Comparing Capacitive and LC Compensators for ... - Science Direct

Electric Power Systems Research, 17 ( 1 9 8 9 ) 57 - 64

57

Comparing Capacitive and LC Compensators for Power Factor Correction and Voltage Harmonic Reduction R. K. H A R T A N A a n d G. G. R I C H A R D S

Department o f Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803

(U.S.A.) (Received D e c e m b e r 16, 1 9 8 8 )

ABSTRACT

Capacitors designed for compensation o f local loads may inadvertently serve as series LC filters for harmonic currents injected by nonlinear loads at remote points in a distribution system. This may cause unexpected voltage harmonic distortion, even at loads which do not themselves produce harmonics. The problem is made more difficult by the fact that system harmonic impedance is constantly changing, causing intermittent resonance between source impedance and compensating capacitor. The remedy explored in this paper is insertion o f a reactor in series with the local compensating capacitor. The reactor must be chosen to minimize voltage distortion at the bus while holding the fundamental p o w e r factor (displacement factor) at a desired value. The size and corresponding cost o f compensating equipment must be constrained. Plots o f simulation tests show that a series reactor may provide a more cost effective remedy for voltage harmonics than merely detuning the original pure capacitive compensation.

INTRODUCTION

Voltage distortion at load buses in power distribution systems shortens the life expectancy of equipment and can interfere with communication circuits and computers. Such distortion may be caused b y local compensating capacitors that form a series resonance with source reactance at some harmonic frequency, providing a low impedance path for harmonic currents inserted at neigh0378-7796/89/$3.50

boring buses. It m a y also be caused b y harmonic currents from the local load that encounter a parallel resonance from source and compensating capacitor looking back into the distribution system. In either case, the compensating capacitor may need to be altered to reduce the problem. In ref. 1 it was shown that harmonic currents and voltages at linear loads m a y be reduced b y changing the value of compensating capacitors, b u t at the cost of reducing the fundamental frequency power factor (displacement factor). However, a relatively high power factor (including harmonics) can be obtained with harmonic source distortion if an optimal reactor is put in series with the compensating capacitor [2]. Such an arrangement, the series LC compensator, may actually be less expensive than purely capacitive compensation for certain harmonic conditions. However, source harmonic impedance is constantly changing because of switching and load changes within the distribution system. Therefore, fixed solutions for o n e harmonic source condition may not be optimal for another. In ref. 3 it is shown that compensating capacitance can be altered to achieve an improved p o w e r factor in the presence of source harmonics where harmonic sources and source impedances vary over time. Reference 4 explores the relationship between distribution system losses and the extent of variation of source harmonics. Reference 5 treats the problem of time varying harmonics b y assuming normally distributed source and load harmonic impedances, and optimizing the expected value of the power factor using LC compensation. However, the problem of reducing voltage O Elsevier S e q u o i a / P r i n t e d in T h e N e t h e r l a n d s

58 harmonic distortion while maintaining a given displacement factor has not been considered. The purpose of this paper is to present a method of reducing voltage total harmonic distortion (THD) at buses with capacitor compensation where it is desired to maintain a given displacement factor. A series reactor, XL, will be selected that will minimize expected THD for a specified range of source impedance values, while constraining the total size and cost of the compensation network. The result, the expected value of THD as a function of compensator cost for a given displacement factor, is compared with the performance and cost of purely capacitive compensation.

CIRCUIT DESCRIPTION

Figure l(a) is a single-phase equivalent circuit of a bus experiencing voltage harmonic distortion at harmonic n because of a nonlinear voltage source, vsn, and harmonic current sources within the load itself, iLn. Figure l(b) is the same bus, except that a series reactor, XL, has been added to minimize voltage THD while Xc has been altered

Rtn

Xtn

)Isn I

) I£n

IIcn --~X~n

v

R£n

V,~n

sn

b

iX£n

~

ILn

(a)

Rtn 4V~

Xtn

~Isn I V

sn

•I£n cn

ILn

V~n ~/n

X£n

(b) Fig. 1. Single-phase equivalent circuit for the nth harmonic: (a) with capacitive compensation, ( b ) w i t h LC compensation.

to maintain the displacement factor. The Th4venin voltage source representing the utility supply and the harmonic current source are

v~(t) = y_v~.(t)

(1)

tl

and iL(t) = E ib,(t)

(2)

n>l

where n is the harmonic order. The source and load impedances are also functions of harmonic order: Ztn = Rt, + jXtn

(3)

Z~, = R~, + jX~,

(4)

