IEEE Transactions on Power Systems, vol. 3, No. 3, August 1988. OPTINAL POWER FACTOR CORRECTION llohamed Hostafa Saied !Senior Member. IEEE.
IEEE Transactionson Power Systems, vol. 3, No. 3, August 1988
844
OPTINAL POWER FACTOR CORRECTION llohamed Hostafa Saied
!Senior Member. IEEE Electrical and Computer Engineering Department College of Engineering and Petroleun Kuwait University KUWAIT
This paper discusses the different parameters affecting the economic feasibility of power fat:!ox- cor-rection. It will be shown that t.he specific costs of the transmission and compensation elements as weii as those of the eiectric enerqy and power losses have a decisive infiuence on the achievabie overall saving. After formulatiny an objective function repre-senting this overall saving, takin? also the load fac'or into account. an analytical approach to the determination of the optimai size of the compensation equipment is presented. A qeneraiized chart is given which enables the designer to know whether the power factor cor'rection i s feasible o r not under any circumstances. Also. the most suitable degree of reactive power compensation can be directiy found.
-Cl':
The results obtained from a digitai program are aiso given to indicate the sensitivity of the optimal compensation factor to changes in system parameters such as the specific enercy l o s s cost and the annual rate of interest and depreciation. Moreover. the good agreement of the resuits of the diqital computation with those determined using the suggested simple generaiizeti chart, couia be realized.
INTRODUCTION The shunt reactive power compensation, usuai iy called power factor correction. is one of the common t.echniques presently used to reduce the fixed and running costs of supplying iow power factor loads. In the recent. IEEE Power Society 1975-1984 Cumulative Index [l]. there are over 70 papers neaiing wirh the t.opic of reactive power control, from which about 10 papers are addressin? the concept of power f a c t o r correction as a measure t o imurove the network voltage profiles and minimize the system reai power losses. such as i 2 ] . Also. severai interestin: papers investigate the proper choice of size and location o f shunt capacitors of primary distriburion feeders [ 3 j , according to some criteria such a s minimizing the network losses [ 4 ] . The computer-based procedure of applying voltage dependent continuous-time control of r.ciic:t ive power is presented and analyzed in [ 5 ] . The basic idea of the reactive power compensation is to prevent the reactive component of the loati current from flowing along the transmission eiements. i.e. lines, cables and transformers, simply by connecting a reactive compensating eiement o f opposite
a7 SM 499-7 A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society f o r presentation at the IEEE/PES 1987 Summer Meeting, San Francisco, California, July 12 - 17, 1987. Manuscript submitted March 11, 1986; made availahle f o r printing April 15, 1987.
reactance polarity in parallel with load. The flow of these reactive currents will then be mainly confined to the load and compensator. As a result, the transmission elements can be chosen to transmit the load active current component. Accordingly. the required sizes of these elements will be smaller. This represents a reduction of the system fixed costs. Although the currents flowing through these elements wiil be smailcr. it is generally not quite definite whether the power loss. the lost energy, and hence the system running costs will increase o r decrease. This is because of the higher resistances of the smaller crnss sections of the transmission eiements. For an economical evaluation of the reactive power compensation. the cost of the shunt compensating element should also be taken into account. It can be assumed proportionai to the compensated portion of the load reactive current component. From the above consideration. the mutuai interaction between the different. c o s t items can be clearly recognized. The duty of the engineer is. therefore, to choose the proper size of the compensator. that yields the highest possibie saving in the system total annual cost. As wjil be shown later. there are some situations, in which the concept of reactive power compensation wiil have an adverse economical effect. i.e. it will increase the system annual cost as compared to the uncompensated case. .The purpose of this paper is to recognize. under any given circumstances. whether a power factor correction is recommended o r not. and if s o , does an opt.ima1 value of the compensator size exist? The last step is to identify the economically best value.. if any. METBOD OF ANA.LYSIS
Consider the sample power network shown in Fig.(l). It includes a balanced three-phase load of a fuil-load active power rating Po at a lagging power factor COSI#J~. The load is assumed to have a load fact o r ( L F ) . The load phase angle bo is taken constant. i.e. load independent. The rated load line voltage is assumed V. The load is supplied generally via a threephase transformer T and a section of an overhead line o r cable of length I . Without compensation. the load apparent power S o is qiven by T r a nf o m e r
