A new honey bee mating optimization algorithm for non-smooth economic dispatch

Energy 36 (2011) 896e908

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A new honey bee mating optimization algorithm for non-smooth economic dispatch Taher Niknam a, Hasan Doagou Mojarrad a, Hamed Zeinoddini Meymand a, Bahman Bahmani Firouzi b, * a b

Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, P.O. 71555-313, Islamic Republic of Iran Islamic Azad University, Marvdasht Branch, Marvdasht, Islamic Republic of Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 May 2010 Received in revised form 10 December 2010 Accepted 12 December 2010 Available online 26 January 2011

The non-storage characteristics of electricity and the increasing fuel costs worldwide call for the need to operate the systems more economically. Economic dispatch (ED) is one of the most important optimization problems in power systems. ED has the objective of dividing the power demand among the online generators economically while satisfying various constraints. The importance of economic dispatch is to get maximum usable power using minimum resources. To solve the static ED problem, honey bee mating algorithm (HBMO) can be used. The basic disadvantage of the original HBMO algorithm is the fact that it may miss the optimum and provide a near optimum solution in a limited runtime period. In order to avoid this shortcoming, we propose a new method that improves the mating process of HBMO and also, combines the improved HBMO with a Chaotic Local Search (CLS) called Chaotic Improved Honey Bee Mating Optimization (CIHBMO). The proposed algorithm is used to solve ED problems taking into account the nonlinear generator characteristics such as prohibited operation zones, multi-fuel and valvepoint loading effects. The CIHBMO algorithm is tested on three test systems and compared with other methods in the literature. Results have shown that the proposed method is efficient and fast for ED problems with non-smooth and non-continuous fuel cost functions. Moreover, the optimal power dispatch obtained by the algorithm is superior to previous reported results.
2010 Elsevier Ltd. All rights reserved.

Keywords: Economic dispatch Chaotic local search Improved honey bee mating optimization Multi-fuel effect

1. Introduction Optimization problems are widely encountered in various fields in science and technology. Sometimes such problems can be very complex because of the actual and practical nature of the objective function or the model constraints. A typical optimization problem minimizes or maximizes an objective function subject to complex and nonlinear characteristics with heavy equality and/or inequality constraints. Economic dispatch is a computational process, where the total required generation is distributed among the generation units in operation by minimizing the selected cost criterion, while satisfying the total demand and the transmission losses and operational constraints. Economic dispatch is used in real-time energy management power system control by most programs to allocate the total generation among the available units, unit commitment and in some other operation functions. Many different

* Corresponding author. E-mail addresses: [email protected], [email protected] (T. Niknam), [email protected] (H.D. Mojarrad), [email protected] (B.B. Firouzi). 0360-5442/$ e see front matter
2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.12.021

mathematical techniques such as gradient method, linear programming algorithm, lambda iteration method, quadratic programming, nonlinear programming algorithm, Lagrange relaxation algorithm, etc., have been employed for solving the ED problems. However, for effective implementation of these methods, the formulation needs to be continuous. The basic ED considers the power balance constraint apart from the generating capacity limits. However, a practical ED must take prohibited operating zones, valve-point loading effects, and multi-fuel options [1e4] into consideration to provide the completeness for the ED problem formulation. The resulting ED is a non-convex optimization problem, which is a challenging one and cannot be solved by traditional methods. Taking the valve-point effects and multiple fuels into account will increase the number of local minimum points in the fuel cost function and make the problem of finding the global optimum more difficult. Evolutionary algorithms (EA) are search methods inspired by natural selection and survival of the fittest in the biological world and behavior of social insects such as ants and bees. With the development of computer science and technology, many evolutionary algorithms such as the genetic algorithm (GA) [5],

T. Niknam et al. / Energy 36 (2011) 896e908

particle swarm optimization (PSO) [6], simulated annealing (SA) and tabu search (TS) [7] were proposed to solve the economic dispatch problem, which resulted in optimal power system unit’s generation scheduling. One of the recently proposed evolutionary algorithms that have shown great potential and good perspective for the solution of various optimization problems is honey bee mating optimization (HBMO) [8e17]. The HBMO algorithm has been used to solve a single reservoir optimization problem [10,12], data mining [11,13], state estimation in distribution networks [14], multi-objective distribution feeder reconfiguration [15,16]. Original HBMO often converges to local optima. In order to avoid this shortcoming, in this paper, we propose a new HBMO algorithm to solve ED problems. In the proposed algorithm first the original HBMO is combined with a chaotic local search (CLS) and then a new

897

fuel cost function of the generating units with valve-point loadings is represented as follows:

       FCj Pj ¼ aj Pj2 þ bj Pj þ cj þ ej sin fj Pjmin  Pj 

where, ej, fj are non-smooth fuel cost coefficients of generator ‘j’. Pjmin is the minimum power generation limit of generator ‘j’. Also, there are usually many generating units supplied with multiple fuels. In this case, unlike the conventional cost function, the cost function of each unit should be presented with a few piecewise functions reflecting the effect of fuel type changes. The piecewise quadratic cost function considering multi-fuel is shown in Fig. 1. The fuel cost function for the jth unit reflecting the multiple fuel option and valve-point loading is as follows:

    8  2 þb P aj;1 Pj;1  if Pj;min  Pj  Pj;1 ; Fuel Type 1 > j;1 j;1 þ cj;1 þ ej;1 sin fj;1 Pjmin  Pj > >    <     2 aj;2 Pj;2 þ bj;2 Pj;2 þ cj;2 þ ej;1 sin fj;1 Pjmin  Pj  if Pj;1  Pj  Pj;2 ; Fuel Type 2 FCj Pj ¼ > > /   >   :  2 aj;Nft Pj;N þ bj;Nft Pj;Nft þ cj;Nft þ ej;Nft sin fj;Nft Pjmin  Pj  if Pj;Nft 1  Pj  Pj;max ; Fuel Type Nft ft

method is proposed to improve the mating processing. The proposed approach is called the chaotic improved honey-bee mating optimization (CIHBMO). The main contributions of this paper are: 1. presenting an improved version of the honey bee mating optimization (HBMO) algorithm 2. combining IHBMO with the chaos local search 3. applying the proposed approach to the ED problem with nonsmooth cost functions. The paper is organized as follows: Section 2 presents the ED formulation. Section 3 contains a brief overview of the original HBMO and proposes the CIHBMO algorithm. Section 4 describes how the CIHBMO algorithm can be used to solve the ED problem. Section 5 reports different experimental results and compares the proposed algorithm with other methodologies.