The approach will be to minimize voltage harmonic distortion on the load by adjusting Xc and XL while keeping the displacement factor constant and constraining the cost of the compensating circuit to a series of fixed values. This can be accomplished by minimizing a total cost function which represents a combination of THD and compensator cost: K = cf THD 2 + M

(5)

In this equation, M is the cost of Xc and XL based on voltampere requirement, THD 2 is the voltage total harmonic distortion squared, and cf is an arbitrary factor to convert THD 2 to cost. It serves as a Lagrange multiplier in the optimization and need not be found explicitly. THD is squared to facilitate the minimization calculation. Although the total cost K generated by (5) may have little meaning because of the difficulty in assigning a cost factor to THD, minimizing K for a range of c~ will provide a series of THDs, each with its corresponding m i n i m u m compensator cost. Minimum cost and THD can then be plotted with c~ as a parameter. Since source and load harmonics and impedances are randomly time varying, (5) has to be modified to accommodate a probabilistic cost and THD. K becomes E(K), the expected value of cost, which is to be minimized. On the right-hand side of (5), M must represent the most severe of the varying loadings to which Xc and XL may be subjected, rather than the average. Thus, for the probabilistic case, (5) becomes

E(K) = cfE(THD 2) +

Mma x

(6)

59 COST FUNCTION CALCULATION

With the LC compensator connected at the load terminals as shown in Fig. l ( b ) , the load voltage and compensator current for each harmonic frequency nco0 are, respectively, Vs.(R~n + jXcQ~) - - ILn [ j ( r / X b

v~. =

--

Zc/rl)(Rt~n + j X t 2 n ) ]

(7)

Arn + jAin

and Icn =

Ysn(Rf~n + jX~n) .... ILn(Rt~n + jXt~.)

A~n + jAin

(8)

where XL = c~oL,

X c = 1/cooC

Rt;~n = Rt,,R~. -- Xt.X~n

XtQ~ = R t . XQn + Rf~nXtn Rc~ n = - - X ~ n ( n Z L - - Xc/gl )

Xc~. = R ~ . ( n X L -- X c / n ) Arn = Rt~n -- (Xf~n + X t n ) ( n Z L - - Z c / n ) Ain = XtQn + (R~n + R t n ) ( n X L - - X c / n )

The voltage harmonic distortion compensated load terminals is THD = (.>~ 1V~n2)I/2/V~,

(9)

at the (10)

The compensator cost in the second term of (5) is defined as M = UbSL + UcSc

ables. Because distribution system harmonic generators are generally current sources, a positive correlation exists b e t w e e n source harmonic impedance X t . and Rtn , and source harmonic voltage Vs., which is a product of source impedances and Norton equivalent current sources. In the most extreme case, source voltage, source resistance and reactance are linearly correlated to each other at each harmonic order n > 1, that is, V~. = g . X t . and R t n = h n X t n , where gn and h. are constants for n > 1, and Xtn is a random variable which varies linearly with frequency. Then the expected value of VQn2 (from eqn. (7)) can be written as ( d"Xtn2 ) E(Yfln 2) = E c . Xtn2 + bnXtn + an

where a . , bn, and c. are functions of h . , R~n, X~n, n X b and X c / n ; and dn is a function of I b . , gn, hn, R~n, Xf.n, n X L and X c / n . By definition,

(11)

where UL and Uc are, respectively, the cost of reactor and capacitor per k V A R and are considered to be constant parameters. For reactors and capacitors, ratings are defined [6] as

and

E(V~. 2) = f f(Xt. ) V~n2 dXt.

where f(Xtn) is the probability density function of X t . . If the source reactance Xtn is assumed to have a uniform distribution function with minimum value a . and maxim u m value/3., eqn. (15) becomes

2\1/2

where I~n is given in (8). In (12) and (13), the harmonic voltage magnitudes are added linearly to emphasize the effect of peak (as opposed to RMS) voltage on c o m p o n e n t rating or cost. For randomly time varying voltage source harmonics and source impedances, E(THD 2) must be expressed as a function of Xc and XL and of the statistics of the random vari-

(15)

_oo

dn

E ( V ~ 2) -

[

(14)

~n - - OLn

~n

X o~¢n

Xtn 2 dXtn CnXtn 2 + bnXtn + an

(16)

By integration, E(V~. 2) for n > 1 can be calculated, and the expected value of THD 2 can be expressed as

E E(V~nb E(THD 2) -

n>l

V~ 2

(17)

60

M m a x in eqn. (6) is calculated in the same way as M in eqn. (11), except that Icn in eqns. (12) and (13) assumes its maximum magnitude:

s m a x t - ~Arn --~-7"~in2-

Icn :

+

COST

FUNCTION

To minimize E(K) in (6) it is necessary to find optimum values, X c and XL, which cause the differential of E(K) to vanish, that is,

/max

dE(K)-

R 2 s L- [t T - ~ +

ANALYSIS

~E(K)

~E(K)

dXc+ - -

. A,.

