-
Electrical Power Systems Power Engineering and Power System
Paper ID 1349
A Multi-objective Bees Algorithm for Multi-objective Optimal
Power Flow Problem
Sumetha Anantasate and Pornrapeepat Bhasaputra
Department of Electrical and Computer Engineering, Thammasart
University, 99 Moo 18 Pahonyotin Rd., Pathumthani, 12120, Thailand.
Email: [email protected]
ABSTRACT This paper proposes a new application of bees
algorithm (BA). In this application, the multi-objective bees
algorithm (MOB A) is developed for solving the multi-objective
optimal power flow (MOOPF) problems on small and large scale power
system. The objectives of MOOPF are to minimize the total fuel cost
of generation, environmental pollution and power system loss by
considering many constraints i.e. limits on generator real and
reactive power outputs, bus voltages, transformer tap-setting and
power flow of transmission lines. Proposed crowded selection
mechanism and fuzzy mechanism techniques are added into selection
process of proposed MOBA. Also, the algorithm based on fuzzy set
theory is used to extract the best compromise solution. In order to
prove the proposed MOBA method, the simulation result of weight
aggregation is used for comparison. The standard IEEE-30 and
IEEE-1I8 bus system is selected to test the algorithm. The result
shows that Pareto front of proposed MOB A are diverse and well
distributed. The proposed MOB A have good performance and
efficiency to solve multi-objective optimal power flow.
Keywords: Bees Algorithm, Multi-objective Optimal Power Flow,
Multi-objective Bee Algorithm
1. INTRODUCTION Optimal power flow (OPF) is important problem
for
power system engineering. Its objective is to find optimal
solution of power flow subjected to various constraints. Many OPF
studies has dealt with single objective cases and applied solution
techniques based on traditional optimization methods [1-3].
However, due to the fact that real life problems involve several
objectives and the decision maker would like to find solution,
which gives compromise between the selected objectives.
Traditionally, MOOPF was treated as a single objective optimization
problem. The objective function will be formed as a weighted sum of
all objectives using suitable scaling/weighting factors. This
approach has the disadvantage of finding only a single solution and
it cannot find trade-off between the different objectives in single
run [4]. Generating multiple solutions using this approach requires
several runs with different weighting factors and hence elongates
the running time [5]. As an alternative to this approach, recent
studies consider the OPF as a true multi-objective optimization
problem in which the objectives are treated simultaneously and
independently [5-9]. This, however, makes the problem more
complicated, whereas, traditional optimization techniques have
several weakness and drawbacks such as: linearization, continuity,
differentiability, local optimal and constraints handling.
Therefore, new optimization techniques such as genetic algorithms
(GA), particle swarm optimization (PSO), and differential evolution
(DE) are recently introduced and also applied in the field of power
systems with promising success [7-14,16].
This paper presents a MOBA with proposed crowded
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selection mechanism and fuzzy mechanism to solve the MOOPF
problem on both small and large scale power system with many
constrains. The MOOPF problem is formulated as a nonlinear
constrained multi-objective optimization problem where the fuel
cost, emission and loss are treated as competing objectives.
Proposed crowded selection mechanism is also applied to manage the
size and diversity of the Pareto set. The proposed MOBA has been
tested on IEEE 30 bus and IEEE 118 bus standard system to
demonstrate its effectiveness and performance.
2. PROBLEM STATEMENT The MOOPF problem is to minimize several
objective
functions with several equality and inequality constraints. It
can be written as follows.
2.1 Problem objectives of multi-objective optimal power flow
Aggregating the objectives and constraints, the problem can be
mathematically formulated as a nonlinear constraint multi-objective
optimization problem as follows.
