Mathematical and computer simulation techniques in reactive power compensation

Math1 Comput. Modelling, Vol. 14, pp. 312-371, 1990 Printed in Great Britain MATHEMATICAL AND POWER COMPENSATION

0895-7177/90 $3.00 + 0.00 Pergamon Press plc

COMPUTER

S Etminan, R H Kitchin (Sunderland R Sotudeh (Teesside Polytechnic)

SIMULATION

TECHNIQUES

IN

REACTIVE

Polyteclmic)

Abstract Heavy industrial loads having random, erratic and unbalanced changes in thier real and reactive power requirement may cause considerable disturbances at the point of common coupling in the supply system. Static VAr compensaton have been employed in many industrial applications throughout the world to compensate for tbe disturbance described and for the low lagging power factor associated with heavy inductive loads. The principal requirement of such compensators are the accurate computation of reactive power for each phase of the load and tbe rapid introduction of the computed swing in reactive power. Various alternative compensator designs have been used including schemes which, although effective, also introduce undesirable harmonic distortions. This paper describes and compares the performance of two types of high speed static reactive power compensators which have been simulated on a inductive randomly varying power system load. Computed bus voltage variations are processed by flicker meter software, developed by the Electricity Council for use in industry.

Electric

Arc Simulation;

Reactive

Power Compensation;

Computer Simulation and mathematical techniques can be used to create a reactive power compensator software simulation package, with built in options to increase overall system flexibility. Such developments would provide ideal testing grounds for assessing the operational capabilities of newly designed compensator systems, and also be used to assess the simulated system can in different capabilities environments. Load and compensator parameters can be varied, and the overall system response to such changes can be assessed; such flexibility would not be easily available in a hardware orianted model. educational purposes, such a For computer model would provide an effective tool in the different operational features of such conveying The present trend in educational tools, in systems. power systems along with many other scientific and engineering areas, is towards development of software models or integration of software and hardware models while introducing various computer control of the integrated features.

For the purpose of modelling and siulation the arc characteristics can be represented as a randomly varying resistance. The single phase model, shown on Fig 2, is used to simulate the electrical characteristics of the electric arc furnace. Within the simulation package, the arc furnace resistance variations can be simulated in a There can be computer generated number of ways. random furnace resistance changes, or furnace resistance variations generated by the user, and in addition furnace The resistance values can be read from a data file. furnace resistance can also be changed in a square or a sine wave manner with the frequency and the magnitude of the oscillation defined by the user.

proximity of large furnaces help to load reactive power increase the useful networks. Such reflected as light lamps and to a sensitive electronic

For the purpose of electric arc furnace simulation two The first is an mathematical approaches can be used. inductive load where the heavy analytical approach, undergoes a change in impedence, and the next is the numerical approach; the latter has been chosen for the Load-Compensator simulation package. Analytical

In order to provide an ideal test bed for performance evaluation of the compensators under investigation, a Load-Compensator computer simulation package has been Fig 1 gives an overview of the developed developed. package which consists of number of units, namely an arc unit, a furnace simulator, a VAr measurement compensator unit, and an U.I.E. flicker meter. Arc Furnace

Power Measurement.

various shapes are charged into the furnace for initial melting folowed by either refining or production of steel alloys. The arc is a column of ionized gas characterised by a high current, high temperature and low voltage. The arc is very unstable while running on to cold scrap charge, and under the action of the electromagnetic changes its route and length. forces present, Such changes in the arc path are some of the reasons for arc resistance variations, which result in furnace current fluctuation causing voltage dips at the point of common coupling.

INTRODUCTION

Reactive power compensation in the industrial loads such as electric arc reduce the disturbances caused by the variations; and in addition help to power rating of the transmission disturbances, at their severest, are flicker and mainly affect incandescent lesser extent television sets and equipment.

Reactive

Aooroach

The computer model is based on the electrical characteristics of an electric arc furnace undergoing a The various system parameters are change in impedence. defined below. es = Es Sin(wt) il = E,/Z2.[

Simulator

Sin[w(t+tl)-U2)] -R12.uL1

+ Sin(U2-wtl)e

The electric arc furnace1 utilises number of engineering sciences such as materials, control. electrical. mechanical, The electric arc furnace plays a very structures, etc. important role as a robust and resilient tool in the hostile Metal pieces of and rugged conditions of a steel works.

