nodes from which a collapse mayoriginate. It can be shown that the derived theory is exactwhen two conditions are fulfilled, i.e. that the stability limitis reached for L ¼ 1. The first condition requires that allgenerator voltages, amplitudes and phase angles remain

unchanged. The second condition calls for nodal currentswhich respond directly to the current I j and are indirectlyproportional to the voltage V j at the node j underconsideration. The stability margin in this case is obtainedas the distance of L from a unit value, i.e. (1ÀL). AnL-index value away from 1 and close to 0 indicatesimproved system stability.

While the different methods give a general picture of theproximity of the system voltage collapse, the L-index gives ascalar number to each load bus. Among the various indicesfor voltage stability and voltage collapse prediction, theL-index gives fairly consistent results [13, 14]. The L-indicesfor given load condition are computed for all load busesand the maximum of the L-indices gives the proximity of

the system to voltage collapse. The L-index is chosen as thebasis for optimisation as it is defined in terms of networkparameters and generator/load complex voltages, whichmakes it possible to relate with other physical reactivepower control variables in the system.

3 Approach

The major blocks in the approach adopted are shown inFig. 1. At the beginning of the reactive power optimisationin AC/DC power systems, a satisfactory initial operatingcondition for the DC system is selected based on the controlstrategies, viz., constant power control, constant currentcontrol, and constant voltage control applicable at the DC

terminals.A solution for DC system is first obtained in block 2 and

then the voltage, active and reactive power requirements at

238 IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 2, March 2006

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the DC terminals are computed. Defining these require-ments at the AC side of the converter/inverter transformers,an AC power-flow solution is obtained in block 3. Now theterminal transformer taps are computed and their rangechecked for the satisfactory solution of the AC/DC systemin block 4. If the transformer tap range is not satisfactory or

the voltage stability has to be further improved, reactivepower optimisation for the AC system is carried out inblock 5 with suitable terminal conditions. At this stage,a check for the AC/DC system satisfactory condition isperformed in block 6. If the solution is not satisfactory,modifications in the initial conditions of the DC system areaffected with suitable changes in the firing angles in block 7and the process of blocks 2–5 is repeated. Finally, thenearest practical possible tap settings are selected for thetransformers at the AC/DC terminal and the final AC/DCpower-flow solution is obtained in block 8.

4 Description of model

4.1 Converter representation A general AC/DC terminal and its equivalent circuit inFig. 2. The basic equations describing the converter with its

firing angle, tap controls and the DC network aresummarised based on the per-unit system selected asfollows:

AC system base quantities

P acbase ¼ 3-phase power

V acbase

¼line-to-line RMS value

I acbase ¼ P acbase= ffiffiffi

3p V acbase

DC system base quantities

P dcbase ¼ P acbase; V dcbase ¼ K bV acbase; I dcbaseð

ffiffiffi3

p = K bÞ I acbase

where K b ¼ ð3 ffiffiffi

2p

=pÞnbnb is the number of series-connected bridges in a terminal

The direct voltage and power at the converter aregiven by

V dc ¼ aV ac cosaÀ Rc I dc ð4Þ

P dc ¼ V dc I dc ð5Þwhere Rc is commutation resistance, a is the transformer tapsetting and a the firing angle. Neglecting the losses in theconverter and its transformer and equating the expressionfor powers on the AC and DC sides, the equation for powerfactor angle c À xð Þ is given by

V dc ¼ aV ac cos cÀ xð Þ ð6Þand for the reactive power flowing from the AC bus into theconverter terminal is

Qdc ¼ P ac tan c À xð Þ ð7Þwhere c is the alternating voltage angle and x thealternating current angle.

A practical operating scheme for a DC system usinglocal terminal controls is to have the DC-system voltagedetermined at one terminal and the other terminals areprovided with scheduled power or current settings. To keepthe reactive power consumption of the converter and thelosses low the firing angles should be small. But to maintainphase control and reliable commutation, a minimumcontrol angle should be maintained.