Fig.(l): Sample Power Network
0885-8950/88/0800-0844$01 .000 1988 IEEE
845
so
=
so
=
J Po 2 oo2
Using the above definitions, the different. annual cost items can be derived as foliows:
+
r) r
(1)
Po/cos Qo 1)
Annual Line Cost
where po is the rated load reactive power demand. Assuming cp to be the specific capital cost per VA and per unit length of the transmission line o r cable. then
Now consider the situation after connecting the shunt capacitors of a constant three-phase reactive power Qc. At full-load. the overall reactive power demand at the line receivina-end (load terminals) is (O*,-Qc}, so that the resultant apparent power with compensation, Soc. along the transmission elements is equal to
Annual line cost = a cp P S, without compensation Cpo = a c p ~ ~ , =
(6)
and sol:
(2)
=-
Annual line cost = a cp 8 S o , with compensation c1 C = a cp P Po~l+(1-k)2tan%o
(7)
where a is the annual rate of interest and aepreciation. It is also equal t o the A/P factor of the standard interest tables. 2)
Annual Transformer Cost If CT represents the specific transformer capital cost per VA, it follows
a ) at full load
Annual transformer= a CT S o cost without compensation cT0 = a CT p0
b ) PO. If, on the contrary, YtO, then this means that this method will increase the system total annual cost.
Pft) dt
0 =
(l+tan+,)
2 T tan%,
T P&s'k2Po
-2k(LF)PE T tan$,
(13)
where Prms is the effective value of the load active power demand, assumed = A Po. In this expression A is a factor depending on the load duration curve. It represents the load demand effective value in per unit. based on the peak active power Po. In the literature. there are some approximate relations between A and the load factor (LF). analyticaily as in [6,7] o r in tabulated form as in [ 8 ] . For example. the following approximate relations betweeen the load factor and A2 are derived in [ 7 ] : A2
=
6(LF)/6
0.5>LF>O
h2
=
[1+(LF)+4(LF)2]/6
1>LF>0.5
From the above relations, the following expressions can be derived f o r the annual energy loss cost:
The Optimization Procedure The above expression for Y will be considered as the objective function for the following optimization procedure. Required is to find the optimal value of the compensation factor k, if any, that results the maximum possible annual saving Y. Apart from the possibility of using sophisticated digital optimization programs, which wouid obviously limit the applicability of this approach. it would be of practical advantage. if an analytical solution could be found. Since Y is a function of the single independent variable k , then the optimal condition is characterized by
Introducing the new variable x, then follows
dY =
.Po[
loss
ax -
G
dx
2
cr +
3/2 (1+x2)
A2 [ x3+x . [2- cos Q0
P$T(RTo+~Rpo)A2ce
cost without rompensation Ceo
v2 cos Lo
(14)
From Eqs.(l2) and (131, it follows:
-
{
This will result
where
. [- A 2
+
x = (1-k)
tan 8 ,
cos%o
is the new indeuendent variabie.
where ce is the power per Wh.
sys
I t can be seen that E q . ( 1 3 k = O (no compensation),
em specific energy cost reduces t o Eq.(14) if
J I
1-2(LF) )tanZbo] +2( (LF)-l)tan@,
= 0
847
It is noted that if the load factor LF=1 and hence A=l. then A = l and B=O. Eq.(19) can be solved iteratively to get the optimal compensation factor, or the following graphical procedure can be easily used: Applying a linear regression analysis to the expression (l+x2)3/2 in the range 1>x>O. with emphasis for small values of x between 0 and 0.7. it follows: ( l + ~ 2 ) ~z/ ~(0.8786+1.15578 x ) . The correlation coefficient for this approximation is 0.9547. The following table gives the original and approximated functions:
Substituting in Eq.(19) results for the optimal condition x3 + F1 x - F2
= 0
(21)
where F1 and F2 are dimensionless system constants characterizing the considered case study, given by
and
F1 = A - 1.15578 D
(22)
F 2 = B + 0.8786 D
(23)
To demonstrate the applicability of this approach, the solution of the above equation is given in the generalized chart depicted in Fig.(3). The optimal value of x. denoted as x*. can be directly found for any given nrimericai values of the system constants F1 ana F2. According to the definition of x=(l-k)tanto, x*=O means perfect compensation at full load (i.e. k =l), while x*=tanlbo (or k*=O) represents zero compensation. The symbol k* denotes the optimal value of the compensation factor k. The generalized chart eiven in Fig.(3) is valid for finding the optimal power factor correction of any power network having the configuration shown in Fig.(l). If one of the transmission elements (transformer or line) does not exist, then the only modification is to put its specific fixed cost as well as its ohmic resistance equal to zero in the evaluation of the relevant values of the system constants F1 and F2. Depending on the location of the system characterizing point P having the coordinates (F1.F2) in Fig.(3). the following useful information can be directly gained: Case I :
0
I Fig.(3):
3
2
1
F1 = A
-
4
5
1.15578 D
Generalized chart giving the optiaal coapensation parameter x*= (I-k*1 tan+, . The points P1 and P2 will be discussed later.