Minimize [f(x,u),e(x,u)'osJx,u)] (I) Subject to:
g(x,u)=O (2) h(x,u)s,O (3)
where f(x,u),e(x,u)'osJx,u) are objective functions of fuel
cost, environment and total real power loss respectively. g(x,u) is
the equality constraints , h(x,u) is the system
inequality constraints. u is the vector of control variables
consisting of real power outputs Pc; except at slack bus, generator
voltage Vc; and transformer tap settings T. N is the number of
generators. NT is the number of regulating transformers. Hence, u
can be expressed as:
T u =[PG2,oo"PGN,VGl,oo"VGN,7j,oo.,TNr] (4)
x is the vector of state variables consisting of slack
bus real power Pc;], load bus voltage VI"' generator reactive
power outputs Qc;, and transmission line loading Sf. Therefore, x
can be expressed as:
xT = [Pc']' "], ... , "Nl ,Qc;], .. ,QC;N' Sf]'" "S/V1] (5)
where N, is the number of load buses. N L is the number of
transmission lines.
2.1.1 Objective of fuel cost The total US$/h fuel cost f(x,u)
can be expressed as:
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N f(x,u) = Iii i=1
where ii is the fuel cost of the ith generator.
(6)
The fuel cost curve is modeled by quadratic function as:
f, = a; +b;Pc;; +c;Pc5; (7) where ai' bi and C i are the cost
coefficients of the /h
generator, and PGi is the real power output of the /h
generator.
2.1.2 Objective of emission The environmental pollutants such as
sulphur oxides
SOx and nitrogen oxides NO, caused by fossil-fueled units can be
modeled separately. However, for comparison purposes, the total
ton/h emission e( x, u) of these pollutants can be expressed
as:
N e(x,u) = Iei
i=1 where ei is the emission of the i1h generator.
(8)
The emission curve is modeled by quadratic function as:
ei = 1O-2(ai + PiPGi + riPei) + i exp(AiPGi) (9) where ai' Pi'
Yi, i and Ai are coefficients of the /h generator emission
characteristics.
2.1. 3 Objective of total real power loss The total real power
loss in transmission lines can be
calculated as: Nr
oss = L gk [2 + 2 - 2Vf cos( - eJ) ] (10) k=l gk is the
conductance of the eh line that connects bus i to bus
j . V, and V) are the voltage magnitudes at bus i and
j . 0; and Of
are the voltage angles at bus i and j .
2.2 Problem constraints
2.2.1 Equality constraints Power balance is equality constraint.
The total power
generation must cover the total demand PD and total real
power
loss in transmission lines ()SS Hence, N
L Pc;; - ) - ()SS = 0 (11 ) ;=1
As a matter of fact, the power loss in transmission lines can be
calculated by different methods such as B matrix loss formula
method and power flow method. The second method has
been adopted in our implementation where calculation of ()SS
implies solving the load flow problem with equality constraints on
real and reactive power at each bus as follows.
NE PGi - PDi - V; I Vj [cij cOS(Oi - OJ) + Bij sin(Oi - OJ )]= 0
j=l
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(12)
Electrical Power Systems Power Engineering and Power System
HB QGi - QDi - V, I Vj [cij sine 0i - OJ) - Bij cos( 0i - OJ)] =
0 j=l
(13) where Pc;; , Qc;; are real and reactive power generated at
the /h PDi, QDi are demand real and reactive power generated at
the /h bus. Gij and Bij are the transfer conductance and
b th b d th
b N h b susceptance etween 1 us an ] us . B IS t e num er of
system buses.
2.2.2 Inequality constraints (1) Generation constraints
For stable operation, generator voltages, real power outputs and
reactive power outputs and reactive power outputs are restricted by
the lower and upper limits as follows: TTmin < v. < TTmax . 1
N r Gz - Gi - r Gz , 1 = , ...... , Dmin < p < Dmax . 1 N TGz
- Gi - TGz , 1 = , ...... , Qmin < Q < Qmax . = 1 N Gz - Gi -
Gz , 1 , ...... , where :;; are the voltage magnitudes at generator
bus
(2) Transformer constraints:
(14)
(IS)
(16)
Transformer tap settings are restricted by the minimum and
maximum limits as follows:
1jmin :::; 1j :::; 1jmax , i = 1, ...... , NT (17) where NTis
the number of regulating transformers.