4.2 Load model A composite load model, a combination of the ZIP modeland exponential model, is considered. Active and reactivepower loads are modelled as a function of voltage at thebus. The functions considered are

P Li ¼ P Loið A0 þ A1V þ A2V 2 þ A3V ep Þ ð8ÞQ Li ¼ Q Loið R0 þ R1V þ R2V

2 þ R3V eqÞ ð9Þ

4

6

system data

DC system solution

AC system solution

reactive power optimisation

check for AC/DC system

satisfactory

solution

modify DC system initial

settings

AC/DC system power-flow

solution

check for AC/DC system

satisfactory

solution

no

yes

no

1

3

2

5

7

8

yes

Fig. 1 Major blocks showing reactive power optimisation inAC/DC system

AC

system

1 : a

P ac

Q ac

α firing angle V dc

I dc

AC bus

I ac ∠ξ

Fig. 2 Equivalent circuit of DC terminal




where A0, R0, A1, R1, A2, R2, A3, R3 denote the portion of total load proportional to constant power, constantimpedance, constant current and exponential of voltageswith ep, eq given values.

4.3 AC/DC power-flow solution method Considering a DC system where

m represents the total number of DC terminals

p represents the number of terminals with constant powercontrol

c represents the number of terminals with constant currentcontrol and

m ¼ ( p+c+1) the terminal with voltage control

it is assumed that 1, 2,y, p are the constant power controlterminals, p+1, p+2,y, p+c are the constant currentcontrol terminals, and m the voltage controlled terminal.The algebraic sum of the direct currents flowing into theDC network must be zero and therefore

Xmk ¼ 1

I dck ¼ 0 ð10Þ

The direct voltages at terminals other than the voltage-

controlled terminal are given by,V bus½ ¼ Rbus½ I bus½ þ V m½ ð11Þ

where

V bus½ t ¼ ½V dc1 ; V dc2 ; . . . ; V dcp ; V dc

P þ1; . . . ; V dcp þc

I bus½ t ¼ I dc1 ; I dc2 ; . . . ; I dcp ; I dc P þ1; . . . ; I dcp þc

h i

V m½ t ¼ V dcm ; V dcm ; . . . ; V dcm ; V dcm ; . . . ; V dcm

Â Ã Rbus½ is the bus resistance matrix of the DC network withvoltage controlled terminal as reference

V

dc

m is the scheduled voltage at the voltage controlledterminal

I dcp þ1; . . . ; I dcp þc are the scheduled currents at the controlled

terminals

I dc1 ; . . . ; I dcp are computed currents at the power controlled

terminals I dc ¼ P dc=V dcÀ Á

Using an iterative technique the solution of these equations(11) is obtained for the values of direct currents, voltagesand powers at all the DC terminals.For the terminals withpower control and current control it is common practice toco-ordinate the tap control with phase control so that theterminal will operate at some direct voltage below its own

minimum firing (ignition or extinction) angle characteristicto avoid frequent mode shifts from occurring with normalalternative voltage fluctuations. Thus the direct voltageequation for the terminals with power control and currentcontrol is modified as

V dc ¼ M baV ac cosa À Rc I dcc ð12Þ

where M is a coefficient typical of 0.97 for 3% voltage

margin. Substituting the values of a; Rc; V dc; I dcand M , the

values of aV ac for all the terminals are obtained from (4)and (11). Substituting the values of aV ac into (6) the powerfactor angles (cÀx) at all the terminals are obtained. Theactive and reactive powers flowing from the AC bus to theconverter terminals are computed from (5) and (7),

respectively. Now the AC power-flow solution is obtainedwith the defined values of P ,Q at the AC/DC terminals.This solution provides the voltage conditions at all the AC

buses. Knowing the values of aV ac, i.e. the product of converter station transformer tap and AC-bus voltage fromthe DC-system solution and values of V ac from the ACsystem solution, the tap settings of the converter transfor-mers are determined. If the tap settings violate the limits,modifications such as a change in scheduled voltage V dc atthe voltage-controlled DC terminal, a change in controlangle a and optimisation of the reactive power schedulein the AC system to obtain improved values of V ac at theAC/DC terminals are effected and the procedure to obtainAC/DC system solution repeated.

5 Description of reactive power optimisation

Minimisation of voltage stability index in a system formsthe basis for the reactive power optimisation problem. Themodel uses linearised sensitivity relationships to define theproblem. The constraints are: the linearised networkperformance equations relating to control and dependentvariables and the limits on the control variables. Then themodel selected for the reactive power optimisation useslinearised sensitivity relationships to define the optimisationproblem. The objective is to minimise the voltage stability

objective function v L

¼ P L2 in the system. The control

variables are

transformer tap settings T

generator excitation settings V

switchable VAr compensator settings Q.