(18)

Jmax

where Vs, max is a priori known. The expressions for E(THD:) and Mma x are then substituted in eqn. (6) which is minimized with respect to Xc and XL for a range of cf values.

dXL=0

(19)

OX L

OX C

However, Xc and XL are related because the displacement factor of the load is to remain constant. To keep the fundamental frequency compensating current constant, XL -- Xc = constant

(20)

so that (19) reduces to 0 C) -0 aD

0 0 -0 r'-

t3 0 --C:)

if) rrcr . ....1

o

co~_

w

..................... / ................................................... i!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D k--

#--

r-~ m

o

-::,,=,

IJ_l --~ I - - c~(..) I,I n X

I-LD

LIA ___....__.-----

I.l-I o

i

O_

~x ol,

o

!

a

E(~2)

I Ill

I O O

~II l

i

I l\l

~

! ~

Ill '

i

I

kk_

i

i

i

l

I

i

,

1 m

0.25

O. 50

O. 75

1.DO

.25

50

thick curves,

o = 10%.

XL (OHMSl Fig. 2.

Plot o f E(K) and E ( T H D 2) versus XL: thin curves,

o = 30%;

1.75

O

i

61 TABLE 1 System parameters and source harmonics Parameters and harmonics

Case 1

Case 2

Rt 1 (~)a Xtl ( ~ ) E(Xts) ( ~ ) E(Xt7 ) (~'~) E ( X t 11) ( ~ ) E ( X t ,3) ( ~ ) R~l (~)b X~ 1 (~)b Vsl (V)

0.01154 0.1154 0.577 0.8078 1.2694 1.5002 1.7421 1.696 2400

0.01154 0.1154 0.577 0.8078 1.2694 1.5002 1.7421 1.696 2400

E(Vss) (%Ys,) E(VsT) (%Vs,) E(Vs H) (%Vs,) E(G,3) (%G,)

5 3 2 1

ILS (%Is 1) IL7 (%Is 1) /Lll (%Is 1) ILl3 (%Is l) UL ($/kVAR) Uc ($/kVAR)

5 3 2 1 2 2

1

7 2 1 5 3 2 1 2 2

a R t n = h n X t n , where h n = 0.1 for n > 1. bR~n is frequency independent and X~. n = n X ~ l , where n is the order of the harmonic.

dE(K)-

OE(K) 3Xc

3E(K) + - 0 3XL

(21)

Since E(K) is generally nonlinear and multimodal in the variables X c and XL, the condition in (21) is n o t sufficient. Figure 2 is a plot of E(K) and E ( T H D 2) versus XL for the parameter values and load listed in Table 1, with a displacement factor of 0.78, cr= 5 X l 0 s , and source harmonics and impedance varying with standard deviations o = 30% and 10%, with uniform probability density functions. This Figure shows a series of E(K) minima separated b y resonance peaks. These resonance peaks are obtained b y setting the imaginary part of the expected impedance seen from the Th4venin source to zero, resulting in a quadratic equation in Xc and XL for any given harmonic order n, A ( n X L - Xc/n) 2 + B(nX L --Xc/n) + C = 0

(22)

where A

= Xtn

+ Zg. n

(23a)

B = R~n 2 + X~. 2 + 2X~nXt.

(23b)

C = X t n ( R ~ n 2 + X ~ n 2)

(23c)

and b y taking the solution of quadratic equation (22) where the square r o o t of the discriminant is positive. (The other solution corresponds to resonance between the load and the combination of source impedance and compensator.) Note that for sufficiently large RQn and/or X ~ , eqn. (22) reduces to n X L - Xc/n + Xt, = 0

(24)

which then represents only the resonance condition between source impedance and compensator. Note that the resonance peaks broaden for increasing variance of Xt. Immediately to the right of each E(THD ~) peak are local minima where Xc and X b act as a series filter, effectively grounding the corresponding harmonic. At these points, one harmonic voltage is eliminated at the bus, b u t large harmonic currents are attracted d o w n the distribution line through the compensator, so that compensator cost (which subsumes a worst-case condition) is relatively high. An E(K) minimum is found when XL is such that compensator cost combined with THD, weighted by c~, satisfies eqn. (21). To find the minimum value of E(K), it is necessary to find the local minimum b e t w e e n each pair of potential resonance points. Then, b y comparing the local minima, a global minimum can be found. It should be noted that the E(K) global minimum does not necessarily occur in the range of XL enclosed by the lowest order resonance lines. A different content in the voltage or load harmonic sources may result in a global minimum that occurs between higher order resonance peaks. OPTIMIZATION ALGORITHM

