نویسندگان
دانشکده برق و کامپیوتر، دانشگاه تبریز، تبریز، ایران
چکیده
کلیدواژهها
موضوعات
عنوان مقاله [English]
نویسندگان [English]
In this paper, a new approach based on imperialist competitive algorithm (ICA) has been proposed to solve the combined heat and power economic dispatch (CHPED) problem. In order to avoid trapping in local optimum and improve the solution quality of the original ICA, a new assimilation policy has been addressed with varying coefficients during iterations. CHPED problem is a nonconvex and nonlinear optimization problem which has various constraints. Unlike previous methods, valve point effects are considered in some case studies and the effect of valvepoint in cost function considered with adding an absolute sinusoidal term to conventional polynomial cost function. To evaluate the effectiveness of the proposed method, three different test cases with small, medium and large scales have been applied to investigate the performance of the proposed method on the CHPED problems. Each case study is including different test systems. Numerical results demonstrate the superiority of the proposed framework and reveal that MICA can find better solutions in comparing with the other methods.
Keywords:
کلیدواژهها [English]
1 Introduction ^{[1]}
Economic load dispatch (ELD) is one of the fundamental optimization problems in power system analysis. The purpose of ELD is to determine the optimal scheduling of power generations to match total power demand at minimal possible cost while satisfying the power generators and system constraints. The cost of generation, particularly in thermal power plants is excessive, hence, suitable planning of unit outputs can contribute to significant saving in operating cost. Over the years, a wide variety of optimization techniques [16] have been adapted to solve ELD problems, each of them has advantages and disadvantages. However, the growing trend of energy consumption in recent years has been the world's energy crisis and with rising fuel prices and environmental concerns of the electricity industry, the optimal utilization of multiple combined heat and power (CHP) units has been become a fundamental problem in Electric Power System. The purpose of combined heat and power, also known as the simultaneous production, is the concurrent production of electricity and useful heat. CHP is an efficient and reliable approach to generating power and thermal energy from a single fuel source. This can greatly increase the effectiveness and reduction of operational energy costs. CHP also contribute to global climate change by reducing greenhouse gas emissions. Complication arises if both of heat and power demands are required to meet simultaneously. The utility of cogeneration unit over conventional power generating unit and heatonly unit is that it satisfies both heat and electricity demands in an economical way. It makes the CHPED problem more complex than the conventional ELD problem. Conversion from fossil fuels and coal to electricity is a complicated process and most of the heat energy is wasted through this conversion process. For this reason, efficiency achieved by most of the conventional power plants is only about 50–60%. CHP unit reduce fuel and primary energy consumption without compromising the quality and reliability of the energy supply to consumers. The best CHP system can increase the efficiency up to 80% or more at the point of use. Moreover, significant reduction of environmental pollutants like CO_{X}, SO_{X} and NO_{X} can be achieved by CHP system. Consequently, it provides a costefficient means of generating lowcarbon or renewable energies [7].
Several optimization algorithms have been employed for solving the CHPED problems in recent two decades which can be divided into two main groups: mathematical approaches and metaheuristics. Nonlinear optimization dual programming procedure that basically follows a twolevel strategy [8], Lagrangian relaxation approach [9] and BranchandBound algorithm [10] are considered as mathematical based methods which have been used to solve different CHPED problems. However, these algorithms are not able to solve discrete input and output modules, and/or nonconvex characteristics of generator fuel to be used. In order to overcome the disadvantages of the above methods, a variety of techniques based on artificial intelligence have been proposed for solving CHP problems [11]. Generally, metaheuristics can optimize different problems without considering the complexity and constraints of the problem. Given that the CHPED problem is nonconvex intrinsically, hence, the use of these methods seems reasonable.
Evolutionary Programming (EP) [12] was the first algorithm which has been developed for handling the CHPED problem in cogeneration systems. In this algorithm, new techniques for satisfying heat and power constraints has been suggested. A Genetic Algorithm (GA) based method entitled selfadaptive realcoded genetic algorithm (SARGA) has been proposed in [13] for solving CHPED problem. The proposed method uses a novel methodology to access constraints and has been tested on a simple cogeneration system and does not require any penalty parameters. Improved Genetic Algorithm (IGA) and IGA with multiplier updating (IGAMU) [14] are other algorithms based on genetic algorithm which are proposed for solving CHPED. In the proposed IGAMU method, it is assumed that the cost functions of heat generation plants and conventional power units are linear. In addition, recently, Haghrah et. Al. [15] have proposed a new version of real coded genetic algorithm with improved Mühlenbein mutation. The obtained results show that the suggested mutation is effective but unfortunately the selected test studies are partly small and the operation of this method on largescale test systems has not been investigated. In Ref. [16] Harmony Search Algorithm (HSA) is proposed to handle the CHPED problem. In order to better evaluate the HS method, a new case study has been introduced. However, this framework has not regarded the valvepoint effect in the formulation. Ant Colony Search Algorithm (ACSA) is another metaheuristic algorithm which has been proposed in [17] for solving the CHPED problem. Despite the acceptable ability search on small test systems, ACSA has some weaknesses such as premature convergence, access constraints and so on. Therefore, it has been proposed to improve the deficiencies of the algorithm by combining other methods. The first implementation of the cuckoo optimization algorithm for handling the complicated CHPED problem has been developed in [18]. Despite the acceptable results of this approach, the fiveunit system is the largest case study which has been investigated by this approach and the capability of this method on largescale test systems are not investigated. MohammadiIvatloo et al. have developed timevarying acceleration coefficients particle swarm optimization (TVACPSO) [19] to solve the nonconvex CHPED problems. In pursuance of conducting different constraints, appropriate penalty functions have been incorporated into objective function. In Ref. [20] a firefly algorithm is proposed to handle the reserve constrained combined heat and power dynamic economic emission dispatch problem. A noteworthy point of this work is that the ramp rate limits, valvepoint effect, and spinning reserve requirements have been considered concurrently. Besides, enhanced simultaneous idea to fulfill the constraints has been presented in this work which biases the optimization towards the feasible region without imposing any limits on the objective function. Optimal planning including sitting and sizing of CHPs among arbitrary buses has been considered by Pazouki et al. in Ref. [21]. Ref. [22] can be used as a suitable survey for studies related to shortterm scheduling of combined CHP. By and large, CHPED problem is a nonlinear, nonconvex, and nonsmooth problem and each method not only must face these challenges but also find better optimal solution as global as possible in a reasonable time.
Recently, a stateoftheart evolutionary algorithm inspired by social phenomenahuman, has been proposed by Atashpaz et al., named Imperialist Competitive Algorithm (ICA) [23]. This approach has been developed effectively to solve various real problems [2426] and seems adaptable enough to improve its exploitation and exploration abilities. The algorithm is basically based on the empire framework, which supposes that stronger empires “imperialist” try to extend their power over the other weak countries “colony”. However, original ICA does not have effective performance in confronting with complex or largescale optimization problems, and is likely to be trapped in the local optima. In addition, ICA has not been, to author’s knowledge, used for solving economic dispatch of cogeneration systems. On account of all aforementioned reasons, a modified ICA (MICA) algorithm is proposed in this paper. In the proposed ICA, the effect of the most powerful empire is also considered in assimilation policy and this process is modeled by moving all the colonies toward the relevant imperialist and the most powerful empire. Additionally, to encourage the individuals to wander through the entire search space and enhancement of the global exploration ability, TimeVarying Coefficients (TVC) as the parameter automation strategy are proposed to incorporate in the assimilation concept. The effectiveness of the proposed strategy has been validated by numerous test cases. In accordance with the numerical results, MICA not only is able to solve various CHPED problems, but also provides economic benefits comparing to the other optimization approaches in an acceptable computational time.
This paper is organized as follows: First, the characteristics of CHP units are expressed and the CHPED problem is formulated by considering the valve point effects and losses. ICA algorithms are briefly presented in the next section and then the proposed algorithm is introduced. In the next section, how to apply the proposed algorithm on CHPED problem is described. Then the performance of the proposed approach on several systems is studied and the results of other methods are compared with the results of classical ICA algorithm as well as other approaches. Finally, the conclusion of the paper is outlined.
2. Mathematical Formulation of CHPED Problem
2.1. Objective Function
The system under consideration has poweronly units, CHP units, and heatonly units. Fig. 1 shows the heatpower feasible operation region of a combined cycle cogeneration unit. The feasible operation region is enclosed by the boundary curve ABCDEF. Along the boundary curve BC, the heat capacity increases as the power generation decreases while the heat capacity decreases along the curve CD. The power output of the power units and the heat output of heat units are restricted by their own upper and lower limits. Usually the power capacity limits of cogeneration units are functions of the heat unit productions and the heat capacity limits are functions of the unit power generations [27].
Fig (1): Feasible Operation Region for a Cogeneration Unit.
The CHP dispatch problem of a system is to determine the unit heat and power production so as to minimize the total production costs while satisfying various constraints. The mathematical model of the CHPED problem can be expressed as follows:
(1) 
where , and are the number of conventional thermal units, cogeneration units and heat only units, respectively. , and are the fuel cost of conventional thermal unit i, the cost function of the cogeneration unit j and the cost of heatonly unit k for 1 h period. and are the heat and power output, respectively. The quadratic cost function of conventional units can be expressed as follows:
(2) 
where , and are the cost coefficients of the ith conventional thermal unit.
For a practical system, steam valve admission effects lead to the ripple in the production cost of generating unit. In order to model this effect more accurately, a sinusoidal term is added to the quadratic cost function [23]. Considering that the valvepoint effect increases the nonsmoothness and local optimal points of the solution space. So, the cost function with the valvepoint effects can be represented as:
(3) 
where and are the are the coefficients of generator reflecting valvepoint effects.
The production cost of cogeneration and heatonly units are expressed as follows:
(4) 

