Economic dispatch of power systems using an adaptive charged system search algorithm

Accepted Manuscript Economic dispatch of power systems using an adaptive charged system search algorithm P. Zakian, A. Kaveh

PII: DOI: Reference:

S1568-4946(18)30524-6 https://doi.org/10.1016/j.asoc.2018.09.008 ASOC 5086

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Applied Soft Computing Journal

Received date : 17 November 2017 Revised date : 30 August 2018 Accepted date : 6 September 2018 Please cite this article as: P. Zakian, A. Kaveh, Economic dispatch of power systems using an adaptive charged system search algorithm, Applied Soft Computing Journal (2018), https://doi.org/10.1016/j.asoc.2018.09.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights



Charged System Search (CSS) algorithm is improved with an adaptive approach.



Adaptive Charged System Search (ACSS) algorithm is proposed for economic dispatch considering different types of constraints.



A new simple constraint handling approach is developed.



The performances of both ACSS and CSS algorithms are compared herein.



Numerous benchmarks from small to large scale test cases are employed.

*Manuscript Click here to view linked References

Economic dispatch of power systems using an adaptive charged system search algorithm P. Zakiana*, A. Kavehb a

Faculty of Engineering, Arak University, P.O. Box 38156-8-8349, Arak, Iran

b

Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of

Science and Technology, Narmak, Tehran-16, Iran

Abstract In this article, an adaptive charged system search (ACSS) algorithm is developed for the solution of the economic dispatch problems. The proposed ACSS is based on the charged system search (CSS) which is a meta-heuristic algorithm utilizing the governing Coulomb law from electrostatics and the Newtonian laws of mechanics. Here, two effective strategies are considered to present the new ACSS. The first one is an improved initialization based on opposite based learning and subspacing techniques. The second one is Levy flight random walk for enriching updating process of the algorithm. Many types of economic dispatch cases comprising 6, 13, 15, 40, 160 and 640 units generation systems are testified as benchmarks ranging from small to large scale problems. These problems entail different constraints consisting of power balance, ramp rate limits, prohibited operating zones and valve point load effects. Additionally, multiple fuel options and transmission losses are included for some test cases. Moreover, simple constraint handling functions are developed in terms of penalty approach which can readily be incorporated into any other meta-heuristic algorithm. Results indicate that the ACSS either outperform or perform well in comparison to the CSS and other optimizers in finding optimized fuel costs.

Keywords: Economic dispatch; Adaptive charged system search (ACSS); Power generation; Ramp rate limits; Valve point load effects; Engineering cost optimization.

1. Introduction *

Corresponding author’s E-mail address: [email protected] (P. Zakian). The second author’s E-mail address: [email protected] (A. Kaveh).

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Economic dispatch is a critical part of designing power systems aiming at allocating optimum generation values to minimize the fuel cost upon reaching simultaneously satisfying several equality and inequality constraints. These problems have been considered as important optimization issue since 1970s. Early attempts for solving such problems are focused on gradient based techniques and dynamic programming. Nevertheless, in order to converge to global optimum, gradient based methods require some criteria for objective functions such as differentiability, convexity and continuity. Non-convexity and discontinuity of economic dispatch problem arise from considering some constraints consisting of prohibited operating zones (POZs), ramp rate and valve point load effect constraints, making the gradient based methods hard to be modified as those were performed in the literature. However, there are some investigations, applying the mathematical programming for economic dispatch problem [1-4]. On the other hand, many non-gradient based algorithms like evolutionary and metaheuristic ones were employed for the economic dispatch; particle swarm optimization [5-10], biogeography-based optimization [11, 12], firefly algorithm [13], teaching learning-based algorithm [14], cuckoo search algorithm [15, 16], honey bee mating optimization [17], hybrid grey wolf optimizer [18, 19], chaotic bat algorithm [20] and lighting flash algorithm [21], among others [22, 23]. As another kind of economic dispatch, standard charged system search (CSS) has been used for multi-objective environmental economic dispatch which is beyond the scope of this paper [24]. Meta-heuristic algorithms are problem independent methods for finding optimal (or sub-optimal) results. Although they cannot warrant optimality as a shortcoming, they are very widespread due to their extensive usage, simple applicability and global search ability. Based on inspiration source, meta-heuristics are classified to nature behavior-based and physics-based algorithms. However, this terminology is sometimes utilized for evolutionary and genetic algorithms interchangeably. Recent advances of meta-heuristic algorithms in civil engineering were presented in [25], which can also be applied to mechanical and electrical engineering due to problem independency nature of these algorithms. Notwithstanding there are many papers on solving economic dispatch problems, it is necessary to perform a study dealing with various types of constraints and suitable constraint handling approach, which also contains either small or large scale problems. Clearly, industrial application of a comprehensive study becomes more practical rather than a study focusing on limited or a few kinds of economic dispatch problems. Even though metaheuristics cannot warrant optimal design, a practical optimization problem can rapidly be

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formulated in terms of these algorithms such that an acceptable and reasonable design can be obtained as an initial design for industrial application. The aim of our work is to present the CSS and develop a new so-called adaptive charged system search (ACSS) algorithm for solving the economic dispatch problems, apart from presenting some simple constraint handling functions for penalty approach which can be used by any meta-heuristic algorithm. The CSS is a meta-heuristic algorithm inspired from the laws of Gauss and Coulomb of electrostatics, and the Newtonian laws of classical mechanics in physics. Here, two useful remedies are employed to establish the ACSS. The first one improves initialization stage based on opposite based learning (OBL) and subspacing approaches. The second one enhances updating process of the algorithm with a random walk called Levy flight. Different kinds of generating units involving 6, 13, 15, 40, 160 and 640 generators are solved as benchmarks covering small and large scale problems. These problems are subjected to various constraints including power balance, ramp rate limits, POZs and valve point load effects. Furthermore, some test cases consider multiple fuel options and transmission losses. The outcomes demonstrate desirable performances of the CSS and the ACSS. However, the ACSS outperforms the CSS and significant number of other algorithms in solving such problems.

2. Economic dispatch formulation In the economic dispatch problem, the fuel cost of thermal power plant should be minimized for a given load demand subjected to imposed constraints. Objective function, constraints and a new simple constraint handling approach are explained in this section.

2.1. Objective function The quadratic fuel cost function of the thermal units is the objective function expressed as [26]: Minimize Ng

Ng

j 1

j 1

F (P)   F j ( Pj )   (a j  b j Pj  c j Pj2 );

PR

Ng

(1)

in which Ng , Fj(Pj) and Pj are the total number of generating units, the fuel cost for the jth generating unit (in $/hr) and the power generated through the jth generating unit (in MW), respectively; aj, bj and cj denote cost coefficients for the jth generator. 3

For the case where the valve-point loading effect is taken into account, the objective function is rewritten as follows:

Minimize (2)

Ng

F (P)   F j ( Pj ) j 1

Ng

  (a j  b j Pj  c j Pj2 )  e j sin( f j ( Pjmin  Pj )) ;

PR

Ng

j 1

in which ej and fj are constant parameters for the valve-point effect of generators. For the case of multiple fuel options, the fuel cost of the jth generator is calculated by a j1  b j1 Pj  c j1 Pj2 ;  2 a j 2  b j 2 Pj  c j 2 Pj ; F j ( Pj )    a  b P  c P 2 ; jk j jk j  jk

fuel1 , Pjmin  Pj  Pj1 fuel2 , Pj1  Pj  Pj 2

(3) fuelk , Pjk 1  Pj  Pjmax

so that k discrete regions exist when a generator with k fuel options is undertaken.

