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Maximizing Electrical Power Saving Using Capacitors Optimal Placement

Article  in  Recent Advances in Electrical & Electronic Engineering · February 2020

DOI: 10.2174/2352096513666200212103205

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Recent Advances in Electrical & Electronic Engineering, 2020, 13, 000-000 1

RESEARCH ARTICLE

Maximizing Electrical Power Saving Using Capacitors Optimal Placement

Ayman Agha1, Hani Attar2,*, Audih Alfaoury3 and Mohammad R. Khosravi4

1
Electrical Engineering Department, Faculty of Engineering, Philadelphia University, Amman, Jordan; 2Department of
Energy Engineering, Zarqa University, Jordan; 3Electrical Engineering Department, Al-Balqa Applied University, Salt,
Jordan; 4Department of Electrical and Electronic Engineering, Shiraz University of Technology, Iran

Abstract: Background: Low power factor is regarded as one of the most dedicated issues in large
scale inductive power networks, because of the lost energy in term of a reactive power. According-
ly, installing capacitors in the network improves the power factor and hence decreases the reactive
power.
Methods and Objectives: This paper presents an approach to maximize the saving in terms of finan-
cial costs, energy resources, environmental protection, and also enhance the power system efficien-
cy. Moreover, the proposed technique tends to avoid the penalties imposed over the electricity bill
(in the case of the power factor drops below the permissible limit), by applying a proposed method
that consists of two stages. The first stage determines the optimal amount of compensating capaci-
ARTICLE HISTORY
tors by using a suggested analytical method. The second stage employs a statistical approach to as-
Received: October 15, 2019
sess the reduction in energy losses resulting from the capacitors placement in each of the network
Revised: December 21, 2019 nodes. Accordingly, the expected beneficiaries from improving the power factor are mainly large
Accepted: January 06, 2020
inductive networks such as large scale factories and industrial field. A numerical example is ex-
DOI: plained in useful detail to show the effectiveness and simplicity of the proposed approach and how it
10.2174/2352096513666200212103205
works.
Results: The proposed technique tends to minimize the energy losses resulted from the reactive
power compensation, release the penalties imposed on electricity bills due to the low power factor.
The numerical examples show that the saved cost resulted from improving the power factor, and
energy loss reduction is around 10.94 % per month from the total electricity bill.
Conclusion: The proposed technique to install capacitors has significant benefits and effective pow-
er consumption improvement when the cost of the imposed penalty is regarded as high. The trade-
off in this technique is between the cost of the installed capacitors and the saving gained from the
compensation.
Keyword: Energy loss reduction, shunt capacitors, maximized saving, power factor improvement, power systems, energy policy.

1. INTRODUCTION authorities. As a result, valuable penalties over the electricity

bill are charged due to this low power factor, which is
The structure of the industrial plant networks usually
considered as a part of the electricity tariff applied by the
starts from a single supplying point to feed the primary
electricity supplier [1].
substation; and hence it feeds the lateral substation(s). The
flowing of the electrical current through the feeder's The low power factor creates many undesired technical
conductors is regarded as the main reason for the power and economic effects on the electrical networks. The location
losses, mainly in large electrical networks such as industrial of capacitors adjacent to loads has well-known benefits [2],
plants, because the loads in industrial plants are mostly such as optimizing investment cost, releasing the power
inductive as a result from connecting a significant number of capacity, improving the voltage profile, enhancing the
induction motors to it. These huge inductive loads minimize system stability and quality, minimizing the annual fee, and
the average network power factor to a value -in general- maximizing the saving in terms of money. However, it is
below the specified limit imposed by the electricity essential to understand that the installation of the capacitors
requires an additional cost; therefore optimizing the size of
the capacitors is a critical issue, i.e. the price of the annual
*Address correspondence to this author at the Department of Energy Engi-
neering, Zarqa University, International Street, Zarqa 13132, Jordan;
installed capacitors should not be more than the annual saved
E-mails: hattar@ze.edu.jo; attar_hani@hotmail.com value from improving the power factor.

2352-0965/20 $65.00+.00 © 2020 Bentham Science Publishers

2 Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 Agha et al.