(5) 
where , , , , and are the cost coefficients of cogeneration units and , and are the cost coefficients of the kth heatonly unit.
2.2. Constraints
The necessary equality and inequality constraints for minimizing the optimizing function (1) are represented as follows:
2.2.1. Equality Constrains
(6) 
(7) 
and represent, respectively, the electric power and heat demand of the system.
2.2.1. Inequality Constrains
(8) 

(9) 

(10) 

(11) 
and represent the minimum and the maximum output power limits of the ith thermal power unit, in MW, respectively. , , and are the linear inequalities that define the feasible operating region of the jth CHP unit. and are the minimum and the maximum outputs heat of the kth heatonly unit, respectively.
3. Methodological Framework
3.1. Imperialist Competitive Algorithm
ICA for the first time is presented in 2007, inspired by the socialhuman phenomenon [23]. Similar to the other evolutionary algorithms, this algorithm also starts with initial random populations. Any individual of an empire is called a country. There are two types of countries; colony and imperialist state that collectively form empires. Imperialistic competitions among these empires develop the basis of the ICA. During this competition, weak empires collapse and powerful ones take possession of their colonies. Imperialistic competitions converge to a state in which there is only one empire and its colonies are in the same position and have the same cost as the imperialist, which represents the best solution of the matching problem.
First, the number of initial countries and the number of variables are determined. Then of most powerful countries are selected to form the empires. The remaining of the population will be the colonies which each of them belongs to an empire. The imperialist countries absorb the colonies towards themselves using the absorption (assimilation) policy. The absorption policy makes the main core of this algorithm and causes the countries move towards to their minimum. The total power of each empire is determined by the power of both parts: the imperialist power plus percent of its average colonies power. Pursuing assimilation policy, the imperialist states tried to absorb their colonies and make them a part of themselves. More precisely, the imperialist states made their colonies to move toward themselves along different socialpolitical axis such as culture, language and religion. In the ICA, this process is modeled by moving all of the colonies toward the imperialist along different optimization axis. This movement is shown in Fig. 2 in which the colony moves toward the imperialist by units and is reached from the previous position “” to the new position “”. If the distance between colony and imperialist is shown by, is a random variable with uniform (or any proper) distribution. New position of the colony is given as follows:
(12) 

(13) 
where is a number greater than 1 and in the most implementation a value of about 2 results in good convergence.
Fig (2): Moving Colonies toward their Relevant Imperialist (Assimilation Policy in the ICA Algorithm).
If a colony reaches a better point than an imperialist in its movement towards the imperialist country (equal to having more power than the country), it will replace with imperialist country. This causes the algorithm to continue with the imperialist country in a new location and in this time it is the new imperialist country which begins to apply assimilation policy for its colonies. The imperialistic competition consists in the dispute between empires in order to conquer the colonies of other empires. This event makes the most powerful empires increase their powers, while the weakest empires tend to decrease their power over time. The imperialistic competition can be modeled by choosing the weakest colony from the weakest empire to be disputed among the other empires. After a while, all the empires except the most powerful one will collapse and all the colonies will be controlled by this unique empire. More details on ICA can be found in [23].
3.2. Modified Imperialist Competitive Algorithm
As previously mentioned, assimilation policy in the imperialist competitive algorithm is affected only by the properties of their relevant imperialist whereas in the real world the impact of the most powerful empire on other colonies can also clearly be seen and the strongest imperialist tried to absorb the colonies and make them a part of itself. Indeed, the relevant imperialist and the most powerful imperialist attract the colony along different optimization axis as language and culture. In the proposed method, in addition to considering the effect of the central imperialist, the influence of the strongest empire in the various socialpolitical aspects is also considered. To enrich the searching behavior and to avoid being trapped into local optimum, assimilation process in the ICA algorithm is changed and a new absorbing process is introduced. In the modified ICA, the assimilation policy is modeled by moving all the colonies toward the relevant imperialist and the most powerful empire. This movement is shown in Fig. 3, in which a colony moves toward the relevant imperialist by units and also moves toward the most powerful imperialist by units and finally reach from the previous position to a new position .
If the distance between the colony and relevant imperialist is shown by and the distance between colony and the strongest empire is shown by ,, are random variables with uniform (or any proper) distribution and are defined as following:
(14) 