2.2. Optimization constraints The inequality and equality constraints of the problem are the power balance equality, and power generation limits as represented by these two equations as: Ng

P j 1

j

 PD  PL

(4)

Pjmin  Pj  Pjmax

(5)

wherein Pj, PD, Pjmin and Pjmax are the generation of the jth generator unit (in MW), the total power demand (in MW), the minimum and maximum power generation limits of the jth generator. Also, PL indicates the line losses (in MW) that is determined through B coefficients, given by:

4

Ng Ng

Ng

j 1 i 1

j 1

(6)

PL   Pj B ji Pi   B0 j Pj  B00

where Pi and Pj stand for the power injection at the ith and jth buses, respectively, and Bij denotes the loss coefficients which are often assumed to be constant for normal operating conditions. Vibrations in the shaft bearings or steam valve operation induce the prohibited zones which are considered as another constraint (i.e., POZ). Admissible operating zones of the jth unit is defined by the following criteria: Pjmin  Pj  Pjl,1   Pj  Pju, k 1  Pj  Pjl,k ,  u max  Pj , n j  Pj  Pj

k  2,3,n j , j  1,2,n

(7)

l u with nj , Pj ,k and Pj ,k 1 being the number of prohibited zones, lower and upper power

outputs of the kth prohibited zone of the jth generator, respectively. Ramp rate limits are imposed for sake of the physical limitations of shutting down and starting up of generators, which are implemented by the following two conditions:

Limitation of the increase in generation:

Pj  Pj0  UR j

(8)

Limitation of the decrease in generation:

Pj0  Pj  DR j

(9)

0 in which Pj , DR j and UR j are the previous output power, the down-ramp and the up-ramp

limits, respectively, for the jth generator. Combining relevant equations leads to the generation limits, that is Pj  Pj  Pj

(10)

5

wherein Pj  max( Pjmin , Pj0  DR j )

(11)

Pj  min( Pjmax , Pj0  UR j )

(12)

After some mathematical manipulations, the problem may be expressed as:

Minimize (13)

Ng

F (P)   F j ( Pj ) j 1

Ng

  (a j  b j Pj  c j Pj2 )  e j sin( f j ( Pjmin  Pj )) ;

PR

Ng

j 1

such that Ng

P j 1

j

(14)

 PD  PL

max( Pjmin , Pj0  DR j )  Pj  Pjl,1 Pju, k 1  Pj  Pjl, k ;

k  2,3, n j , j  1,2, N g

Pju, n j  Pj  min( Pjmax , Pj0  UR j )

2.3. A simple penalty approach for constraint handling This section develops some functions for constraint handling of the economic dispatch problems in terms of the penalty technique. Penalty approach is a commonly used method for constraint handling due to its simplicity, efficiency and ease of implementation. Thus, in order to impose the constraints, one can optimize the objective function governed by:

Minimize

Obj (P)  F (P)  f penalty (P)

(15)

in which Obj (P ) is the objective function (i.e., the penalized fuel cost), F(P) is the fuel cost, and fpenalty(P) is the penalty function for constraint handling [27], given by: 6

n

f penalty(P)  (1  1  ) 2 ,    max{ 0, i }

(16)

i 1

where  is the sum of the violated constraints and n is the total number of constraints. Here, 1 is taken as a constant value drawn from [1, 2] depending on the constraints’ status of the

problem and  2 is an ascending function of iteration, varying from 1.5 to 3, linearly. Power balance is ideally described as: Ng

P j 1

 PD  PL  0

j

(17)

However, the above equation cannot be exactly satisfied due to the errors within digits of real numbers existing in computers. Therefore, one can relax the equation in the following form: Ng

P P j 1

j

D

 PL  

(18)

such that the tolerance  can be selected as a small number (e.g., less than 10-3) relying on the problem. Obviously, exact satisfaction of this constraint becomes possible only when  is equal to zero (or converges to zero). Alternatively, the following two equations are suggested to be more efficient for the usage, instead of Eq. (18): Ng

1 

P j 1

j

PD  PL

(19)

1  0

and Ng

 2 1

P j 1

j

PD  PL

(20)

0

Eqs. (19) and (20) should be satisfied simultaneously, wherein the equality sign is dropped, clearly.

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For POZ constraint, the following function is suggested here: nj  Pj  Pmin Pj  Pju, k  max{ l , 0}; if P j  POZ   3    Pmax  Pmin  Pj , k  Pju, k k 1 j 1  otherwise 0; Ng

(21)

Indeed, the above function calculates relative distance of an obtained Pj from lower bounds of problem’s domain and evaluates the amount of constraint violation. Operator max, assigns zero value once Pj does not belong to the kth POZ. It can be found that the presented functions smoothen the convergence curve of an optimizer compared to those of the literature, while desirable solutions are attained. It is worth mentioning that either X or x variables of optimizers in the forthcoming sections are the same as P, and hence they should be replaced with P as optimization variables. However, the next sections are presented based on x for maintaining the generality of the algorithm definition.

3. Adaptive charged system search (ACSS) The CSS is a recently developed meta-heuristic optimization algorithm proposed by Kaveh and Talatahari [28]. This algorithm, which is inspired by the Coulomb law of physics and Newtonian laws of mechanics have been successfully applied to various nonlinear structural optimization problems with different complexity in sense of constraints, variables and convexity [25, 27, 29, 30]. The CSS is a population based algorithm in which each agent is known as a charged particle (CP) supposed to be a sphere with uniform charge density and radius of a. Every CP is under the effects of particles’ force field. The resultant force is calculated by using the electrostatics laws and the quality of the movement is attained using Newtonian mechanics laws. In contrast to a good CP, a bad CP must induce more force.

In this section, a new so-called ACSS is proposed to enhance the performance of the CSS. Two remedies are employed for this enhancement, which includes a modified initialization and a modified random walk.

The basic steps of the ACSS algorithm are summarized as follows:

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Step 1: Initialization. The arrays or initial positions of CPs are specified randomly in the search space, while the initial velocities of CPs are assumed to be zero. The values of the fitness function for the CPs are calculated, and the CPs are sorted in an ascending order. The best CP among the entire set of CPs is taken as Xbest and its corresponding fitness is fitbest. Analogously, the worst CP will be taken as fitworst. Two main features of meta-heuristics are problem independency and weak dependency on initialization. Nevertheless, some algorithms like the CSS and harmony search uses a memory for storing best solutions from the beginning of initialization part. Thus, an improved initialization can help the procedure, particularly for huge search spaces existing in large scale optimization problems. In contrast to the CSS which uses one search space for initialization, the ACSS uses an initialization space that is 4-fold as that of the CSS, i.e. three new spaces are added to that of the CSS. Firstly, the ACSS is initialized like the CSS. Secondly, opposition based learning (OBL) concept is employed to initialize the second space. The OBL is a new concept in soft computing for accelerating convergence of various optimizers, which was originally developed by Tizhoosh [31]. The OBL utilizes a population as well as its opposite counterpart to nominate better potential solutions. Numerous studies admitted and proved that employing the OBL increases the chance of finding global optimum [31, 32]. However, the OBL is herein utilized for initialization part to eliminate additional computational cost. A simple definition for an opposite number is that if x is a real number within [a, b], then its opposite counterpart will be equal to a+b-x. As another point, optimal solutions have usually a tendency to be near to domain boundaries, and hence one can divide a domain to upper bound and lower bound subdomains to encounter this phenomenon. Therefore, the third and the forth spaces are the initializations from lower bound and upper bound subdomains. Consequently, these four spaces are introduced as initialization part of the ACSS as follows: Space 1 (ordinary; similar to the CSS): 1 xiInitial  xi,min  rand  ( xi ,max  xi,min ) , i  1,2,..., nv ,j

(22)

Space 2 (according to the OBL): 2 1 xiInitial  xi,max  xi ,min  xiInitial , ,j ,j

i  1,2,..., nv

9

(23)

Space 3 (lower bound): 3 xiInitial  xi , min  rand  ( ,j

xi , min  xi , max 2

 xi , min ) , i  1,2,..., nv

(24)

Space 4 (upper bound): 4 xiInitial  ,j

xi , min  xi , max 2

 rand  ( xi , max 

xi , min  xi , max 2

) , i  1,2,..., nv

(25)

Now, best solutions from the four spaces are stored in Charged Memory (CM) without any change in size of the CM with respect to the CSS. No additional computational efforts are inserted to the algorithm during the iterative process as discussed earlier. In other words, the four spaces are only used for the initialization stage. CM is a matrix wherein a number of the best CPs and their related values of the fitness function are saved.

