داودی, الناز, بابائی, ابراهیم. (1397). حل مساله پخش بار اقتصادی نیروگاههای تولید هم زمان برق و حرارت با استفاده از الگوریتم اصلاح شدهای بر پایه الگوریتم بهینهسازی رقابت استعماری. هوش محاسباتی در مهندسی برق, (), 0-0. doi: 10.22108/isee.2018.90070.0

الناز داودی; ابراهیم بابائی. "حل مساله پخش بار اقتصادی نیروگاههای تولید هم زمان برق و حرارت با استفاده از الگوریتم اصلاح شدهای بر پایه الگوریتم بهینهسازی رقابت استعماری". هوش محاسباتی در مهندسی برق, , , 1397, 0-0. doi: 10.22108/isee.2018.90070.0

داودی, الناز, بابائی, ابراهیم. (1397). 'حل مساله پخش بار اقتصادی نیروگاههای تولید هم زمان برق و حرارت با استفاده از الگوریتم اصلاح شدهای بر پایه الگوریتم بهینهسازی رقابت استعماری', هوش محاسباتی در مهندسی برق, (), pp. 0-0. doi: 10.22108/isee.2018.90070.0

داودی, الناز, بابائی, ابراهیم. حل مساله پخش بار اقتصادی نیروگاههای تولید هم زمان برق و حرارت با استفاده از الگوریتم اصلاح شدهای بر پایه الگوریتم بهینهسازی رقابت استعماری. هوش محاسباتی در مهندسی برق, 1397; (): 0-0. doi: 10.22108/isee.2018.90070.0

حل مساله پخش بار اقتصادی نیروگاههای تولید هم زمان برق و حرارت با استفاده از الگوریتم اصلاح شدهای بر پایه الگوریتم بهینهسازی رقابت استعماری

مقالات آماده انتشار، اصلاح شده برای چاپ ، انتشار آنلاین از تاریخ 28 آذر 1397

در این مقاله یک روش جدید بر پایه "الگوریتم رقابت استعماری (ICA)" برای حل مساله "پخش بار اقتصادی تولید هم زمان برق و حرارت "(CHPED) پیشنهاد میشود. به منظور جلوگیری از به دام افتادن ICA در بهینههای محلی و بهبود کیفیت شبیهسازی، سیاست همسانسازی جدیدی معرفی میشود که به طور انطباقی در هر تکرار تغییر میکند. مساله CHPED یک مساله بهینهسازی غیرخطی و غیرمحدب میباشد که دارای قیود مختلفی میباشد. بر خلاف روشهای قبلی، اثر تلفات و موقعیت شیر در بعضی مثالها در نظر گرفته شده و به وضوح در تابع متعارف هزینه به صورت عبارت سینوسی دقیقی فرموله شده است. به منظور ارزیابی کارآیی روش پیشنهادی سه نوع مثال مختلف با ابعاد کوچک، متوسط و بزرگ که هر کدام دارای سیستمهای آزمایشی مختلفی میباشند با هدف پیادهسازی بر روی روش پیشنهادی به کار رفته است. قابل ذکر است در حل مسأله CHPED دو سیستم جدید با ابعاد بزرگ با در نظر گرفتن اثرات موقعیت شیر در این مقاله در نظر گرفته شده است. نتایج عددی نشان از برتر بودن و کیفیت بالای حل الگوریتم پیشنهادی در مقایسه با سایر روشها دارد.

A modified imperialist competitive algorithm for combined heat and power dispatch

نویسندگان [English]

Elnaz Davoodi^{}؛ Ebrahim Babaei^{}

چکیده [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 non-convex and non-linear optimization problem which has various constraints. Unlike previous methods, valve point effects are considered in some case studies and the effect of valve-point 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.

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, which each of them has advantages and disadvantages. However, the growing trend of energy consumption in recent years leads to 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 the operational energy costs. CHP also contributes to global climate change by reducing greenhouse gas emissions. Complication arises from the demand of heat and power 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 of the 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 of 50–60%. CHP units reduce fuel and primary energy consumption without compromising the quality and reliability of the energy supply to consumers. Best CHP systems 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 systems. Consequently, it provides a cost-efficient means of generating low-carbon or renewable energy [7].

Several optimization algorithms have been employed for solving the CHPED problems in recent two decades which can split them 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 the generators cannot be considered. 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 in recent years [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 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 the 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 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 (HS) is proposed to handle the CHPED problem. In order to better evaluation of 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 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, failure to meet the constraints and so on. Therefore, it has been proposed to improve the deficiencies of the algorithm by combining with 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, 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 been 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. 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 [21]. Ref. [22] can be used as a suitable survey for researches related to a short-term scheduling of the CHP systems. By and large, CHPED problem is a non-linear, non-convex, and non-smooth problem, hence each method not only must face these challenges but also find better optimal solution as possible as a global solution 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) algorithm [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, to the best of our knowledge, ICA has not been applied 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 imperialist is also considered in assimilation policy and this process is modeled by moving all the colonies toward the relevant imperialist and the most powerful imperialist. 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. 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 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 problems is described. Then the performance of the proposed approach on several systems is studied and the results of other methods are compared with those of classical ICA as well as the 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 heat and power production of the units in such a way that the total production costs is minimized and the corresponding constraints are satisfied. 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 heat-only unit k for 1 h period. P and H 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 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 valve-point effects 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:

(3)

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

The production cost of cogeneration and heat-only 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 heat-only unit.