(15) 
Then new position of the colony can be defined by:

(16) 
and is also given as follows:
(17) 
where constant pulls the colonies towards best local position whereas pulls it towards the best global position. A proper choice for these coefficients can be a value of about 2 for and about 10 for in most of implementations. However, depending on the optimization problem, the coefficient may need to be changed. So, convergence and solution quality of the algorithms depends on the proper choice of coefficients. Relatively higher value of , compared with the , results in the roaming of individuals through a wide search space. On the other hand, a relatively high value of the social component leads particles to a local optimum prematurely. Therefore, setting the parameters is a key factor to find accurate and efficient solutions.
In populationbased optimization methods, the policy is to encourage individuals to roam through the entire search space during the initial part of the search, without clustering around local optima. During the latter stages, convergence towards the global optima is encouraged to find the optimal solution efficiently [2830].
So, the MICA technique with the timevarying coefficients (TVC) is proposed in this paper, to solve CHPED problems.
Fig (3): Assimilation Policy in the MICA Algorithm by Affecting of the most Powerful Empire and Relevant Imperialist.
The idea behind TVC is to enhance the global search in the early part of the optimization and to encourage the colonies to converge towards the global optima at the end of the search. This is achieved by changing the coefficients with time in such a manner that is reduced while the social component is increased as the search proceeds.
This modification can be mathematically represented as follows:
(18) 

(19) 
, , and are initial and final values of and , respectively.
The flowchart of the proposed MICA is shown in Fig. 4.
Fig (4): The Flowchart of the Proposed MICA Algorithm.
4. Application of MICA to the Problem
This section describes the procedural steps for the implementation of the proposed algorithm to the CHPED problem described above. In the CHPED problem, the real power output of the thermal conventional and cogeneration units and the heat output of cogeneration and heatonly units are considered as decision variables and are used to form the objective function of the problem. MICA approach implementation for solving CHPED can be summarized as the following steps:
Step 1: Input to the necessary data:
At this stage, the required data to solve the CHPED problem and necessary parameters for MICA are defined.
Step 2: Generate the initial population:
The problem independent variables are initialized somewhere in their feasible numerical range. The independent variables such as real power output of number of generating units, number of heat only units and real power and heat output of the CHP units are initialized randomly within their specified operating limits as follows:
(20) 

(21) 
where is a random generated number between 0 and 1, and has uniform distribution. In order to meet the equality constraints of the power demand and heat demand, power generation of the power generating unit without considering the loss and heat output of the heat generating unit are evaluated as follows:
(22) 

(23) 
Step 3: Calculate the fitness of the initial population:
In order to assess the situation of each country, the objective function using (1) is defined. The objective function should be minimized satisfying all constraints.
Step 4: Determine the initial empires and colonies:
Based on the cost function, initial empires and their colonies are identified.
Step 5: Update the new position of colonies:
MICA colonies in the new position will be updated as follows:


(24) 

(25) 
Step 6: Evaluate the colonies:
At this stage, the new position of the colonies is evaluated using the objective function and if the colonies have reached a higher status than their imperialists, their places are changed with each other. Then the algorithm continues with the new imperialists, and at this time the new imperialists start to do assimilation policy on their colonies.
Step 7: Imperialistic competition:
Any empire that is not able to succeed in this competition and can’t increase its power (or at least prevent decreasing its power) will be eliminated from the competition. The imperialistic competition will gradually result in an increase in the power of powerful empires and a decrease in the power of weaker ones. Weak empires will lose their power and ultimately will collapse.
Step 8: Check stopping criteria:
At this step exit condition of the loop is checked. If the convergence criteria are met, the loop breaks and the best imperialism results are shown as optimal solution, otherwise, it returns to Step 5. In the proposed method, if all empires collapse and only one empire remained or the maximum number of iterations is reached then, the stop condition is met.
5. Simulation Results and Discussion
To evaluate the performance and ability of the modified ICA, this algorithm is implemented and tested on several types of systems to validate the efficiency and scalability of the proposed method. In order to demonstrate the scalability of the proposed method, scale up is conducted based on a 24unit system contain 13 thermal units, 6 cogeneration units and 5 heatonly units. The commercial software MATLAB has been used for implementing the proposed algorithm, which has been performed on an Intel Core i77500U, 2.70 GHz laptop with 12 GB of RAM memory.
5.1. Setting the Simulation Parameters
In all experiments, the number of the initial countries and maximum iteration of both ICA and MICA algorithm are 80 and 1000, respectively. They are assumed unless the information is clearly mentioned in the case study.
In the ICA algorithm, and coefficients have been considered 2 and 0.02, respectively. In the proposed algorithm, the initial and final values of and have also been chosen 0.5 and 2.5, respectively and coefficient is selected equal to 0.02. It should be noted that due to the random nature of the ICA and MICA methods, 30 independent experiments have been carried out to compare the convergence characteristics and the quality of problem solving.
5.2. Case Study 1: Small Scale Systems (4 unitSystem without Considering Transmission Losses, 5 UnitSystem without Considering Transmission Losses with Three Different Scenarios)
A. 4Unit Test System
The first test case is a simple system proposed by [9] and is presented to demonstrate the quality and performance of the proposed method. The studied system consists of a conventional poweronly unit, two CHP units and a heatonly unit. All of constraints and cost functions of the conventional thermal units (unit 1) and a heatonly (unit 4) are assumed linear and are shown in the Eqs. (26) to (28), respectively. CHP units’ data as the cost function parameters are presented in [9]. The system power demand and heat demand are 200 MW and 115 MWth, respectively. The fuel cost function for the units in the system have already been mentioned in (4).
(26) 

(27) 