between the two straight lines x*=O and x*=tan@o. then there exists an optimal value for the compensation parameter x*. The relevant value of x* can be readily obtained according to the location of the point (Fi9F2). using the given famil$ of str3ight lines representing different values of x . From x , the optimal reactive power compensation factor k* is given by k*=l-(x*/tang,) .
If the point P lies close to the horizontal F1-axis. then x*-0 ( o r k*=l) is the optimal value of x. This means that the annual saving Y increases steadily with the reactive power compensation factor k. This occurs if cr is relatively small and if the load factor (LF) is close to 1. In this case, full reactive power compensation yielding a resultant unity power factor at full-load, is recommended.
An Analytical Solution For n-1 A closed form solution for the optimal value of x
is available for the special case, in which the load If P lies above the straight line representing x*=tan@,. determined by the "intrinsic" load power factor. then the reactive power compensation will not resuit in any economical advantaee. This will happen especially for large values of the compensator specific cost, cr (note that the parameter D in the expressions for F 1 and F2 is directly proportional to cr). For this case, no power factor correction is recommended. Case 11: If the characterizing point P , i.e.(Fl,F2) l i e s
active power demand is constant with time. i.e. the load factor LF=I. For this case. it follows that also the per unit load RMS value h = l . Substituting in the definitions of the coefficients A and B . it follows A = l and B=O. Then Eq.(19) reduces to (24) from which the solution for x is easily found t o be X;LF=l)=
D
/
6
(25)
848 APPLICATION AND RESULTS The presented approach is applied to the sample three-phase power network shown in Fig.(l). The following relevant numerical data [6.10] are assumed: Load Parameters V=llx103 V, 4x106 W , cos1#1~=0.8 lagging. The analysis will be done for the case in which the load factor LF=0.2. as well as the special case LF=l. Overhead Line Parameters p=3000 m. c p = 4 . 2 ~ 1 0 -$/m/VA. ~ Rpo=3.125x10-5 ohm/m Transformer Parameters q = l .25x1W3 $/VA
and
R T ~ = O024 . ohm
Power System Parameters co=0.03 $/W
and
~ ~ = 0 . 7 5 x l O$/Wh -~
The annual rate of interest and depreciation a is taken and for the specific cost of the reactive power compensation the value cr=l.5 ~ 1 0 $/VAr - ~ is assumed. 10%.