 

CM  xi , j

(26)

nv*CMS

nv and CMS are respectively the number of variables and the charged memory size usually equal to the integer part of the quarter number of CPs. If the objective function values of variables are also stored in this matrix, the number of rows will be nv+1. xi , max and xi ,min are upper and lower bounds of variable domain i, respectively. Here, rand is an uniformly distributed random number drawn from [0,1]. Step 2: Force determination. Calculate the force vector for every CP as:

q q Fj  q j  ( 3i rij .i1  2i .i2 )ar pij ( X i  X j ) rij i ,i  j a

 j  1,2,..., Ncp  i1  1, i2  0  rij  a  i1  0, i2  1  rij  a

(27)

where Fj and Ncp are the resultant force acting on the jth CP and the total number of CPs, respectively. ar is a sign factor defined by:

1 ar   0

rand  0.8 otherwise

(28)

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The charge magnitude for each CP, qi, is expressed incorporating its solution quality as:

qi 

fit (i)  fitworst fitbest  fitworst

i  1,2,..., N cp

(29)

in which fitbest and fitworst are the best and the worst finesses of all the particles, fit(i) denotes the fitness of the agent i. The separation distance rij between two charged particles is determined as follows:

rij 

norm ( X i  X j ) X Xj norm ( i  X best )   0 2

(30)

in which Xi and Xj are the positions of the ith and jth CPs, respectively, Xbest is the position of the best current CP, and ε0 is an infinitesimal number for singularity prevention. Here, pij is the probability of moving each CP towards the others whereby the following function is defined:  fit (i )  fitbest  rand  fit ( j )  fit (i ) 1 fit ( j )  fit (i ) pij   0 otherwise 

(31)

As already mentioned, each CP is defined as a charged sphere with radius a and a uniform charge density. A proper choice for a is obtained considering the size of the search space as:

a  c0  max ( xi ,max  xi ,min );

i  1,2,..., nv

(32)

with c0 being a constant coefficient which is herein chosen to be a magnitude near to 0.001. Step 3: Solution and updating procedure. Each CP moves to its new position where is a function of the resultant force of the CP, old velocity and the old position. As mentioned before, here Levy flight algorithm is proposed to improve the random exploration. Levy flight is an efficient random walk which was recently implemented successfully in some

11

optimization algorithms [33-35]. Levy motion is a non-Gaussian random process whose random walks are based on Levy distribution as a power law formula L( s)  s

1  

(33)

where  is a stability parameter in range (0, 2). In mathematical illustration, a simple version of Levy distribution may be ruled as:

     1 exp    3/ 2 L( s,  ,  )   2  2( s   )  ( s   )  0 

(34)

if 0    s  , if s  0

in which  parameter is shift parameter,   0 parameter is a scale controlling the distribution.



is the skewness parameter within interval [-1,1].

Here, the influence of the local best solution, Xbest, is incorporated through Levy flight for improving the algorithm. By adding Levy flight to updating process of the CSS, we have new position of the particles as follows: X j ,new  rand j1  F j  rand j 2  kv V j ,old  rand j 3 . . Levy(  )  X j ,old

(35)

where new velocity is specified as:

V j ,new  X j ,new  X j ,old

(36)

The step size corresponds to the scale of the problem of interest and a non-trivial scheme of forming step size is described as: rand j 3 . . Levy(  )  0.01

u v

1/ 

(37)

( X best  X j ,old )

where u and v are randomly selected numbers with normal distribution, that are u  N (0, u2 ) ,

(38)

v  N (0, v2 ).

in which randj1, randj2 and randj3 are random numbers uniformly distributed in the domain [0,1]. In this procedure, the effects of local best particles are directly implemented in updating process. Additionally, the ACSS does not have the acceleration coefficient ka, and 12

instead it has only the same

kv of the CSS, but with values cv equal or less than those of the

CSS. kv is a decreasing function known as the velocity coefficient to stabilize the influence of the previous velocity and to control exploration procedure. Hence, the function is determined as:

k v  cv  [1  (

iter )] itermax

(39)

where iter is the number of current iteration and itermax is the maximum number of iterations. cv

has constant values that should be adjusted based on the optimization problem.

Exploration is an ability of an algorithm to search during the iterations and it should be reduced by increasing the iterations for two reasons. The first reason is to use exploitation ability of the algorithm for improving the obtained solutions further. The second reason stands for the convergence. Hence, Eq. (39) provides the reduction of search ability, whereas the ACSS considers solution changes during iterations as the exploitation ability; and these changes are supplied by the first term and the new term defined in Eq. (35). During updating process, similar to the CSS, if a new CP exits from the permissible search space, a harmony search-based handling approach can be used to correct its position based on the allowable search space of the undertaken problem. According to this strategy, any component of the solution vector violating the variable boundaries is regenerated from the CM or from a randomly selecting value that belongs to the possible range of values. Additionally, if there are some new CP vectors better than the worst ones in the CM, then those are substituted by the worst ones in the CM, while the worst ones are eliminated. Step 4: Termination criterion. Steps 2–3 must be repeated upon reaching a pre-defined stopping criterion. In this paper, maximum number of iterations is taken as a stopping criterion. As it will be demonstrated in the next section, exploration and exploitation of the ACSS is better than those of the CSS ones after implementing these two strategies. In summary, all the steps of the ACSS are identical to those of the CSS, except the steps 1 and 3. Fig. 1 indicates the main steps of ACSS. 4. Simulations Six test cases consisting of 6-units, 13-units, 15-units, 40-units, 160-units and 640-units are solved to validate the merits of the proposed ACSS method in practical non-convex economic dispatch under various constraints. In addition, the CSS is also employed for the optimization 13

examples in order to provide a comprehensive study. Both algorithms utilize the proposed penalty functions for the procedure. Parameter settings of the CSS and the ACSS for each test case are listed in Table 1. To have a fair competition, the CSS and the ACSS use the same number of particles and maximum number of iterations. Due to random nature of the metaheuristic algorithms, 50 independent computer runs are accomplished for each algorithm to achieve useful statistical results. Finally, the results are also compared with those of previously solved algorithms in literature.

4.1. Test Case 1: 6-units system A system composed of six generators meeting a load demand of 1263 MW is considered as the first test case, including ramp rate limits, transmission loss and POZ. Although this is a small system, the existence of ramp rate limits, transmission loss and POZs makes it a practical and real-life problem. More information about the system data is found in [7, 8]. Table 2 discloses optimal cost and generations attained by the ACSS and the CSS. Optimal cost and its corresponding transmission loss for the ACSS and the CSS are equal to 15443.5562 $/hr, 15448.3972 $/hr, 12.4699 MW and 12.7318 MW, respectively. This table also reveals that generation limits are satisfied by the generations whose values do not fall within POZ. The ACSS outperforms other optimizers and achieves cheapest fuel cost (i.e., the best fuel cost of 50 runs) as admitted by Table 3 showing statistical results wherein bold faced characters stand for the best solutions. The convergence characteristics of the ACSS and the CSS for the best run are shown in Fig. 2. As it is visible, with an identical maximum number of iteration and number of particles, the ACSS converges to a better solution for the generation cost than the CSS. Furthermore, Fig. 3 visualizes final solution of each run for the both algorithms during 50 runs

4.2. Test Case 2: 13-units system The second test case comprises of thirteen thermal units with the load demands of 2520 MW. Valve point effect and transmission losses are included in this model. The results of the ACSS and the CSS are provided in Table 4 and compared with those from the literature in Table 5. It can be observed that the proposed ACSS provides better generation schedule in comparison with other algorithms. Optimal cost and the transmission loss for the ACSS and the CSS are obtained as 24510.8385 $/hr, 24710.1658 $/hr, 36.2546 MW and 39.1426 MW, 14

respectively, which also confirms the constraint holding. Also, the ACSS reaches to the minimum fuel cost compared to other algorithms in Table 3. The statistical performances of the ACSS and CSS over 50 runs show the superiority of the ACSS. Additionally, the best convergence curves of the both algorithms are drawn in Fig. 4. Optimized generation costs of all 50 runs of these algorithms are illustrated in Fig. 5 for better comparison.