Different approaches and techniques have been utilized To maximize the annual saving associated with real ener-
to solve the problem of the capacitor sizing, and the gy loss, and the cost of power factor penalties, compared
allocation in the electrical systems, [3-7] that used analytical with the cost involved with capacitor banks installation, the
methods with or without statistical approaches. The objective function can be mathematically formulated as fol-
analytical method that is based on loss sensitivity factors lows:
were used in a few studies [8-11]. The authors carrying out a
Maximize the objective function:
study [12, 13] used spicific dynamic programming methods
or fuzzy with dynamic programming-based as in previous Max. {S} (1)
studies [14, 15]. A linear and nonlinear programming-based
Subject to:
approaches were explained in [16] and [17], respectively.
!
Q ! =   !!! Q !" (2)
In previous studies [18-23], both of fuzzy or fuzzy based
techniques were investigated and managed to solve the Where: {S} is the saving function in Jordanian Dinar or
optimam capacity sizing matter. Genetic Algorithm was used US Dollar [JD or $] (1JD=1.4$) per time unit, taking into
in several studies [24-28], and a neural network method was consideration that more S is positive, the more saving in JD
used in a study [29] as well as different techniques to achieve is achieved, which is the proposed as the objective function
the same goal. in this paper, (Qci) is the reactive power injected at the (ith)
bus, and (Qc) is the total amount of reactive power compen-
Generally, most of the research works and techniques
sators in [kVAR].
focused on obtaining the main objective and other related
benefits resulted from this objective, however, in recent Eqs. (1) and (2) have the constraints shown in Eqs. (3) to
years, some approaches, such as in a sudy [30-32], have been (5):
utilized to obtain multi-objective optimization in the same V(min.)≤Vi ≤ V(max.) (3)
technique, such as; proposing one method that minimizes the
power and energy losses, improves the voltage profile, and Qc (min.)≤Qci≤ Qi (4)
minimizes the investment cost simultaneously. cosφ(a) ≥cosφ(min.) (5)
Bacterial Foraging Optimization Algorithm to achieve Where the objective function is as in Eq. (6)
the optimum solution were used in several studies [33-36], !
while the Flower Pollination Optimization Algorithm was S = K ! ∙   ∆E! + K !".!"#.   − K ! ∙   !!! Q !" (6)
studied in [37, 38]. And;
More information and beneficial background regarding ΔE= ΔPd (max.)⋅τ (7)
this essential technique can be found in a few other studies
[39-43]. Where:

The approach presented in this paper aims to minimize Ke: A cost of unit energy loss [JD/kWh].
the energy losses resulted from the reactive power compen- Kc: The cost of the capacitor [JD/1 kVAR].
sation, releases the penalties imposed on electricity bills due
K !".!"#. : The penalty imposed on electricity bill due to
to the low power factor, and as a result, enhances the system
the low power factor [JD/month].
efficiency and reduces the annual cost of electrical bill i.e.,
the price of the bill. ΔEi: The active energy loss reduction in (ith) line in
[kWh].
The advantage of the proposed approach over the others
is its implementation simplicity, and the ability to be applied cosφ(a): The monthly average power factor after compen-
straightforwardly without the need for extensive input data sation.
or special mathematical tools, which makes it possible to be cosφ (min.): The minimum allowed power factor value by
performed directly by the plant’s day duty electrical engi-
electricity authority.
neers.
Qc (min.): The minimum standard size of the capacitor
The rest of the paper is organized as follows: In Section
[kVAR].
2, the problem formulation is presented, wherein Section 3,
the electrical network model is illustrated. In Section 4, the Qi : The maximum load of reactive power at (ith) bus
mathematical model of the proposed method is formulated [kVAR].
and solved. The solution algorithm, along with numerical V(min.): The minimum permissible voltage magnitude at
examples demonstrate the way the proposed method, is im- the (ith) bus [V].
plemented and explained in Sections 5 and 6, respectively.
Results and conclusions are present in Section 7. V(max.): The maximum permissible voltage magnitude at
the (ith) bus [V].
2. PROBLEM FORMULATION Vi: The voltage magnitude at the (ith) bus [V].
The rating of capacitor banks, by nature, changes in a ΔPd(max.): The active power loss reduction in (ith) line in
discrete manner that makes the formulation of the objective [kW].
function to be a nonlinear optimization problem.
τ: The maximum loss time [h].
Optimal Capacitor Placement Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 3

Fig. (1). Industrial Plant SLD.

