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В1СНИК ПРИАЗОВСЬКОГО ДЕРЖАВНОГО ТЕХН1ЧНОГО УН1ВЕРСИТЕТУ
Вип. №19
2009 р.
УДК 621.311
Khalil Т.М.1, Gorpinich A.V.2
THE EFFECT OF MUTUAL COUPLING, LOAD UNBALANCE AND HARMONICS ON CAPACITOR PLACEMENT IN DISTRIBUTION NETWORKS
Рассмотрено влияние высших гармоник, несимметрии и взаимоиндукции на выбор мощности и мест установки батарей конденсаторов в распределительных электрических сетях. Предложен бинарный метод оптимизации, основанный на моделировании скоплений частиц (по аналогии с биологическими системами). При составлении целевой функции учитывались стоимость батарей конденсаторов и потерь мощности, типоразмеры батарей конденсаторов, ограничения по уровню напряжения и его искажений в узлах электрической сети.
Introduction
The problem of capacitor placement for loss reduction in electric distribution systems has been extensively researched over the past decades. The objective of capacitor placement is to achieve the loss reduction weighted against capacitors costs keeping the operational and power quality constraints within required limits. In reality, distribution networks are unbalanced systems due to mutual coupling between phase conductors and unbalanced loading on different phases. Moreover, a considerable amount of harmonic distortion exists in distribution system.
Most of the capacitor placement techniques assume the distribution system to be balanced and the supply as sinusoidal. Limited publications have taken into account system unbalance and the presence of harmonics [1-3] when solving the capacitor placement problems. Consideration of three-phase system and harmonics complicate the capacitor placement problem compared with the balanced sinusoidal case. In this paper, the work reported in [3] has been extended to include the effect of mutual coupling and load unbalance on capacitor placement in distribution system.
Problem formulation
The objective function is to minimize the total annual costs due to capacitor placement and power losses [3] with constraints that include limits on voltage, total harmonic distortion and size of installed capacitors (see equations (l)-(5)).
H m-1
Ploss = (^j Ploss(i,i+1) ) 5 (1)
h=1 ¿=0
6max = LQo i (2)
F = K>Ploss+fdK°Q°-, (3)
j=i
/Я/) < THl)m.. ; (4)
V ■ <\V- \<V (5)
mm —\ i I— max '
where Ploss- total power losses; <2^ax - maximum allowable capacitor size to be placed; L - integer; <2o - smallest capacitor size; F - total annual cost function: К'' - equivalent annual cost per unit of power losses; Kc, - capacitor annual cost/kvar; QCj - shunt capacitor size placed at bus /: Vmm , Vtm:. -minimum and maximum permissible rms voltage; THDmax - maximum permissible total harmonic distortion; J, m- shunt capacitor buses and number of buses.
Considering investment costs, there are a finite number of standard capacitor sizes that are integer multiples of smallest size. The cost per kilovar varies from one size to another. Generally, large sizes are cheaper than smaller ones (see table 1).
1 Pryazovskyi State Technical University, Master of Science, Eng., Ph.D. student
2Pryazovskyi State Technical University, Candidate of Science, Eng.
Table 1 - Yearly cost of: ~ixed capacitor size
ID 1 2 3 4 5 6 7 8 9
Capacitor size (kvar) 150 300 450 600 750 900 1050 1200 1350
Capacitor cost ($/kvar) 0.5 0.35 0.253 0.22 0.276 0.183 0.228 0.17 0.207
ID 10 11 12 13 14 15 16 17 18
Capacitor size (kvar) 1500 1650 1800 1950 2100 2250 2400 2550 2700
Capacitor cost ($/kvar) 0.201 0.193 0.187 0.211 0.176 0.197 0.17 0.189 0.187
ID 19 20 21 22 23 24 25 26 27
Capacitor size (kvar) 2850 3000 3150 3300 3450 3600 3750 3900 4050
Capacitor cost ($/kvar) 0.183 0.18 0.195 0.174 0.188 0.17 0.183 0.182 0.179
System model at fundamental and harmonic frequencies The distribution system has been modeled considering mutual coupling effect between phases. A direct approach for unbalanced three-phase distribution load flow solutions which presented in [4] has been used. In this approach, the special topological characteristics of distribution networks have been fully utilized to make the direct solution possible. Two developed matrices (the bus-injection to branch-current matrix and the branch-current to bus-voltage matrix) and a simple matrix multiplication are used to obtain load flow solutions. Due to the distinctive solution techniques of the proposed method, the time-consuming decomposition and forward/backward substitution of the Jacobian matrix or admittance matrix required in the traditional load flow methods are no necessary. For the harmonic flow study, linear load is represented by parallel combination of resistance and inductance to account for the respective active and reactive power at fundamental frequency.