4.3. Test Case 3: 15-units system Here, a system consisting of fifteen generators with a load demand of 2630 MW is considered [7, 8], which involves ramp rate limits, transmission loss and POZ constraints. Table 6 reports optimal costs and generations obtained by the ACSS and CSS, whereby the optimal costs are 32678.1290 $/hr and 32693.3116 $/hr, and the corresponding transmission losses are equal to 29.4543 MW, 29.1089 MW, respectively; while all the constraints are satisfied. Convergence curves of the optimizers are compared in Fig. 6. The statistical results of these algorithms as well as those of the literature are compared in Table 7 confirming the cheapest fuel cost attained by the ACSS. The favorable performance of the ACSS is apparent during 50 independent runs whose details are illustrated in Fig. 7 along with results of the CSS.

4.4. Test Case 4: 40-units system Here, a system with forty thermal units with a load demand of 10500 MW considering valve point loading effect and the transmission loss is examined. Formation of coefficient matrix B for this system is performed through multiplication of arrays of 6 generating units up to 40 units [7, 8]. The results obtained by the present algorithms are indicated in Table 8 exhibiting the optimal costs of 136653.7317 $/hr and 136679.0228 $/hr corresponding to 959.2408 MW and 956.9544 MW transmission losses achieved through the ACSS and the CSS without constraint violations of defined functions, respectively. Table 9 shows that the ACSS outperforms all the optimizers with the expectation of OIWO [32] which has a slightly better solution. The convergence characteristics and final solutions of the ACSS and the CSS over 50 trials are presented in Figs. 8 and 9.

4.5. Test Case 5: 160-units system In order to provide more realistic evaluation, a benchmark having 160 thermal generators with load demand of 43200 MW is employed to verify the usefulness and performance of the 15

ACSS and the CSS for solving economic dispatch challenges. This system is formed by duplicating the 10-unit system [36], which is along with both valve point effects and multiple fuel options, 16 times. No transmission loss is imposed in this test case. As it can be observed from Tables 10, 11 and 12, the ACSS finds the minimum fuel cost (i.e., 9979.8281 $/hr) among other existing methods reported, while CSS also provides fine cost, but slightly more expensive that is 9980.7184 $/hr. Convergence histories of the ACSS and CSS as well as results of 50 trials are shown Figs. 10 and 11 admitting better performance of the ACSS versus the CSS. In addition, Tables 10 and 11 list generations obtained by both algorithms, confirming the suitable constraint handling.

4.6. Test Case 6: 640-units system Here, a severely large scale system is examined to validate the effectiveness of the ACSS, the CSS and the constraint handling functions. There are only two references [21, 36] that tackled such large scale problems of economic dispatch. This system contains 640 generation units meeting the load demand of 172800 MW [36]. Formation of this system is similar to the previous test case, but one should duplicate 10 units system 64 times. Due to the curse of dimensionality and drastically increased number of local minima, this problem is enlightened as a realistic problem. Table 13 pinpoints optimized fuel costs of CSO, LFA, the CSS and the ACSS, which are 39964.0603 $/hr, 39957.7748 $/hr, 39960.7064 $/hr and 39950.7907 $/hr. Similarly, the ACSS has the best solution compared to other algorithms. Fig. 12 demonstrates that the ACSS and the CSS have suitable convergence rate. For sake of dimensionality, generations obtained for this problem are not reported in the literature. However, their values in scattering form are illustrated in Fig. 13. Optimized costs of 50 runs are entirely plotted in Fig 14. Bar chart depicted in Fig. 14 demonstrates stability and successfulness of the ACSS against the CSS similar to bar charts of the previous simulations. Moreover, statistical assessments show that average values and standard deviations of fuel cost obtained by the ACSS is less than those of the CSS for all six simulations involving this one, which is evident as another demonstration of the ACSS suitability. Computing times of optimization for all the examples are listed in Table 14. All the simulations are implemented through Intel® Core™ i7-4790K CPU @4.00 GHz. Obviously, the ACSS slightly takes more time to optimize a generation system. This is because of initialization part of the ACSS in which number of objective function evaluation is four-fold (four spaces) relative to that of the CSS (one space). However, since only the initialization 16

part needs more computational effort, there is a negligible deference between elapsed time of both algorithms for a certain problem, while a better result is attained by the ACSS. On the other hand, statistical indices consisting of average, standard deviation, maximum and the best values corresponding to optimized fuel cost are summarized in Table 15 possessing results of the CSS and the ACSS.

5. Conclusions In this research, a new algorithm so-called ACSS is proposed for economic dispatch problems which are categorized as non-convex, multi-modal, non-smooth, discontinuous and nonlinear optimization problems with various inequality and equality constraints like power balance, valve point effect, POZ and ramp rate limits. As a physics-based optimizer, the CSS is also utilized for this kind of problems. The robustness and feasibility of the presented algorithms are investigated on six test systems with 6, 13, 15, 40, 160 and 640-units. Results of simulations demonstrate that minimum values achieved by the ACSS outperform other algorithms in almost all the test cases constituting different types of systems from small to very large scale ones. Furthermore, some simple penalty functions are formulated for constraint handling for which benchmarks are selected to be subjected to power balance, ramp rate limits, POZs and valve point load and transmission loss constraints. According to the results, the ACSS is a good rival for other optimizers in these problems. Although the ACSS and the CSS perform well, their solutions are sometimes near together and computational effort of the ACSS is a little greater than that of the CSS due to the enhanced initialization step of the ACSS; nevertheless the performance of the ACSS is better and more promising in these test cases and can be extended and applied to other engineering disciplines as well as optimal design of other power systems and economic dispatch problems like environmental economic dispatch.

17

Nomenclature a

Radius of charged particles

ar

Sign factor

aj, bj and cj

Cost coefficients for the jth unit

Bij

Loss coefficient

B

Loss coefficient matrix

c0

Constant coefficient for finding a

cv

Constant value of velocity coefficient

DR j

Down-ramp limit

F (.)

Fuel cost function

Fj

Force vector of the jth particle

f penalty (.)

Penalty function

fit(.)

Fitness function

fitworst(.)

Fitness function corresponding to the worst solution

fitbest(.)

Fitness function corresponding to the best solution

i1 and i2

Constant binary values

iter

Current iteration

itermax

Maximum number of iterations

kv

Linearly descending function for discounting velocity

L(.)

Power formula for Levy flight

Levy(.)

Levy flight function

N cp

Total number of charged particles

Ng

Total number of generation units

N (0,  u2 )

Zero mean normal distribution with variance of  u2

18

Obj(.)