(28) 
Computational results obtained from this example by ICA and MICA are compared with other methods like LR [8], genetic algorithms [910] as IGA_MU, IGA and SARGA, TAVACPSO algorithm [29] and the comparing results are presented in Table 1. LR and SQP methods are converged to 9257.1 $ and other algorithms have also reached 9257.07 $. Table 1 shows the satisfactory solution results of this problem by MICA. Despite the closeness of the MICA to results of procedures such as IGA_MU (9257.07 $), the best result for solving this problem is for MICA (9257.0652 $) which is most likely to be a global optimal point for this problem. Also, according to the results, it can be seen that the proposed algorithm satisfies all constraints and all CHP units work in the defined feasible region.
Table (1): Comparison of Obtained Results by Different Methods for Case Study 1 Section A.
Methods 
Output 

Total Cost ($) 

LR [8] 
0 
160 
40 
40 
75 
0 
9257.1 
IGA_MU [10] 
0 
160 
40 
39.99 
75 
0 
9257.07 
SQP [7] 
0 
160 
40 
40 
75 
0 
9257.1 
SARGA[9] 
0 
159.99 
40.01 
39.99 
75.00 
0 
9257.07 
MADS [32] 
0 
160 
40 
40 
75 
0 
9257.07 
IGA [10] 
0 
160 
40 
39.99 
75 
0 
9257.09 
TAVACPSO [29] 
0 
160 
40 
40 
75 
0 
9257.07 
BD [22] 
0 
160 
40 
40 
75 
0 
9257.07 
ICA 
0 
160 
40 
40 
75 
0 
9257.07 
MICA 
0 
159.9996 
40.0004 
39.9910 
75.0089 
0 
9257.0221 
B. 5Unit Test System with Three Different Scenarios
In order to further test the proposed MICA in facing smallsize systems and comparing it with the other methods, another experiment with different scale, power and heat demands have been conducted. The system of Section B first is presented by Vasebi et al. in 2007 [16]. It consists of a conventional power unit, three cogeneration units and one heatonly unit. Experimental system for three different power and heat demand is considered. Power and heat demanded for three different scenarios are respectively: 300MW and 150 MWth (scenario I), 250 MW and 175 MWth (scenario II) and 160 MW and 220 MWth (scenario III). Cost functions and constraints of conventional and heatonly units are expressed in Eqs. (29)(31), respectively. The cost function parameters of the CHP units are presented in [16].
(29) 

(30) 

(31) 

Table (2): Comparison of Obtained Results by Different Methods for Case Study 1 Section B.
Scenarios 
Demand 
Methods 
Output 

I 
300 
150 
CPSO [19] 
135.0000 
40.7309 
19.2728 



TVACPSO [19] 
135.0000 
41.4019 
18.5981 



GSA [31] 
135.0000 
41.7806 
18.1736 



EMA [32] 
135.0000 
40.7163 
19.2837 



BD [27] 
135.0000 
40.7687 
19.2313 



ICA 
134.9963 
40.7309 
19.2728 



MICA 
135.0000 
40.7472 
19.1560 
II 
250 
175 
CPSO [19] 
135.0000 
40.3446 
10.0506 



TVACPSO [19] 
135.0000 
40.0118 
10.0391 



GSA [31] 
135.0000 
39.9998 
10.0000 



EMA [32] 
135.0000 
40.0000 
10.0002 



BD [27] 
135.0000 
40.0000 
10.0000 



ICA 
129.7710 
40.4355 
14.0021 



MICA 
135.0000 
39.9994 
9.9997 
III 
160 
220 
CPSO [19] 
35.5972 
57.3554 
10.0070 



TVACPSO [19] 
42.1433 
64.6271 
10.0001 



GSA [31] 
39.2183 
60.1454^{a} 
10.0000 



EMA [32] 
42.1433 
64.6378 
10.0000 



BD [27] 
42.1454 
64.6296 
10.0000 



ICA 
35.5789 
57.3554 
10.0070 



MICA 
41.6965 
63.8884 
10.1123 
Output 

Total Cost ($) 

105.0000 
64.4003 
26.4119 
0.0000 
59.1955 
13692.5212 
105.0000 
73.3562 
37.4295 
0.0000 
39.2143 
13672.8892 
105.0000 
74.0890 
37.3336 
0.0000 
38.5713 
13,671.1490 
105.0000 
73.7022 
36.7183 
0.0000 
39.5829 
13,672.7407 
105.0000 
73.5957 
36.7759 
0.0000 
39.6284 
13672.83 
105 
64.4003 
26.4119 
0.0000 
59.1878 
13692.4191 
104.9969 
73.6467 
36.7714 
0.0036 
39.5781 
13668.8326 
64.6060 
70.9318 
39.9918 
4.0773 
60.0000 
12132.8579 
64.9491 
74.8263 
39.8443 
16.1867 
44.1428 
12117.3895 
64.9807 
74.9844 
40.0000 
17.8939 
42.1095 
12,117.3700 
64.9997 
74.9980 
40.0001 
14.0624 
45.9394 
12,117.0785 
65.0000 
75.0000 
40.0000 
14.4029 
45.5971 
12116.60 
65.7911 
75.1881 
27.3526 
22.3190 
50.1401 
12253.1006 
64.9008 
75.0003 
40.0003 
14.4391 
45.5601 
12113.5990 
57.0587 
89.9767 
40.0025 
30.0232 
60.0000 
11781.3690 
43.2295 
96.2593 
40.0001 
23.7407 
60.0000 
11758.0625 
50.6296 
92.8700^{a} 
40.0000 
27.1044 
60.0000 
11745.5546 
43.2188 
96.2653 
40.0000 
23.7338 
60.0000 
11757.9124 
43.2250 
96.2614 
40.0000 
23.7386 
60.0000 
11758.06 
57.0587 
89.9767 
40.0025 
30.0232 
59.9976 
11781.2024 
43.3026 
95.6224 
40.0485 
24.2298 
60 
11754.9219 
This problem has already been solved using different evolutionary methods. The minimum cost obtained by other methods for solving this problem for comparing with performance of MICA is given in Table 2 that shows the best solution obtained from the proposed method and reported solutions by other authors. It is observable from Table 2 that after the implementation of the MICA on the scenario I, the total cost is equal to 13668.8326 $. This cost is much less than best results of CPSO [19] (13692.5212 $), TVACPSO [19] (13672.8892 $), GSA [31] (13,671.1490), EMA [32] (13,672.7407), BD [27] (13672.83) and ICA (13692.4191 $) methods. Scenario II in the previous scenario has less power and much heat demand than previous scenarios. The best total cost obtained by applying the MICA on Scenario II, is equal to 12113.5990 $, which is significantly lower than the results of the CPSO [19] with the best cost 12,132.8579 $ and ICA with the cost of 12253.1006 $. Also, despite being close to the results of TVACPSO [19], GSA [31], EMA [32] and BD [27], the obtained cost is less than the cost of them. Table 2 presents the optimal heat and power dispatches of scenario III. According to the results, in spite of multiple local optimal points, the proposed method is able to find a minimum cost of 11754.9219 $ which is 0.23%, 0.027%, 0.08%, 0.025% and 0.027% better than CPSO [19], TVACPSO [19], EMA [32], BD [27] algorithms respectively and also 0.22% is better than the result of ICA. It is clear from the results that the proposed MICA method can avoid the shortcoming of premature convergence and can approach to the global optimum.
5.3. Case Study 2: Medium Scale Systems (24 UnitSystem Considering Valve Point Effects and48 UnitSystem Considering Valve Point Effects)
A. 24Unit Test System Considering Valve Point Effects
In this case study, the simulation consists of mediumscale experiments to demonstrate the validity and efficiency of the proposed algorithm. Conventional thermal units based on the 13unit standard ELD test system which has a lot of local minimal and is one of the challenging ELD test cases [19, 33], which has been proposed by Mohammadi et. al [19].
Table (3): Optimal Dispatch Results of ICA and MICA Methods for Case Study 2 Section A.
Output 
Methods 
Output 
Methods 