Particle swarm optimization The Particle Swarm Optimization (PSO) method was first introduced by Kenney and Eberhart [5] in 1995. It was developed through simulation of a simplified social system, and has been found to be robust in solving continuous nonlinear optimization problems [5-8]. One of reasons that PSO is attractive is that there are very few parameters [9]. There are different versions of PSO that aim to widen its applicability. Kennedy and Eberhart [10] proposed the first discrete version.
In PSO algorithm, each member is called "particle", which represents a candidate solution to the problem at hand, and each particle flies around in the multi-dimensional search space with a velocity, which is constantly updated by the particle's own experience and the experience of the particle's neighbors. The basic PSO technique is the real valued PSO, whereby each dimension can take on any real valued number. On the other hand, in binary PSO each dimension of the particle can only take on the discrete values of 0 or 1.
Basic particle swarm optimization Particle Swarm Optimization is a stochastic optimization algorithm that simulates the social behaviors of bird flocking or fish schooling and the methods by which they find roosting places, foods sources or other suitable habitat. The PSO algorithm searches in parallel using a group of individuals. In the basic PSO technique, suppose that the search space is ¿/-dimensional [3].
1. Each member is called particle, and each particle (/-th particle) is represented by d-dimensional vector and described as Xt = [xn, xi2 ... xid\.
2. The set of n particle in the swarm are called population and described as pop — \X\, ... Xn \.
3. The best previous position for each particle (the position giving the best fitness value) is called particle best and described as I'li, = [pbiU pbi2 ... pbid].
4. The best position among all of the particle best position achieved so far is called global best and described as GB=[gbh gb2 ... gbd],
5. The rate of position change for each particle is called the particle velocity and described as
Vi=lVil>Vi2 -Vid\-
At iteration k the velocity for ¿/-dimension of 7-th particle is updated by:
41 =Hi-4d+<V2(g$ (6)
where i = 1,2 ... n: n - size of population; w - inertia weight; c\ and c2 - acceleration constants; r\ and r2 - two random values in range [0,1]. The 7-th particle position is updated by
Binary particle swarm optimization In 1997, Kennedy and Eberhart [10] have adapted the PSO to search in binary spaces by applying a sigmoid transformation to the velocity component to squash the velocities into a range [0,1], and force the component values of the locations of particles to be 0's or l's (see equation (8)). The equation for updating positions (equation (7)) is then replaced by equation (9).
sigmoid ) = - 1
l + exp(-v*+1)
xk+l - ■
(8) (9)
11, if rand < sigmoid (vk [O otherwise
For the capacitor placement problem, a binary PSO will be used as follows. To select the optimal capacitor size QCj to be placed at bus j choose a combination of capacitor sizes (/¿-size) from table 1 as an example
QCj =6j -szx +b2 ■sz2 +... + br ■szr +... + bR ■szR, (10)
where j e J ; J - set of candidate buses to capacitors placement; br={0,1}; szr - capacitor size from table 1; R - number of chosen capacitor sizes; QCj < Q'mi:.; <2^ax - maximum allowable capacitor size to be placed at any bus.
Thus, the candidate buses are ./-buses, and the capacitor Qc placed at candidate bus j consists of small capacitor sizes (/?-size according to equation (10)).
The population of n particles at iteration k represented by:
popk=[Xk,Xk2,...,Xk,...,Xk].
Each particle i represented in ./-dimensional by:
X-i = YXi\ ■> Xi2 ■> ■ ■ ■ ■> Xij 5 • • • 5 AiJ j
Each dimension j represented in /^-dimensional by:
Xij = \.Xij\ > Xi¡2 5 • • • 5 Xijr ■>■■■■> AijR -
Therefore, each particle / represented in (J, R) dimensions
,4].
xk 1
X* =
k k
XiU i\2
k k
Xi2\ Xi22
k k
xvi X'j 2
xk xiJ 1 xk iJ 2
.k
ilr .k i2r
.k
■ilR .k
i2R
vi/r
Vi]R
iJR
For particle i. the capacitor size at bus j and iteration k represented by:
'ij\ ' SZ\
- Xjj 2 • sz2
- xjjr ■ szr
' XijR ' SZR
The dimension x ijr indicates whether capacitor size szr to be placed at bus j and iteration k for particle i or not. In other words, xkijr is a binary value such that xkijr = 1 if capacitor size szr is placed at bus j at
iteration k for particle i. and xkr =0 if it is not placed.