+ Es/Z1 .Sin(wtl-Ul)e vb = v,-i& 312

I

-R12.‘/Ll

Proc. 7th Int. Conf. on Mathematical and Computer Model@ where tl is the time at which the load impedance changes from Zl to 3 due to a change in the load r&stance from Rl1 to R12. A computer simulation accomplished. Numerical

of the analytical

model above was

Aooroach

approach proves difficult and almost The analytical impossible to undertake when there is compensation in For this reason the numerical parallel with the load. solution for the simulation of the electric arc furnace load A predictor-corrector was looked into in some detail. numerical approach was thought best and for thii reason Euler Trapezoidal method with very fine steps of about 1 degree was chosen. The differential equations obtained as shown below. Vb = Ll.dil/dt

describing

Sampling the load current cross over.

at voltage zero

2- Integrating the voltage and delayed current product, over one supply period. 3- The cross product of delayed current parameters.

voltage and

the load system were of

S
- Rl.il) solution,

i.e. the predictor

. (vb[o] - Rl.il[O])

il[l I = il[O] + h . Dil[O] = es - La

. (il(l] - il[O])/h - &.il[l]

The next part is the corrector, which checks the values of Vb[2] with its previous values and if not within the given tolerance limits, the procedure below is repeated with the new values of il[l] and Vb[I]. DllP

l-

4- The cross product of rate of change voltage and current parameters.

The first part of the numerical is given below.

Vb]l]

There are a number of different VAr measurement techniques available, each having certain characteristics in terms of speed of response, accuracy and number of adequate and current needed for voltage samples A range of calculation of the load reactive power flow. VAr measurement algorithms have been simulated, as detailed in reference 2. The simulated algorithms use the following alternative techniques to evaluate the reactive power.

+ Rl.il

dil/dt = l/Ll.(Vb

Dil[O] = VLl

313

1 = l/L1 . (%[ll - 4i~Pl)

il[2] = il[2] = il[O] + h/2 Vb[2] = es - Ls

. (Dil[O] + Dil[Z])

. (il[2] - il[O])/h i Rs.il[2]

In the numerical approach, at every step of the calculated Vb must satisfy a limit simulation, lE-3% before next value of Vb is calculated. predictor-corrector step length is about 1 degrees, The values of keep the errors to a minimum. analytical and numerical techniques were compared they were of good agreement.

the Of The to the and

easurement VAr M

have been used for techniques VAr measurement in place of bus voltage variation compensator control, The VAr measurement techniques employed, sensing. monitor both voltage and current parameters yielding a more accurate response and helping to achieve more control in compensating for industrial loads. Successful minimisation of light flicker is directly related to the accuracy, the speed of response and the rate of convergence to the steady state, of the VAr measurement Such parameters will be affected by technique employed. harmonics and transients generated due to random and erratic variations in the load impedance, as is the case for an electric arc furnace.

This type of compensator consists of a bank of capacitors connected in parallel3. The compensator has a maximum delay of one cycle in responding to changes in reactive power demand. Each single phase of the compensator has in series with the capacitor a thyristor switch comprising of two inverse-parallel connected thyristors and an inductor, which is used to limit the rate of rise of current to within the thyristors’ capabilities. The inductor can also be used to tune each single phase of the compensator, to sink harmonics of different orders. The total amount of compensator susceptance is adjusted by the number of parallel capacitors switched into the system. This susceptance varies in a step wise manner, and in order to reduce the step sizes, the total compensator susceptance can be divided into smaller capacitors, or alternatively the parallel susceptances can increase in a binary pattern which would help achieve smaller step sizes, combined with greater range in compensator susceptance variations. Fig 2 shows the single phase network diagram of the simulated Binary-Weighted Thyrsitor Switched Capacitor compensator. The complete compensator consists of four similar capacitor units connected in parallel. For transient free switching the capacitors are pre-charged to the positive or negative voltage peak, and are switched into system when the voltage across the thyristors is zero (i.e. at voltage peaks). The thyristors’ conduction is turned off by inhabiting the thyristor firing pulses, as the capacitor current reaches the natural zero. However, due to leakage the capacitors slowly lose their charge, causing transients at the time of switching. In order to avoid this situation the capacitors’ charge is topped up through switching the appropriate thyristors at the right voltage peak. By achieving no high current transients, on switching, the life of the thyristors and the parallel capacitors is increased. Disintegration of the capacitor dielectric occurs if the units are left on a given DC charge for a long period. For this reason the unused capacitors are repolarized to avoid danger of dielectric disintegration. For the purpost of repolarization an additional circuit is included in parallel with the capacitors, as shown on Fig 2.