The constraint in Eq. (3) is used to maintain the voltage For simplicity and without losing the generality of the
magnitude level after the compensation within the permissi- proposed work, the calculation is performed for one phase
ble value. The condition in Eq. (4) is used to overcome the only as the other two phases are assumed to be symmetric;
minimum standard unit size of the compensating capacitor, which means that all of the three phases have the same line
i., e., (Q !"(!"#.) ≤ Q !" ) and to avoid over compensation parameters, voltage magnitudes, and load.
(Q !" ≤ Q ! ), which means wasting money by installing larger
and more expensive capacitors than needed and other un- 4. THE PROPOSED METHOD
wanted effects over the buses’ voltage. Determining the
maximum capacitors required to sustain the power factor The installation of shunt capacitor banks is a capital in-
limit is one of this paper’s achievements. While the condi- vestment; thus the value of the capacitors should be sized in
tion in Eq. (5) is the monthly average value of the power such a manner to realize the aims of compensation.
factor after the compensation that avoids the penalties im- In this paper, the main objective function is to achieve
posed on the electricity bill as a result of the low power fac- the maximum annual cost saving in the industrial network
tor. Eq. (6) is the key objective function equation, which is plant by reducing the electrical bill price; to achieve this tar-
regarded as the saving indicator in money that arises from the get; the formulated problem in section-II is split up into the
cost of energy losses reduction resulted from the capacitor following five stages:
compensation and the power factor penalties against the ca-
pacitor bank cost. Finally, Eq. (7) shows energy losses in the Stage one: Determining the optimal capacitor bank val-
ues that should be installed in each substation to realize the
network, i. e., the power losses over a certain period of time,
objective function.
which is what this paper is based on, as shown in the follow-
ing sections. Stage two: Determining the equivalent loading time
(T(w/m)eq) that also called, maximum load utilization time
3. NETWORK MODELING T(max.) for the desired period i. e., the analyzed period that set
to be one month in this paper as an example, taking into con-
The general industrial plant network configuration is sideration that the period could be set practically for one
shown in Fig. (1). year, longer than one year, or shorter than one year.
Where: Stage three: Determining the maximum loss time (τ) that
i=1,2, …, n, is the number of buses in the network. also called, equivalent losses time and then Eq. (7) is ap-
plied.
P and Q: are the active [kW] and reactive power [kVAR],
respectively. Stage four: Determining the optimum size of compensat-
ing capacitors in each (SBB-i) and at the main point of sup-
A and L: are the line cross sectional area [mm2] and the ply (MPS), as shown in Fig. (1), and then applying Eqs. (18)
length [m], respectively. and (30), respectively.
T(w/m)eq : The equivalent operating time (loading) per Stage five: The calculated objective function in Eq. 6 is
month [h/month]. tested to make sure that the constraints mentioned in Eqs. (2)
In the case when a Secondary Distribution Board current to (5) are completely fulfilled.
feeder (SDB-i) acts as a primary feeder for further sup-
feeders; the procedures of the proposed technique remain Optimal Capacitor Sizing
unchanged, i.e., one primary feeder supplies the below sub-
feeders. The flowing electrical current (Ii) in each line resistance
(RLi) causes the active power losses (ΔPi) as in Eq. (8).
4 Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 Agha et al.

ΔP!(!"#.) = I!.!"# ! ∙ R !" (8) optimal value of compensating capacitors is regarded as a