The classic ED minimizes the following fuel cost function associated with dispatchable units: Ng Ng   P P Min FðXÞ ¼ FCj Pj ¼ aj Pj2 þ bj Pj þ cj j¼1 j¼1 i h X ¼ P1 ; P2 ; .; PNg

(3)

where, aj;Nft ; bj;Nft ; cj;Nft ; ej;Nft ; fj;Nft are fuel cost coefficients of generator ‘j’ for the fuel type Nft and Pj;max is the maximum power generation limit of generator ‘j’. In the ED problem, the following constraints should be satisfied.  Power balance constraint: Ng X

Pj ¼ PD þ PLoss

(4)

j¼1

where, PD is the total load of consumers and PLoss is the total transmission network losses. Loss of transmission network is expressed as a quadratic function of the generators’ power outputs as shown in (5).

PLoss ¼

Ng X Ng X i¼1 j¼1

2. Economic dispatch formulation

(2)

Pi Bij Pj þ

Ng X

B0i Pi þ B00

(5)

i¼1

where, Bij is the ijth element of the loss coefficient square matrix. B0i and B00 are the ith element of the loss coefficient vector and the loss coefficient constant, respectively.

(1)

where, F(X) is the total generation cost. FCj(Pj) is the fuel cost function of generator ‘j’. Pj is electrical output power of generator ‘j’ and aj, bj, cj are fuel cost coefficients of generator ‘j’. Ng is the number of generators and X is the vector of control variables. Inclusion of valve-point loading effects makes the modeling of the fuel cost function of the generators more practical. A sinusoidal term is incorporated in (1) to model the fuel cost function more accurately and superpose a ripple like inputeoutput curve. This raises the non-linearity and increases the number of local optimum solutions in the search space. Also, the solution procedure can be easily trapped in local optimum solutions the optimal value. The

Fig. 1. Fuel cost function of a thermal generating unit supplied with multiple fuels.

898

T. Niknam et al. / Energy 36 (2011) 896e908

 Power output constraints:

3. The basic concept of the honey bee mating optimization

The electrical output power of each unit must be between its minimum and maximum values.

Pjmin  Pj  Pjmax

(6)

 Prohibited operating zone constraints: Faults in the generators themselves or in the associated auxiliaries such as boilers, feed pumps, etc., may cause instability in certain ranges of generator power output. Consequently, discontinuities are produced in cost curves corresponding to the prohibited operating zones. So, there is a quest to avoid operation in these zones in order to economize the production. The prohibited operating zones constitute the following constraint for ED:

8 >  Pj  PjLBz P > < jmin UBz1  Pj  PjLBz Pj ˛ Pj > > : P UBz  P  P j

j

j ¼ 1; 2; .; Ng

(7)

jmax

PjLBz

where and are PjUBz lower and upper bounds of the zth prohibited zone of unit j; z is the index of prohibited zones.

Honey bees are social insects that work together in a highly structured social order and create hives. The honey bee community consists of three structurally different forms: the queen, drones and workers. In the original HBMO algorithm a drone mates with a queen probabilistically using an annealing function as follows [8,9,17]:

  PrðDÞ ¼ exp  DðFÞ=Vqueen ðtÞ

where Pr(D) is the probability of adding the sperm of drone D to the spermatheca of the queen, DðFÞ is the absolute difference between the fitness of D and the fitness of the queen and Vqueen(t) is the velocity of the queen at time t. After each transition in space, the queen’s velocity decreases according to the following equations:

Vqueen ðt þ 1Þ ¼ a  Vqueen ðtÞ

Define the input data

Generate the initial population based on state variable and chaos initialization Sort the initial population based on the objective function values

Select the queen

Run CLS1 and CLS2 simultaneously

Yes

Is the new best solution better than the previous one?

No

Generate queen’s speed and also select the population of the drones from the sorted initial population.

Generate the queen's spermatheca matrix (Mating flight) by using the annealing function

Do the improved breeding process (Eq. 11)

Calculate the objective function value for the new generated solutions

Select the new population based on the new broods and initial population

No

(9)

where a is a positive real parameter called decreasing factor. It controls the amount of speed reduction after each transition and each step. It changes between zero and one.

Start

Substitute the new best solution with the previous solution

(8)

Is the termination criteria satisfied? Yes Stop and print results

Fig. 2. Flowchart of the proposed CIHBMO algorithm.

Keep the previous best solution

T. Niknam et al. / Energy 36 (2011) 896e908 Table 1 The parameters used in the proposed algorithm. Parameter NBrood NDreone NSperm Vmax Vmin

a

Table 3 Comparison of fuel costs for 10 generators for power demand of 2700 MW.

Explanation

Value

Number of queen Number of drones Number of broods Size of the queen’s spermatheca Velocity of queen at the start of a mating flight Velocity of queen at the end of a mating flight Decreasing factor

1 100 100 50 Randomly between [0.5,1] Randomly between [0,1] 0.98

3.1. Improved honey bee mating optimization In the original HBMO algorithm as shown in (10), the broods are generated by mating between the queen and one drone.

  XBrood;j ¼ Xqueen þ b  Xqueen  Di i h XBrood;j ¼ xbroodj;1 ; xbroodj;2 ; .; xbroodj;Ng

(10)

1Ng

where Di is the ith drone stored in the spermatheca and b˛½0; 1 is called mating factor. Traditional mating process combines the features of two parent structures to form two similar offsprings. Its purpose is the maintenance and exchange of queen’s place. But this cannot guarantee the convergence to the optimal point and sometimes causes premature convergence to local minima. In this paper, a new approach is proposed to improve the broods as described in the following: At first four drones ðXs;z1 ; Xs;z2 ; Xs;z3 ; Xs;z4 Þ among the sorted drones (in spermatheca) are selected randomly such that z1sz2sz3sz4. Then a changing drone position vector is calculated as:

i h Xchange ¼ x1change ; x2change ; .; xNg change

1Ng

¼ Xs;z1 þ rand1ð:Þ

    t  Xs;z4  Xs;z2  Xs;z3 þ rand2ð:Þ  Xqueen

( xtþ1 brood;j

¼

899

ð11Þ

j

xchange if ðrand3ð:Þ  rand4ð:ÞÞ or j ¼ randpermð:Þ xtqueen;j otherwise

(12) where, randpermð:Þ is a randomly chosen index from 1, 2, ., Ng. rand1(), rand2(), rand3() and rand4() are random numbers between zero and one. The utilization of improvements in the breeding process can be useful to escape more easily from local minima compared to the traditional mating.