Penalized fuel cost function or objective function

P

Power vector generated for all units

Pj

Power generated for the jth unit

Pj

Upper bound of power generated for the jth unit

Pj

Lower bound of power generated for the jth unit

PD

Total power demand

PL

Line loss

qi

Charge magnitude for the ith particle

rand

Uniformly distributed random number

R

Ng

Ng-dimensional real space

rij

Distance between two particles

UR j

Up-ramp limit

u and v

Two random numbers with normal distribution

Vj

Velocity vector of the jth particle

Xj

Position or solution vector of the jth particle

Xbest

The best solution vector

xij

The ith array of the jth particle

Greeks

0

An infinitesimal number



Tolerance value

 u2

Variance of u



Shift parameter



Sum of violated constraints

19

1 and  2

Coefficients for penalty function



Skewness parameter



Scale factor



Stability parameter

Abbreviations ACSS

Adaptive Charged System Search

CM

Charged Memory

CP

Charged Particle

CSS

Charged System Search

OBL

Opposition Based Learning

POZ

Prohibited Operating Zone

20

References [1] S. Pan, J. Jian, L. Yang, A hybrid MILP and IPM approach for dynamic economic dispatch with valve-point effects, International Journal of Electrical Power & Energy Systems, 97 (2018) 290-298. [2] Z.L. Wu, J.Y. Ding, Q.H. Wu, Z.X. Jing, X.X. Zhou, Two-phase mixed integer programming for non-convex economic dispatch problem with spinning reserve constraints, Electric Power Systems Research, 140 (2016) 653-662. [3] A. Rabiee, B. Mohammadi-Ivatloo, M. Moradi-Dalvand, Fast Dynamic Economic Power Dispatch Problems Solution Via Optimality Condition Decomposition, IEEE Transactions on Power Systems, 29 (2014) 982-983. [4] M.S.P. Subathra, S.E. Selvan, T.A.A. Victoire, A.H. Christinal, U. Amato, A Hybrid With Cross-Entropy Method and Sequential Quadratic Programming to Solve Economic Load Dispatch Problem, IEEE Systems Journal, 9 (2015) 1031-1044. [5] M. Basu, Modified particle swarm optimization for nonconvex economic dispatch problems, International Journal of Electrical Power & Energy Systems, 69 (2015) 304-312. [6] T. Niknam, H.D. Mojarrad, M. Nayeripour, A new fuzzy adaptive particle swarm optimization for non-smooth economic dispatch, Energy, 35 (2010) 1764-1778. [7] G. Zwe-Lee, Closure to "Discussion of 'Particle swarm optimization to solving the economic dispatch considering the generator constraints'", IEEE Transactions on Power Systems, 19 (2004) 2122-2123. [8] G. Zwe-Lee, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Transactions on Power Systems, 18 (2003) 1187-1195. [9] S. Duman, N. Yorukeren, I.H. Altas, A novel modified hybrid PSOGSA based on fuzzy logic for non-convex economic dispatch problem with valve-point effect, International Journal of Electrical Power & Energy Systems, 64 (2015) 121-135. [10] K.K. Mandal, S. Mandal, B. Bhattacharya, N. Chakraborty, Non-convex emission constrained economic dispatch using a new self-adaptive particle swarm optimization technique, Applied Soft Computing, 28 (2015) 188-195. [11] A. Srinivasa Reddy, K. Vaisakh, Discussion of “Bhattacharya, A., & Chattopadhyay, P. K. (2010). Solving complex economic load dispatch problems using biogeography-based optimization. Expert Systems with Applications, 37(5), 3605–3615”, Expert Systems with Applications, 39 (2012) 13547-13548. [12] A. Bhattacharya, P.K. Chattopadhyay, Solving complex economic load dispatch problems using biogeography-based optimization, Expert Systems with Applications, 37 (2010) 3605-3615. [13] X.-S. Yang, S.S. Sadat Hosseini, A.H. Gandomi, Firefly Algorithm for solving nonconvex economic dispatch problems with valve loading effect, Applied Soft Computing, 12 (2012) 1180-1186. 21

[14] S. Banerjee, D. Maity, C.K. Chanda, Teaching learning based optimization for economic load dispatch problem considering valve point loading effect, International Journal of Electrical Power & Energy Systems, 73 (2015) 456-464. [15] M. Basu, A. Chowdhury, Cuckoo search algorithm for economic dispatch, Energy, 60 (2013) 99-108. [16] T.T. Nguyen, D.N. Vo, The application of one rank cuckoo search algorithm for solving economic load dispatch problems, Applied Soft Computing, 37 (2015) 763-773. [17] T. Niknam, H.D. Mojarrad, H.Z. Meymand, B.B. Firouzi, A new honey bee mating optimization algorithm for non-smooth economic dispatch, Energy, 36 (2011) 896-908. [18] T. Jayabarathi, T. Raghunathan, B.R. Adarsh, P.N. Suganthan, Economic dispatch using hybrid grey wolf optimizer, Energy, 111 (2016) 630-641. [19] M. Pradhan, P.K. Roy, T. Pal, Grey wolf optimization applied to economic load dispatch problems, International Journal of Electrical Power & Energy Systems, 83 (2016) 325-334. [20] B.R. Adarsh, T. Raghunathan, T. Jayabarathi, X.-S. Yang, Economic dispatch using chaotic bat algorithm, Energy, 96 (2016) 666-675. [21] M. Kheshti, X. Kang, Z. Bie, Z. Jiao, X. Wang, An effective Lightning Flash Algorithm solution to large scale non-convex economic dispatch with valve-point and multiple fuel options on generation units, Energy, 129 (2017) 1-15. [22] S.D. Beigvand, H. Abdi, M. La Scala, Hybrid Gravitational Search Algorithm-Particle Swarm Optimization with Time Varying Acceleration Coefficients for large scale CHPED problem, Energy, 126 (2017) 841-853. [23] A. Meng, H. Hu, H. Yin, X. Peng, Z. Guo, Crisscross optimization algorithm for largescale dynamic economic dispatch problem with valve-point effects, Energy, 93 (2015) 21752190. [24] S. Özyön, H. Temurtaş, B. Durmuş, G. Kuvat, Charged system search algorithm for emission constrained economic power dispatch problem, Energy, 46 (2012) 420-430. [25] A. Kaveh, Advances in Metaheuristic Algorithms for Optimal Design of Structures, Springer International Publishing, 2016. [26] A.J. Wood, B.F. Wollenberg, Power Generation, Operation, and Control, Wiley, 2012. [27] A. Kaveh, P. Zakian, Optimal design of steel frames under seismic loading using two meta-heuristic algorithms, Journal of Constructional Steel Research, 82 (2013) 111-130. [28] A. Kaveh, S. Talatahari, A novel heuristic optimization method: charged system search, Acta Mechanica, 213 (2010) 267-289. [29] A. Kaveh, P. Zakian, Seismic design optimisation of RC moment frames and dual shear wall-frame structures via CSS algorithm, Asian Journal of Civil Engineering (BHRC), 15 (2014) 435-465.