These variables have their upper and lower limits. Changesin these variables affect the distribution of reactive powerand therefore change the reactive power at generators, thevoltage profile and thus voltage stability of the system. Thedependent variables are

reactive power outputs of the generators Q

voltage magnitudes of the buses other than the generator

buses (V ).These variables also have their upper and lower limits.

Consider an AC system where n represents the number of total buses in the AC system, g the number of generators,t the number of on-load tap changing (OLTC) transfor-mers, s the number of switchable VAr compensator buses,and r ¼ n À (g+s), the number of remaining buses; it isassumed that 1, 2,y, g are the generator buses,g+1,y, g+s are the switchable VAr compensator buses,and g+s+1,g+s+2,y, n are the remaining buses.

In this approach an initial AC/DC load flow is firstobtained and the AC terminal connecting the DC systemare treated as switchable VAr compensator buses where

desired voltage limits are specifiedminimise v L ¼ C x

subject to

bmin b ¼ Sx bmax and xmin x xmaxð13Þ

where C is the row matrix of the linearised loss sensitivitycoefficients, S is the linearised sensitivity matrix relatingthe dependent and control variables, b the column vectorof the linearised dependent variables, x the column vector of the linearised control variables, bmax and bmin are thecolumn vectors of the linearised upper and lower limits onthe dependent variables and xmax and xmin are the columnvectors of the linearised upper and lower limits on the

control variables. The linear programming technique is nowapplied to these problems to determine the optimal settingsof the control variables [15].


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The control vector in incremental variables is defined as

x ¼ DT 1; . . . ;DT 2;DV 1; . . . ;DV g;DQgþ1; . . . ;DQgþ sÂ Ãt

and the dependent vector in incremental variables as

b ¼ DQ1; . . . ;DQg;DV gþ1; . . . ;DV gþ s;DV gþ sþ1; . . . ;DV nÂ Ãt

The upper and lower limits on both the control anddependent variables in linearised form are expressed as

bmax ¼ bDQmax1 ; . . . ;DQmax

g ;DV maxgþ1 ; . . . ;DV maxgþ s ;DV maxgþ sþ1; . . . ;DV maxn ct

bmin ¼ bDQmin1 ; . . . ;DQmin

g ;DV mingþ1; . . . ;DV mingþ s ;DV mingþ sþ1; . . . ;DV minn ct

xmax ¼ bDT max1 ; . . . ;DT maxg ;DV max1 ; . . . ;DV maxg ;DQmaxgþ1; . . . ;DQmax

gþ sct

xmin ¼ bDT min1 ; . . . ;DT ming ;DV min1 ; . . . ;DV ming ;DQmingþ1; . . . ;DQmin

gþ sct

where

DT min ¼ T min À T actual;DT max ¼ T max À T actual

DQmin ¼ Qmin À Qactual;DQmax ¼ Qmax À Qactual

DV min ¼ V min À V actual;DV max ¼ V max À V actual

5.1 Computation of sensitivity matrix The sensitivity matrix S relating the dependent and controlvariables is evaluated in the following manner [13].Considering the fact that the reactive power injection at abus does not change for a small change in the phase angle of the bus voltage, the relation between the net reactive powerchange at any node due to change in the transformer tapsettings and the voltage magnitudes can be written as

DQg

DQ s

DQr

24

35 ¼

A1 A2 A3 A4

A5 A6 A7 A8

A9 A10 A11 A12

24

35

DT t DV gDV sDV r

2664

3775 ð14Þ

Then, transferring all the control variables to the right-handside and the dependent variables to the left-hand side andrearranging,

DQg

DV sDV r

24

35 ¼ S ½

DT t DV gDQ s

24

35 ð15Þ

5.2 Computation of voltage stability objective function sensitivities withrespect to control variables The sensitivities of the voltage stability objective function vLwith respect to the real and reactive power injections at all

the buses except the swing bus (angle reference bus) are firstcomputed and these values are used to compute theobjective function sensitivities with respect to the controlvariables. Considering the fact that the real power injectiondoes not change for a small change in voltage magnitude of the bus and reactive power injection at a bus does notchange for a small change in the phase angle of the busvoltage, the relation between the sensitivities of the objectivefunction with respect to the real and reactive powerinjections at all the buses except the swing (angle referencebus) bus is given by

@ v L@ d2::

@ v L@ P n

2

66664

3

77775

@ P 2@ d2

Á Á Á @ P n@ d2

:Á Á Á

:: Á Á Á :