Evolution method

Minimum cost

Maximum cost

Mean cost

CPU time (s)

CIHBMO CHBMO IHBMO HBMO CGA-MU [22] IGA-MU [22] PSO-LRS [22] DE [22] NPSO [22] NPSO-LRS [22] DSPSO-TSA [24] PSO [22] APSO [22] CPSO [25] PSO-GM [25] CBPSO-RVM [25] GA [24] TSA [24] TS [26] CCPSO [23] ARCGA [22] RGA [22] ED-DE [27] ACO [26] Proposed ACO [26]

623.59601 623.8905 623.7620 624.2547 624.7193 624.5178 624.2297 624.5146 624.1624 624.1273 623.8375 624.3045 624.0145 624.1715 624.3050 623.9588 624.5050 624.3078 624.01 623.8266 623.8281 624.5079 623.8790 623.90 623.70

623.6673 624.0748 623.9586 624.4829 627.6087 630.8705 628.3214 624.5458 627.4237 626.9981 623.8625 624.5054 627.3049 624.7844 625.0854 624.2930 624.7419 624.8285 624.51 623.8291 623.8550 624.5088 623.8894 624.35 623.90

623.6341 623.9876 623.8263 624.3962 633.8652 627.6087 625.7887 624.5246 625.2180 624.9985 623.9001 625.9252 624.8185 624.5493 624.6749 624.0816 624.8169 635.0623 624.96 623.8273 623.8431 624.5081 623.8807 624.78 624.09

2 2 1.4 1.4 26.64 7.32 0.88 3.3852 0.35 0.52 3.44 11.04 e e e e 18.37 9.71 18.51 3.2 0.85 4.1340 e 8.35 5.16

3.2. Chaotic honey bee mating optimization Honey Bee Mating Optimization (HBMO) has gained much attention and widespread applications in different fields. However, it often converges to local optima. Generally, a probabilistic method has a large possibility of exploring the search space freely in the beginning and slowly few valleys while the run is progressed. In order to avoid this shortcoming, we propose a chaotic Improved HBMO(IHBMO) method that combines IHBMO with chaotic local search (CLS). There are two CLS procedures. In the first CLS method, CLS is based on the logistic equation (CLS1), while the latter CLS is based on the Tent equation (CLS2) [18e20]. To apply the chaotic local search on the IHBMO algorithm, the following steps should be repeated: Step 1: Generate an initial chaos population randomly for CLS1 or CLS2 Step 2: Determine the chaotic variables Step 3: Mapping the decision variables Step 4: Convert the chaotic variables to the decision variables Step 5: Evaluate the new solution with decision variables

Table 2 Optimum dispatch result of the proposed algorithm for power demand of 2700 MW. Generator

Output power (MW)

Fuel type

Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Load (MW) Total cost ($)

218.105 211.6596 280.6571 239.6864 279.9344 239.661 287.7275 239.552 427.1485 275.8687 2700 623.59601

2 1 1 3 1 3 1 3 3 1 Fig. 3. Convergence behaviors of CIHBMO, CHBMO, IHBMO and original HBMO for 10-unit system.

900

T. Niknam et al. / Energy 36 (2011) 896e908 Table 4 Comparison of fuel costs for 10 generators for different power demand. Methods

Total generation cost ($/h)

Proposed CIHBMO DE RGA PSO ARCGA

Pload ¼ 2400 (MW)

Pload ¼ 2500 (MW)

Pload ¼ 2600 (MW)

481.73574

526.24671

574.39252

482.5114 482.5275 482.5088 481.7434

527.0189 527.0360 527.0185 526.2589

575.1610 575.1753 575.1606 574.4054

j

where, cxi indicates the jth chaotic variable. Nchaos is the number of individuals for CLS. rand(.) is a random number between 0 and 1. In CLS2, the Tent equation can be defined by the following equation:

i h Cxi ¼ cx1i ; cx2i ; .; cxNg ; i ¼ 0; 1; 2; .; Nchoas i 1Ng 8 j < 2cxj ; if 0 < cxi  0:5  i cxjiþ1 ¼ j ¼ 1; 2; .; Ng : 2 1  cxj ; if 0:5 < cxj  1 i i j

cx0 ¼ randð:Þ

ð14Þ

In the CLS1 and CLS2, the best solution (Queen’s place) is consid0 ) for CLS. X 0 is scaled into ½0; 1 ered as an initial solution (Xcls cls according to the following equation:

i h Ng 0 ¼ x1 Xcls ; x2 ; .; xcls;0 cls;0 cls;0 1Ng i h Ng Cx0 ¼ cx10 ; cx20 ; .; cx0 j cx0

¼

xjcls;0 Pj;min Pj;max Pj;min ;

(15)

j ¼ 1; 2; .; Ng

The chaos population for CLSs is generated as follows:

i h Ng i Xcls ¼ x1cls;i ; x2cls;i ; .; xcls;i ; i ¼ 1; 2; .; Nchoas 1Ng   j j xcls;i ¼ cxi1  Pj;max  Pj;min þ Pj;min ; j ¼ 1; 2; .; Ng Fig. 4. Distribution of generation cost of system test 1 for CIHBMO, CHBMO and IHBMO (20 trials).

(16)

The objective function is evaluated for all individuals of CLS. The best solution among them is replaced with one drone selected randomly. 4. Applying the CIHBMO to the ED problem

Step 6: Evaluate fitness function for all individuals of CLS1 or CLS2 Step 7: Replace the best solution among them with one drone selected randomly Step 8: Termination criteria.

In this section the proposed algorithm is applied to solve the economic dispatch problem. To apply the CIHBMO, the following steps have to be taken. Step 1: Define the input data

A more detailed description relating to theoretical and implementation aspects of the proposed chaotic local search can be shown as follows: The procedures of CLS1 and CLS2 are the same. The only difference is in the chaotic variables. In CLS1, the logistic equation is defined by the following equation:

i h Ng Cxi ¼ cx1i ; cx2i ; .; cxi ; i ¼ 0; 1; 2; .; Nchoas 1Ng   j j j cxiþ1 ¼ 4  cxi  1  cxi ; j ¼ 1; 2; .; Ng cxji ˛½0:1; cxj0 ;f0:25; 0:5; 0:75g j

cx0 ¼ randð:Þ

(13)

In this step, the input data including the cost coefficients of the generators, output generator constraints, transmission loss matrix coefficients and loads, the velocity of queen at the start of a mating flight (Vmax), the velocity of queen at the end of a mating flight (Vmin), the decreasing factor (a), the number of iterations (Itermax), the number of drones (NDrone), the size of the queen’s spermatheca (NSperm) and the number of broods (NBrood) are defined. Step 2: Generate the initial population In this step, an initial population based on state variables and chaos initialization is generated, randomly. It is formulated as

