1 Introduction

Economic dispatch problem considering air pollution was taken into account by the enactment of the Clean Air Act of 1990. Accordingly, all utilities are required to take the rate of SO2, NOx, and CO2 emissions of their generating units into consideration when dispatching them (Talaq and EI-Hawary 1994). Since then, much research has been conducted in this field and many approaches have been proposed to reduce the emissions. These methods can be divided mainly into three groups:

  • Installation of equipment for cleaning emissions in site of generating units;

  • Replacement of old equipment with new ones;

  • Operation of generating units considering environmental pollutants.

Various approaches are available to take into account the pollution of generating units through environmental economic dispatch (EED) problem. In recent years, several swarm intelligent algorithms have been proposed and improved such as Genetic algorithms (GA) (Holland 1975), Particle swarm optimization (PSO) (Shi and Eberhart 1998), Differential evolution (DE) (Storn and Price 1997) and Artificial bee colony (ABC) (Karaboga 2005a, b), etc., and have been introduced for optimization problem. ABC is one of swarm intelligent algorithms inspired by the foraging behaviors of honeybee colony. ABC was first introduced by Karaboga (Karaboga and Basturk 2007; Gao et al. 2012) and simulates the intelligent foraging behavior of honey bee swarms. Since the ABC is simple in concept, easy to implement and it has fewer control parameters. It has attracted the attention of many researchers and has been widely used in solving numerical optimization (Liao et al. 2013). However, the convergence speed of ABC algorithm will decrease as the dimension of the problem increases. To address these issues, several methods have been proposed to improve the algorithm to overcome these drawbacks (Zhu and Kwong 2010; Yan et al. 2012). For clustering, several methods based on Evolution algorithm (EA) have been proposed such as combining K-means with ABC for clustering (Zou et al. 2010; Ebrahimian et al. 2005). The experimental study of the colony algorithm using an improved artificial bee colony (ABC) algorithm in this paper has been used to solve the ED problem, considering the objective function which consists of fuel cost of units, the constraints of the valve-point effect, the transmission losses, the balance of supply and demand in the system, the production limits the up-ramp and down-ramp rates, and the pollution issues. The resulting algorithm is implemented on the case study systems and the obtained results were compared with those of other algorithms. This algorithm has fast convergence and is less likely to be trapped in local minima compared to other algorithms (Balamurugan and Subramanian 2008).

In this paper, EED problem is solved using a hybrid approach carried out in an objective function consisting of cost and pollution considering transmission system power loss. Improved Artificial Bee Colony (ABC) algorithm, as a most novel approach employed on non-linear models, is used to solve mathematical model. The rest of the paper is presented as follows. In Sect. 2, problem formulation is provided. Section 3 presents the proposed algorithm for solving the EED problem. In Sect. 4, case studies are presented. Finally, the paper is ended with conclusions and feature works in Sect. 5.

2 Problem formulation

The aim of solving EED problem considering pollution is to simultaneously manage fuel cost and pollution from fossil fuel consumption of generating units. This issue is considered as an optimization problem in which the objective function consists of fuel cost and pollution from generating units. In addition, various constraints are taken into account to solve this problem.

2.1 Objective function

With EED problem, operational cost of generating units is expressed as the output power. Considering that fuel cost is the main cost factor of generating units, operational cost function of generating units is expressed as the input fuel cost. It is typically written as quadratic function in terms of output active power of generating unit. Thus, production cost function of generating units is given by:

$$FC=\sum\limits_{i=1}^M {{a_i}P_{{Gi}}^{2}+{b_i}{P_{Gi}}+{c_i}}$$
(1)

where M is the number of generating units, a i , b i , and c i are cost coefficients of ith generating unit, P Gi denotes ith generating unit’s active power, and FC is cost function of production in $.

Considering that SO2 and NOx are the main components for emissions from generating units, it is necessary to minimize the amount of these gases in order to reduce the pollution. Investigations revealed that output power is the most influencing factor in producing emissions by generating units. There is a nonlinear relationship between pollution of a generating unit and its output power. This can be modeled as a quadratic function in terms of output power. Thus, emission function of generating units is given by:

$$FE=\sum\limits_{i=1}^M {{\alpha _i}P_{{Gi}}^{2}+{\beta _i}{P_{Gi}}+{\gamma _i}}$$
(2)

where α i , β i , and γ i are emission coefficients of ith generating unit, FE denotes total emission in kg. The EED problem’s objective function, comprising of fuel cost and emission of generating units, should be minimized as:

$$\begin{aligned} FT & =w \times \sum\limits_{i=1}^M {\left( {{a_i}P_{{Gi}}^{2}+{b_i}{P_{Gi}}+{c_i}} \right)} \\ & \quad +(1 - w) \times \sum\limits_{i=1}^M {{h_i}\left( {{a_i}P_{{Gi}}^{2}+{b_i}{P_{Gi}}+{\gamma _i}} \right)} \\ \end{aligned}$$
(3)

where w is weighting factor of fuel cost, h i is emission penalty factor from the viewpoint of utility. The value of emission of interest may be different. In reported research, various techniques have been proposed to define this factor (Muralidharan 2006). Typically, penalty factor for emission of each generating unit is defined as fuel cost divided by emission amount and then multiplied by maximum output power of that generating unit (AlRashidi and El-Hawary 2006):

$${h_i}=\frac{{{a_i}P_{{Gi}}^{{^{{_{{\max }}^{2}}}}}+{b_i}P_{{Gi}}^{{\max }}+{c_i}}}{{{\alpha _i}P_{{Gi}}^{{_{{\max }}^{2}}}+{\beta _i}P_{{Gi}}^{{\max }}+{\gamma _i}}}$$
(4)

where h i is emission penalty factor and Pmax Gi is the maximum output power of ith generating unit.

2.2 Constraints

Constraints of EED problem are given as bellow:

2.2.1 Power supply and demand balance in system

Total produced power by all generating units should be equal to total system demand.

$$\sum\limits_{i=1}^{{N_i}} {{P_{mi}}+\sum\limits_{h=1}^{{N_h}} {{P_{mh}} - {P_{md}} - {P_{lm}}=0} }$$
(5)

where P mh is the produced power of hth hydro system in mth sub-branch, P md is total load demand in mth sub-branch, P Lm is total active power loss in transmission lines in mth sub-branch. They should be calculated as in Karaboga (2005):

$${P_{lm}}=\sum\limits_{i=1}^{{N_i}+{N_h}} {\sum\limits_{}^{{N_i}+{N_h}} {{P_{mi}}{B_{ij}}{P_{mj}}} }$$
(6)

2.2.2 Production constraint

For each generating unit, the maximum and minimum produced power, reactive power, and voltage are defined by:

$${p_i}^{{\min }} \leq {p_i} \leq {p_i}^{{\max }}$$
(7)

3 Improved Artificial Bee Colony (ABC) algorithm

3.1 Standard ABC algorithm

Artificial Bee Colony is a new swarm intelligence algorithm proposed by Karaboga (Karaboga and Bahriye 2008; Palanichamy and Babu 2008) which is motivated from the intelligent food foraging behavior of Honey Bee. Since the development of ABC it has been applied to solve different kinds of problems. The ABC algorithm is developed based on inspection the behaviors of real bees on finding nectar and sharing the information of food sources to the bees in their hive. The main advantages of the ABC algorithm over other optimization methods for solving optimization are simplicity, high flexibility, strong robustness, few control parameter, ease of combination with other methods, ability to handle the objective with stochastic nature, fast convergence.

Based on their experience and position, onlookers choose appropriate food sources. Scouts select food sources randomly and without their experience. Each selected food source indicates a possible solution for the problem. The amount of nectar in food sources indicates the fitness of the problem solution. The number of employed bees is equal to the number of onlookers and equal to the random initial population size. It is initialized with the size of Ne, where Ne is the number of food sources and equal to employed bees’ number. Each solution Xi = (Xi1, …, Xin) is n-dimension vector. Then, this population enters into search process for employed bee, onlookers, and scouts (Dhillon et al. 1993). The main steps of coding for the algorithm are given below:

  • Initializing for initial solutions;

  • Calculating initial solutions in objective function;

  • Initial iteration;

  • Finding new solutions based on new food sources Vij in neighborhood of Xij to produce new solutions by Eq. (8); use SI not CGS as primary units. Avoid combining SI and CGS units. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity in an equation:

    $${v_{ij}}={x_{ij}}+{\Phi _{ij}}({x_{ij}} - {x_{kj}})$$
    (8)

    where K is the obtained solution in neighborhood of i, and Φ is a random number in (−1 to 1).

  • Selecting the best food source or best solution between Vij and Xij;

  • Calculating the value of possibility for solutions Xij based on the following relationship

    $${P_i}=\frac{{fi{t_i}}}{{\sum\limits_{i=1}^{{N_e}} {fi{t_i}} }}$$
    (9)

In fact, in order to obtain fitness of solutions, the following relationship is used:

$$fi{t_i}=\left\{ \begin{aligned} & \frac{1}{{1+{f_i}}}\quad & {f_i} \geq 0 \\ & 1+abs({f_i})\quad & {f_i} \leq 0 \\ \end{aligned} \right.$$
(10)

Solutions are in the range (−1 to 1).