(3) Security constraints Theses incorporate the constraints of
voltage
magnitudes of load as well as transmission line loadings as
follows:
Vrnin <
V <
Vrnax
. - 1 N Li - Li - Li , I - , , I
S < smax . -1 N I - I , I - , ...... , L I I
(18)
(19)
where N, ,N '"
respectively.
is the number of buses and transmission
3. PROPOSED MOBA TECHNIQUE
The computation flow chart of the proposed MOBA method is
depicted in Fig. 1. Upon having the Pareto-optimal solution,
fuzzy-based mechanism [6-ll] to extract the best compromise
solution is imposed to present one solution to decision maker.
Fig.2 shows mechanism of MOB A for two objectives. MOBA will move
continually close to Pareto front in next iteration.
4. EXPERIMENTAL RESULTS AND DISCUSSION
The proposed approach is tested on IEEE 30 bus, 6 generators, 41
branch and IEEE 118 bus, 54 generators, 146 branch systems. The
system data and limit of voltage, transformer taps and transmission
line loading of IEEE 30 bus is taken from [15]. The IEEE 30 bus has
a total of 15 control variables as follow: Five unit active power
outputs, six generator bus voltage magnitudes, four transformer tap
settings. They are treated as continuous controls. The system data
of IEEE ll8 bus is taken from [17].The lower voltage magnitude
limits at all buses are 0.95 p.u. and the upper limits are 1.1
p.u.
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transmission line loading can be found in [18].
I
Initialize bees population Set parameter
iteration counter
I
Objective function evaluation I
I Search for nondominated solutions from initial population I
Set selected bees site set = non dominated set I Determine elite
nondominated bees using fuzzy mechanism I I and other selected
nondominated bees I
Iteration = iteration+1 I I Generate recruit bees around elite
nand aminated bees and I "------------' other selected non
dominated bees I
+ Evaluate the objective function of union set
Updating nondominated set I I Acquire Pareto set using crowding
distance and crowded I selection mechanism
t
I Assign scout bees to search new potential solution Form
population of bees for next iteration No :-:,:i>
'-----------C cc;Tiet? Yes
Stop
Fig.1 Computation flow chart of the proposed MOB A technique
C'\;.- recruited bees /P
\ \ }hec selected bees ___ _
;f/=/ .,;; _,o
\ \."- ".,
,, best compromise elite bee / I \ i V- recruited bees // ;' \_
, .... .. t----- / // .. /' "" . (J \ \ :?' ." --'. ,/ " :'- other
selected bees
""'-. ', .......... : ./). .. -.. selected bees on Pareto front
in first iteration [! _ - recruited bees
selected bees on Pareto front in next iteration Pareto front
solution
L-____________________________________________ --+
Fig.2 Mechanism of proposed MOBA technique
I
In this paper, the IEEE 118 bus has a total of 53 control
variables as fifty three unit active power outputs and they are
treated as continuous controls. In order to demonstrate efficiency
of MOBA, Bees algorithm will be verified to solve single and
multi-objective optimal power flow on small and large scale power
system in many cases as follow.