22

[30] A. Kaveh, S. Talatahari, Charged system search for optimal design of frame structures, Applied Soft Computing, 12 (2012) 382-393. [31] H.R. Tizhoosh, M. Ventresca, Oppositional Concepts in Computational Intelligence, Springer Berlin Heidelberg, 2008. [32] A.K. Barisal, R.C. Prusty, Large scale economic dispatch of power systems using oppositional invasive weed optimization, Applied Soft Computing, 29 (2015) 122-137. [33] H. Haklı, H. Uğuz, A novel particle swarm optimization algorithm with Levy flight, Applied Soft Computing, 23 (2014) 333-345. [34] B. Mandal, P.K. Roy, Optimal reactive power dispatch using quasi-oppositional teaching learning based optimization, International Journal of Electrical Power & Energy Systems, 53 (2013) 123-134. [35] X.S. Yang, Cuckoo Search and Firefly Algorithm: Theory and Applications, Springer International Publishing, 2013. [36] A. Meng, J. Li, H. Yin, An efficient crisscross optimization solution to large-scale nonconvex economic load dispatch with multiple fuel types and valve-point effects, Energy, 113 (2016) 1147-1161. [37] W.T. Elsayed, E.F. El-Saadany, A Fully Decentralized Approach for Solving the Economic Dispatch Problem, IEEE Transactions on Power Systems, 30 (2015) 2179-2189. [38] I. Ciornei, E. Kyriakides, A GA-API Solution for the Economic Dispatch of Generation in Power System Operation, IEEE Transactions on Power Systems, 27 (2012) 233-242. [39] D.C. Secui, A new modified artificial bee colony algorithm for the economic dispatch problem, Energy Conversion and Management, 89 (2015) 43-62. [40] A.I. Selvakumar, K. Thanushkodi, A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems, IEEE Transactions on Power Systems, 22 (2007) 42-51. [41] S. Pothiya, I. Ngamroo, W. Kongprawechnon, Application of multiple tabu search algorithm to solve dynamic economic dispatch considering generator constraints, Energy Conversion and Management, 49 (2008) 506-516. [42] K. Bhattacharjee, A. Bhattacharya, S.H.n. Dey, Oppositional Real Coded Chemical Reaction Optimization for different economic dispatch problems, International Journal of Electrical Power & Energy Systems, 55 (2014) 378-391. [43] A. Srinivasa Reddy, K. Vaisakh, Shuffled differential evolution for large scale economic dispatch, Electric Power Systems Research, 96 (2013) 237-245. [44] J.G. Vlachogiannis, K.Y. Lee, Economic Load Dispatch-A Comparative Study on Heuristic Optimization Techniques With an Improved Coordinated Aggregation-Based PSO, IEEE Transactions on Power Systems, 24 (2009) 991-1001.

23

[45] R. Azizipanah-Abarghooee, T. Niknam, M. Gharibzadeh, F. Golestaneh, Robust, fast and optimal solution of practical economic dispatch by a new enhanced gradient-based simplified swarm optimisation algorithm, IET Generation, Transmission & Distribution, 7 (2013) 620-635. [46] K. Bhattacharjee, A. Bhattacharya, S.H.n. Dey, Chemical reaction optimisation for different economic dispatch problems, IET Generation, Transmission & Distribution, 8 (2014) 530-541. [47] J.P. Zhan, Q.H. Wu, C.X. Guo, X.X. Zhou, Fast lambda-Iteration Method for Economic Dispatch With Prohibited Operating Zones, IEEE Transactions on Power Systems, 29 (2014) 990-991. [48] Q. Niu, H. Zhang, X. Wang, K. Li, G.W. Irwin, A hybrid harmony search with arithmetic crossover operation for economic dispatch, International Journal of Electrical Power & Energy Systems, 62 (2014) 237-257. [49] V. Hosseinnezhad, M. Rafiee, M. Ahmadian, M.T. Ameli, Species-based Quantum Particle Swarm Optimization for economic load dispatch, International Journal of Electrical Power & Energy Systems, 63 (2014) 311-322. [50] C. Chao-Lung, Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels, IEEE Transactions on Power Systems, 20 (2005) 16901699.

24

Table 1. Parameter tuning details for the CSS and ACSS.

Test case 1 2 3 4 5 6

CSS Number of particles 20 20 20 50 80 80

ACSS

Maximum iteration

c0

ca

cv

400 800 800 800 1200 4000

0.005 0.0008 0.002 0.0008 0.002 0.001

0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5 0.5

Number of particles 20 20 20 50 80 80

Maximum iteration

c0

cv

400 800 800 800 1200 4000

0.005 0.0008 0.002 0.0008 0.002 0.001

0.5 0.5 0.5 0.5 0.5 0.5

Table 2. Optimized cost and generation solved by the CSS and ACSS for 6-units system with 1263 MW demand (test Case 1). Unit 1 2 3 4 5 6 Fuel cost ($/hr) Transmission loss (MW)

Pjmin

Pjmax

POZ

100 50 80 50 50 50

500 200 300 150 200 120

[210, 240]; [350, 380] [90, 110]; [140, 160] [150, 170]; [210, 240] [80, 90]; [110, 120] [90, 110]; [140, 150] [75, 85]; [100, 105]

Generation CSS ACSS 459.7104 440.1305 170.0896 174.7059 255.7824 261.6509 134.0250 139.2963 181.1244 172.5325 74.9999 87.1538 15448.39722 15443.5562 12.7318

12.4699

Table 3. Comparison of optimized costs and statistical indices for 6-units system (test Case 1). Algorithm

Best fuel cost ($/hr)

Mean fuel cost ($/hr)

Maximum fuel cost ($/hr)

Standard deviation of fuel cost ($/hr)

DE [37]

15,449.58

15,449.62

15,449.65

NA

GA-API [38]

15,449.78

15,449.81

15,449.85

NA

MABC [39]

15,449.90

15,449.90

15,449.90

6.04E-08

NPSO-LRS [40]

15,450

15,454

15,452

NA

MTS [41]

15,450.06

15,451.17

15,450.06

0.9287

GA Binary [38]

15,451.66

15,469.21

15,519.87

NA

TS [41]

15,454.89

15,472.56

15,454.89

13.7195

SA [41]

15,461.10

15,488.98

15,461.10

28.3678

CBA [20]

15,450.24

15,454.76

15,518.66

2.965

CSS

15448.3972

15557.8899

16616.3788

256.9598

ACSS

15443.5562

15458.2023

15490.6899

14.8632

Present study

25

Table 4. Optimized cost and generation solved by the CSS and ACSS for 13-units system with 2520 MW demand (test Case 2). Generation

Pjmin

Pjmax

CSS

ACSS

1

0

680

628.3185

628.2684

2

0

360

357.3048

299.1925

3

0

360

298.6112

299.1765

4

60

180

109.8678

159.7181

5

60

180

159.9011

159.6840

6

60

180

159.7173

159.7334

7

60

180

158.0921

159.7327

8

60

180

159.7307

159.7139

9

60

180

159.7305

159.7445

10

40

120

111.6364

77.4184

11

40

120

77.3971

109.5427

12

55

120

92.9501

92.2618

13

55

120

85.8851

92.0676

24710.1658

24510.8385

39.1426

36.2546

Unit

Fuel cost ($/hr) Transmission loss (MW)

Table 5. Comparison of the optimized costs and statistical indices for the 13-units system (test Case 2).

Algorithm

Best fuel cost ($/hr)

Mean fuel cost ($/hr)

Maximum fuel cost ($/hr)

Standard deviation of fuel cost ($/hr)

OIWO [32]

24,514.83

24,514.83

24,514.83

NA

ORCCRO [42]

24,513.91

24,513.91

24,513.91

NA

SDE [43]

24,514.88

24,516.31

NA

NA

ICA-PSO [44]

24,540.06

24,561.46

24,589.45

NA

BBO [42]

24,515.21

24,515.32

24,516.09

NA

DE/BBO [42]

24,514.97

24,515.05

24,515.98

NA

CSS

24710.1658

24969.5320

25393.8096

142.4635

ACSS

24510.8385

24697.5081

25005.6027

115.8008

Present study

26

Table 6. Optimized cost and generation solved by the CSS and ACSS for 15-units system with 2630 MW demand (test Case 3).