@ P 2@ dn

Á Á Á @ P n@ dn

2

66664

3

77775

@ v L@ P 2::

@ v L@ P n

2

66664

3

77775 ð16Þ

@ v L@ V 2::

@ v L@ V n

266664

377775 ¼

@ Q2

@ V 2V 2 Á Á Á @ Qn

@ V 2V 2

: Á Á Á :: Á Á Á :

@ Q2

@ V nV n Á Á Á @ Qn

@ V nV n

266664

377775

@ v L@ Q2

::

@ v L@ Qn

266664

377775 ð17Þ

Knowing the terms @ v L=@ d, @ v L=@ V , @ P =@ d and @ Q=@ V ;the sensitivities of the objective function with respect to thereal and reactive power injections at all the buses except theswing bus @ v L=@ P k , @ v L=@ Qk , k

¼2yn (bus 1 is considered

as a reference bus) can be computed.

5.3 Objective function sensitivities withrespect to transformer taps Considering a transformer connected between buses k andm with taps on bus k , the real and reactive power injectionsin to the buses k and m are P k , Qk , P m and Qm.Computation of the sensitivity with respect to thetransformer tap is based on the approximation that thesepower injections into end-buses k and m do not change withtransformer tap

@ v L

@ T km ¼@ v L

@ P k À@ P km

@ T km þ

@ v L

@ Qk À@ Qkm

@ T km

þ @ v L@ P m

À @ P mk @ T km

þ @ v L

@ Qm

À @ Qmk

@ T km

!ð18Þ

The values @ v L=@ P k , @ v L=@ P k , @ v L=@ P k and @ v L=@ P k areobtained from the solution of (16) and (17).

5.4 Objective function sensitivities withrespect to generator excitation voltage A change in the excitation voltage of a generator results inthe modified VAr injection into the system at the generatorexcitation voltage are given by

@ v L@ V k

¼ @ v L@ Qk

@ Qk

@ V k ð19Þ

where k ¼ 2,3,y, g. The values of @ v L=@ Qk are obtainedfrom the solution of (16) and (17) and @ Qk =@ V k is given by

@ Qk

@ V k ¼ Qk

V k À Bkk V k ð20Þ

5.5 Objective function sensitivity withrespect to a excitation voltage of reference bus generator A change in the excitation voltage of swing bus generator

(reference bus 1) results in modified reactive powerinjections at all the other generator buses and in reactivepower injection errors at all the load buses connected to theexcitation voltage of the swing bus generator is given by

@ v L@ V 1

¼Xr

@ v L@ Qr

À @ Qr

@ V 1

!þXk

@ v L@ Qk

@ Qk

@ V 1ð21Þ

where r is the set of all the load buses connected to bus 1and k ¼ 2,y, g. The values of @ v L=@ Qr and @ v L=@ Qk areobtained from the solution of (16) and (17). Values for@ Qr =@ V 1 are computed as

@ Qr

@ V 1 ¼Y r 1V r sin

ðdr

Àyr 1

Þ ð22

ÞAnd the values for @ Qk =@ V 1, k ¼ 2,y, g are taken from thematrix S (Y r 1; yr 1 are Y -bus magnitude and angle).


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5.6 Objective function sensitivities withrespect to switchable VAr compensators The values @ v L=@ Qgþk , k ¼ 1,2,y, s are obtained from thesolution of (19) and (20).

5.7 Computational procedure This Section presents the computational steps followed inthe program developed for the optimisation of reactivepower allocation in an AC/DC power system. In the day-to-day operation of the power systems the following steps

are used to obtain the optimal reactive power allocation inthe system for improvement of voltage stability.

Step 1:

Input the data relating to

DC system

(i ) Network

(ii ) Power, current and voltage schedule at the terminals

(iii ) Firing angle setting

(iv) Converter transformer tap ranges.

AC system

(i ) Network

(ii ) Scheduled load and generation

(iii ) Upper and lower limits and step-size for transfor-mers tap settings, generator excitation settings andswitchable VAr compensator settings

(iv) Upper and lower limits on the generator reactivepowers and voltage magnitudes at buses other thanthe generator buses.

Form the network matrices.

Step 2:

Set the initial/modified values for scheduled firing

angle a voltage V

dc

, current I

dc

and power P

dc

for theconverter terminals.