The complex power (S) resulted from the electrical cur- minimum value.
rent is shown in Eq. (9). Based on the above, the optimal size of compensating
S = VI ∗
(9) capacitors in Eq. (18) should not be less than the value in
Eq.(5), i. e., the power factor after installing the capacitor
By manipulating Eqs. (9), (10) is obtained: should be more than the limit set by the electrical authorities.
!
I ∗ =   (10) Therefore, the total minimum amount of compensating
!
capacitors Qc(T.min.) must take Eq. (5) into consideration; so, it
Where: I ∗ is the conjugate value of the current (I). So, must not be less than the value proved in previous works [18,
when taking into consideration that  I ∗ ∙ I ∗ = I ! , and the fact 22], and as of Eq. (19):
that the complex power is S=P+jQ; Eq. (11) is obtained:
Qc(T.min.) = Pmax.(MPS).(tgϕb - tgϕa) (19)
! !! !
! !!"!
I =   (11) Where:
!!
Substituting Eq. (11) in to Eq. (8) gives the maximum P(max.-(MPS)): The maximum value of the active power
power losses at the (ith) line before adding the capacitors drawn from the main point of supply (main substation) in
(ΔPi (max.-b)) as in Eq. (12). [kW].
!! !
! !!! ϕa,ϕb: The load angle before, and after compensation in
ΔP!(!"#.!!) = I!.!"# ! ∙ R !" = ∙ R !" (12)
!! degrees, respectively.
The installing of a capacitor (Qci), at the end of the (ith) Qc(T.min.): The total minimum amount of capacitors
line, changes the flow of the reactive power. The maximum [KVAR].
power losses after adding the capacitor is shown in Eq. (13).
!!
! !(!! !  !!" )
! 4.1. Determination of Equivalent Working Hours
ΔP!(!"#.!!) = I!.!"# ! ∙ R !" = ∙ R !" (13)
!!
The first author of this paper presented in his previous
The reduction in the active power losses (ΔPd(max.)) re- works [39-40] a formula to determine the equivalent working
sulted from the installation of the capacitors is simply found hours per month (T(w/m)eq.) that is also called, maximum load
from the difference between Eqs.(12) and (13) as shown in utilization time. The expression of (T(w/m)eq.) is shown in Eq.
Eq. (14): (20).
!!! !!!! !!! !(!! !  !!" )! !∙!!   ∙  !!" !  !!!" !!  (!")
∆P!(!"#.) =
!!
−
!!
∙ R !" =
!!
∙ R !" (14) T(!/!)!". = 2 ∙ D   ∙
!!
+  
!!!!
∙ + 2 ∙ (365 −
!/! ! ! !!(!)
Substitute Eq. (14) into Eq. (7) to calculate the reduced !!  (!")
amount of energy during the time period (τ), and then substi- D !/! )∙ (20)
!!(!)
tute the resulting equation into the objective function in Eq.
The equivalent working hours for (N- months) is given in
(6) gives Eq. (15)
Eq. (21).
!∙!!   ∙  !!" !  !!
!"
S = K!   ∙ R !" ∙ τ! +   K !".!"#. − K ! ∙ Q !" (15) T(eq.) = N· T(w/m) (21)
!!

To obtain the maximum saving concerning the capacitors Where:

size; Eq. (15) is differentiated concerning (Qc) and then Ap(w): The sum of the consumed active energy (day and
!!
equalized to zero, i.e., = 0, that gives Eq. (16). night) during the time of operation per month [kWh/N].
 !!!"
!" ! !∙!!   ∙  !!" !  !!!"
= !!   ∙ R !" ∙ τ! +  K !".!"#. − !! ∙ !!" =0 (16) Ap(aw): The consumed active energy after the time of op-
!"!" !"!" !!
eration per month [kWh/N].
yields;
D(w/y): The number of working days over the year (ex-
!(∙!!   !  !!" ) cluding holidays, shutdowns, weekends, etc.)
K!   ∙ R !" ∙ τ! − !! = 0 (17)
!!
n(h/m): The number of hours per month (i.e. year’s hours /
It is useful to notice that K !".!"#. in Eqs. (16) And (17) is
12 month =730 [h].
a constant value, so, the first and second order differentiation
is equal to zero, which means that K !".!"#. does not affect the ns :The number of working shifts during one day [/].
calculations. T(w/m)eq: The equivalent working time over the month in
By manipulating Eq. (17), the optimal value of compen- hours [h/month].
sating capacitors (Qc.opt.) that should be installed in the (ith) N: The number of months (N=1, 2…12, where N= 12 for
bus (substation) is shown in Eq. (18). one full year).
!!"   ∙!!
Q !" !"#.  ! Q ! − (18) 4.2. Determination of Maximum Loss Time
!∙!! ∙!!"  ∙ !!