4.1 MOBA for multi-objective optimal power flow on IEEE 30
bus
The parameter of MOBA with following setup: initial population
of bees = 40, the number of bees around el ite bee = 8, the number
of bees around other selected bees = 4, the number of scout bees
=20, patch size = 0.1, maximum iteration = 100. Crowded setting
value = 0.04. The maximum size of the Paretooptimal front = 50
solutions. In order to show performance of
The 8th Electrical Engineering/ Electronics, Computer,
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Electrical Power Systems Power Engineering and Power System
MOBA, it will be compared with weight aggregation method (WA)
[14]. In this paper, all the objectives are combined into one
objective by using appropriated weight with minimization of bees
algorithm. To generate the trade-off surface by weighted
aggregation method, bees algorithm has run 20 times with different
values of weight in the range [0, 1]. Case 1.1: Fuel cost and
emission minimization
Fig. 3 shows distribution of the Pareto optimal solution for
fuel cost and emission minimization. The best compromise solution
obtains by MOB A is 836.97 $/hr and 0.2441 ton/hr, which shows 7.24
% reduction in fuel cost and 1.24% increment in emission from base
case. Minimum point of fuel cost 803.221 $/hr and emission 0.2058
ton/hr of MOBA method are very close to the optimized values with
single objective approach. The results and corresponding control
settings are shown in Table I. Case 1.2: Fuel cost and loss
minimization
Fig. 4 shows the Pareto optimal front obtained with fuel cost
and loss minimization. The best compromise solution obtains in
848.228 $/hr and 5.2061 MW, which shows 5.99% reduction in fuel
cost and 12.50% reduction in loss from base case. Minimum point of
fuel cost 802.869 $/hr and loss 3.300 MW of MOB A method are almost
close to the optimized values using single objective optimization.
The results and corresponding control settings are shown in Table
1. CaseI.3: Emission and loss minimization
Fig. 5 shows the Pareto optimal front obtained with emission and
loss minimization. The best compromise solution obtains in 0.20556
ton/hr and 3.3556 MW, which shows 14.22% reduction in emission and
43.60% reduction in loss from base case. Minimum point of emission
0.20517 ton/hr and loss 3.2656 MW of MOBA method are very close to
the optimized values using single objective optimization. Further,
the best compromise solution of MOBA is also close to the optimized
values using single objective optimization. The results and
corresponding control settings are shown in Table I. Case 1.4: Fuel
cost, emission and loss minimization
With simultaneous minimization of all the objectives, the best
compromise solution of MOB A obtains that fuel cost is reduced by
0.56%, the transmission loss is reduced by 22.33 % and emission is
reduced by 9.26 % from the base case values. Minimum point of fuel
cost 802.71 $/hr, emission 0.2051 ton/hr and loss 3.464 MW of MOBA
method are close to the optimized values using single objective
optimization and it uses CPU time not much more than other case of
MOBA. Fig 6 shows the optimized Pareto optimal front of MOB A and
WA of all the objectives. The optimized objective function values
and corresponding control settings are shown in Table I.
From Fig. 3-6. it can be observed that the nondominated
solutions obtained by MOBA are diversed and well distributed over
the Pareto front. It shows that weight aggregation method does not
guarantee uniformly distributed solutions in the Pareto front and
it also uses CPU time more than MOBA. Further, all the nondominated
solutions cannot be obtained and some of the solution are
inferior.
4.2 MOBA for multi-objective optimal power flow on IEEE 118
bus
The large power system IEEE 118 bus is used to test MOB A method
for minimization of fuel cost and emission. The parameter of MOB A
with following setup: initial population of bees = 80, the number
of bees around elite bee = 10, the number of bees around other
selected bees = 5, the number of scout bees = 40, patch size = 0.1,
maximum iteration = 300. Crowded setting value = 0.04. The maximum
size of the Pareto-optimal front = 50 solutions.
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950 0 (a2ll5B4. 8441IE3) 8 o '3
o D o *
MOBA Best compromise solution Minimum point
-;::- 900 o o Weigth aggregation
.c
W u
0* o 11
o LL 850
'\ (a24413.II:lB 8Cll)-WA EIb(a24417. 836.8783) -MIHA
li1 (a364B.1Il3.221)
800L----------------=O------" 0.2 0.25 0.3 0.35 0.4
Emission(ton/hr) Fig 3. Pareto front of MOBA and WA with fuel
cost and
emission minimization
960 0 (3i ll[Lqili53S6
940 * 920
900 Ui 8 880 860 LL
840
820
o *
o JiO
o MOBA D Best compromise solution o Minimum point * Weigth
aggregation
ii (5D1271!l55 VIA ffib (520.1IIII.22Iln MOI!\ 0
,p 0 O * *
800 L- __ ____
____
__
__ 0
_0 __ (LOyO1Il2_.II-J)
3 4 5 6 7 8 9 10 Loss in MW
Fig 4. Pareto front of MOBA and WA with fuel cost and loss
minimization
Fig. 7 shows distribution of the Pareto optimal solution for
fuel cost and emission minimization. The best compromise solution
obtained by MOB A is 16,252.97 $/hr and 3.043 tonlhr. which shows
16.20 % reduction in fuel cost and 94.73% reduction in emission
from base case. Minimum point of fuel cost 16233.63 $lhr and
emission 2.7093 ton/hr of MOBA method are very close to the
optimized values with single objective approach. Table 2 shows the
control settings and two objective function values with
optimization of MOB A and W A on IEEE 118 bus system. The bold
values indicate the best compromise solution.
