A Multi-Objective Bees Algorithm for Multi-Objective OPF - S. Anantasate and P. Bhasaputra [2011]

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

The 8th Electrical Engineering/ Electronics, Computer, Telecommunications and Information Technology (ECTI) Association of Thailand - Conference 2011

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


(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



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.


10

2 1000

Fuel cost{$'hr)


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

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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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A Multi-Objective Bees Algorithm for Multi-Objective OPF - S. Anantasate and P. Bhasaputra [2011]