Solve for the direct voltage and current at all the DCterminals. Compute the values of aV ac, power factorangles cÀx, power factor, and active and reactivepowers flowing from the AC bus into converters at allthe AC/DC terminals.

Find the equivalent active and reactive loadings at theAC terminals including the local reactive powercompensation at the terminals.

Step 3:

Perform the AC power flow (or output of the stateestimation) to obtain the values of L-indices of all load

buses. Find the tap settings of all the converter station

Fig. 3 AC/DC system of two-terminal DC and 96 AC buses




transformers and check for the satisfactory rangeof the converter transformer tap settings. If yes, go tostep 6.

Step 4:

Compute the column vectors bmax, bmin, xmax, xmin andmodify the column vectors xmax and xmin to reason-ably small ranges. compute the sensitivity matrix S ,the row vector C of the objective function. Solve theoptimisation problem using the linear programmingtechnique and obtain the AC power-flow solution.

Find the tap settings of all the converter stationtransformers and check for the satisfactory range of the converter transformer tap settings. If yes, go tostep 6.

Step 5:

Find the suitable modified settings for the DCscheduled voltages and scheduled firing angles andgo to step 2.

Step 6 :

Set the converter station transformer tap settings tothe nearest practical possible settings. Compute themodified converter control (ignition or excitation)

angles a. Compute the modified power factor, activeand reactive powers flowing from the AC bus intoconverters at all the terminals.

Step 7 :

Perform the AC power-flow solution with theoptimum settings of the reactive power controlvariables. Check for satisfactory limits on the depen-dent variables, voltage profile and voltage stabilityL-indices. If no, go to step 2.

Step 8:

AC–DC system final results.

6 Typical system studies and results

6.1 System considered An AC/DC system of two-terminal DC and 96 AC buses,typical of Indian grid equivalent system including thevoltage levels of 220 and 400 kV as shown in Fig. 3 has beenconsidered for studies. There are 20 generators in the systemconnected at buses 1–13, 15–19, 95 and 96. The AC/DCconverter stations are connected at buses 29 and 32. TheDC system data is given in Table 1. There are 20 generators,18 tap regulating transformers and 95 transmission lines inthe system. About 30 buses are considered as switchableVAr compensator buses. These buses have bulk loads andlow power factor compared with other load buses. The

system has about 12345.8 MW, 6410.0 MVAr peak loadand 8631.07 MW, 4289.67 MVAr light load. The generatorsP max, Qmax and Qmin are given in Table 2. Results obtainedfor the two cases, viz. peak load and light load have beenpresented.

Table 1: DC system data

SE RE

Transformer secondary (kV) 219.00 216.00

MVA rating 465.00 460.00

X c p.u. 0.19000 0.19000

Tap max ( p.u.) 1.10 1.10

Tap min ( p.u.) 0.90 0.90

Tap step ( p.u.) 0.0125 0.0125

P specified (MW) 1540.00 1500

Commutating resistance (p.u.) 0.00535 0.00541

R dc line p.u. 0.00137

Table 2: Generator limits

Generator no. P max (MW) Q max MVAr Q min MVAr

1 1142.0 731.0 À350.02 1800.0 1116.0 À500.0

3 900.0 558.0 À250.0

4 1467.0 910.0 À450.0

5 1800.0 1116.0 À500.0

6 275.0 206.0 À100.0

7 227.0 170.0 À80.0

8 594.0 368.0 À150.0

9 476.0 330.0 À150.0

10 400.0 248.0 À100.0

11 48.0 24.0 À10.0

12 410.0 135.0

À50.0

13 289.0 96.0 À40.0

15 297.0 99.0 À40.0

16 180.0 90.0 À40.0

17 66.0 39.0 À10.0

18 945.0 586.0 À200.0

19 396.0 297.0 À100.0

95 241.0 120.0 À50.0

96 756.0 469.0 À200.0

Table 3: DC system results of peak load condition (initial)

Sending end Receiving end

V ac p.u. 1.01944 0.98023

Tap 0 .9750 1.01250

I dc p.u 17.79944 17.79944

V dc p.u. 0.87232 0.84788

Power factor 0.87835 0.85411

c, g (deg.) 13.04031 18.00000

P dc MW 1552.685 1509.177

Q dc MVAr 845.000 918.972

Table 4: DC system results of peak load condition (final)