0.212
E 0.21 C-o 'iii . 0.208 w
0.206
o MOBA D o
*
*
(32BCli.02117IB) * 'D (333. 0201ll-WA \
* " " "
(35B.0 2IliCll)-MOI!\ (H2U2ll517)
*
Best compromise solution Minimum point Weigth aggregation
* * *
*
*
*
"
0.204 L-__ ____ ____ ____ ____ ____ ---i 3.2 3.4 3.6 3.8 4 4.2
4.4
Loss in MW
Fig 5. Pareto front of MOB A and WA with emission and loss
minimization
It's so difficult to tind out optimized solution with
multi-objective solution for large power system. However, MOB A can
search the best compromise solution with fuel cost and emission
from Pareto front. From Fig. 7, it shows that the non-dominated
solutions obtained by MOB A are distributed over the Pareto
front.
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10
2 1000
Fuel cost{$'hr)
Electrical Power Systems Power Engineering and Power System
850 800 0.2
0.25 0.3
Ernission(bnlhr)
0.4
Fig 6. Pareto front of MOB A and WA in three objectives with
fuel cost. emission and loss minimization
Table 1. Optimization results of MOBA and W A based OPF of IEEE
30 bus for two and three objectives
Control varillble
PG4(MW)
VG2(p.U.)
VG5(p.U.)
Tn
Tn
Fuel Cost (S/hr)
Emission (ton/hr)
Transmiss ion losses
(MW) CPU time
(sef,:)
elise 1.1
MOBA WA
61.579 4 5Um
27 .8376 27 . 406(,
,,3JJ702 3 4 ,34255)
o Minimum point * Weigth aggregation
1.66
1.64 'Olli 164i1l1!115) lA *
'0 *
*
'\h (3043.1S25297IB) -MIHA
Three objel1ives
Clisel.4
MOBA WA
6 4 . 408 5 4 . 255
31.12 2 4('.908
31,37 4 .; 4 .56 2
2 S ,379 2RJ179
3 2 . 209 33 .148
1.023 1.012
0.975 UXl 4
0,99 4 09 S7
I .OO!) 1 (EO
0.971 0.968
1.031 UX12
0.910 1.08 2
IJJ7 4 09 46
UJ66 1,040
1.009 0.979
856.7221 897.2797
0.2303 0.2175
5.6436 4.6211
141.6 93 4 , 2
1 .62 L- _____ "--'-l'IlDo=(3"'2B4C'7"'. IB=Z3i3.61""27",) __
______ ____ ----.i 2 3 4 5
Emission(ton/hr) 6 7
Fig 7. Pareto front of MOB A with fuel cost and emission
Page 855
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Table 2. Optimization results of BA and MOB A based OPF of IEEE
lIS bus
Control variahle Base case Two objective
WA MOBA
PG1(MW) 5.000n 25.0445 26.0993 PG3(MW) [9.0000 28.6617 20.8682
PG4(MW) 14.0000 [6.6833 24.1961 PG5(MW) [7.0000 14.2081 25.8346
PG6(MW) 185.0()OO 151.7827 [55.9767 PG7(MW) 207.0000 114.7857 [
17.2121 PG8(MW) [ 1.0000 29.6550 27.2357 PG9(MW) 55.0000 58.3293
91.1048 PGu(MW) IS.OOOO [2.8375 2:J.O]I] PGu(MW) [8.0000 28.8219
24.9476 PG12(MW) 206.0()OO 105.2:J71 [04.0977 P(;n(MW) 184.0000
116.8009 113.9675 P(;u(MW) 17.0000 22.8642 14.7711 PC1;;(MW) 8.0000