ICA 
MICA 
ICA 
MICA 

628.3188 
628.3185 
117.4848 
80.9924 

63.1622 
299.1955 
45.9155 
40.0010 

0.0014 
299.1614 
9.9991 
10.0013 

179.9162 
109.8665 
42.1170 
34.9784 

179.9147 
109.8665 
125.2766 
104.8124 

179.9162 
109.8662 
80.1160 
75.0082 

179.9162 
59.9999 
125.2754 
104.8013 

179.9162 
109.8658 
80.1074 
75.0094 

179.9162 
109.8572 
39.9993 
40.0048 

39.9999 
39.9928 
23.2354 
19.9947 

50.0917 
77.0322 
415.9857 
470.3287 

55.0004 
54.9986 
60.0009 
60.0099 

55.0005 
54.9932 
60.0009 
60.0099 

117.4866 
81.0122 
120.0009 
120.0099 

45.9255 
39.9995 
120.0009 
120.0099 

Total Cost ($) 

ICA 

59431.8103 

MICA 

57823.1426 
The system consists of 13 poweronly units, 6 cogeneration units and 5 heatonly units. Power and heat demands are 2350MW and 1250MWth, respectively. All data units is presented in [19]. Detailed solutions to solve this CHPED problem by ICA and MICA are presented in Table 3. Table 4 shows also Comparison of the best, average and worst results obtained from this method and the results of other algorithms. It can be observed that the total cost obtained by this method (57823.1426 $) is significantly less than the cost of the procedures TLBO [5] (58006.9992 $), OTLBO [5] (57856.2676 $), CPSO [29] ((59736.2635 $, TVACPSO [29] (58122.7460 $) and ICA (59431.8103 $) which indicates the ability of the algorithm in dealing with the variousscale problems. Obtained results by the proposed method were respectively 0.27%, 0.012%, 3.26%, 0.47% and 2.74% which are less than the obtained costs of the TLBO [5], OTLBO [5], CPSO [29], TVACPSO [29] and ICA methods. Also, the worst result obtained by the proposed MICA method is 57954.9118 $, which is less than the best result of ICA that is59431.8103 $. This demonstrates the high efficiency of the proposed absorption strategy, which is used in the classical ICA algorithm and improves dramatically the local and global search ability. Also, it is noteworthy that the worst result obtained from the proposed method is better than the best results of the TLBO [5], CPSO [32] and TVACPSO [29].
Table (4): Comparison of Obtained Results by Different Methods for Case Study 2 Section A.
Methods 
Best Cost ($) 
Average Cost ($) 
Worst Cost ($) 
TLBO [7] 
58006.9992 
58014.3685 
58038.5273 
OTLBO [7] 
57856.2676 
57883.2105 
57913.7731 
GSA [31] 
58,121.8640 
58,122.7460 
59,736.2635 
EMA [32] 
57,825.4792 
57,832.7361 
57,841.1469 
CPSO [19] 
59736.2635 
59853.4780 
60076.6903 
TVACPSO [19] 
58122.7460 
58198.3106 
58359.5520 
ICA 
59431.8103 
59780.9874 
60188.2073 
MICA 
57823.1426 
57845.9767 
589421.0543 
B. 48Unit Test System Considering Valve Point Effects
To better illustrate the validity of MICA, another system that its number of units is twice as much as the number of units in the previous section used in this section. The system contains 26 thermal units, 12 CHP units and 10 heatonly units. The total power demand of this case is 4700 MW and heat demand is 2500 MWth. The unit characteristics of this system are similar to case study 2 section A and obtain by duplicating data of the previous section. Table 5 shows accurate heat and power dispatches using of ICA and MICA algorithms. Also, the best, worst and average optimal solutions by MICA and other algorithms can be seen in Table 6 and are compared with the obtained results of the TLBO [7], OTLBO [7], CPSO [19], CSA [34] TVACPSO [19] techniques and classical ICA. According to the presented results, the final cost of MICA is 116530.8610 $ which is significantly lower than other approaches, so, it is clear that the proposed method does not suffer from premature convergence and is able to find optimal solutions than other methods. However, the obtained power and heat results from all of methods confirm that the equality and inequality constraints are fulfilled by MICA and the proposed approach is operated in a bounded heat versus power plane.
Table (5): Optimal Dispatch Results of ICA and MICA Methods for Case Study 2 Section B.
Output 
Methods 
Output 
Methods 

ICA 
MICA 
ICA 
MICA 

538.5665 
628.3185 
10.9034 
10.0881 

76.4028 
225.3250 
37.9270 
39.3100 

68.6531 
224.7586 
109.3835 
82.0192 

135.5216 
159.7814 
61.1959 
40.1102 

161.9568 
109.9063 
111.9550 
81.2957 

145.3224 
159.7354 
55.3394 
45.6646 

120.3936 
109.8683 
22.9130 
13.8709 

147.8076 
109.8734 
54.7853 
30.3870 

135.9726 
159.7348 
108.0122 
107.5957 

112.0880 
40.9267 
87.6785 
125.4914 

108.2171 
41.1113 
104.7373 
105.2105 

74.2195 
55.1796 
88.4668 
82.6853 

65.2589 
92.4469 
40.3868 
40.0381 

248.0558 
448.7989 
21.3115 
21.9595 

299.2280 
225.5280 
120.5256 
105.3725 

299.6861 
75.5055 
92.0441 
75.0960 

142.5869 
160.1166 
122.1583 
104.9665 

138.8223 
110.1600 
88.2426 
79.8908 

141.4212 
159.7385 
45.4608 
41.6593 

142.9812 
159.7790 
28.9895 
17.9036 

119.5467 
159.7420 
417.0202 
436.0609 

139.5290 
160.1720 
59.5362 
60.0009 

77.8037 
40.2144 
59.9175 
60.0009 

81.7250 
40.3064 
119.9926 
120.0009 

110.3924 
92.6548 
118.4968 
120.0009 

111.6903 
92.4681 
418.2604 
436.0611 

95.1582 
85.9808 
59.7099 
60.0009 

54.6874 
98.4890 
59.9816 
60.0009 

86.2998 
81.7305 
119.2701 
120.0010 

55.6011 
48.9018 
119.7994 
120.0010 

Total Cost ($) 