Assuming that only arm 0 of the compensator switched into the system the analytical state equations are derived as shown below.

was space

314

Proc. 7th Int. Co& on Mathematical and Computer Modelling

es = 4

. +l+i,)

+ Rl.il

+ Ll. +

es = La.+,+&) + L,

2 = l/C

dil/dt = l/L1 di,/dt = l/h dVc/dt = l/C

+ Rs.(il+ic)

+ Rs.(il+ic)

.

il[l] i,[l]

. i, with

the

differential

functions

4 0 -(&+RI)

-Rs

the state space matrix, L,

0

&+I_.)

0

-R,

0

-(Rs+%)

-1

1

0

Dil[l] DiJl]

section

1

- R&O] is given below.

(vb[l] - Rl.il[l]) (vb[l] - VJl] - E&[l]) (Dil[O] + Dif[l]) (DiJO] + DiJl])

DV,[Z] = l/C i,[Z] V,[2] = V,[O] + h.DV,[Z]

1 +

= l/L1 = l/L,

il[Z] = if[O] + h/2 i,[2] = i,[O] + h/2

=

4

i,

DVJl] = l/Lc. i,[l] V,[l] = V,[O] + h.DV,[l]

the corrector

DVC

.

= it[O] + h.Dil[O] = i,[O] + h.d&[O]

for the above equations.

D’c

1

0

0

.

(Vb[O] - V, - R&[O])

i,[l] = il[l] + iJl] i,[O] = il[O] + i,.[O] Vb[l] = es - f&[O]/h

Dil

is the predictor

. (vb[o] - Rl.il[O])

Dil[O] = l/L1 Di,[O] = l/Lc

&+Ll).Dil + Ls.Di - -&+Rl).il - Rs.ic + es Ls.Dil + &+L,).DF c -= -Rs.if - (Rs+Rr.).ic - V, + es Dw = l/C ic

(Ls+Ll)

. (Vb - V, - &.ir.) . i,

The first part of the numerical technique section, and this is shown below.

+ Vc

+ WC

Rearranging the equations on the left hand side

Creating

. (vb - Rl.il)

i,[2] = il[2] + i,.[2] Vb[2] = es - L&PI

.e,

0

V, The above state space matrix is in the following

form.

+ i,[O])/h - Rs.il[2]

Vb values are checked for convergence at the end of each corrector run, and if not within the defined limits, the corrector is repeated with the following values.

F.X’= Al .X +Bl.e, By multiplying

vb[ll = vl$l V,Ul = V,Pl

both sides with inverse F-l,

il[l] = il[2] i,_[l] = i,[2]

X’ = A.X + B.e, where A = Al

F-l

B = Bl

. F-l

Using the above state space equations, the i,, and V, can be obtained by for different and es.

when Vb has converged and satisfied the defined accuracy percentage defined the corrector operation is complete for the current sample period, and the predictor gets the calculated parameters and uses them for initial condition values, as shown below. values of il. values of Rl

The analytical approach is somewhat difficult to simulate, since the arrangement of the state space matrix varies as the compensation arms are switched in and out of the The numerical approach discussed in the next system. section provides an alternative solution. Numerical

aooroach

The numerical approach, uses the Euler Trapxoidal predictor-corrector technique, in which it uses the differential equations topredict an initial value for the current in each phase of the connected arms of the compensator and the load; and then uses this predicted value to approximate the bus bar voltage Vb. In the corrector section the current and voltage values are adjusted until Vb satisfies the percentage accuracy limit. With the same arms, as in the analytical case, switched into the system, the differential equations in terms of the bus bar voltage are derived as shown below.

Vb[O] = Vb[2]

V,Pl = V,Pl

il[O] = il[Z] i,[O] = i,[2] Comoarison

of the Analvtical

and the Numerical

Aooroach

Using the analytical approach, the state space matrix changes in terms of number of rows and columns, as the number of compensation arms switched in to system increase or decrease, to respond to reactive power Different available in the load system. changes mathematical packages were looked in to some detail, to find an appropriate package for implementation of the analytical technique, with no success. The changes in the number of rows and columns of the state space matrix makes it very difficult to simulate the Although the analytical approach is analytical technique. more accurate than the numerical approach, it cannot be implemented.