  • Generating new solutions (new sources) Vi based on watching bees from the solutions Xi and determining their possibilities Pi;

  • Selecting the best solution (the most gluttonous bee) between Xij and Vij;

  • Determining wasted sources and replacing them with stochastic sources or stochastic sources produced by leader bee Xi using the following relationship:

    $${X_{ij}}={X_{j\min }}+rand(0,1)*({X_{j\max }} - {X_{j\min }})$$
    (11)

3.2 Improved ABC algorithm

Improved ABC (or IABC) algorithm is based upon gravitational force between objects. The steps toward implementation of this algorithm are as follows:

3.2.1 Initial framing

Ne value is chosen as initial solutions in search space of the algorithm. And, their fitness value is studied based on the objective function. Indeed, random selection of these solutions is done in search space and indicates the employed bees.

3.2.2 Movement of onlookers

The investigation of selected foods’ possibility based on Eq. (13) and selection of one food source are completed in order to use roulette wheel for each onlooker and to determine nectar value for each of them on the basis of developed gravitational counterforce among onlookers. They are obtained Eqs. (10)–(15) (Karaboga and Bahriye 2008).

$${P_i}={\raise0.7ex\hbox{${fi{t_i}}$} \!\mathord{\left/ {\vphantom {{fi{t_i}} {\left( {\sum\limits_{n=1}^{{N_e}} {fit{}_{i}} } \right)}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\left( {\sum\limits_{n=1}^{{N_e}} {fit{}_{i}} } \right)}$}}$$
(12)

Counterforce between two objects (masses) m 1 and m 2 is given by the following relationship and depicted in Fig. 1:

Fig. 1
figure 1

Counterforce between two objects

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$$F_{{12}} = G\frac{{m_{1} m_{2} }}{{r_{{21}}^{2} }}\hat{r}_{{21}}$$
(13)
$$\hat{r}_{{21}} = \frac{{r_{2} - r_{1} }}{{\left| {r_{2} - r_{1} } \right|}}$$
(14)

where F 12 , r 12 , and G are counterforce, unit vector, and gravitational constant, respectively.

Likewise, based on the fitness values of bees, the following relationships are provided.

$${F_{i{k_j}}}=G\frac{{F({\theta _i}) \times F({\theta _k})}}{{{{({\theta _{kj}} - {\theta _{ij}})}^2}}}.\frac{{{\theta _{kj}} - {\theta _{ij}}}}{{\left| {{\theta _{kj}} - \theta ij} \right.\left. {} \right|}}$$
(15)
$${X_{ij}}(t+1)={\theta _{ij}}(t)+{F_{i{k_j}}}\left[ {{\theta _{ij}}(t) - {\theta _{kj}}(t)} \right]$$
(16)

where F(θi) and F(θj) are the fitness expressed for employed bees. Equation (16) expresses the resulting effect for new supply source. Considering counter effect of all bees on the selected bee, Eq. (16) is extended to Eq. (17) (Zou et al. 2010):

$${x_{ij}}(t+1)={\theta _{ij}}(t)+\sum\limits_{k=1}^n {{F_{i{k_j}}}\left[ {{\theta _{ij}}(t) - {\theta _{kj}}(t)} \right]}$$
(17)

3.2.3 Movement of scouts

If function fitness is not corrected in following iterations of the algorithm, it will be named Limit and the corresponding sources are called obsolete. With the aid of scouts’ movement, obsolete sources are recovered and replaced with the new sources. The movement process will be as follows:

$${\theta _{ij}}={\theta _{ij\min }}+r.({\theta _{ij\max }} - {\theta _{ij\min }})$$
(18)

3.2.4 Replacement

If food sources found become better in the next steps compared to the earlier steps, this value will be stored in bee memory.

3.2.5 Program termination

The program is iterated until all iterations are terminated. If a satisfactory value is obtained, the program will terminate. Otherwise, the second step is restarted. Figure 2 illustrates the proposed algorithm’s flowchart.

Fig. 2
figure 2

IABC algorithm’s flowchart

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4 Case study

In this section, the results obtained from implementation of proposed algorithm are studied and analyzed. EED problem was solved in order to fulfill numerical studies and show the effectiveness of the proposed algorithm in two case tests, i.e. IEEE 6 bus power system and IEEE 11 bus power system. The obtained results were compared with those of other techniques.