T. Niknam et al. / Energy 36 (2011) 896e908

901

Table 5 Optimum dispatch result of the proposed algorithm for different load demand. Pload (MW)

2400

Generator

Output power (MW)

Fuel type

Output power (MW)

2500 Fuel type

Output power (MW)

Fuel type

Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Total cost ($/h)

189.3021 202.252 254.4432 233.1022 240.4016 233.6143 252.1561 233.2366 320.435 241.0568 481.73574

1 1 1 3 1 3 1 3 1 1

206.8087 206.2129 265.5338 235.7896 258.3695 236.5704 268.7633 236.0584 332.0593 253.8341 526.24671

2 1 1 3 1 3 1 3 1 1

217.078 211.1644 278.6406 239.552 276.3377 238.9891 285.3565 239.552 343.6837 269.646 574.39252

2 1 1 3 1 3 1 3 1 1

3 X1 6 X2 7 7 X initial population ¼ 6 4 . 5 XNinitial i h Ng X0 ¼ x10 ; x20 ; .; x0 ;   xj0 ¼ randð:Þ  Pj;max  Pj;min þ Pj;min ; j ¼ 1; 2; .; Ng i h Ng Xi ¼ x1i ; x2i ; .; xi ; i ¼ 1; 2; .; Ninitial   j j j xi ¼ 4  xi1 1  xi1 ; j ¼ 1; 2; .; Ng

2600

2

2

(17)

3 D1 6 D2 7 7 X Drone ¼ 6 4. 5 D NDrone i h N j ¼ 1; 2; .; NDrone Dj ¼ Pj1 ; Pj2 ; .; Pj g

(20)

where Dj is the jth drone. Step 8: Generate the queen’s spermatheca matrix (Mating flight)

where rand(.) is a random function generator between 0 and 1. Ninitial is the number of initial population. Step 3: Sort the initial population based on the objective function values The initial population is ascendingly sorted based on the value of the objective function. Step 4: Select the queen

At the start of the mating flight, the queen flies with her maximum speed. A drone is randomly selected from the population of drones. The mating probability is calculated based on the objective function values of the queen and the selected drone. A number between 0 and 1 is randomly generated and compared with the calculated probability. If it is less than the calculated probability, the drone’s sperm is stored in the queen’s spermatheca and the queen speed is decreased. Otherwise, the queen speed is decreased and another drone from the population of drones is selected until the speed of the queen reaches its minimum or the queen’s spermatheca is full.

2

The individual that has the minimum objective function should be selected as the queen (Xqueen ).

Xqueen

i h ¼ x1queen ; x2queen ; .; xNg queen ;

(18)

3 QS1 6 QS2 7 7 X Queen Spermatheca ¼ 6 4. 5 QSNsperm h i Ng j ¼ 1; 2; .; Nsperm QSj ¼ qs1j ; qs2j ; .; qsj

(21)

where QSj is the jth individual in the queen’s spermatheca. Step 5: Run CLS1 and CLS2 simultaneously Step 9: Breeding process In this paper, the proposed algorithm is the hybridization of the IHBMO and both of the chaotic local search methods. The best 0 ) for CLSs. If solution (Xqueen ) is considered as an initial solution (Xcls the best solution found by chaotic local searches is better than the queen, it is considered as the queen.

In this step, a population of broods is generated based on mating between the queen and the drones stored in the queen’s spermatheca according to the proposed method (see Section 3.1). Step 10: Calculate the objective function value for the new generated solutions

Step 6: Generate the queen speed The queen velocity is as

Vqueen ¼ Vmax

(19)

Step 7: Select the population of drones The population of drones is selected from the sorted initial population as

Table 6 Prohibited zones of units for test system 2. Generator

Zone 1 (MW)

Zone 2 (MW)

Zone 3 (MW)

2 5 6 12

[185,225] [180,200] [230,255] [30,40]

[305,335] [305,335] [365,395] [55,65]

[420,450] [390,420] [430,455] e

902

T. Niknam et al. / Energy 36 (2011) 896e908

Table 7 Comparison of fuel costs for 15 generators for power demand of 2630 MW. Evolution method

Best cost ($)

Worst cost ($)

Average cost ($)

CPU time (s)

CIHBMO CHBMO IHBMO HBMO PSO [29] GA [29] ESO [30] DE [30] SPSO [28] PC_PSO [28] SOH-PSO [28,29] MTS [24] SA [31] SCA [31] APSO [32] CSO [31] CPSO [29] BF [29] MDE [29] TSA [24] DSPSO-TSA [24]

32548.585876 32555.7052 32552.4613 32637.6219 32858 33063.54 32640.86 32588.865 32798.69 32775.36 32751.39 32716.87 32786.4 32867.025 32742.77 32588.9189 32834 32784.5 32704.9 32917.87 32715.06

32548.585876 32589.7629 32554.6649 32676.07 33031 33337 32710 32641.419 e e 32945 32796.13 33028.95 33381.0607 e 32796.7792 33318 e 32711.5 33245.54 32730.39

32548.585876 32571.4438 32552.8961 32663.19 32989 33228 32620 32609.851 e e 32878 32767.4 32869.51 33138.3020 32976.6812 32679.8775 33021 32796.8 32708.1 33066.76 32724.63

3.1 3.1 2.8 2.8 26.59 49.31 e e 0.0913a 0.0967a 0.0936a 3.65 e 10.7 e 1.68 13.13 e e 25.75 2.30

a

Results obtained from reference.

The objective function is evaluated for each individual of the new generated solutions. If the new best solution is better than the queen, it replaces the queen. Step 11: Run CLS1 and CLS2 simultaneously In this step, the best solution (result of the previous step) is 0 ) for CLS1 and CLS2. considered as an initial solution (Xcls

performance of the proposed algorithm described in the previous sections using computer simulations. The proposed algorithm has been applied to ED problems in three different test cases. The first case study is a 10-generator system with combined valve-point loading effects and multi-fuel options without transmission loss. The second one is a 15-generator system with quadratic cost function, prohibited operation zones and transmission loss. The third test system consists of 40 generators with valve-point loading effects.