26.0805 29.6589 P(;u(MW) 65.0000 68.2292 75.6887 P(;17(MW) 18.0000
23.0656 29.1121 P(;ls(MW) 36.0000 73.5878 78.5419 P(;n(MW) 14.0000
26.7466 28.4430 P(;ll(MW) 22.0000 28.7399 27.6930 P(;ll(MW) 62.0000
71.4399 64.7255 P(;12(MW) 147.0000 107.1828 87.5004 P(;lJ(MW)
118.0000 14Ul773 108.0801 P(;u(MW) 34.0000 99.9643 81.1743
P(;l;;(MW) 30.0000 87.3179 76.8545 PG2.MW) 94.0000 129.4397
[61.1776 PG27(MW) 59.0000 85.1664 99.8662 PG28(MW) 35.0000 98.7168
90.8175 PGH(MW) [47.0000 112.4144 [ 1O.838[ PGJ1(MW) 2l:J.OOOO
134.9295 139.6653 PGJ1(MW) 59.0000 67.7884 49.0584 PGJ2(MW) [5.0000
23.2944 28.8848 PGJJ(MW) [0.0000 7.1937 15.5798 PGJ4(MW) [2.0000
[5.7448 18.9486 PGJs(MW) 44.0000 85.8022 68.6155 PGJ.(MW) 72.0000
92.9233 87.6878 PGJ7(MW) [60.0000 151.3263 [53.3762 P(;3s(MW)
2UlOOO 23.2848 23.3394 P(;3,(MW) 129.00I.lO 122.8043 125.1085
P(;4u(MW) 253.001.lO 169.7999 167.0270 P(;41(MW) 13.l.lOOO 19.2624
13.2810 P(;42(MW) 3UlOOO 38.1.lO35 49.0079 P(;4J(MW) 118.001.lO
101.2031 101.8323 P(;4t(MW) 135.00I.lO 158.8497 167.0281 P(;4;;(MW)
128.00I.lO 100.3756 103.6806 P(;4,(MW) 8.OI.lOO 10.8291 14.2857
P(;47(MW) 57.1.lO00 74.(l397 56.1425 P(;4s(MW) 66.1.lO00 58.5912
60.2343 P(;4,(MW) lUlOOO 16.5566 17.7541 P(;51(MW) 26.1.lO00
35.1045 29.2784 PG51MW) 45.0000 77.6158 64,4491 PG52(MW) 63.0000
43.0695 74.3375 PG5J(MW) 54.0000 645138 80,4152 PG54(MW) 28.0000
44.9864 38,4692
Fuel cost(S/hr) [9)95.12 16,482.89 16,252.97 Emissiou(tou/hr)
57.7688 2.9005 3.0430
CPU timc(scc) 450.2 2.102.5
5. CONCLUSION The paper has employed proposed MOBA based
optimization approach to solve the MOOPF problem with many
constraints in IEEE 30 and IEEE lIS bus system. Crowded selection
mechanism is also added to manage the size and to get good
diversity of the Pareto front. Fuzzy mechanism is combined for
selection process in this algorithm. The comparison of MOB A and WA
shows that MOBA is capable of giving good Pareto front and
maintains diversity. Fuzzy set theory is used to extract the best
compromise solution from Pareto front. The proposed approach has
performance to solve both small and large scale power system for
multiobjective optimization problem. In the future. efforts will be
made to incorporate with many objective functions to the problem
structure will be attempted by the proposed methodology.
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Electrical Power Systems Power Engineering and Power System
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