ICA 


119499.3441 



MICA 


116530.8610 



Table (6): Comparison of Obtained Results by Different Methods for Case Study 2 Section B.
Solution technique 
Best Cost ($) 
Average Cost ($) 
Worst Cost ($) 
TLBO [7] 
116739.3640 
116756.0057 
116825.8223 
OTLBO [7] 
116579.2390 
116613.6505 
116649.4473 
CPSO [19] 
119708.8818 
NA* 
NA* 
CSA [34] 
116843.300 
117245.6 
117636.1 
TVACPSO [19] 
117824.8956 
NA* 
NA* 
ICA 
119499.3441 
119709.2169 
119954.2551 
MICA 
116530.8610 
116582.1498 
116594.7520 
^{*} Not available in the referred literature.
5.4. Case Study 3: Large Scale Systems (72 UnitSystem Considering Valve Point Effects and 96 UnitSystem Considering Valve Point Effects)
In order to better assess the ability of the proposed algorithm and test its performance in dealing with largesized problems, two largescale systems considering valve point effects is implemented which are presented for the first time in in Ref. [11]. and include 72 and 96 units.
A. 72Unit Test System Considering Valve Point Effects
The new proposed system consists of 39 conventional thermal units, 18 cogeneration units and 15 heatonly units. The system data is based on a 24unit system and is achieved by the triple repetition of that system. Power and heat system demands are 7050MW and 3750MWth, respectively. Power and heat dispatching results by ICA and MICA algorithms are described in Table 7. The total cost obtained by the proposed MICA and ICA are 173642.8608 $ and 181846.7024 $, respectively, which the best cost of MICA is 8203.8416 $ and is less than the achieved cost by ICA algorithm. The results show that the MICA is an accurate and efficient algorithm for dealing effectively with the difficult CHP problems and without using the proposed assimilation strategy, the classical ICA can easily trap in local optimal and shows poorer results compared with MICA which uses a new assimilation strategy. It is noteworthy that the CHPED problem is more complex by increasing the size, particularly in the nonsmooth and nonconvex problems. Therefore, the overall performance of each algorithm decreases by increasing of the problems size. However, the proposed algorithm has acceptable performance and achieves good optimal solutions in solving the 72 units with 90 variables problem. The presented results in Table 8 confirmed these issues. The initial population and maximum iterations for this problem are assumed 100 and 2000, respectively.
B. 96Unit Test System Considering Valve Point Effects
The new 96unit system is a large system to demonstrate the effectiveness of the proposed algorithm in solving the practical problems with large dimensions. The system consists of 52 conventional power generation units, 24 CHP units and 20 heatonly units that simply is achieved by expanding the 24unit system data. The degree of complexity of the CHP dispatch problem is related to the systemsize. The mutual dependencies of heatpower capacity make it hard to find a feasible region, not to mention the optimal capacity. The larger systemsize increases the nonlinearity as well as the number of equality and inequality constraints in the CHP dispatch problem [35]. The Number of decision variables of this problem is 120 that indicates the complexity of the problem. The best results obtained by the ICA and MICA are presented in Table 8. The results show that the proposed algorithm is able to satisfy all constraints of the problem and is able to find a more optimal solution than ICA. So, it can be said the proposed algorithm has high resolution quality, especially in solving the largesized problems. In order to demonstrate the convergence of ICA and MICA methods for solving CHPED problems, convergence characteristics of these methods, for example, for case study 3 section B, is shown in Fig. 5. To solve this problem, the initial population of 100 and a maximum of 2000 iterations are assumed.
Table (7): Optimal Dispatch Results of ICA and MICA Methods for Case Study 3 Section A.
Output 
Methods 
Output 
Methods 

ICA 
MICA 
ICA 
MICA 

190.7873 
564.9184 
87.2573 
80.9909 

206.3452 
299.1993 
83.4423 
39.9908 

156.7992 
224.3995 
112.6863 
80.9995 

115.6038 
109.8666 
40.5056 
39.9980 

115.1675 
109.8665 
12.4469 
9.9926 

145.2583 
109.8666 
17.7242 
34.9911 

160.2530 
109.8664 
97.7451 
80.9911 

144.9226 
109.8666 
103.6941 
39.9909 

113.8495 
109.8666 
103.1622 
80.9995 

83.2285 
39.9905 
60.1970 
39.9980 

114.8828 
39.9907 
15.3798 
9.9926 

57.5228 
55.0001 
62.7017 
34.9905 

90.3005 
55.0000 
111.8438 
104.8062 

448.7558 
628.3185 
93.0473 
75.0072 

299.2448 
299.1993 
139.2308 
104.8038 

299.2128 
299.1993 
110.9482 
75.0082 

63.6316 
109.8666 
47.0245 
40.0004 

98.5883 
109.8666 
4.1395 
20.0036 

172.9762 
159.7331 
108.3105 
104.8005 

178.9187 
109.8639 
112.5018 
75.0007 

177.1266 
109.8660 
122.5504 
104.8053 

160.7478 
109.8666 
75.4359 
75.0069 

40.2516 
39.9951 
41.0415 
40.0011 

62.9404 
40.0001 
2.4728 
20.0002 

118.3770 
54.9953 
114.1956 
104.8006 

92.4453 
54.9918 
129.9838 
75.0007 

448.8109 
628.3188 
117.2352 
104.8053 

218.5126 
299.1992 
92.4001 
75.0069 

59.4582 
299.0638 
18.4776 
40.0011 

162.2253 
109.8661 
32.5910 
20.0002 

109.4150 
109.8665 
428.4352 
470.4922 

171.3456 
159.7331 
59.9965 
60.0100 

175.8149 
109.8666 
59.9343 
60.0100 

156.6411 
109.8659 
119.9932 
120.0100 

160.9804 
109.8665 
119.9852 
120.0100 

42.8263 
39.9953 
433.1106 
470.2647 

46.0230 
39.9962 
1.8680 
60.0099 

90.1878 
55.0001 
59.9679 
60.0100 

97.6710 
54.9916 
119.9944 
120.0100 

93.5506 
81.0011 
119.9991 
120.0100 

60.9055 
39.9983 
396.4278 
470.2644 

142.3551 
80.9968 
60.0005 
60.0100 

81.6472 
39.9995 
59.9714 
60.0100 

26.3897 
9.9910 
120.0002 
120.0099 

0.1563 
34.9980 
119.3565 
120.0100 

Total Cost ($) 