Step 12: Check the termination criteria If the termination criterion is satisfied, finish the algorithm; otherwise return to Step 3 until convergence criterion is met. Fig. 2 depicts the schematic representation of the proposed algorithm to solve the ED problem. 5. Numerical examples and results In order to verify the feasibility and effectiveness of the proposed method for solving ED problem, we assess the

Table 8 Optimum dispatch result of the proposed algorithm for power demand of 2630 MW. Generator

Output power (MW) (with loss)

Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Pg11 Pg12 Pg13 Pg14 Pg15 Ploss (MW) Pload (MW) Total cost ($)

454.9999 454.9999 130 130 234.2005 460 464.9999 60 25 30.9939 76.7014 79.9999 25 15 15 26.8959 2630 32548.585876

Output power (MW) (without loss) 455 455 130 130 271.5806 460 465 60 25 25 43.4193 55 25 15 15 e 2630 32256.754239

Table 9 Comparison of fuel costs for 40 generators for power demand of 10500 MW. Evolution method Minimum cost Maximum cost Mean cost

CPU time (s)

CIHBMO CHBMO IHBMO HBMO FAPSO-DE [38] FAMPSO [2] FAPSO-NM [36] FAPSO [36] PSO [29] MPSO [34,35] HGA [34.35] DEC-SQP [34,35] NPSO [33] PSO-SQP [34,35] DE [34,35] MDE [29] HDE [34,35] BF [29] IFEP [37] EP-SQP [30,31] SOH-PSO [29] ED-DE [27] PSO-GM [25] CBPSO-RVM [25] CPSO [25] PSO-LRS [22] NPSO-LRS [22] GA [29] APSO [22] TS [26] ACO [26] Proposed ACO [26]

16.8 16.8 15.2 15.2 31 37.4 40 87 152 e 84.21 14.26 e 733.97 72.94 e e e 1167.35 997.73 e e e e e e 16.81 320.31 5.05 238.35 92.54 52.45

121412.57 121639.38 121517.8 122841.4 121412.57 121412.57 121418.3 121712.4 123930.45 122252.27 121418.27 122174.16 121704.74 122094.67 121416.29 121414.79 121698.51 121423.63 122624.35 122323.97 121501.14 121412.50 121845.98 121555.32 121885.11 122035.7946 121664.4308 121996.40 121663.5222 122288.38 121811.37 121532.41

121412.63 121939.742 121711.8526 123654.751 121412.92 121415.78 121419.8 121873.17 124312.63 e e e 122995.09 e 121431.47 121466.04 e e 125740.63 e 122446.3 121517.80 123219.22 123094.98 123767.36 123461.6794 122981.5913 122919.77 122912.3958 122424.81 121930.58 121606.45

121412.5919 121851.7724 121589.1827 123371.6284 121412.628 121413.38794 121418.803 121778.246 124155 e 121784.04 122295.13 122221.37 122295.13 121422.72 121418.44 122304.30 121814.94 123382 122379.63 121853.57 121460.70 122398.38 122281.14 122469.64 122558.4565 122209.3185 123807.97 122153.6730 122590.89 122048.06 121679.64

T. Niknam et al. / Energy 36 (2011) 896e908

903

Table 10 Optimum dispatch result of the proposed algorithm and other methods for power demand of 10500 MW. Generator

Output power (MW) CIHBMO

Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Pg11 Pg12 Pg13 Pg14 Pg15 Pg16 Pg17 Pg18 Pg19 Pg20 Pg21 Pg22 Pg23 Pg24 Pg25 Pg26 Pg27 Pg28 Pg29 Pg30 Pg31 Pg32 Pg33 Pg34 Pg35 Pg36 Pg37 Pg38 Pg39 Pg40 Total cost($)

Minimum

Mean

Maximum

110.80182 110.80003 97.399913 179.7331 87.799785 140 259.59965 284.59966 284.59965 130 94 94 214.75979 394.27937 394.27937 394.27937 489.27937 489.27937 511.27937 511.27937 523.27937 523.28065 523.27937 523.27937 523.27937 523.27937 10 10 10 87.799891 190 190 190 164.80149 194.39276 200 110 110 110 511.27937 121412.57

110.8006 110.8233 97.39991 179.7331 87.79998 140 259.5997 284.5997 284.5997 130 94 94 214.7598 394.2794 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794 523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10 10 10 87.79966 190 190 190 164.8002 200 194.3734 110 110 110 511.2794 121412.5919

110.7982 110.803 97.39991 179.7331 87.83428 140 259.5997 284.5997 284.5997 130 94 94 214.7598 394.2804 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794 523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10 10 10 87.80023 190 190 190 164.8004 200 194.3599 110 110 110 511.2794 121412.63

SOH_PSO [28]

NPSO_LRS [37]

NPSO [37]

MDE [29]

CBPSO-RVM [25]

FAPSO-NM [36]

110.8 110.8 97.4 179.73 87.8 140 259.6 284.6 284.6 130 94 94 304.52 304.52 394.28 394.28 489.28 489.28 511.28 511.27 523.28 523.28 523.28 523.28 523.28 523.28 10 10 10 97 190 190 190 185.2 164.8 200 110 110 110 511.28 121501.14

113.9761 113.9986 97.4241 179.7327 89.6511 105.4044 259.7502 288.4534 284.646 204.812 168.8311 94 214.7663 394.2852 304.5187 394.2811 489.2807 489.2832 511.2845 511.3049 523.2916 523.2853 523.2797 523.2994 523.2865 523.2936 10 10.0001 10 89.0139 190 190 190 199.9998 165.1397 172.0275 110 110 93.0962 511.2996 121664.4308

113.9891 113.6334 97.55 180.0059 97 140 300 300 284.5797 130.0517 243.7131 169.0104 125 393.9662 304.7586 304.512 489.6024 489.6087 511.7903 511.2624 523.3274 523.2196 523.4707 523.0661 523.3978 523.2897 10.0208 10.0927 10.0621 88.9456 189.9951 190 190 165.9825 172.4153 191.2978 109.9893 109.9521 109.8733 511.5671 121704.7391

110.831 110.815 97.399 179.734 87.808 140 259.6 284.604 284.601 130 168.799 168.799 214.759 394.28 394.28 304.519 489.279 489.28 511.28 511.279 523.279 523.28 523.28 523.28 523.281 523.279 10 10 10 92.645 190 190 189.999 164.831 164.802 164.805 109.999 109.999 109.999 511.278 121414.79

114 114 97.4859 179.7331 97 140 300 300 286.0079 130 94 94 214.7598 304.5196 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794 523.2796 523.2794 523.2797 523.2802 523.2795 523.2794 10 10 10 97 190 190 190 200 166.8603 200 110 110 110 511.2794 121555.32

111.38 110.93 97.41 179.33 89.22 140 259.62 284.66 284.66 130 168.82 168.82 214.75 394.28 304.54 394.3 489.29 489.29 511.28 511.29 523.33 523.48 523.33 523.33 523.33 523.33 10 10 10 88.7 190 190 190 165 166 165 110 110 110 511.3 121418.3

To show the advantages of the proposed CIHBMO method, the results obtained from the proposed method are compared with those of other methods available in the literature. Also, since the proposed algorithm is the hybridization of the improved HBMO and chaotic methods, it is necessary to find the relative strength of each constituent. So we have implemented the following four different testing strategies applied to all test systems.

algorithm in each case study, 50 independent runs were made for each of the optimization methods involving 50 different initial trial solutions for each optimization method. The implementation of the proposed algorithm needs many parameters to be defined as shown in Table 1. The value of Itermax(maximum iteration) is varied depending on the problem dimension and solution space complexity.