Then the values of the parameters a and b can be calcuiated. giving a
=
2 . 6 5 5 9 ~ 1 0 -S/VA ~ and
It follows: D
=
cr/(a+b)
b
=
=
0.256 (dimensionless)
3 . 1 9 7 ~ 1 0 -%/VA ~
In the following analysis, the MKS system of units will be used throughout. --____ Case Study 1: (LF=0.2) For this value of the load factor LF. it can be found that the per unit RMS value of the active load duration curve. A. can be calculated approximately by [7]:
A2
z
5(LF)/6
zz
0.1667
(dimensionless)
This agrees also quite well with the corresponding value in [8!. Fig.(4) gives the contributions of the different cost items to the total annual cost. in terms of the independent variable x , or indirectly the reactive power compensation factor: k=l-(x/tan@,), as indicated on the upper horizontal axis. It is noted that x=O (i.e. k=ll represents perfect reactive power compensation. while x=tan$,, 0.75 in this example, means no compensation. Startinrj from the uncompensated condition (x=taiiQ0=O.75). as x decreases the annual line and transformer fixed costs as well as the power l o s s cost decrease steadily, while t.he energy loss cost d r o p s slightly and then increases to $763 at full compensation. This is mainly due to the relatively low value of the load factor, s o that the load will be overcompensated most. of the 1-ime and the resultant current will include a leading component, which increases with k . The annual cost of the react.ive power compensator increases linearly with k. and reaches its maximum value of $450 at full power factor correction. From the curve of the total annual cost, it is noticed that Lhere is a n optimal value of x*=0.66. col-responding to an optimal reactive power compensation factor k*=0.117 at which the total annual cost has a minimum o f $1584. Beyond this value of k , the total cost will increase until it reaches $2275 at full compensation (i.e.43.5%
X-
Full Cornpcnsallon x.0,K.l
Fig.(4):
Effect of the reactive power compensation on the total annual cost and the different cost components, for load factor LFs0.2. The total cost is also given for LF=l.
higher). This shows clearly the importance of the proper choice of the degree of compensation. The optimal degree of compensation k* (or x*) can be directly determined from the generalized chart given in Pig.(3). For LF=0.2, it is possible to calculate the following constants using the corresponding definitions: A =
hence:
1.2196, B
=
0.6555, D
=
0.256
F1
=
1.2196
-
1.15578xO.256
F2
=
0.6555
+
0.87860~0.256= 0.8804
=
0.9237
Plottinq the point P I of coordinates (0.9237. 0.8804) on the chart of Fig. (3), the value x*=0.67 is readily obtained. To check the accuracy of this approach. especially of the regression analysis for the function (1+x2)3/2.
849 the following equationn. which is based on the exact Eq.(19), gives the recursive relation to find the value of X* iteratively: x = D ( I + x ~ ) ~ / ~ /-A x3/A + B/A
(26)
Substituting the above values of the parameters A, B and D: x
0.21(1+x2)3/2 - 0.82 x3 + 0.5375
=
Solving this equation iteratively, and after few iterations, the solution converges to x*=0.6620. This is only 1.194% smaller than the previously value of x*=0.670. found from the chart in Fig. (3). Case Study 2: (LF=ll The situation is more interesting if the load factor LF=1. From Fig.(4), the minimum annual cost of $2712 occurs at the optimal value x*=0.265, corresponding to the optimal degree of compensation k*=0.647. These optimal values can be obtained directly as follows. Since LF=l, the analytical solution according to Eq.(25) is valid, from which
-
x*
=
D/dl-D2
=
0.256/dl-0.2562 = 0.265
The result can also be found using the chart of Fig.(3). For LF=l, and h=l. the following constants can be calculated A
=
1, B
=
0,
D
=
0.256
Then F1
=
1 - 1.15578xO.256
Fa
=
0 +
0.7041
(dimensionless)
0.87860~0.256= 0.2249
(dimensionless)
=
These are the coordinates of the point P2 plotted in Fig.(3), from which the optimal value is x =0.265. This corresponds to an optimal compensation factor k*=O.6467. The two case studies discussed above prove the accuracy and validity of the suggested approach. Also, the simple application of the generalized chart in Fig.(3) is demonstrated.