The value of the second derivative is checked and found Electric bus (node); is a point where two or more pow-
!!!
that it is less than zero  ( ! > 0). Therefore, the obtained ered lines are electrically connected, concerning to their
 !!! maximum loss time (τ), is as follows:
Optimal Capacitor Placement Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 5

These buses are load nodes supplying the electric power k= 0.16; in USA for rural power grid.
to (SDB-1… SDB-n) as in Fig. 1, presented above. Equation
k= 0.33; in Polish source.
(7) can be represented as Eq. (22) and illustrated, in Fig. 2.
! Also, the relation between (LSF) and (LF) can be pre-
ΔE =   !
ΔP t   ∙ dt = ΔP(!"#.  )     ∙ τ (22) sented in an exponential form as presented in Eq. (27).
LSF = (LF)! (27)
The relation between (τ) and (Tmax.) can be also presented
in terms of curves considering different values of power fac-
tor.
The (LF) is defined as “the ratio of the average load
(P(avg.)) in [kW] or the average current (Iavg) in "A" in Eq.
(28) to maximum power (P(max.)), or maximum current [42-
43]:
! !"#. ! ! ! ! !"#.  ∙ ! !"#. ! !"#.
LF =
!
  =
!
∙ =
! !
∙
!
=  
!
(28)
!"#. !"#. !"#.

4.3. Capacitors Optimal Sizing

The optimal size of compensating capacitors in each
(SDB-i) is obtained by applying Eq. (18), and then standard-
Fig. (2). Electrical Energy Losses behavior. izing the obtained value to the nearest standard value from
the capacitors catalog. The sum of the compensating capaci-
tor in all nodes is as in Eq. (29)
In Fig. 2, the straight-line curve represents the values of
Q ! !"#. =   Q !" !"#. + Q !" !"#. + ⋯ +   Q !" !"#. (29)
the active power losses (ΔP(t)) when these values are rear-
ranged in descending order from (ΔP(max.)) to (ΔP(min.)), the However, if the sum of the compensating capacitors in
dotted line-curve in Fig. 2 represents the equivalent rectangle Eq. (29) is less than the total minimum amount of the com-
curve where (ΔP (max.)) is the vertical axis and (τ) is the hori- pensating capacitors Qc(T.min.) as in Eq. (19), then the bal-
zontal axis. anced reactive power compensators shall be installed over
the primary supplying point at the primary substation, as in
Based on Fig. 2, clearly, the amount of energy losses is
Eq. (30).
the same in both curves, which means that the areas under
these two curves are equal, accordingly, from Fig. 2, (τ) rep- Q ! !"# = Q ! !.!"#. −   Q ! !"#. (30)
resents the maximum loss time used in Eq. (7).
Taken into account the definition of load loss factor 4.4. Cost Function
(LSF): “It is a factor, that when multiplied by the energy The annual net revenue in terms of money (SR) achieved
losses at the time of peak (ΔP(max.)), and the number of load from the application of capacitors can be determined by us-
periods (T) will give overall average energy loss” [41], as in ing Eq. (6). The solution of Eq. (6) is explained in the solu-
Eq. (23). tion algorithm section, and illustrated in details in the numer-
ΔE = ∆P!"#. ∙ T  · LSF (23) ical example in section VI.
By manipulating Eq. (23), Eq. (24) is obtained: 5. SOLUTION ALGORITHM
!!
LSF ∙  T = (24) The algorithm of the maximum cost saving method due
∆!(!"#.)
to reactive power compensation is presented in the flowchart
Where (T) is the year’s hours (8760 h), and according to shown in Fig. 3.
Eqs. (7) and (23); (τ ) over the analyzed period (one year) is
given in Eq.(25): 6. NUMERICAL EXAMPLE
τ = LSF· T = LSF · 8760 (25) In this section a numerical example to show the saving
The relationship between load factor (LF) and (LSF), was percentage by comparing the bill's price, after and before
introduced by Buller, and Woodrow using a completely em- applying the capacitors, taking into consideration of the ca-
pirical approach shown in [41] as in Eq. (26): pacitors' implementation costs is proposed.
LSF = 1 − k ∙   LF ! + k ∙ LF (26)
6.1. Industrial Plant Data
The value of (k) in Eq. (26) varies from zero to one, ac-
cording to the load curve profile confirmed by the original 1) Plant input Network Data
country. For instance, it is equal to: The network under investigation is an industrial plant
k= 0.2; in Great Britain and Australia. network configured in Fig. 1, fed by copper L meters cables
that has 'A' mm2 cross-section each and loaded with apparent
k= 0.3; in USA for urban power grid. power as stated in Table. 1.
6 Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 Agha et al.