V ac p.u. 1.04919 1.01259

Tap 0.95000 0.97500

I dc p.u 17.70209 17.70209

V dc p.u. 0.86755 0.84324

Power factor 0.87088 0.85411

c, g (deg.) 15.11808 18.00000

P dc MW 1543.243 1500.000

Q dc MVAr 870.950 913.384




6.2 Case 1: Peak load condition In this case the DC terminal at bus 32 is considered asreceiving end power control. The initial power flow resultsfor this case show a low voltage profile in the system withthe voltages of about 39 buses not being within acceptablelimits (0.95–1.05 p.u.). There are 12 generators exceedingthe maximum Q limits and no generator Q is exceeding theminimum limit. The minimum singular value beforeoptimisation is 0.25821. The maximum stability L-index is0.667 at bus number 56. The sum of squares of voltage

stability L-indices P L2 is 6.1078. The proposed algorithm

for reactive power optimisation has been applied to improvethe situation. The step-size taken for both the regulatingtransformers and generators excitations is 0.0125 p.u. The

total number of switchable VAr compensator buses selectedfor the compensation is about 30. The compensation at theselected places initially it is assumed to be zero. After fouriterations of the VAr optimisation the voltages at all thebuses have been brought within the satisfactory operablelimits (0.95–1.05 p.u.). After optimisation all the generatorsreactive power outputs Q are brought within the limits,while initially some of the generators Q were exceedingmaximum limits. After the optimisation the minimumsingular value has been increased to 0.29648, the maximumvoltage stability L-index is reduced to 0.457 at bus number

56 and the sum of square of voltage stability L-indices P L2

is reduced to 4.1060. The summarised results, initial andafter optimisation (final), for the DC system are presentedin Tables 3 and 4, while the AC system results are presentedin Tables 5–9. The load bus voltage profiles and voltagestability indices before and after optimisation are shown inFigs. 4 and 5, respectively.

6.3 Case 2: Light-load condition This case study refers to the light-load condition. The DCsystem terminal controls in this case are considered sameas in the case of peak-load condition. The initial power-flowresults for this case show an over-voltage profile in thesystem even with the reduced local reactive powercompensation at the converter stations. There are two

generators exceeding the minimum Q limits and nogenerator Q is exceeding the maximum limit. The maximumvoltage stability L-index is 0.337 at bus number 56. The sum

Table 6: Transformer taps ( p.u.) (initial settings all are 1.0)

From To Final From To Final

38 20 1.0125 41 46 1.0125

25 22 1.0125 42 43 0.9875

24 23 1.0125 58 57 1.0125

33 73 0.9625 63 62 1.0125

34 90 0.9625 66 67 0.9875

35 36 0.9875 79 71 0.9875

37 49 1.0125 86 85 0.9875

40 54 0.9875 88 89 1.0125

Table 5: System-grid totals

Initial Final

Total P gen. (MW) 12 741.37 12 670.77

Total Q gen. (MVAr) 6133.22 3869.79

Total P load (MW) 12 345.80 12 346.20

Total Q load (MVAr) 6410.0 6430.30

Total comp. (MVAr) 1350.00 2475.00

Total P loss 395.56 324.60

Total Q loss 482.17 1817.55

% P loss 3.10 2.56

Reduction in loss (MW) – 70.96

Table 7: Generators exceeding Q max (MVAr) limits

Generator no. Max. MVAr Initial Final

6 206.0 351.30 190.30

9 330.0 434.32 274.80

10 248.0 286.18 161.00

11 30.0 60.33 26.80

12 135.0 196.45 67.30

13 96.0 166.41 92.10

15 99.0 104.15 52.70

16 160.0 215.49 150.60

17 80.0 150.80 72.20

18 586.0 587.69 388.80

19 297.0 340.39 243.60

95 120.0 169.67 81.60

Table 8: Generation excitation (p.u.) (initial settings allare 1.0)

Voltage Bus Final Voltage Bus Final

V1 1 1.0125 V11 11 1.0000

V2 2 1.0250 V12 12 1.0000

V3 3 1.0250 V13 13 1.0125

V4 4 1.0125 V15 15 1.0125

V5 5 1.0125 V16 16 1.0125

V6 6 1.0250 V17 17 1.0375

V7 7 1.0375 V18 18 1.0375

V8 8 1.0125 V19 19 1.0000

V9 9 1.0250 V95 95 1.0125

V10 10 1.0125 V96 96 1.0125

Table 9: Switchable VAr compensator compensation forpeak load condition (final)