Unit

Pjmin

Pjmax

Pj

Generation POZ

Pj

CSS

ACSS

1

150

455

280

455

-

454.5672

454.9941

2

150

455

180

380

[185,225]; [305,335]; [420,450]

379.9010

379.9797

3

20

130

20

130

-

129.9512

129.9845

4

20

130

20

130

-

129.9999

129.9344

5

150

470

150

170

[180,200]; [305,335]; [390,420]

169.5655

169.9251

6

135

460

280

460

[230,255]; [365,395]; [430,455]

459.9664

459.8311

7

135

465

230

430

-

429.9179

429.9816

8

60

300

60

160

-

81.1328

68.4119

9

25

162

25

162

-

94.6923

59.1246

10

25

160

25

160

-

108.6934

159.7769

11

20

80

20

80

-

79.8733

79.6671

12

20

80

20

80

[30,40]; [55,65]

79.7393

79.8250

13

25

85

25

85

-

25.2144

25.0364

14

15

55

15

55

-

16.3491

16.5617

15

15

55

15

55

-

18.2129

15.0816

32693.3116

32678.1290

29.1089

29.4543

Fuel cost ($/hr) Transmission loss (MW)

Table 7. Comparison of the optimized costs and statistical indices for the 15-units system (test Case 3). Algorithm

Best fuel cost ($/hr)

Mean fuel cost ($/hr)

Maximum fuel cost ($/hr)

Standard deviation of fuel cost ($/hr)

EGSSOA [45]

32680.1038

32680.1038

NA

NA

RCCRO [46]

32698.9950

32698.995

NA

NA

F  I [47]

32701.0000

32701.0000

NA

NA

ACHS [48]

32706.6500

32706.6500

57938.3559

35.94

SQPSO [49]

32706.6740

32708.44

121709.5582

NA

IPSO [49]

32709.0000

32784.5

NA

NA

PSO-MSAF [49]

32713.0900

32759.64

NA

NA

DSPSO-TSA [49]

32715.0600

32724.63

NA

NA

HGWO [18] Present study

32679

32685

NA

NA

CSS

32693.3116

32798.3524

32971.0800

65.5745

ACSS

32678.1290

32727.6967

32761.3126

25.4748

27

Table 8. Optimized cost and generation solved by the CSS and ACSS for 40-units system with 10500 MW demand (test Case 4). Generation

Pjmin

Pjmax

CSS

ACSS

1

36

114

112.7233

112.7793

2

36

114

112.4337

3

60

120

4

80

5

47

6

Generation

Pjmin

Pjmax

CSS

ACSS

21

254

550

540.2442

537.7546

110.7495

22

254

550

548.7549

539.4229

118.7760

119.9182

23

254

550

523.5580

523.2798

190

179.7516

189.1786

24

254

550

523.3497

523.3515

97

87.8706

94.3408

25

254

550

529.3652

549.5497

68

140

137.1662

139.9832

26

254

550

534.1253

526.5445

7

11

300

299.6621

294.8122

27

10

150

10.2920

10.5487

8

13

300

295.2148

293.9738

28

10

150

10.2609

10.8687

9

13

300

297.9689

293.9425

29

10

150

10.0577

10.2360

10

13

300

279.6354

279.7199

30

47

97

87.7957

88.1288

11

94

375

168.7756

168.7818

31

60

190

189.9908

189.8971

12

94

375

94.0296

94.3136

32

60

190

189.0700

189.9930

13

12

500

483.8818

484.0383

33

60

190

189.9454

189.8272

14

12

500

484.1531

484.0470

34

90

200

199.8047

199.9436

15

12

500

484.0329

484.0472

35

90

200

195.6955

199.3383

16

12

500

484.0609

484.1362

36

90

200

164.8319

165.5486

17

22

500

489.5026

489.4333

37

25

110

109.9802

109.9994

18

22

500

489.2927

489.1807

38

25

110

109.9770

109.6160

19

24

550

526.3295

511.3114

39

25

110

108.1248

106.4176

20

242

550

511.3491

511.4544

40

242

550

Unit

Unit

545.1178

548.8320

Cost ($/hr)

136679.0228

136653.7317

Transmission loss (MW)

956.9544

959.2408

Table 9. Comparison of optimized costs and statistical indices for 40-units system (test Case 4).

Algorithm

Best fuel cost($/hr)

Mean fuel cost($/hr)

Maximum fuel cost($/hr)

Standard deviation of fuel cost ($/hr)

OIWO [32]

136,452.68

136,452.68

136,452.68

N/A

ORCCRO [42]

136,855.19

136,855.19

136,855.19

N/A

BBO [42]

137,026.82

137,116.58

137,587.82

N/A

DE/BBO [42]

136,950.77

136,966.77

137,150.77

N/A

CSS

136679.0228

136993.6115

137447.4131

171.2636

ACSS

136653.7317

136930.9946

137444.7894

132.0025

Present study

28

Table 10. Optimized cost and generation solved by the CSS for 160-units system with 43200 MW demand (test Case 5). 1

225.4489

33

284.4835

65

287.1704

97

290.1451

129

418.7564

2

209.1642

34

239.0704

66

238.5631

98

240.7240

130

283.7613

3

275.9085

35

287.6384

67

290.2655

99

417.9284

131

226.6511

4

238.4743

36

236.1666

68

238.4382

100

280.2701

132

213.5759

5

281.5424

37

285.8012

69

431.3422

101

227.9961

133

284.2205

6

235.4944

38

237.6007

70

276.5662

102

210.7754

134

241.2985

7

293.7817

39

425.3957

71

220.7751

103

282.0682

135

283.7740

8

240.2878

40

280.4086

72

207.0328

104

238.9846

136

239.3298

9

415.9022

41

209.6329

73

287.9055

105

277.4166

137

282.9138

10

277.1770

42

205.7442

74

239.9123

106

240.5213

138

242.7415

11

223.7105

43

280.6949

75

276.8489

107

291.4997

139

416.3466

12

209.1907

44

240.8286

76

236.4615

108

240.2623

140

290.3343

13

282.5911

45

277.5245

77

285.1997

109

420.9637

141

220.7351

14

241.9707

46

236.3427

78

237.5347

110

271.7846

142

212.6776

15

275.9506

47

279.5422

79

404.9807

111

218.8103

143

285.1654

16

238.1881

48

236.1615

80

281.7649

112

209.7024

144

238.0022

17

274.2186

49

421.0867

81

225.9467

113

285.1267

145

291.1152

18

241.5942

50

292.4419

82

211.1270

114

236.1403

146

239.5997

19

415.9052

51

218.6074

83

284.4960

115

283.2613

147

285.5108

20

269.8586

52

214.7075

84

242.8530

116

237.2032

148

242.1054

21

218.9682

53

278.5036

85

283.1083

117

283.6290

149

419.6082

22

202.6922

54

244.8395

86

238.0350

118

237.6129

150

277.8593

23

278.7044

55

268.6105

87

280.1757

119

419.1786

151

221.4577

24

239.4827

56

243.4841

88

241.9396

120

288.5168

152

212.7213

25

281.7231

57

288.8525

89

418.8862

121

218.8763

153

281.5128

26

239.5048

58

235.6068

90

282.5275

122

206.3130

154

239.9290

27

286.7009

59

420.5548

91

216.6602

123

278.3208

155

277.3469

28

235.7417

60

285.8120

92

209.2608

124

240.1758

156

241.4952

29

415.7548

61

226.2291

93

278.7158

125

289.0857

157

287.1929

30

273.4515

62

213.8201

94

241.0010

126

238.9851

158

239.9058

31

214.1058

63

281.6045

95

279.3965

127

300.1948

159

413.0954

32

214.6055

64

241.9306

96

246.3874

128

240.4617

160

264.8349

Fuel cost ($/hr)

9980.7184

29

Table 11. Optimized cost and generation solved by the ACSS for 160-units system with 43200 MW demand (test Case 5). 1