For numerical studies, parameters related to IABC are provided in Table 1. The precise selection of these parameters can be effective in reaching optimal solution. For instance, increasing the population size up to a definite value leads to improved quality and at the same time reduces the algorithm’s speed. Thus, in order to select population size, a reasonable trade-off should be made between accuracy and speed of the algorithm. Groups of bees should be also selected in a way that the number of available bees in each group is not high in order to avoid quality reduction. It must be also not too small to prevent the movement of bees and, in turn, to trap in local minima. Due to affectability of movement of each member in the group, the maximum useful iteration of local search is equal to the number of group size. The maximum iteration of algorithm is also selected in a way to obtain problem solution with appropriate accuracy at the least possible time. Further, tests carried out on two case systems are described and the numerical results of the proposed algorithm will be presented.

Table 1 Required parameters for the algorithm implementation
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4.1 6-Unit system

Coefficients of the fuel cost and generating units’ power limits as well as emission function coefficients of each generating unit in this system are presented in Tables 2 and 3 (Dhillon et al. 1993). Formulation coefficients of transmission network losses in this system are expressed by Eq. (16) (Eusuff and Lansey 2003). EED problem for this system in two modes, i.e. with and without losses, in load levels varying between 500 and 1100 MW are calculated (Fig. 3).

Table 2 Fuel cost coefficients for 6-unit system (Dhillon et al. 1993)
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Table 3 Emission function coefficients for 6-unit system (Dhillon et al. 1993)
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Fig. 3
figure 3

Variations of fuel cost and pollution functions of 6-unit system

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In modes of with/without power loss, the obtained results were compared with those of λ-iteration algorithm (Dhillon et al. 1993), recursive algorithm (Dhillon et al. 1993), simplified recursive algorithm (Palanichamy and Babu 2008), differential evolutionary (Palanichamy and Babu 2008), PSO (Palanichamy and Babu 2008), SA (Xuebin 2009), shuffled frog leaping algorithm (Muralidharan 2006), and provided in Tables 4 and 5 for the mode without power loss and Tables 6 and 7 of the mode with power loss. For the comparison purpose, importance weighting factors of each objective function is taken 5.0.

Table 4 Fuel cost comparison in various loads in 6-unit system without fuel cost power loss ($, h−1)
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Table 5 Emission comparison in 6-unit system without power loss
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Table 6 The best results obtained in various loads in 6-unit system with power loss
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Table 7 Fuel cost coefficients in 11-unit system (Palanichamy and Babu 2008)
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As seen in Tables above the proposed algorithm greatly outperforms the other techniques. In addition, ABC algorithm-based technique has outstanding results both in terms of fuel cost and emission reduction.

4.2 11-Unit system

This system’s characteristics consisting of fuel cost function’s coefficients, permissible production limits, and pollution function’s coefficients of each generating unit are provided in Tables 7 and 8 (Palanichamy and Babu 2008). In this condition, EED is defined for this system on various load levels between 1000 and 2500 MW. In order to compare the results of the proposed algorithm and other techniques, transmission network losses are neglected. As seen in Tables 9 and 10, the proposed algorithm has high accuracy. Compared to the other computational algorithms in this system, the proposed algorithm has the minimum fuel cost and minimum pollution on each load level (Table 11; Figs. 4, 5, 6).

Table 8 Pollution function coefficients in 11-unit system (Palanichamy and Babu 2008)
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Table 9 Fuel cost comparison in various loads in 11-unit system
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Table 10 Pollution comparison in various loads in 11-unit system
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Table 11 11-unit system
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Fig. 4
figure 4

Variations of fuel cost and pollution functions of 11-unit system

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Fig. 5
figure 5

Interaction curve among objectives of 6-unit system

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Fig. 6
figure 6

Interaction curve among objectives of 11-unit system

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As seen in Tables 9 and 10, the proposed algorithm has high accuracy and it reaches the least possible fuel cost and pollution on each load level compared to the other computational algorithms.

5 Conclusion

In this paper, Artificial Bee Colony (ABC) algorithm was used in order to solve economic dispatch problem considering reduction of costs related to operation and pollution in two standard systems. In fact, one approach for accurate prediction of power production cost in power systems is to model objective functions appropriately and precisely. Thus, in this paper, by taking into account these functions and employing a proper algorithm, this goal was realized. In order to show the effectiveness of the algorithm, obtained results of load flow calculations for case system using improved ABC algorithm were compared with those of various algorithms. Results of economic dispatch indicate that the proposed algorithm is highly effective in dealing with much more complicated problems. In the presented objective function, simultaneous minimization of production cost and reduction of transmission system losses were considered, leading to more successful achievements in finding optimal points near to global one. Thus, implementation of proposed algorithm in practical power systems is significantly effective in achieving more precise operating costs in these systems. For the purpose of ED problem, we will implement ABC algorithm on large standard and practical systems.