375

Proc. 7th Inr. Conf on Mathemaiical and Computer Modelling Simula to i n Comoensator

Caoacit

of

r

An alternative new proposed compensator configuration anti-polarized Switched Capacitor referred to as Compensator has been designed and simulated. The designed compensator, in presence of rapid, random and erratically changing loads such as electric arc furnaces, is expected to have an improved performance than the Conventional type; however the new configuration does number of capacitor units and require an increased which would result in an increased thyrsitor switches, The single phase network diagram of the capital costs. new compensator design is shown on Fig 3. Each arm of the compensator consists of three branches, two capacitive branches and one inductive branch. Each capacitive branch consists of a capacitor in series with, a thyristor switch comprising of two inverse-parallel connected thyristors. The capacitors, of the two capacitive branches, are charged to opposite polarities of the Vbus peak, so that in each arm one of the capacitors is charged to positive and the other charged to negative voltage peak. The compensator can be switched in at Vbus peak regardless of voltage polarity. The maximum switching time of the Anti-Polarized compensator is therefore one half cycle whereas the maximum switching time of the conventional T.S.C. compensator is one cycle, a potential improvement of a half cycle. The compensator control system is designed so that the two capacitors of each arm are always maintained at opposite polarities. The inductive branch of each arm has the same fundamental frequency reactance as that of the capacitive branches. The inductive branch, in order to cancel the effect of excess capacitance in the system, is switched in when one of the capacitive branches is being repolarixed. Analvtical

auuroach

Using the above state space matrix, the values of i], ilb, ica* &b* “cat and Vcb can be obtained by for different values of Rl and es. The analytical approach is somewhat difficult to simulate, since the arrangement of the state space matrix varies as the compensation arms are switched in and out of the The numerical approach discussed in the next system. section provides an alternative solution. Numerical

aooroach

The numerical approach, uses the Euler Trapxoidal predictor-corrector technique, in which it utilizes the system differential equations to predict an initial value for the current in the load and each connected arm of the compensator; and then uses this predicted value to In the corrector approximate the bus bar voltage Vb. section the current and voltage values are adjusted until With the Vb satisfies the percentage accuracy limit. same arms, as in the analytical case, switched into the system, the differential equations in terms of the bus bar voltage are derived as shown below. dilldt = l/L1 . difddt = bLlb di,,/dt = l/Lc dVca/dt = l/C di,,/dt = l/L, dV&dt = l/C

(vb - Rl.it) . (v,, - Rlb.ilb) (Vb - V,, i,,

- Rc.ica)

. (v, - vc,, - +i&) icb

The first part of the numerical technique section, and this is shown below. Dit[O] = l/L1 . Dilb[O] = l/Lib Di,,[O] = l/Lc Dicb]O] = l/Lc

(vb[o] - Rl.il[O]) . (vb[o] - Rtb.itb[O] . (“b]O] - “ca]O] - Rz.ica]O]) (vi+] - v&o] - +&,,[o])

il[l] For the analytical consider that the switched on; this compensating and similar procedure state space matrix equations.

solution of the Anti-Polarized TSC, Arm ‘0’ of the compensator is fully means that the compensator is both repolarizing at the same time. Using as that of the Conventional one, the is obtained from the system differential

Dit Dilb Dica D’cb

=

Dvca Dvcb

-(&+Rl) -% ‘2 -(Rs+Rtb) -0”s 0

‘20 0

-%

-%

+s

0

-R,

0

-(Rs+Rc) -F 0

-Rs -(Rs+Rc) 0 1

Similar to the Conventional TSC, matrix is in the following form.

+

the

above

F.X’=Al .X +Bl .es By multiplying

both sides with inverse F-l,

state

1 es 0 0 space

= l/Lc. ica[l] = L&c. icb[l]

Vca[ll Vcb]ll

= Vca]Ol + h.DVca]ll

is]01 = i,]ll =

4Pl + ilb]O] + &a]01 + icb]O] 4Pl + ilb]ll + &aPl + kb[ll

vb[l] 1 1 1

O 0O 0

= il[O] + h.Dit[O] = ilb[O] + h. Dilb[O] ica[l] = ica[O] + h. Dica[O] icb[l] = icb[O] + h. Dicb[O]

ilb[l ]

DV,,[l] DV,b[l]

= v,b[ol + h.DVcb[ll

= es - L&[O]/h

the corrector

- Rs.is[O]

section is given below.

Dil[l] = l/L1 . (vb[ll - Rl.il]ll Dtlb[ll = l/Lib . tvb[ll - &b.ill$I) DIcaPl = 114 . (vb[ll - Vca - RcAaPl) Di,btll = l/L, (vl,[l] - v,b - s.&b[l]) il[Z] = il[O] + itb[2] = ilb[O] ica[2] = ica[O] icb[2] = icb[O] DV,,[Z] DVcb[2]

h/2 . (Dil[O] + Dil[l]) + h/2 . (Dilb[O] + Dilb[l]) + h/2 . (Dica[O] + Diia[l]) + h/2 . (Di&[O] + Di&[l])

= l/L,. = l/Lc.

ica[2] ich[2]

X’ = A.X + B.e, vcaizl v,b[zl

where

= VcaPl = v,b[o]

is the predictor

+ h.DVcaPl + h.DV,b[21

A = Al

F-L

is[2] = il[2] + ilb[2] + ica[2] + icb[2]