Since the cost function E(K) is constrained b y a constant displacement factor, that is, XL -- Xc = constant, optimization of E(K) reduces to minimization of eqn. (6) with respect to one of the variables (XL or Xc). XL is used as the variable and Xc is replaced b y (XL -- constant). The algorithm using the m e t h o d of steepest descent can be summarized as follows: Step 1. Select a value for the displacement factor. Step 2. Select a starting value for c~. Step 3. Start at an initial point X~°) in the range of values enclosed b y the lowest

62

order resonance peaks which will n o t create a resonance condition as stated in eqn. (24)• Step 4. Calculate the gradient vector at

Step 6. If the range of X L values is not the last one, go to the next range and return to step 3. If the range of XL is the last one, compare all the local minima and find the global minimum cost. Repeat the above process for a given range of c~ values and plot the expected THD versus the minimum compensator cost with cf as parameter. Finally, repeat the whole process for different values of the displacement factor.

X~): N(X~)) dE(K)}dXL(X(~)) Step 5. If

(25)

liN(X~ )) II < e

(26)

=

where e is a preselected small positive number ( e < l ) , then terminate the iterative procedure, o u t p u t the o p t i m u m value XL and go to step 6. If the stopping criterion (26) is n o t satisfied, then generate a new point given by X(~+I) = X ~ ) + pN(X~))

SIMULATED

EXAMPLES

Two cases of an industrial plant were simulated using the optimization method• The numerical data in case 1 were primarily taken from an example in ref. 1. The load consists of an inductive three-phase load

(27)

where p (> 0) is the step size which has to be judiciously selected. Then, replace X~ ) by X~+I) and return to step 4•

0 . 8 6

0.815 8

O Displac~nent

:

F a c t o r

=

0.85

®

& ............................

. ~

..........

...............................

............

.............................................................................................

.................................................................................................................................................

n0.75 .

.

.

.

.

O

i .

.

.

.

.

.

.

.

.

.

.

.....

......................................................................................................................................

UJ LJ C_ X

Displacement Factor = 0.75 .

.

.

.

.

.

.

.

.

.

.

0.8

.......

.

.

J

.

.

.

.

................

i .

~.

.

........................

........................ ..................................... ....................

.....................

0.85

.

500.

.

0.9 .

i

lO00.

.

9S

1500.

i

;'000.

COMPENSATOR Fig. 3. E x p e c t e d only.

THD versus compensator

2500.

COST

3000.

.

3500.

.

.

.

.

.

.

.

.

.

u,O00.

.

.

.

.

.

.

.

.

.

'1500.

.

.

.

.

.

5000.

[OOLLARS)

c o s t f o r c a s e 1. T h e circles i n d i c a t e t h e e x p e c t e d T H D b y c a p a c i t o r

63 which is 5100 kW with a displacement factor of 0.717 and harmonic current sources as listed in Table 1. The 60 cycle supply bus voltage and Th4venin impedance are 4.16 kV line to line and 0.01154 + j0.1154 ~2, respectively. Fundamental parameters and load harmonics were assumed to be time-invariant quantities. The voltage source harmonics and Th4venin impedances for n > 1 were assumed to be randomly time-varying quantities with their expected values as listed in Table 1 and their standard deviations a equal to 30%. The source and load in case 1 were arbitrarily chosen to have the same harmonic c o n t e n t as suggested in ref. 1. These harmonic magnitudes are generally independent. In case 2, the fifth harmonic voltage source was decreased to 1% and the seventh harmonic voltage source was increased to 7% of Vs,. In both cases, the range of cf values is 0 - 106.

Displacement Factcr= 0.775 O

The parameter values of case 1 were then applied to the optimization algorithm with U c and UL taken equal to 2 $/kVAR. This generated several plots of m i n i m u m expected THD versus compensator cost for a range of c~ values with different displacement factors (varying from 0.75 to 1) as displayed in Fig. 3. Note t h a t for each displacement factor the expected THD decreases with increased compensator cost until saturation occurs, at which point a cost increment results in a much smaller decrease in THD. The circles indicate the m i n i m u m expected THDs that can be achieved by pure capacitive compensation with the corresponding displacement factors. For higher displacement factors than those shown, the expected THDs and the corresponding compensator costs are much higher, and some of t h e m are off the Figure's scale. It is observed that

O

D.78 i

[ [

o

............................................................................... .................