The particle best, global best and the particle velocity are represented also in (J, R) dimensions. It should be noted that according to above mentioned method each bus j capacitor sizes will be same for all three phases.
Numerical example
A 9-bus simple feeder [1] as shown in figure 1 is selected for computer simulation to demonstrate the effect of mutual coupling, load unbalance and supply harmonics on the capacitor placement in distribution system. The loads at different buses for balanced loading conditions and branch data are listed in table 2 and table 3. Line impedances are calculated considering the effect of grounding as (3x3) matrix and the impedance values are shown in table 4.
1 2
3 9
S
Fig. 1 - Nine-bus test feeder
Table 2 - Bus data
Bus no. 1 2 3 4 5 6 7 8 9
P (kW per phase) 1840 980 1790 1598 1610 780 1150 980 1640
Q (kvar per phase) 460 340 446 1840 600 110 60 130 200
Table 3 - Brunch data
From bus no. 0 1 2 3 4 5 6 7 8
To bus no. 1 2 3 4 5 6 7 8 9
Length (mile) 0.63 0.88 1.7 0.84 2.3 1.05 1.50 3.5 3.9
Table 4 - Impedance matrix including mutual coupling (Z (fi/mile))
7 ^aa 0.7433+y'1.2092 Zab 0.1566+/0.4790
Zbb 0.7526+/1.1758 7 ^ac 0.1536+/0.3865
7 ^cc 0.7472+/'1.1959 Zbc 0.1587+/0.4370
Kp was selected to be 168 $/kW, and the voltage limits on the rms voltage were selected as Vmm = 0.9 p.u., and Vmax = 1.1 p.u. It was assumed that the substation voltage contains 3 % and 2 % of 5-th and 7-th harmonic, respectively. Commercially-available capacitor sizes with real costs/kvar were used in the analysis. It was decided that the largest capacitor size <2^ax should not exceed the total reactive load, i.e., 4186 kvar. The yearly costs of capacitor sizes are shown in table 1. Optimum shunt capacitor sizes have been evaluated for the following cases.
1. Harmonic frequencies, mutual coupling and load unbalance are ignored.
2. Harmonic frequencies are considered (maximum THD limit is equal to 5 %), mutual coupling and load unbalance are ignored.
3. Harmonic frequencies are ignored, mutual coupling is considered and load unbalance is ignored.
4. Harmonic frequencies are considered (maximum THD limit is equal to 5 %), mutual coupling is considered and load unbalance is ignored.
5. Harmonic frequencies and mutual coupling are ignored and load unbalance is considered (upper limit is equal to 5 %).
6. Harmonic frequencies are considered (maximum THD limit is equal to 5 %), mutual coupling is ignored and load unbalance is considered (upper limit is equal to 5 %).
7. Harmonic frequencies are ignored, mutual coupling and load unbalance are considered (upper limit is equal to 5 %).
8. Harmonic frequencies (maximum THD limit is equal to 5 %), mutual coupling and load unbalance are considered (upper limit is equal to 5 %).
9. Calculation of the rms voltage, THD, power losses and benefits for the test feeder with capacitor sizes and places obtained in case 1.
Capacitive kvar required for the above cases are summarized in table 7. In the reported results, a 5 % unbalance indicates that load at each load node of phase B is 5 % higher than the load of phase A and load at phase C is lower by same amount, thus keeping the total three-phase loads as in the balanced loading situation [1].