214.268

33

291.9626

65

292.9496

97

285.6129

129

421.5624

2

212.6793

34

237.6197

66

239.7741

98

239.369

130

286.5058

3

275.0183

35

277.5796

67

295.7297

99

418.766

131

228.139

4

240.5571

36

242.434

68

232.7897

100

282.7089

132

206.681

5

282.9312

37

268.7951

69

400.6568

101

228.3892

133

273.7678

6

239.6396

38

238.0258

70

284.2461

102

213.2776

134

242.3984

7

291.2515

39

408.0642

71

219.187

103

278.6197

135

279.8217

8

236.4216

40

279.2847

72

211.5703

104

240.4455

136

239.1137

9

415.7503

41

219.1277

73

287.3125

105

271.7961

137

292.72

10

279.3368

42

209.0977

74

239.9122

106

238.4288

138

238.2915

11

219.5157

43

276.4192

75

269.1157

107

291.2352

139

425.6296

12

207.0238

44

238.6559

76

236.9499

108

241.1322

140

278.5768

13

280.8557

45

281.3615

77

292.6246

109

416.8663

141

213.9796

14

241.4962

46

241.6483

78

241.6136

110

281.7058

142

208.4535

15

297.8222

47

293.9326

79

410.8757

111

222.7451

143

282.7126

16

240.5972

48

243.6695

80

279.2151

112

209.2799

144

238.2957

17

281.937

49

410.5883

81

223.8146

113

283.7238

145

285.7127

18

242.9996

50

269.2088

82

213.8941

114

240.59

146

238.8698

19

418.0456

51

219.4954

83

281.5377

115

275.4555

147

296.2243

20

276.1291

52

208.5633

84

237.4954

116

235.2048

148

237.2087

21

228.7081

53

292.676

85

270.6179

117

287.6958

149

420.793

22

207.2608

54

238.6369

86

234.5311

118

238.1612

150

284.1882

23

292.937

55

273.7589

87

288.6932

119

421.474

151

230.7755

24

236.9547

56

239.2354

88

240.71

120

291.2906

152

210.7684

25

266.2163

57

282.8888

89

406.8015

121

221.8048

153

275.721

26

241.1209

58

236.8202

90

284.1697

122

205.4888

154

238.8366

27

295.1151

59

406.2003

91

221.2529

123

288.5126

155

284.0802

28

238.972

60

279.9292

92

206.4977

124

236.4141

156

238.4307

29

416.2495

61

230.3091

93

285.9168

125

279.9917

157

273.809

30

286.6951

62

213.5865

94

237.7624

126

240.5795

158

241.2571

31

216.9942

63

290.9767

95

283.2253

127

288.7929

159

426.762

32

209.8755

64

236.9565

96

242.327

128

241.5232

160

287.7188

Fuel cost ($/hr)

9979.8281

30

Table 12. Comparison of optimized costs and statistical indices for 160-units system (test Case 5). Algorithm

Best fuel cost ($/hr)

Mean fuel cost ($/hr)

Maximum fuel cost ($/hr)

CGA_MU [50] IGA_MU [50] RCCRO BBO DE/BBO CBA [20] LFA [21] Present CSS study ACSS

10143.7236 100042.47 10004.20 10008.71 10007.05 10002.8596 9980.2096 9980.7184 9979.8281

NA NA 10004.21 10009.16 10007.56 10006.3251 9984.9959 9983.3509 9983.0980

NA NA 10004.45 10010.59 10010.26 10045.2265 9988.7855 9985.8878 9985.6966

Standard deviation of fuel cost ($/hr) NA NA NA NA NA 9.5811 1.9379 1.0995 1.1472

Table 13. Comparison of optimized costs and statistical indices for 640-units system (test Case 6). Algorithm

Best fuel cost ($/hr)

Mean fuel cost ($/hr)

Maximum fuel cost ($/hr)

CSO [36] LFA [21] Present CSS study ACSS

39964.0603 39957.7748 39960.7064 39950.7907

39968.03007 39969.2850 39998.5797 39977.3189

39974.1858 39974.6649 40064.1107 40018.4618

Standard deviation of fuel cost ($/hr) 1.9075 4.0725 26.2312 16.6828

Table 14. Computational time for 50 runs of every example. Example No. 1 2 3 4 5 6

Generation system

Elapsed time (min) CSS 1.04 1.65 2.50 9.45 38.57 234.71

6-units 13-units 15-units 40-units 160-units 640-units

31

ACSS 1.08 1.73 2.56 9.62 41.23 279.57

Table 15. Summary of statistical indices for fuel cost optimized by the CSS and the ACSS for every example. Example No.

Generation system

1

6-units

2

13-units

3

15-units

4

40-units

5

160-units

6

640-units

Algorithm CSS ACSS CSS ACSS CSS ACSS CSS ACSS CSS ACSS CSS ACSS

Best fuel cost ($/hr) 15448.3972 15443.5562 24710.1658 24510.8385 32693.3116 32678.1290 136679.0228 136653.7317 9980.7184 9979.8281 39960.7064 39950.7907

32

Mean fuel cost ($/hr) 15557.8899 15458.2023 24969.5320 24697.5081 32798.3524 32727.6967 136993.6115 136930.9946 9983.3509 9983.0980 39998.5797 39977.3189

Maximum fuel cost ($/hr) 16616.3788 15490.6899 25393.8096 25005.6027 32971.0800 32761.3126 137447.4131 137444.7894 9985.8878 9985.6966 40064.1107 40018.4618

Standard deviation of fuel cost ($/hr) 256.9598 14.8632 142.4635 115.8008 65.5745 25.4748 171.2636 132.0025 1.0995 1.1472 26.2312 16.6828

Start Determine the numbers of particles and iterations

Initialize a space of particles using Eq. (22)

Initialize a space of particles using Eq. (23)

Initialize a space of particles using Eq. (24)

Create a space of particles by employing the best particles from the four spaces in the previous step

Save a part of the best particles in the Charged Memory (CM)

Determine the forces and compute relevant equations (Eq. (27) to Eq. (32))

No

Update the position and the velocity vectors using Eq. (35) and Eq. (36)

Is the termination criterion satisfied? Yes End

Fig. 1. Flowchart of the ACSS algorithm.

33

Initialize a space of particles using Eq. (25)

Fig. 2. Convergence curves of the CSS and ACSS for the 6-units system (test Case 1).

34

6-Units System 16800 16600

Fuel Cost ($/hr)

16400 16200 16000 15800 15600 15400 15200 15000 14800 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 3. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 6-units system (test Case 1).

35

Fig. 4. Convergence curves of the CSS and ACSS for 13-units system (test Case 2).

36

13-Units System 25400

Fuel Cost ($/hr)

25200 25000 24800 24600 24400 24200 24000 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 5. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 13-units system (test Case 2).

37

Fig. 6. Convergence curves of the CSS and ACSS for 15-units system (test Case 3).

38

Fuel Cost ($/hr)

15-Units System 33000 32950 32900 32850 32800 32750 32700 32650 32600 32550 32500 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 7. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 15-units system (test Case 3).

39

Fig. 8. Convergence curves of the CSS and ACSS for 40-units system (test Case 4).

40

40-Units System 137600

Fuel Cost ($/hr)

137400 137200 137000 136800 136600 136400 136200 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 9. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 40-units system (test Case 4).

41

Fig. 10. Convergence curves of the CSS and ACSS for 160-units system (test Case 5).

42

Fuel Cost ($/hr)

160-Units System 9986 9985 9984 9983 9982 9981 9980 9979 9978 9977 9976 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 11. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 160units system (test Case 5).

43

Fig. 12. Convergence curves of the CSS and ACSS for 640-units system (test Case 6).

44

Fig. 13. Optimized generation solved by the CSS and ACSS for 640-units system (test Case 6).

45

Fuel Cost ($/hr)

640-Units System 40080 40060 40040 40020 40000 39980 39960 39940 39920 39900 39880 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Run No. CSS

ACSS

Fig. 14. Comparison of fuel costs obtained by the CSS and ACSS during 50 runs for 640units system (test Case 6).

46