Bus MVAr Bus MVAr Bus MVAr

21 15 54 90 75 20

22 15 55 60 76 60

23 15 56 60 80 20

36 90 57 90 82 20

43 60 60 30 83 60

44 30 62 30 85 20

46 60 71 30 90 10

48 30 72 30 92 20

49 30 73 40 93 10

52 30 74 20 94 30




of the square of voltage stability L-indicesP

L2 is 1.989.The proposed algorithm has been applied to improve thesituation. The step-size taken for both the regulatingtransformers and generators excitations is 0.0125 p.u. Thetotal number of switchable VAr compensator buses selected

for the compensation is about 30. The compensation at theselected places initially is assumed to be zero. After oneiteration of the optimisation the voltages at all the buses

initial

final

1.1

1.0

0.9

0.8

20 30 40 50 60 70 80 90 100

bus number

v o l t a g e

p r o f i l e

Fig. 4 Bus voltage profile before and after optimisation (peak load condition)

final

initial

0.7

0.6

0.5

0.4

0.3

0.2

0.1

020 30 40 50 60 70 80 90 100

bus number

v o l t a g e

s t a b i l i t y

i n d e x

Fig. 5 Voltage stability indices before and after optimisation(peak load condition)

Table 10: DC system results for light-load condition (initial)


V ac p.u. 1.05378 1.06011

Tap 0.91250 0.9000

I dc p.u 11.80502 11.80502

V dc p.u. 0.85971 0.84350

Power factor 0.89372 0.88415

c, g (deg.) 16.39157 18.000

P dc MW 1014.892 995.755

Q dc MVAr 509.457 526.167

Table 11: DC system results for light-load condition (final)

SE RE

V ac p.u. 1.03959 1.04258

Tap 0.92500 0.912500

I dc p.u 11.76930 11.76930

V dc p.u. 0.85711 0.84095

Power factor 0.89082 0.88415

c, g (deg.) 17.01140 18.00000

P dc MW 1008.759 989.737

Q dc MVAr 514.506 522.987

Table 12: Transformer taps ( p.u.) (initial settings all are 1.0)

TR From To Final

T1 38 20 0.9875

T2 25 22 0.9875

T3 24 23 0.9875

T4 33 73 0.9875

T5 34 90 1.0125

T6 35 36 0.9875

T7 37 49 0.9875

T8 40 54 0.9875

T9 41 46 1.0125

T10 42 43 0.9875

T11 58 57 0.9875

T12 63 62 0.9875

T13 66 67 0.9875

T14 79 71 0.9875

T15 86 85 0.9875

T16 88 89 0.9875

Table 13: Generation excitation (initial settings all are 1.0)

Voltage Bus Final Voltage Bus Final

V1 1 1.0125 V11 11 0.9875

V2 2 0.9875 V12 12 0.9875

V3 3 0.9875 V13 13 0.9875

V4 4 0.9875 V15 15 0.9875

V5 5 0.9875 V16 16 0.9875

V6 6 0.9875 V17 17 0.9875

V7 7 0.9875 V18 18 0.9875

V8 8 0.9875 V19 19 0.9875

V9 9 0.9875 V95 95 0.9875

V10 10 0.9875 V96 96 0.9875

Table 14: Switchable VAr compensator compensation forlight-load condition

Bus Final

55 20

56 20




have been brought within the satisfactory operable limits(0.95–1.05 p.u.). After optimisation all the generatorsreactive power outputs Q are brought within the limits,while some of the generators Q limits were exceeding theminimum limits before the optimisation. After the optimi-sation the maximum voltage stability L-index is marginallyreduced to 0.329 at bus number 56 and the sum of square of

voltage stability L-indicesP

L2 is reduced to 1.982, whichshows no significant change in the objective function. Butthe voltage profile of the system has improved to remain

within the limits (0.95–1.05p.u.). The summarised results,initial and after optimisation (final), for the DC system arepresented in Tables 10 and 11, while the AC system resultsare presented in Tables 12–16. The load bus voltage profilesand voltage stability indices before and after optimisationare shown in Figs. 6 and 7, respectively.

7 Conclusions

An algorithm for optimum allocation of reactive power inAC/DC system with an objective of improving the voltagestability of the system has been presented. Although the DCsystem does not carry any reactive power it requires it forthe operation. A sequential approach has been adopted

which makes the available algorithms for AC system to beadvantageously used. The developed algorithm has beentested on typical sample systems and results for a practicalreal-life equivalent system of a 96-bus AC and a two-terminal DC system are presented. The proposed algorithmis demonstrated to give encouraging results for improvingthe operational conditions of the system under both peak-load and light-load conditions.