(1) HBMO: The original HBMO. This strategy is selected to analyze the performance of chaotic and proposed method on breeding process in HBMO environment. (2) CHBMO: The chaotic HBMO is applied without the improved mating process. (3) IHBMO: improved HBMO is applied without chaotic local search. (4) CIHBMO: The proposed method is integrated with both of the chaotic local searches and the improved mating process.

5.1. Test system 1: 10-unit system

Each optimization method was implemented in MATLAB 7 programming language and executed on a Pentium IV, 3-GHz, 2-GB RAM processor. Due to the stochastic nature of the evolutionary

The proposed algorithm has been applied to the ED problem with 10 generators, where both multiple fuels and valve-point loading effects were considered. The input data are given in Table A1.1 in Appendix 1 [21]. The total load demand is 2700 MW and transmission line losses were not considered. In this test case, Itermax ¼ 100. The best power output and the fuel options for the 10 generators obtained by different strategies are given in Table 2. The computational results of the minimum, mean, maximum fuel costs and CPU time among the 50 runs of solutions satisfying the system constraints are listed in Table 3. For comparison, the computational

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T. Niknam et al. / Energy 36 (2011) 896e908

Table 11 Results obtained by the CIHBMO, CHBMO, IHBMO and HBMO algorithm for different number of runs. Evolutionary algorithm

Number of run

Minimum cost

Mean cost

Standard deviation (SD)

CIHBMO CHBMO IHBMO HBMO CIHBMO CHBMO IHBMO HBMO CIHBMO CHBMO IHBMO HBMO

30

121412.57 121705.5281 121608.4493 123116.89 121412.57 121681.92 121577.031 122979.78 121412.57 121639.38 121517.8 122841.4

121412.5911 121886.0482 121712.78 123591.16 121412.5913 121821.72 121662.701 123221.821 121412.5919 121851.7724 121589.1827 123371.6284

0.0213 9.371 5.56 22.1281 0.0216 6.031 4.762 17.041 0.0235 4.272 2.175 13.8422

40

50

Table 12 Frequency of convergence for test systems out of 50 trials.

1:4 6 1:2 6 6 0:7 6 6 0:1 6 6 0:3 6 6 0:1 6 6 0:1 6 Bij ¼ 0:001*6 6 0:1 6 0:3 6 6 0:5 6 6 0:3 6 6 0:2 6 6 0:4 6 4 0:3 0:1

1:2 1:5 1:3 0:0 0:5 0:2 0:0 0:1 0:2 0:4 0:4 0:0 0:4 1:0 0:2

0:7 1:3 7:6 0:1 1:3 0:9 0:1 0:0 0:8 1:2 1:7 0:0 2:6 11:1 2:8

B0i ¼ 0:001*½ 0:1

0:2

2:8

B00 ¼ 0:0055

0:1 0:0 0:1 3:4 0:7 0:4 1:1 5:0 2:9 3:2 1:1 0:0 0:1 0:1 2:6 0:1

0:3 0:5 1:3 0:7 9:0 1:4 0:3 1:2 1:0 1:3 0:7 0:2 0:2 2:4 0:3 0:1

0:1 0:2 0:9 0:4 1:4 1:6 0:0 0:6 0:5 0:8 1:1 0:1 0:2 1:7 0:3

0:3

0:1 0:0 0:1 1:1 0:3 0:0 1:5 1:7 1:5 0:9 0:5 0:7 0:0 0:2 0:8

0:2

Test system 1

Test system 2

Test system 3

Minimum fuel cost Between minimum and mean Mean fuel cost Between mean and maximum Maximum fuel cost

31 7 4 5 3

50 0 0 0 0

23 8 5 6 8

accordingly. Thus, choosing the initial populations with chaotic methods in CIHBMO and CHBMO that are close to the optimal dispatch has the higher probability of attaining quality solutions. Fig. 4 displays the distribution of total fuel cost for the results obtained by the proposed algorithm, CHBMO, IHBMO and HBMO for 20 different runs. As shown in Fig. 4, the CIHBMO algorithm has the most stable and minimum deviation. In order to show the effect of variation of the load demand on the performance of the proposed algorithm, the total system demand is varied from 2400 MW to 2600 MW with 100 MW increments. Table 4 shows the comparison of the best solutions obtained by the proposed CIHBMO and Differential Evolution (DE) [22], Adaptive-Real-Coded GA (ARCGA) [22], Real GA (RGA) [22], and particle swarm optimization (PSO) [22] for different power demands. As seen from this table, the proposed algorithm gives a lower generation cost than the other methods for all the load levels. Table 5 shows the best dispatch of generators, fuel type and cost among all solutions, found by the proposed algorithm for different load demands.

results from several recently published ED solution methods are also represented in Table 3. Simulation results reveal that the CIHBMO approach can give an optimal generation cost more consistently than other algorithms due to the extra diversification capability provided by the proposed chaotic component and improved mating process. Execution time of each optimization method is very important for its application to real problems. CPU time cannot be directly compared for different methods because of different computers and programming languages that were used. The convergence characteristics of 10 units test system with CIHBMO, CHBMO, IHBMO and original HBMO algorithms for economic dispatch are shown in Fig. 3. The convergence characteristics are drawn by plotting the minimum fitness value from the combined population across iteration index. From Fig. 3, we can see that the proposed methods can reduce the chance of getting stuck in local minima and result in better performance. The CHBMO and IHBMO are almost similar in convergence and show superiority over the HBMO. The CHBMO algorithm performs well due to the extra diversification provided by the chaotic method. In the figure, we can see that without chaotic method, the cost function converges to a much larger value. From Fig. 3, it is concluded that if the initial populations are far away from the optimal dispatch, their complexity increases

2

Range of generation cost

5.2. Test System 2: 15-unit system The system contains 15 thermal units, whose characteristics are given in Table A1.2 in Appendix 1 [28]. Four units have prohibited operation zones. Table 6 lists the prohibited zones of units 2, 5, 6, and 12. These zones result in three disjoint feasible sub-regions for units 2, 5, 6, and two for unit 12. The load demand is 2630 MW. Losses are determined using B matrix loss formula. The B loss coefficient matrix is:

0:1 0:1 0:0 5:0 1:2 0:6 1:7 16:8 8:2 7:9 2:3 3:6 0:1 0:5 7:8

0:2

0:6

0:3 0:2 0:8 2:9 1:0 0:5 1:5 8:2 12:9 11:6 2:1 2:5 0:7 1:2 7:2

0:5 0:4 1:2 3:2 1:3 0:8 0:9 7:9 11:6 20:0 2:7 3:4 0:9 1:1 8:8

0:3 0:4 1:7 1:1 0:7 1:1 0:5 2:3 2:1 2:7 14:0 0:1 0:4 3:8 16:8

3:9

1:7

0:0

0:2 0:0 0:0 0:0 0:2 0:1 0:7 3:6 2:5 3:4 0:1 5:4 0:1 0:4 2:8 3:2

0:4 0:4 2:6 0:1 0:2 0:2 0:0 0:1 0:7 0:9 0:4 0:1 10:3 10:1 2:8 6:7

0:3 1:0 11:1 0:1 2:4 1:7 0:2 0:5 1:2 1:1 3:8 0:4 10:1 57:8 9:4

6:4 

3 0:1 0:2 7 7 2:8 7 7 2:6 7 7 0:3 7 7 0:3 7 7 0:8 7 7 7:8 7 7 7:2 7 7 8:8 7 7 16:8 7 7 2:8 7 7 2:8 7 7 9:4 5 128:3

T. Niknam et al. / Energy 36 (2011) 896e908

In this test case, Itermax ¼ 100. Table 7 lists the comparative performances for the best, average, worst solution and average CPU times for 50 different runs with different algorithms published in literature. From the above comparisons, it can be seen that CIHBMO provided better results than the other reported methods. It is also observed that the worst solution of the proposed algorithm outperforms all the best solutions found by the other approaches. Another points that the original HBMO has obtained poor results for the ED problem. This indicates that the applied conversion to the proposed algorithm is imperative. In other words, the chaotic local search with new mating process improves the convergence behavior of the proposed CIHBMO. Also, the closeness of the best, average and worst results of the CIHBMO can be seen from Table 1. This indicates the robustness of the proposed method to solve the ED problem. Table 8 shows the best dispatch of generators among all solutions found by the proposed algorithm with and without power loss and cost. 5.3. Test System 3: 40-unit system This test system consists of 40 generators with valve-point loading effects and has a total load demand of 10500 MW. The test data for the system of 40 generating units are given in Table A1.3 in Appendix 1 in which the fuel cost functions also take into account the valve-point loading [33]. In this test case, Itermax ¼ 500. This test system has a larger and more complex solution space than the previous one and, therefore, difference between different ED solution methods will be more obvious. By introducing the chaotic local search into the IHBMO algorithm, the CIHBMO algorithms are capable of yielding better solutions than the IHBMO and CHBMO algorithms. The best generation cost reported until now is 121412.57$, which is obtained by Fuzzy Adaptive Modified Particle Swarm Optimization (FAMPSO) [2] and hybrid Fuzzy Adaptive Particle Swarm Optimization-Differential Evolution (FAPSO-DE) [38]. Table 9 shows the comparison of the minimum, mean, maximum cost and mean computation time for the simulation over 50 trial runs obtained by the proposed algorithm with the other recently reported results in the literature. Table 9 clearly signifies the potential of the proposed CIHBMO algorithm. Also, the mean and maximum cost obtained by the proposed algorithm is better than those obtained by FAMPSO and FAPSO-DE. Moreover, the average CPU time of the CIHBMO is better than the FAPSO-DE, which shows the reliability of the algorithm. The minimum, mean and maximum result of the CIHBMO and 6 other ED solution methods for this test case are compared in Table 10 (the mean and maximum results for other solution methods mentioned in this table were not available in the related references). To show the effect of changing in number of run, we consider Table 11 in three items for CIHBMO, CHBMO, IHBMO and HBMO algorithms. By comparing the results obtained from Table 11, it can be inferred that the proposed algorithm is independent of number of runs and gives the best solution which is 121412.57 ($/h). 5.4. Error analysis In order to evaluate the solution quality and error analysis in proposed optimization algorithm, the following criteria are generally considered.

905

Table 13 Results of deviation of the minimum and maximum solutions of the proposed CIHBMO from the corresponding mean result in terms of percentage. Test system

Deviation of the minimum solution from the corresponding mean result

Deviation of the maximum solution from the corresponding mean result

Test system 1 Test system 2 Test system 3

0.00006107748 0 0.00000018038

0.00005323634 0 0.00000031381

 the deviations of the minimum and maximum solutions of the proposed CIHBMO from the corresponding mean result  the frequency of attaining fuel cost within different ranges Since initialization of population is performed using random numbers in case of stochastic simulation techniques, randomness is inherent property of these techniques. Hence the performances of stochastic search algorithms are judged out of a number of trials. Many trials with different initial populations have been carried out to test the consistency of the CIHBMO algorithm. Table 12 shows the frequency of attaining cost within different ranges for 10-, 15- and 40-unit system out of 50 independent trials. Table 12 discloses that CIHBMO has the higher probability of attaining quality solution. In Table 13, the deviations of the minimum and maximum solutions of the proposed CIHBMO from the corresponding mean result for each ED test system are shown, which are calculated as follows: (a)

ðminimum fuel cost  mean fuel costÞ  100 ðmean fuel costÞ

(b)

ðmaximum fuel cost  mean fuel costÞ  100 ðmean fuel costÞ

The results of Table 13 show that the minimum, mean, and maximum solutions of the proposed method are close to each other, which indicates stability of the results of the CIHBMO.

6. Conclusion In this paper, we proposed an improved version of HBMO algorithm combining with a chaotic local search to solve the ED problems. The main objective of the ED problem is to determine the optimal combination of power outputs for all generating units that minimizes the total fuel cost while satisfying load demand and operating constraints. Nonlinear characteristics of the generator such as prohibited operation zones, valve-point loading and multi-fuel effect were considered for practical generation operation in our proposed method. In the paper, in order to make better the performance of algorithm and to enrich the searching behavior we incorporate the chaotic local search (CLS) into improved HBMO in which the mating process has been improved. To validate the proposed CIHBMO, it is tested on three test systems having non-convex solution spaces. The results of the proposed method and those of the previous approaches are compared. The outcome of the comparisons shows the effectiveness of the CIHBMO algorithm in terms of solution quality, convergence, robustness and consistency.