Fig.(5) summarizes the results of a study showing the effect of both the load factor LF and the specific cost cr of the reactive power compensation on the optimal power factor correction parameter k*. Several curves for different values of cr ranging from 0 to 2 . 5 ~ 1 0 - ~$/VAr are given. It can be noted that for every value of cr. there is a certain minimum value of the load factor Wbelow which no economical advantage can be achieved through the procedure of power factor correction. F o r example. if c , = ~ X ~ O -%/VAr, ~ no compensation is recommended if LF50.2. Even in the theoretical case characterized by cr=O, there is an optimal value of k* less than 1, if the load factor LF is less than 1. This is because of the higher losses during the periods of low load active power demand. The load will then be overcompensated, and the resultant receiving-end current will include a capacitive component . Generally, the optimal degree of compensation k* increases with the load factor and decreases with cr. PARMETER STUDY
In this section, the results of a digital computer program based on the direct solution of the unapproximated Eq.(19) will be presented. They will include also the effect of changing some of the problem parameters on the optimal power factor correction. For convenience, this FORTRAN program is listed in the Appendix. The lines 2 to 15 include the numerical values of the power network parameters using the same notations previously defined. At the end, the result is printed out for Case Study 2, discussed before, yielding k*=O.l168. Fig. ( 6 ) shows the optimal compensation factor k* as a function of the load factor (LF) and the transmission line length ( I ) in the range O-10x103 m. Other parameters have the same values as mentioned before. The diagram depicts that k* increases continuously with both variables LF and I . It is also noticed that - based on the assumed parameter values - no power factor correction can be recommended if the line length is zero. i.e.. direct connection of load and transformer. For low values of (LF), improving the power factor by means of parallel compensation is only justifiable if the line is longer than about 1.5 k.. Pig.(6) shows that the optimal compensation factor k* is less sensitive to changes in line length t , if compared to changes in the load factor (LF), especially at low values of LF. Moreover, the largest value of compensation factor k*=0.866 is
.Cr
Load
Fig.(5):
Factor LF-
Effect of the load factor LF and the reactive power specific cost cr on the optimal compensation factor k*
L i n e Length, km
Fig.(6):
Optimal compensation factor k* as a function of load factor (LF), and line length I . Ce S 0 . 7 5 ~ 1 0 - ~ / W ,h U = 0.10.
850 required for LF=1 and P=10x103 m. For comparison, if P=3000 m. then the points P; and P; are given in Fig.(6). for load factors LF=O.2 and 1 , respectively. As a matter of fact, they correspond to the two points P I and P2 plotted in the generalized chart of Fig.(3), describing the previously discussed Case Studies 1 and 2 . The good agreement of the values of k* in Fig.(6) based on the unapproximated E q . ( 1 9 ) , with those obtained using the simple generalized chart based on the reduced form, Eq.(21), is quite clear. This can be also easily checked if the values of k" in Fig.(6)
corresponding to line length of 3 km, at different values of load factor LF, are compared ta those plotted i n Fig.(5), for c ~ - $ ~ . ~ x ~ O - ~ /obtained VAR, using the chart in Fig.(3). The effect of changing the specific energy loss cost Ce from $ 0 . 7 5 ~ 1 0 -to ~ $1.50~10-~/Wh is depicted in Fig.(7). Comparing it to Fig.(6), it can be noticed that higher ce will result higher values of k* for high LF. and vice versa. For example, Fig.(7) shows k*=0.92 at P=10x103 m and LF=I. whereas Fig.(6) indicates k*=0.865 under the same conditions. On the other hand, Fig.(7) shows k*=0.19 for 9=10x103 m and LF=0.2. while Fig.(6) gives k*=0.245 for the same values of line length and load factor.
1 .O(
The last figure serves to discuss the effect of an increase in the annual rate of interest and depreciation, a, from the originally assumed a=0.10 yielding Fig.(6), to a=O.15 for Fig.(8). Considerable quantitative differences between the two figures, regarding the optimal compensation factor k*, are recognizable for cases characterized by either: load factors close to unity, or load factor less than 0 . 4 , o r lines shorter than 3 km. For instance, if LF=0.2. thenthe increase in a from 0.10 to 0.15 will increase the shortest line length of 1.5 km previously mentioned in connection with Fig.(6) for an economically feasible power factor correction, to about 2.0 km, as seen in Fig.(8). It can be also generally observed that a higher a will always result in lower k*, except in systems having low load factors and relatively long lines. ~ L U S I O N S 1.
The significance of the power factor correction as well as its economical assessment is pointed out, and the different affecting parameters are discussed.
2.
The mutual interaction between the different cost components is recognized and an expression for an objective function is derived. It represents the overall reduction in the total annual cost achievable by reactive power compensation.
3.
An equation in the optimal degree of compensation could be derived. It could be solved analytically for the special cases having a demand load factor LF=l.
4.
For all values of LF, a generalized chart is provided, which gives the optimal compensation factor for any case study, simply by plotting a single system characterizing point on that chart.
5.
Two practical case studies have been investigated using both the analytical solution and the generalized chart. They showed the validity, accuracy and the ease of applying the suggested approach.
6.