Fig. (4). Flow chart for the proposed algorithm of the maxim cost saving.

Table 1. Plant Network Data.

SDB-i SDB-1 SDB-2 SDB-3 SDB-4 MPS Note

1
Pi(max.) [kW] 78.57 125.71 47.14 100.00 351.43 *)

1
Qi(max.) [kVAR] 90.20 120.50 50.80 85.2 346.7 *)

1
cosφi(max.) 0.657 0.722 0.680 0.761 0.712 *)

2
Li [m] 150 470 630 250 / *)

A [mm2] 35 70 90 150 / *)
2

3
RLi [Ω] 0.2857 0.1221 0.1273 0.0303 / *)

Table 2. Electricity Bill Data.

Ap (consumed Energy @ day tariff) in [kWh] 888,776

Ap (consumed Energy @ night tariff) in [kWh] 1,333,163

Total Ap (sum of consumed Energy) in [kWh] 2,221,939

AQ in [kVARh] 2,389,040

cosϕb(natural)[/] 0.681

K !".!"#. in [JD] ( 1[JD] ≅ 1.41 [$]) 22,147.7

Total Bill Amount [JD])* 208,625.029

Optimal Capacitor Placement Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 7

Table 3. Plant Operational Data.

SDB-i SDB-1 SDB-2 SDB-3 SDB-4 MPS Note

4
ns [/] 1 2 3 2 / *)

4
D(w/y) [day] 180 300 353 330 / *)

1
Ap(aw)/Ap(w) 0.2 0.45 0.8 0.65 / *)

5
T(w/m)eq. 242 548.5 725.2 628.5 6,317 *) Eq. (20)

Table 4. Optimal Capacitor Sizing Calculation.

SDB-i SDB-1 SDB-2 SDB-3 SDB-4 MPS Notes

5
T(eq.) 2904 6582 8702.4 7542 6,316 *) Eq. (21)
5
LF [/] 0.332 0.751 0.993 0.861 0.721 *) Eq. (26)
5
LSF [/] 0.184 0.626 0.989 0.780 0.586 *) Eq. (24), k= 0.33
5
τ[h] 1608 5486 8665 6839 5136 *) Eq. (23)
5
Δpd(max.) [kW] 0.01164 0.01108 0.00171 0 0.024430 *) Eq. (14)
7
Qc(T.min.)[kVAR] 188.36 *) Eq. (19)
5
∑ Qc(opt.) [kVAR] 144.35 *) Eq.(29)
8
Qci(opt.) [kVAR] 34.40 82.33 27.62 -38.3 44.01 *) Eq.(18 ,
5
Qc(MPS)=Qc(T.min).- Qc(opt.) [kVAR] 188.36-106.36 = 82.14.fixed type compositors. *) Eq. (30)

Q(ci) [kVAR] 47.75 5

112.01 27.62 0.00 44.01 *) Eq. (5),

5
Q(ci-stnd.) [kVAR] 2x25=50 6x20=120 2x15=30 0.00 3x15=45 *) Eq. (4)
1 2 3
Tables’ notes, abbreviations: *) : Measured for one average working day; *) : Obtained from the plant document (single line diagram); *) : Obtained from the cable’s manufacturer
catalog or calculated from the approximate formula; R(li ) [Ω]= Li [m] /Ai [mm2]· [S/m]; *)4: Operational info, (from operation Log-sheet); *)5: Calculated from equation number, *)6:
7
The values of (Ap(day)) and (Ap(night)) are as presented in Table 2, and (Teq.(MPS)), in Table 3. Where; (P(max..eq)=P(max. - MPS)); *) : Obtaining from Eq. (19) and regardless of the result of
8
Eq.(18); *) : The sign (-) means no need to compensate at this node.

Table 5. Summary of Sizing Capacitors.