Table 5 - Summary for Different Cases ( 0 = No & 1 = Yes )
Case no. 1 2 3 4 5 6 7 8 9
Considering harmonics (THDmiBl = 5 %) 0 1 0 1 0 1 0 1 1
Considering the mutual coupling 0 0 1 1 0 0 1 1 1
Considering unbalance (5%) 0 0 0 0 1 1 1 1 1
Table 6 - Results summary
Accounted parameters Power losses (%) Benefits (%)
Harmonics (TIIDmc: = 5 %) Increased Decreased
Mutual coupling without harmonics Decreased Increased
Mutual coupling with harmonics Increased Decreased
Unbalance (5%) Increased Decreased
Table 7 - Capacitive kvar required for different cases
Cases 1 2 3 4 5 6 7 8
Qc 0 450 0 450 450 150 0 0
Capacitor 02 1200 450 450 450 1500 150 1350 0
bank 03 900 750 1350 300 1350 1650 600 1350
placement Ql 2400 1950 2400 1500 2400 1050 2700 1800
at 05 1200 1200 750 1350 0 1200 450 900
each Ql 0 1650 900 600 750 900 0 450
phase 07 450 0 450 0 0 600 750 0
Ô8C 450 750 450 450 750 1350 900 450
Ql 600 0 600 1500 1200 300 450 1650
Total capacitor sizes (3-phase) 21600 21600 22050 19800 25200 22050 21600 19800
Table 8 - Results when ignoring harmonics (BCP - before capacitor placement
& ACP - after capacitor placement'
ID case 1 case 3 case 5 case 7
BCP ACP BCP ACP BCP ACP BCP ACP
' '„„„ (3-phase, p.u.) 0.8032 0.9011 0.69281 0.9001 0.79022 0.90014 0.7036 0.90016
' max (3-phase, p.u.) 0.9794 0.9894 0.99412 0.9968 0.98053 0.99247 0.9927 0.99542
Ploss (3-phase, kW) 3636.8 2880.1 3591 2826.8 3651.4 2910.1 3555 2825.3
Total costs (3-phase, $/year) 610987 487880 603290 479499 613429 494060 597232 479301
Benefits (3-phase, $/year) 123107 123791 119369 117931
Benefits (%) 20.149 20.519 19.459 19.746
Table 9 - Results when considering harmonics (BCP - before capacitor placement _& ACP - after capacitor placement)___
ID case 2 case 4 case 6 case 8 case 9
BCP ACP BCP ACP BCP ACP BCP ACP ACP
' mm (3-phase, p.u.) 0.8035 0.9012 0.6935 0.9173 0.79046 0.9003 0.70435 0.9269 0.8883
' max (3-phase, p.u.) 0.98 0.9899 0.9948 0.9973 0.98113 0.9921 0.99328 0.9959 0.996
THDmax (3-phase, %) 3.4821 4.9285 5.5297 4.9953 3.4855 4.9582 5.5726 4.9878 6.018
Ploss (3-phase, kW) 3639.3 2924.5 3609.1 2990.7 3653.9 2958.9 3571.5 3011.3 2922.1
Total costs (3-phase, $/year) 611407 496046 606320 506823 613849 501994 600006 510122 494936
Benefits (3-phase, $/year) 115361 99497 111855 89885 105070
Benefits (%) 18.868 16.41 18.222 14.981 17.512
From the simulation results showed in Tables 6 to 9 it follows.
1. When harmonic frequencies are considered (maximum THD limit is equal to 5%) the power losses increased and benefits decreased (cases 1, 3, 5 and 7).
2. When mutual coupling is considered and ignoring the harmonic frequencies, the power losses decreased and the benefits (%) increased (cases 1 and 5). When mutual coupling is considered and harmonic frequencies are ignored the power losses decreased and benefits (%) increased (cases 1 and 5). When mutual coupling is considered and harmonic frequencies are considered (maximum THD limit is equal to 5%) the power losses increased and benefits decreased (cases 2 and 6).
3. When load unbalance is considered the power losses increased and benefits decreased (cases 1, 2, 3 and 4).
From comparing simulation results for case 1 and case 9 may be concluded.
1. Power losses and benefits ($/year) in case 9 less than in case 1.
2. Minimum rms voltage in case 9 less than minimum rms voltage limit.
3. THD in case 9 more than maximum THD limit.
Conclusions
The necessity considering harmonics, mutual coupling and load unbalance in capacitor placement problem modeling was investigated in this paper. Test results indicated that ignoring supply harmonics, load unbalance and mutual coupling can cause power quality degradation (for example, in case 9 THl)m .: =6% and Vmm =0.89p.u.). A binary particle swarm optimization was used for discrete optimization of capacitor placement. The objective function was to minimize the total annual costs and power losses due to capacitor placement with constraints including limits on voltages, total harmonic distortion and sizes of installed capacitors. Future work will involve more realistic representation of large-scale distribution system with time-varying harmonics.
References
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Рецензент: Ю.Л. Саенко д-р техн. наук, проф., ПГТУ
Статья поступила 09.02.2009