A parameter study is also done in order to investigate the effect of both the load factor and the specific cost of reactive power compensation, on the optimal degree of power factor correction.
7.
The results obtained from a digital program are also given to indicate the sensitivity of the optimal compensatjon factor k* to changes in system parameters such as the specific energy loss cost ce and the annual rate of interest and depreciation a. Moreover, the good agreement of the results of the digital computation with those determined using the suggested simple generalized chart, could be realized.
8.
It is found that there are some situations for which no power factor correction can be recommended.
I
Line L e n g t h , km
Fig. ( 7 ) :
Optimal compensation factor k* as a function of load factor (LF). and line length P . ce = $ 1 . 5 0 ~ 1 0 - ~ / W,h a = 0.10.
9.
L i n e L e n g t h , km
Fig.(8):
Optimal Compensation factor k* as a function of load factor (LF), and line length P. ce = $ 0 . 7 5 ~ 1 O - ~ / W,h a = 0.15.
It is believed that the presented approach, the analytical solution for unity load factor as well as the generalized chart for the determination of the optimal degree of compensation can be helpful in the planning and design of such systems. Of course, some technical constraints, such as the permissible limit of the voltage regulation, should be also included in assessing the power factor correction.
85 1
REFERENCES "The IEEE-Power Society 1974-1975 Cumulative Index". IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No.10. October 1985, pp.169-170. K.R.C. Mamandur, R.D. Chenoweth: "Optimal Control of Reactive Power Flow For Improvements in Voltage Profiles and f o r Real Power Loss Minimization". IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No.7. July 1981. pp.3185-3194. T.H. Fawzi, S.M. El-Sobki, M.A. Abdel-Haliin: "New Approach for the Application of Shunt Capacitors to the the Primary Distribution Feeders". IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No.1, January 1983, pp.10-13.
APPSUDIX The FORTRAN program written to compute the optimal compensation factor k*, using the unapproximated Eq.(19). is listed below. The symbols in lines 2 to 15 are according to the notations adopted in the analysis. The last line gives the print out for Case Study 2.
1:
REAL LFILVMDA~.LIK
2:
V-
CL-4 .?E-7 RLO-3.12SE-5 CT-l.25E-3
e: 7:
e:
5 . 3 . Grainger, S.H. Lee:
"Optimum Size and Location of Shunt Capacitors for Reduction of Losses on Distribution Feeders". IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100. No.3, March 1981, pp.1105-1118. 3 . 5 . Grainger, S. Civanlar, K.N. Clinard.L.Gale:
"Optimal Voltage Dependent Continuous-Time Control of Reactive Power on Primary Feeders". IEEE Transactions on Power Apparatus and Systems, Vol. PAS-103, No.9, September 1984, pp.2714-2722.
.
11 E3 PO-4.EA COS0.C.P:
3: e! 5:
RTO=O.O24
9: 10:
11: I?:
13: 14:
!5: 16:
l?: 18: 19:
20: 21: 22: 2 2 : 27 24: 25:
26; 27:
M.M. Saied, N.H. Fetih, H. El-Shewy: "Optimal Expansion of Transformer Substations". IEEE Trans on Power Apparatus and Systems, Vol. PAS-101, Nov. 1982, pp.4333-4340. M.M. Saied, H. El-Shewy: "Optimal Design of Power Transformers on the Base of Minimum Annual Cost". Paper No. A 80 054-7, presented at the IEEE Winter Power Meeting, New York, NY, February 3-8, 1980. M.G. Say: "Performance and Design of AC Machines" Book, Pitman Publishing, Third Edition, 1974.
M.M. Saied: "Optimal Long Line Series Compensation". IEEE Transactions on Power Delivery, Vol. PWRD-1, No.2, April 1986, pp. 2 4 8 - 2 5 3 . W.L. Weeks: "Transmission and Distribution of Electrical Energy". Book. Harper I Row Publishers, New York, 1981. pp.206-210.
28: 29: 30: 31: 32:
33: 34: 35:
3t: 2 37:
N-1.0-X/TANO
38: 39: 40: 41:
42: EOF:42
2e
IF(K.LE.O.0 JK-0.0 F R I N l 28, L F I L I X I K FORMAT (2X1F3.1r3Y1F7.112X1F6.4,2X1F6.4) STOP END