Q(ci) Q(ci) std. Q(ci)-std Q(ci) Q(ci)

SDB-i
[kVAR] [kVAR] [kVAR] [JD/1kVAR] [JD]

Qc(MPS) 44.01 3x 15 45 25 1125

Qc1 47.75 2x 25 50 35 1750

Qc2 112.01 6x 20 120 35 4200

Qc3 27.62 2x15 30 35 1050

Qc4 0 0 0 0 0

SUM 231.39 245 245 - 8125

8 Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 Agha et al.

Table 6. Saving Summary.

Reduction in energy [kWh/Month] 7854

Reduction in energy [JD/Month] 373.4

Reduction of PF penalty [JD/Month] 22147.7

PF (final after compensation) (cosφ(a)) [/] 0.960

Total saving [JD/Month] 22817

Total saving in [%] 10.94

2) Electrical Energy Consumption And; the summary of saving due to compensation is

shown in Table 6.
The data shown in Table 2 is extracted from a plant’s
electricity bill for a specific month:
RESULTS AND CONCLUSIONS
3) Plant Operational Data In this paper, a mathematical approach is proposed and
used to find the optimal size of compensating capacitors. The
The plant operational data can be extracted from the installed capacitors in the plant network are used to improve
plants’ technical documents, considering the following:
the power factor when the average power factor per month
a) The cable conductivity (σ) at 20°C is equal to becomes less than the permissible values assigned by the
35x106 S/m. electricity provider. By doing so, the penalty imposed over
the electricity bill is released, and the optimum bill price is
b) The power factor after compensation shall be ≥
0.88. achieved. A statistical approach is produced and used to de-
termine the annual maximum loss period (equivalent losses
c) ns =1, where ns is the number of working shifts per period) to establish the reduction in active energy losses. An
day (each eight working hours is considered as one analytical method is also proposed and used to determine the
shift.) optimal size of compensating capacitors, taking into consid-
d) D(!/!) ≅ 248.7 days, which is the number of work- eration the inequality constraints.
ing days over the year (excluding holidays, shut- The calculations show that the proposed technique to in-
downs, weekends etc.). D(!/!) is calculated based stall capacitors has significant benefits and effective saving
on 5/7 working days per week plus twelve days per when the cost of the imposed penalty is considerably regard-
year are considered as national holidays. ed as high that makes the electricity bill to be remarkably
Plant operational data is summarized in Table 3. expensive. The trade-off in this technique is between the cost
of the installed capacitors and the saving resulted from the
compensation. Clearly, the saving has a direct relation to the
6.2. Capacitors Sizing
operation period of the plant, which adds a third factor to the
The data in Table 4 is calculated based on the proposed price optimization criteria.
technique methodology and equations as below:
Practical numerical exams have been introduced to testify
6.2.1. Check the Cost Function the objective function in terms of the percentage of saving in
the electricity bill. The numerical examples show that the
The cost function is carried out under the following as-
saving in money resulted from improving the power factor,
sumptions:
and energy loss reduction is around 10.94 % per month from
1) The average cost of the unit, Ke = 0.09339 [JD/ the total electricity bill. Moreover, the more the power factor
kWh] is low, the more saving in the electricity bill is achieved,
2) The cost in [JD] for [1kVAR] is equal to Ki = 25 requiring larger capacitor size, which is the trade-off that
must be optimized carefully through the proposed objective
[JD / kVAR] for the fixed type, and 35 [JD / kVAR]
function.
for the regulated type.
The presented example is calculated by a simple and
3) The lifetime of the capacitors is equal to 20 [Years].
straightforward manner, and it does not require specific and
4) The cost of the capacitors is depreciated based on complicated software programs or mathematical tools to
excel double line decline method and is equal to solve the objective function. Accordingly, Excel sheet is
7000 [JD/year], i.e. Kc= 58.33 [JD/month]. sufficient to conduct the calculation; however using
MATLAB software could make it even easier mainly for
6.3. Summary of Compensation wide range of variable cases, and to obtain ready carves.
Extending the calculation over MATLAB software to make
The result of the compensation can be summarized in
full operation sheets or curves is out of this papers’ scope
Table 5 as follows: though it is easy to perform.
Optimal Capacitor Placement Recent Advances in Electrical & Electronic Engineering, 2020, Vol. 13, No. 0 9

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