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T. Niknam et al. / Energy 36 (2011) 896e908

Appendix 1. The test system parameters including fuel cost coefficients and generator capacities are listed in Tables A1.1eA1.3, respectively. Table A1.1 Unit’s data for test system 1 (10 units).

Generator

Generation Min P1 P2 Max F1 F2 F3

Fuel type

Cost coefficients a

1

100

196 1

2

50

3

200

4

332

99

200

213

200

0.1184e3 0.1865e1 0.1365e2

0.1269e1 0.3988e-1 0.1980e0

0.4194e-2 0.1138e-2 0.1620e-2

0.1184e0 0.1865e-2 0.1365e-1

0.1269e2 0.3988e0 0.1980e1

500

1 2 3

0.3979e2 0.5914e2 0.2875e1

0.3116e0 0.4864e0 0.3389e-1

0.1457e-2 0.1176e-4 0.8035e-3

0.3979e-1 0.5914e-1 0.2876e-2

0.3116e1 0.4864e1 0.3389e0

265

1 2 3

0.1983e1 0.5285e2 0.2668e3

0.3114e-1 0.6348e0 0.2338e1

0.1049e-2 0.2758e-2 0.5935e-2

0.1983e-2 0.5285e-1 0.2668e0

0.3114e0 0.6348e1 0.2338e2

490

1 2 3

0.1392e2 0.9976e2 0.5399e2

0.8733e-1 0.5206e0 0.4462e0

0.1066e-2 0.1597e-2 0.1498e-3

0.1392e-1 0.9976e-1 0.5399e-1

0.8733e0 0.5206e1 0.4462e1

265

1 2 3

0.5285e2 0.1983e1 0.2668e3

0.6348e0 0.3114e-1 0.2338e1

0.2758e-2 0.1049e-2 0.5935e-2

0.5285e-1 0.1983e-2 0.2668e0

0.6348e1 0.3114e0 0.2338e2

500

1 2 3

0.1893e2 0.4377e2 0.4335e2

0.1325e0 0.2267e0 0.3559e0

0.1107e-2 0.1165e-2 0.2454e-3

0.1893e-1 0.4377e-1 0.4335e-1

0.1325e1 0.2267e1 0.3559e1

265

1 2 3

0.1983e1 0.5285e2 0.2668e3

0.3114e-1 0.6348e0 0.2338e1

0.1049e-2 0.2758e-2 0.5935e-2

0.1983e-2 0.5285e-1 0.2668e0

0.3114e0 0.6348e1 0.2338e2

440

1 2 3

0.8853e2 0.1530e2 0.1423e2

0.5675e0 0.4514e-1 0.1817e-1

0.1554e-2 0.7033e-2 0.6121e-3

0.8853e-1 0.1423e-1 0.1423e-1

0.5675e1 0.1817e0 0.1817e0

490

1 2 3

0.1397e2 0.6113e2 0.4671e2

0.9938e-1 0.5084e0 0.2024e0

0.1102e-2 0.4164e-4 0.1137e-2

0.1397e-1 0.6113e-1 0.4671e-1

0.9938e0 0.5084e1 0.2024e1

3 370

1 362

1

1 2 3

3

2

3 10

391

138

130

230

3

2

1 9

200

331

99

0.3975e1 0.3059e1

3

1

1 8

407

138

200

0.2697e1 0.2113e1

3

2

2 7

200

338

85

0.2176e-2 0.1861e-2

2

2

1 6

388

138

190

0.3975e0 0.3059e0

1

3

1 5

157 3

1

c

0.2697e2 0.2113e2

2

114 2

b

1 2

250

3 407

3

2

e

f

Table A1.2 Unit’s data for test system 2 (15-units). Generator

Pmin (MW)

Pmax (MW)

a

b

c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

150 150 20 20 150 135 135 60 25 25 20 20 25 15 15

455 455 130 130 470 460 465 300 162 160 80 80 85 55 55

0.000299 0.000183 0.001126 0.001126 0.000205 0.000301 0.000364 0.000338 0.000807 0.001203 0.003586 0.005513 0.000371 0.001929 0.004447

10.1 10.2 8.8 8.8 10.4 10.1 9.8 11.2 11.2 10.7 10.2 9.9 13.1 12.1 12.4

671 574 374 374 461 630 548 227 173 175 186 230 225 309 323

T. Niknam et al. / Energy 36 (2011) 896e908

907

Table A1.3 Unit’s data for test system 3 (40-units). Generator

Pmin (MW)

Pmax (MW)

a

b

c

e

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

36 36 60 80 47 68 110 135 135 130 94 94 125 125 125 125 220 220 242 242 254 254 254 254 254 254 10 10 10 47 60 60 60 90 90 90 25 25 25 242

114 114 120 190 97 140 300 300 300 300 375 375 500 500 500 500 500 500 550 550 550 550 550 550 550 550 150 150 150 97 190 190 190 200 200 200 110 110 110 550

0.0069 0.0069 0.02028 0.00942 0.0114 0.01142 0.00357 0.00492 0.00573 0.00605 0.00515 0.00569 0.00421 0.00752 0.00708 0.00708 0.00313 0.00313 0.00313 0.00313 0.00298 0.00298 0.00284 0.00284 0.00277 0.00277 0.52124 0.52124 0.52124 0.0114 0.0016 0.0016 0.0016 0.0001 0.0001 0.0001 0.0161 0.0161 0.0161 0.00313

6.73 6.73 7.07 8.18 5.35 8.05 8.03 6.99 6.6 12.9 12.9 12.8 12.5 8.84 9.15 9.15 7.97 7.95 7.97 7.97 6.63 6.63 6.66 6.66 7.1 7.1 3.33 3.33 3.33 5.35 6.43 6.43 6.43 8.95 8.62 8.62 5.88 5.88 5.88 7.97

94.705 94.705 309.54 369.03 148.89 222.33 287.71 391.98 455.76 722.82 635.2 654.69 913.4 1760.4 1728.3 1728.3 647.85 649.69 647.83 647.81 785.96 785.96 794.53 794.53 801.32 801.32 1055.1 1055.1 1055.1 148.89 222.92 222.92 222.92 107.87 116.58 116.58 307.45 307.45 307.45 647.83

100 100 100 150 120 100 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 120 120 120 120 150 150 150 200 200 200 80 80 80 300

0.084 0.084 0.084 0.063 0.077 0.084 0.042 0.042 0.042 0.042 0.042 0.042 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.077 0.077 0.077 0.077 0.063 0.063 0.063 0.042 0.042 0.042 0.098 0.098 0.098 0.035

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