A modified imperialist competitive algorithm for combined heat and power dispatch

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 [1-6] 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 heat-only 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 cost-efficient means of generating low-carbon 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. Non-linear optimization dual programming procedure that basically follows a two-level strategy [8], Lagrangian relaxation approach [9] and Branch-and-Bound 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 non-convex 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 non-convex 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 self-adaptive real-coded 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 (IGA-MU) [14] are other algorithms based on genetic algorithm which are proposed for solving CHPED. In the proposed IGA-MU 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 large-scale 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 valve-point 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 five-unit system is the largest case study which has been investigated by this approach and the capability of this method on large-scale test systems are not investigated. Mohammadi-Ivatloo et al. have developed time-varying acceleration coefficients particle swarm optimization (TVAC-PSO) [19] to solve the non-convex 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, valve-point 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 short-term scheduling of combined CHP. By and large, CHPED problem is a non-linear, non-convex, and non-smooth 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 state-of-the-art evolutionary algorithm inspired by social phenomena-human, has been proposed by Atashpaz et al., named Imperialist Competitive Algorithm (ICA) [23]. This approach has been developed effectively to solve various real problems [24-26] 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 large-scale 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, Time-Varying 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 power-only units, CHP units, and heat-only units. Fig. 1 shows the heat-power 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:

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 heat-only 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:

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 valve-point effect increases the non-smoothness and local optimal points of the solution space. So, the cost function with the valve-point effects can be represented as:

where and  are the are the coefficients of generator  reflecting valve-point effects.

The production cost of cogeneration and heat-only units are expressed as follows:

where , , , ,  and  are the cost coefficients of cogeneration units and ,  and  are the cost coefficients of the kth heat-only 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

• Power Production and Demand Balance Constraint

and represent, respectively, the electric power and heat demand of the system.

2.2.1. Inequality Constrains

• Capacity Limit Constraints

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 heat-only unit, respectively.

3. Methodological Framework

3.1. Imperialist Competitive Algorithm

ICA for the first time is presented in 2007, inspired by the social-human 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 social-political 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:

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 social-political 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:

Then new position of the colony can be defined by:

and is also given as follows:

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 population-based 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 [28-30].

So, the MICA technique with the time-varying 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:

, ,  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 heat-only 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:

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:

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:

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 24-unit system contain 13 thermal units, 6 cogeneration units and 5 heat-only units. The commercial software MATLAB has been used for implementing the proposed algorithm, which has been performed on an Intel Core i7-7500U, 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 unit-System without Considering Transmission Losses, 5 Unit-System without Considering Transmission Losses with Three Different Scenarios)

A. 4-Unit 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 power-only unit, two CHP units and a heat-only unit. All of constraints and cost functions of the conventional thermal units (unit 1) and a heat-only (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).

Computational results obtained from this example by ICA and MICA are compared with other methods like LR [8], genetic algorithms [9-10] as IGA_MU, IGA and SARGA, TAVAC-PSO 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.

B. 5-Unit Test System with Three Different Scenarios

In order to further test the proposed MICA in facing small-size 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 heat-only 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 heat-only units are expressed in Eqs. (29)-(31), respectively. The cost function parameters of the CHP units are presented in [16].

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

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 $), TVAC-PSO [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 TVAC-PSO [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], TVAC-PSO [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 Unit-System Considering Valve Point Effects and48 Unit-System Considering Valve Point Effects)

A. 24-Unit Test System Considering Valve Point Effects

In this case study, the simulation consists of medium-scale experiments to demonstrate the validity and efficiency of the proposed algorithm. Conventional thermal units based on the 13-unit 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.

The system consists of 13 power-only units, 6 cogeneration units and 5 heat-only 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 $, TVAC-PSO [29] (58122.7460 $) and ICA (59431.8103 $) which indicates the ability of the algorithm in dealing with the various-scale 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], TVAC-PSO [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 TVAC-PSO [29].

Table (4): Comparison of Obtained Results by Different Methods for Case Study 2 Section A.

B. 48-Unit 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 heat-only 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] TVAC-PSO [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.

Table (6): Comparison of Obtained Results by Different Methods for Case Study 2 Section B.

^* Not available in the referred literature.

5.4. Case Study 3: Large Scale Systems (72 Unit-System Considering Valve Point Effects and 96 Unit-System Considering Valve Point Effects)

In order to better assess the ability of the proposed algorithm and test its performance in dealing with large-sized problems, two large-scale 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. 72-Unit Test System Considering Valve Point Effects

The new proposed system consists of 39 conventional thermal units, 18 cogeneration units and 15 heat-only units. The system data is based on a 24-unit 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 non-smooth and non-convex 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. 96-Unit Test System Considering Valve Point Effects

The new 96-unit 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 heat-only units that simply is achieved by expanding the 24-unit system data. The degree of complexity of the CHP dispatch problem is related to the system-size. The mutual dependencies of heat-power capacity make it hard to find a feasible region, not to mention the optimal capacity. The larger system-size increases the non-linearity 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 large-sized 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.

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