ICA 
181846.7024 

MICA 
173642.8608 
Table (8): Optimal Dispatch Results of ICA and MICA Methods for Case Study 3 Section B.
Output 
Methods 
Output 
Methods 

ICA 
MICA 
ICA 
MICA 

628.3190 
502.0642 
82.1448 
80.9996 

224.5221 
299.1993 
47.0820 
39.9980 

224.4710 
224.3995 
13.9366 
9.9926 

161.4762 
109.8666 
29.9989 
34.9911 

111.3439 
109.8665 
102.2237 
80.9911 

160.8654 
109.8666 
109.4620 
39.9908 

108.5931 
109.8665 
100.4924 
80.9995 

114.3767 
109.8665 
53.8352 
39.9980 

162.2904 
109.8666 
22.4173 
9.9926 

40.1482 
39.9904 
20.7867 
34.9905 

41.7847 
39.9912 
104.0357 
80.9910 

55.7697 
55.0000 
75.0282 
39.9908 

116.8735 
55.0000 
95.4412 
80.9995 

451.1011 
628.3183 
50.8429 
39.9980 

224.4753 
299.1992 
15.9830 
9.9926 

74.2867 
299.1993 
26.4996 
34.9905 

161.5653 
109.8665 
107.5327 
104.8062 

108.8009 
109.8663 
125.9673 
75.0072 

160.7853 
159.7323 
106.3386 
104.8037 

159.7224 
109.8638 
88.6605 
75.0082 

161.8797 
109.8659 
40.0309 
40.0004 

131.5898 
109.8666 
21.9708 
20.0036 

44.9350 
39.9945 
105.1469 
104.8005 

47.4198 
40.0000 
75.5610 
75.0006 

90.9249 
54.9968 
105.4429 
104.8054 

95.4558 
54.9919 
81.1144 
75.0069 

450.1927 
628.3185 
41.6875 
40.0011 

226.7237 
299.1993 
17.7267 
20.0005 

75.1030 
299.1988 
116.7111 
104.8006 

157.4396 
109.8661 
129.2096 
75.0007 

109.8845 
109.8666 
111.6395 
104.8053 

162.1369 
159.7331 
86.9440 
75.0069 

176.7750 
109.8637 
45.3055 
40.0011 

163.4927 
109.8666 
13.5398 
20.0002 

163.1571 
109.8673 
117.7280 
104.8006 

41.0573 
39.9953 
105.2385 
75.0007 

40.2377 
39.9940 
112.9048 
104.8053 

93.0677 
54.9979 
84.3601 
75.0069 

95.0943 
54.9914 
42.5645 
40.0011 

448.9955 
628.3186 
12.3498 
20.0002 

227.0345 
299.1993 
358.1764 
470.5737 

74.9066 
299.1980 
59.9999 
60.0100 

159.2037 
109.8660 
59.9960 
60.0100 

118.0485 
109.8666 
119.9809 
120.0100 

161.7393 
159.7331 
120.0009 
120.0100 

159.6920 
109.8665 
437.2219 
470.2648 

160.2331 
109.8664 
60.0000 
60.0100 

160.4884 
109.8665 
60.0009 
60.0100 

40.9685 
39.9953 
120.0000 
120.0100 

40.0636 
39.9939 
119.9978 
120.0100 

92.0630 
54.9953 
434.3301 
470.2644 

90.7863 
54.9975 
60.0009 
60.0100 

85.8709 
81.0011 
59.9989 
60.0100 

99.0405 
39.9983 
118.8831 
120.0099 

85.0034 
80.9967 
119.9603 
120.0093 

55.8236 
39.9995 
436.1627 
470.2644 

10.0713 
9.9910 
60.0009 
60.0100 

39.3492 
34.9980 
59.6094 
60.0100 

81.6172 
80.9908 
120.0005 
120.0100 

40.6489 
39.9908 
120.0009 
120.0095 

Total Cost ($) 