B = Bl

F-l

Vb[2] = es - Ls.(is[2]

+ i,[O])/h

- Rs.il[2]

Proc. 7th Int. Conf on Mathematical and Computer Modelling

376

vb values are checked for convergence at the end of each corrector run, and if not wiithin the defined limits, the corrector is repeated with the following values vb[I] iI [l] ilb[I] icaP icb[ll VcaPl vcl,[ll

The simulated U.I.E. flicker meter monitors the bus bar and outputs the running flicker voltage fluctuations, The flicker perceptibility and the flicker severity Pst. meter helps quantify improvements made by the two types and helps compare their of simulated compensators performance.

= vb[2] = iI[2] = ilbL2] = icaP1 = i&l = VcaL21 = Vcb[2]

CONCLUSION

when Vb has converged and satisfied the defined accuracy percentage defined the corrector operation is complete for and the predictor gets the the current sample period, calculated parameters and uses fhem for initial condition values, as shown below. vb[o] iI[O] ilb[O] icaP1 icb[ol vca[“l vcb[ol

= vbL2] = iI[2] = ilbL2] = icaP1 = id21 = Vcai21 = v&l

Comoarison

The two compensators are presently undergoing flicker resistance random and for periodic testing, meter and it is intended that comprehensive results variations, will be published in the near future. REFERENCES

of the Analvtical

and the Numerical

Aooroach

analytical The technique proved cumbersome to implement, due to the fact that the state space matrix is always changing its number of rows and columns due to changes in the number of compensator arms switched in and out of the system. The numerical technique giving acceptable accuracy as shown in the previous sections, provided good grounds for simulation of such system. The recorded waveforms of hardware simulation of parts of the system is in good agreement with that of the numerical simulation of the compensator systems as shown in the later sections. U.I.E.

Flicker

The developed package has been used to test the performance of the two simulated reactive power to furnace resistance Their response compensators. Note the Vbus of the variation is shown on Fig 4. maintained at its peak, as the Anti-Polarized TSC, furnace resistance changes from a high resistive to a low Vbus of the the level; as opposed to resistive Conventional TSC.

1. U.I.E., “Arc Furnace Disturbances”, Disturbances Study Committee 1980. 2. Etminan S., Sotudeh R., “Fast Converging Reactive Power Measurement Techniques for High Speed Static VAr Compensators”, 22nd UPEC Sunderland Polytechnic April 1987 3. Sotudeh R., Holmes J.; “High Speed Switched Capacitor Reactive Power Compensator”, PROC 20th UPEC April 1985. 4. International Union for Electra-Heat; “U.I.E. Flickermeter, Functional and Design Specifications”, Disturbances Study Committee, Flicker Measuring Methods WG 1983.

Meter Simulation

The U.I.E. flicker4 meter was developed as a result of The U.I.E. Disturbances Study Committee efforts, in order to establish an internationally agreed functional and design specification for flicker measuring apparatus. The aim of the flicker meter is to simulate the response of the lamp-human eye-brain combination and to carry out an on line statistical calculation of the flicker level. Output parameters consist of instantaneous time series flicker perceptibility. and the flicker severity “Pst”. When the cumulative frequency distribution function of the input is calculated over a IO minutes period, the terminology used is “Pst” and refers lo Short Term calculation of the flicker severity. A

w

supply

Voltage

ES

l”terDcti”e Program

k!iiiii ,pens.ator

Fig

1 - Simple

block

diagram

Control

of

unit

the

Load-Compensator

Simulation

System.

Proc. 7th Int. Conf on Mathematical and Computer Modelling

I

I

I

I

317

Fig 2 - Network diagram of the simulated Conventiona Switched Capacitor compensator and the Arc Furnace model.

Fig 3 - Network diagram of the simulated Anti-Polarized Switched Capacitor compensator and the Arc Furnace Model.

Fie, 4' (a) - Simulated bus bar voltage and the furnace current (il) (b) - The output of algorithm 2.2, (c) - The output of algorithm 1.1, (d) - The output of algorithm 3.3, (e) - The output of algorithm 4.4,

Fig'5 with (a) (b) (C)

n r N N

= = = =

360 2 90 0

Simulated %s bar voltage@b) - No compensation - Conventional TSC - Anti-Polarized TSC

*uPkk.trrm

No compensation

111,

I

Conventional TSC

Anti-Polarized TSC Fig 6 (a) (b) (c)

-

System source current 'is'.with No compensation Conventional TSC Anti-Polarized TSC

Fig 7 - Running perceptible flicker level at the bus bar.