..........................................................

o b-

i

o d " . , hJ b-(-3

Dlsplacement Factor

LLI

=

:

0.775 0.8

..... 02 ................

° il i

. . . . . . . . . . . . . . . . . . .

....................................................... ................................................................................"

'

' ..................... .........................

Z

~.

500.

rico0,

t500.'

2000.'

2500.

COMPENSATOR COST

3000.

3500.

u,o00.

i i

=t5OO.

5000.

[DOLLARS]

Fig. 4. E x p e c t e d T H D versus c o m p e n s a t o r cost for case 2. The circles indicate the e x p e c t e d T H D by capacitor only.

64

the LC compensator achieves a much lower expected THD than a capacitor alone at the same displacement factor, and an LC compensator may cost less than a capacitor. Another simulation with different voltage source harmonic contents {Table 1) was performed (case 2). The optimization result is shown in Fig. 4. The expected THDs are higher than those in Fig. 3 with the same compensator cost. This is to be expected because the voltage source harmonic contents in case 2 are higher than in case 1. Although the prescribed values of the XL in case 1 were below the 5th harmonic, for case 2 the optimal values of XL fell between the 5th and 7th harmonic. The expected THDs achieved by using a pure capacitive compensator (marked by the circles) are much higher than with the LC compensator at the same displacement factor.

CONCLUSIONS

Compensated load buses with or w i t h o u t nonlinear loads may experience voltage harmonic distortion caused by voltage source distortion. Source distortion can be caused by harmonic currents injected into the distribution system by neighboring loads. Simply altering the value of the compensating capacitor to reduce resonance with the source impedance may not be effective because of the time-varying nature of source impedance. The displacement factor can be maintained while reducing voltage harmonics if a reactance is placed in series with the compensating capacitor to form an LC compensator. The optimal reactance value can be found by minimizing a cost function including harmonic distortion and total compensator cost. The cost function is derived from a probabilistic model of the Th~venin source harmonic voltages and impedances.

Compared with pure capacitive compensation, LC compensation may provide a higher displacement factor, or a lower total harmonic distortion, for the same cost. In any case, voltage harmonic distortion levels are reduced below those found with capacitance alone.

ACKNOWLEDGEMENTS

The work reported in this paper is part of a multiarea research project sponsored by the LSU-Utilities Consortium including Louisiana Power and Light Co., Gulf States Utilities Co. and Central Louisiana Electric Co.

REFERENCES 1 R. F. Chu a n d R. H. A v e n d a n o , A direct m e t h o d for i d e n t i f y i n g t h e o p t i m a l p o w e r f a c t o r correct i o n in n o n s i n u s o i d a l s y s t e m s , IEEE Trans., PAS-104 ( 1 9 8 5 ) 959 - 964. 2 G. G. Richards, O. T. Tan, P. K l i n k h a c h o r n a n d N. I. S a n t o s o , Cost c o n s t r a i n e d p o w e r f a c t o r o p t i m i z a t i o n w i t h s o u r c e h a r m o n i c s using LC c o m p e n s a t o r s , IEEE Trans., IE-34 ( 1 9 8 7 ) 2 6 6 270. 3 R. T. Saleh a n d A. E. E m a n u e l , O p t i m u m s h u n t c a p a c i t o r for p o w e r f a c t o r c o r r e c t i o n at buses w i t h lightly d i s t o r t e d voltage, IEEE Trans., PWRD-2 ( 1 9 8 7 ) 165 - 173. 4 D. R a o n i c , D. Cyganski a n d A. E. E m a n u e l , P o w e r f a c t o r c o m p e n s a t i o n at buses w i t h slightly d i s t o r t e d voltage due to r a n d o m h a r m o n i c s , IEEE PES Winter Meeting, New York, 1988, Paper No. 88 WM 093-7. 5 G. G. Richards, P. K l i n k h a c h o r n , O. T. T a n and R. K. H a r t a n a , O p t i m a l LC c o m p e n s a t o r s for n o n l i n e a r loads w i t h u n c e r t a i n n o n s i n u s o i d a l s o u r c e a n d load characteristics, IEEE PES Winter Meeting, New York, 1988, Paper No. 88 WM 202-4. 6 IEEE Stand. 519-1981: IEEE Guide for Harmonic Control and Reactive Compensation o f Static Power Converters, IEEE, New York, 1981.