ICA 


235860.8140 


MICA 


231494.8552 


Fig (5): Convergence Cost Curve of ICA and MICA Methods for the Case Study 3 Section B.
6. Conclusion
Combining cogeneration units to the conventional ED problem increases the complexity of the problem. In order to solve the problem and satisfy all constraints of the CHPED problem, considering valve point effects and losses, a new algorithm based on the colonial competitive algorithm is proposed in this study. The proposed strategy uses a new assimilation policy which considers the impact of the most powerful empire besides the effect of other imperialists. Experimental systems with various units (4, 5, 24, 48, 72 and 96unit) and constraints, with/without losses and valve point effects are applied to validate the modified algorithm. Quality of solutions, convergence characteristics and ability to find the nearglobal (maybe global) optimal solutions of the proposed method are obviously better than the classical ICA and the other stateoftheart algorithms. Besides, the results demonstrate the high potential of MICA in solving nonconvex with differentscale CHPED problems. Satisfactory performance of the MICA in this paper acknowledges that this method can be used as a suitable tool for solving many practical problems of power system in the future.
References
[1]. Q. Qin, S. Cheng, X. Chu, X. Lei, and Y. Shi, "Solving nonconvex/nonsmooth economic load dispatch problems via an enhanced particle swarm optimization," Applied Soft Computing, 2017.
[2]. M. Pradhan, P. K. Roy, and T. Pal, "Grey wolf optimization applied to economic load dispatch problems," International Journal of Electrical Power & Energy Systems, Vol. 83, pp. 325334, 2016.
[3]. C. Shilaja and K. Ravi, "Optimization of emission/economic dispatch using euclidean affine flower pollination algorithm (eFPA) and binary FPA (BFPA) in solar photo voltaic generation," Renewable Energy, Vol. 107, pp. 550566, 2017.
[4]. N. Ghorbani and E. Babaei, "RealCoded Genetic Algorithm with Smart Mutation for Solving Nonconvex Economic Dispatch Problems," Computional Intelligence in Electrical Engineering, Vol. 7, No. 1, pp. 13 22, 2016.
[5]. M. Abdoli and A. Hatami, "Dynamic economic emission dispatch incorporating wind farms using modified coevolutionary particle swarm optimization metaheuristic algorithm," Computional Intelligence in Electrical Engineering, Vol. 5, No. 4, pp. 31 44, 2015.
[6]. K. Zare, M. T. Haque, and E. Davoodi, "Solving nonconvex economic dispatch problem with valve point effects using modified group search optimizer method," Electric Power Systems Research, Vol. 84, No. 1, pp. 8389, 2012.
[7]. P. K. Roy, C. Paul, and S. Sultana, "Oppositional teaching learning based optimization approach for combined heat and power dispatch," International Journal of Electrical Power & Energy Systems, Vol. 57, pp. 392403, 2014.
[8]. F. J. Rooijers and R. A. van Amerongen, "Static economic dispatch for cogeneration systems," IEEE Transactions on Power Systems, Vol. 9, No. 3, pp. 13921398, 1994.
[9]. T. Guo, M. I. Henwood, and M. Van Ooijen, "An algorithm for combined heat and power economic dispatch," IEEE Transactions on Power Systems, Vol. 11, No. 4,pp. 17781784, 1996.
[10]. A. Rong and R. Lahdelma, "An efficient envelopebased Branch and Bound algorithm for nonconvex combined heat and power production planning," European Journal of Operational Research, Vol. 183, No. 1, pp. 412431, 2007.
[11]. E. Davoodi, K. Zare, and E. Babaei, "A GSObased algorithm for combined heat and power dispatch problem with modified scrounger and ranger operators," Applied Thermal Engineering, Vol. 120, pp. 3648, 2017.
[12]. K. P. Wong and C. Algie, "Evolutionary programming approach for combined heat and power dispatch," Electric Power Systems Research, Vol. 61, No. 3, pp. 227232, 2002.
[13]. P. Subbaraj, R. Rengaraj, and S. Salivahanan, "Enhancement of combined heat and power economic dispatch using self adaptive realcoded genetic algorithm," Applied Energy, Vol. 86, No. 6, pp. 915921, 2009.
[14]. C.T. Su and C.L. Chiang, "An incorporated algorithm for combined heat and power economic dispatch," Electric Power Systems Research, Vol. 69, No. 2, pp. 187195, 2004.
[15]. A. Haghrah, M. NazariHeris, and B. Mohammadiivatloo, "Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved Mühlenbein mutation," Applied Thermal Engineering, Vol. 99, pp. 465475, 2016.
[16]. A. Vasebi, M. Fesanghary, and S. Bathaee, "Combined heat and power economic dispatch by harmony search algorithm," International Journal of Electrical Power & Energy Systems, Vol. 29, No. 10, pp. 713719, 2007.
[17]. Y. Song, C. Chou, and T. Stonham, "Combined heat and power economic dispatch by improved ant colony search algorithm," Electric Power Systems Research, Vol. 52, No. 2, pp. 115121, 1999.
[18]. M. A. Mellal and E. J. Williams, "Cuckoo optimization algorithm with penalty function for combined heat and power economic dispatch problem," Energy, Vol. 93, pp. 17111718, 2015.
[19]. B. MohammadiIvatloo, M. MoradiDalvand, and A. Rabiee, "Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients," Electric Power Systems Research, Vol. 95, pp. 918, 2013.
[20]. T. Niknam, R. AzizipanahAbarghooee, A. Roosta, and B. Amiri, "A new multiobjective reserve constrained combined heat and power dynamic economic emission dispatch," Energy, Vol. 42, No. 1, pp. 530545, 2012.
[21]. S. Pazouki, A. Mohsenzadeh, S. Ardalan, and M.R. Haghifam, "Optimal place, size, and operation of combined heat and power in multi carrier energy networks considering network reliability, power loss, and voltage profile," IET Generation, Transmission & Distribution, Vol. 10, No. 7, pp. 16151621, 2016.
[22]. F. Salgado and P. Pedrero, "Shortterm operation planning on cogeneration systems: A survey," Electric Power Systems Research, Vol. 78, No. 5, pp. 835848, 2008.
[23]. E. AtashpazGargari and C. Lucas, "Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition," in Evolutionary computation, 2007. CEC 2007. IEEE Congress on, 2007, pp. 46614667: IEEE.
[24]. A. Fathy and H. Rezk, "Parameter estimation of photovoltaic system using imperialist competitive algorithm," Renewable Energy, 2017.
[25]. H. Illias, K. Mou, and A. Bakar, "Estimation of transformer parameters from nameplate data by imperialist competitive and gravitational search algorithms," Swarm and Evolutionary Computation, 2017.
[26]. M. Mehdinejad, B. MohammadiIvatloo, R. DadashzadehBonab, and K. Zare, "Solution of optimal reactive power dispatch of power systems using hybrid particle swarm optimization and imperialist competitive algorithms," International Journal of Electrical Power & Energy Systems, Vol. 83, pp. 104116, 2016.
[27]. H. R. Abdolmohammadi and A. Kazemi, "A Benders decomposition approach for a combined heat and power economic dispatch," Energy Conversion and Management, vol. 71, pp. 2131, 2013.
[28]. M. Clerc and J. Kennedy, "The particle swarmexplosion, stability, and convergence in a multidimensional complex space," IEEE transactions on Evolutionary Computation, Vol. 6, No. 1, pp. 5873, 2002.
[29]. Y. Ma, C. Jiang, Z. Hou, and C. Wang, "The formulation of the optimal strategies for the electricity producers based on the particle swarm optimization algorithm," IEEE Transactions on Power Systems, Vol. 21, No. 4, pp. 16631671, 2006.
[30]. K. T. Chaturvedi, M. Pandit, and L. Srivastava, "Particle swarm optimization with time varying acceleration coefficients for nonconvex economic power dispatch," International Journal of Electrical Power & Energy Systems, Vol. 31, No. 6, pp. 249257, 2009.
[31]. S. D. Beigvand, H. Abdi, and M. La Scala, "Combined heat and power economic dispatch problem using gravitational search algorithm," Electric Power Systems Research, Vol. 133, pp. 160172, 2016.
[32]. N. Ghorbani, "Combined heat and power economic dispatch using exchange market algorithm," International Journal of Electrical Power & Energy Systems, Vol. 82, pp. 5866, 2016.
[33]. K. Meng, H. G. Wang, Z. Dong, and K. P. Wong, "Quantuminspired particle swarm optimization for valvepoint economic load dispatch," IEEE transactions on power systems, Vol. 25, No. 1, pp. 215222, 2010.
[34]. T. T. Nguyen, D. N. Vo, and B. H. Dinh, "Cuckoo search algorithm for combined heat and power economic dispatch," International Journal of Electrical Power & Energy Systems, Vol. 81, pp. 204214, 2016.
[35]. S. S. S. Hosseini, A. Jafarnejad, A. H. Behrooz, and A. H. Gandomi, "Combined heat and power economic dispatch by mesh adaptive direct search algorithm," Expert Systems with Applications, Vol. 38, No. 6, pp. 65566564, 2011.
[1] Submission date: 18, 05, 2015
Acceptance date: 26, 11, 2017
Corresponding author: Ebrahim Babaei, Electrical Engineering Department Graduate University of Advanced Technology Kerman Iran