8 References

1 Fudeh, H., and Ong, C.M.: ‘A simple and efficient AC-DC load-flowmethod for multiterminal DC systems’, IEEE Trans. Power Appar.Syst., 1982, 101, pp. 4381–4396

2 Tamura, Y., Mori, H., and Iwanmoto, S.: ‘Relationship between

voltage instability and multiple load-flow solutions in electricpower systems’, IEEE Trans. Power Appar. Syst., 1983, 102, (6),pp. 1115–1125

3 Tiranuchit, A., and Thomas, R.J.: ‘A posturing strategy againstvoltage instabilities in electric power systems’, IEEE Trans. PowerSyst., 1988, 3, (1), pp. 87–93

4 IEEE Committee reports on voltage stability of power systems:Concepts, analytical tools and industry experiences, IEEE PESpublication 90TH0358-2 PWR

5 Arrillaga, J., and Smith, B.: ‘AC-DC power system analysis’ (IEEE,London, 1998)

6 Kessel, P., and Glavitsch, H.: ‘Estimating the voltage stability andloadability of power systems’, IEEE Trans. Power Deliv., 1986, 1, (3),pp. 1586–1599

7 Clark, H.K.: ‘New Challenges: Voltage stability’, IEEE Power Eng.Rev., 1990, pp. 33–37

8 Flatabo, N., Ognedal, R., and Carlsen, T.: ‘Voltage stability conditionin a power transmission system calculated by sensitivity methods’,IEEE Trans. Power Syst., 1990, 5, (4)

9 Ajjarapu, V., and Cristy, C.: ‘The continuation power flow: A tool forsteady-state voltage stability analysis’, IEEE Trans. Power Syst., 1992,7, (1)

10 Chebbo, A.M., Irving, M.R., and Sterling, M.J.H.: ‘Voltage collapseproximity indicator: behaviour and implications’, IEE Proc.-C , 1992,139, (3)

11 Gao, B., Morison, G.K., and Kundur, P.: ‘Voltage stability evaluationusing modal analysis’, IEEE Trans. Power Syst., 1992, 7, (4)

12 Lof, P.A., Anderson, G., and Hill, D.J.: ‘Voltage stability indices forstressed power systems’, IEEE Trans. Power Syst., 1993, 8, (1)

13 Thukaram, D., Parthasarathy, K., Khincha, H.P., Udupa, A.N., andBansilal, D.: ‘Voltage stability improvement: case studies of Indianpower networks’, Int. J. Electr. Power Syst. Res., 1998, 44, pp. 35–44

14 Bansilal, D., Thukaram, D., and Parthasarathy, K.: ‘Optimal reactivepower dispatch algorithm for voltage stability improvement’, Int. J.Electr. Power Energy Syst., 1996, 18, (7), pp. 461–468

15 Thukaram, D., Parthasarathy, K., and Prior, D.L.: ‘Improvedalgorithm for optimum reactive power allocation’, Int. J. Electr.Power Energy Syst., 1984, 6, (2)

Table 15: Generators exceeding Q min (MVAr) limits

Generator Min. MVAr Initial Final

12 À50.0 À59.34 À33.20

15 À40.0 À53.33 À36.8

Table 16: System-grid totals

Initial Final

Total P Gen. (MW) 8780.96 8784.35

Total Q Gen. (MVAr) 631.78 757.36

Total P load (MW) 8631.00 8631.00

Total Q load (MVAr) 4288.20 4289.67

Total comp. (MVAr) 1350.00 1390.00

Total P loss 149.97 153.51

Total Q loss 4111.85 3889.21

% P loss 1.71 1.74

initial

final

100

0.98

0.96

0.9420 30 40 50 60 70 80 90

v o l t a g e p

r o f i l e

1.08

1.06

1.04

1.02

1.00

bus number

Fig. 6 Bus voltage profile before and after optimisation (light load condition)

initial

final

1000

0.1 v o l t a g e s

t a b i l i t y

i n d e x

0.3

0.2

20 30 40 50 60 70 80 90

bus number

Fig. 7 Voltage stability indices before and after optimisation (light

load condition)


Optimum Allocation of Reactive Power for Voltage Stability Improvement in ACDC Power Systems

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