2.2. Constraints

The necessary equality and inequality constraints for minimizing the optimizing function (1) are represented as the following:

2.2.1. Equality constrains

Power production and demand balance constraint

(6)

(7)

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

2.2.1. Inequality constrains

Capacity limit constraints

(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 heat-only unit, respectively.

3. Methodological framework

3.1. Imperialist competitive algorithm

ICA for the first time in 2007, inspired by the social-human phenomenon, is presented [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 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 x units and reaches from the previous position “”to the new position “”. If the distance between colony and imperialist is shown by d and x 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 leads to a good convergence.

Fig. 2. Moving colonies toward their relevant imperialist (assimilation policy in ICA).

If a colony reaches a better point than an imperialist in its movement towards the imperialist country (be more powerful than the current imperialist), it will replace with the imperialist country. Then the algorithm continues with the imperialist country in a new location and the assimilation policy is applied 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 of 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 imperialist on the other colonies also clearly seen and the strongest imperialist tried to absorb the colonies and make them as a part of itself. Indeed, 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 country 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 ICA is changed and new absorbing process is introduced. In the modified ICA, the assimilation policy is modeled by moving all the colonies toward the relevant and the most powerful imperialists. 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 reaches from the previous position to a new position .

Fig. 3. Assimilation policy in the proposed MICA considering the effect of the most powerful imperialist.

Define the distance between colony and the relevant imperialist by and the distance between colony and the strongest imperialist by . In addition, consider , as random variables with uniform (or any proper) distribution as follows:

(14)

(15)

Then new position of the colony can be defined by:

(16)

and

(17)

where constant pulls the colonies towards local best position whereas pulls it towards the global best position. A proper choice for these coefficients can be a value of about 10 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 roaming of the 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, MICA with the time-varying coefficients (TVC) is proposed in this paper, to solve CHPED problems. 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.

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. The implementation of MICA for solving CHPED can be summarized as the following steps:

Step 1: Input 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 independent variables of the problem 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, which 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 imperialists and colonies:

Based on the cost function, initial imperialists and their colonies are identified.

Step 5: Update the new position of colonies:

In MICA the new position of colonies 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: Imperialists competitive:

Any empire that is not able to succeed in this competition and can’t increase its will be eliminated from the competition. The imperialistic competition will gradually result in an increase in the power of powerful imperialists and a decrease in the power of weaker ones. Weak imperialists will lose their power and ultimately they 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, returns to Step 5. In the proposed method, if all imperialists collapse and only one imperialists remained or the maximum number of iterations is reached then 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 validity 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 is including of 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.

Fig. 4. The flowchart of the proposed MICA algorithm.

5.1. Setting the simulation parameters

In the most of experiments, the number of the initial countries and maximum iteration of both ICA and MICA algorithm, respectively, 80 and 1000 are assumed; otherwise the information is clearly mentioned in the relevant case study.

In ICA, 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 equivalent to 0.02. It should be noted that due to the random nature of the ICA and MICA, 30 independent experiments have been carried out to compare the convergence characteristics and the quality of problem solution.

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)-(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 those of the other methods like LR [8], genetic algorithms [9-10] as IGA_MU, IGA and SARGA, TAVAC-PSO [29]. The comparing results are presented in Table 1. LR and SQP methods have converged to 9257.1 $ and the other algorithms have also reached to 9257.07 $. Table 1 shows the satisfactory solution results of this problem by MICA. Despite the closeness of the results of MICA to those of the other procedures such as IGA_MU (9257.07 $), the best result for solving this problem belongs to MICA (9257.0652 $) which is most likely to be a global optimal point for this problem. Also, according to the results, can be seen that the proposed algorithm satisfies all constraints and all CHP units work in the defined feasible region.

B. 5-unit test system with three different scenarios

In order to further test of the proposed MICA in facing of small-size systems and comparing with the other methods, another experiment with different scale, power and heat demands have been conducted. The system of Section B, firstly 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. The demand of power and heat 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 the conventional and heat-only units are expressed in Eqs. (29)-(31), respectively. The cost function parameters of the CHP units are presented in [16].

(29)

(30)

(31)

This problem has already been solved using different evolutionary methods. The minimum cost obtained by other methods for solving this problem for comparing to the performance of MICA is given in Table 2 that shows the best solution belongs to the proposed method among the reported solutions. It is observable from Table 2 that after implementation of the MICA on the scenario I, the total cost is equal to 13668.8326 $. This cost is much less than the 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 $). Scenario II has less power and much heat demand than the previous scenarios. 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], obtained cost is less than the cost of them. Table 2 presents the optimal heat and power dispatches of scenario III.

Table (1): Comparison of the 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

Table (2): Comparison of the obtained results by different methods for case study 1 section B.

Scenarios

Demand

Methods

Output

Total Cost ($)

I

300

150

CPSO [19]

135.0000

40.7309

19.2728

105.0000

64.4003

26.4119

0.0000

59.1955

13692.5212

TVAC-PSO [19]

135.0000

41.4019

18.5981

105.0000

73.3562

37.4295

0.0000

39.2143

13672.8892

GSA [31]

135.0000

41.7806

18.1736

105.0000

74.0890

37.3336

0.0000

38.5713

13,671.1490

EMA [32]

135.0000

40.7163

19.2837

105.0000

73.7022

36.7183

0.0000

39.5829

13,672.7407

BD [27]

135.0000

40.7687

19.2313

105.0000

73.5957

36.7759

0.0000

39.6284

13672.83

ICA

134.9963

40.7309

19.2728

105

64.4003

26.4119

0.0000

59.1878

13692.4191

MICA

135.0000

40.7472

19.1560

104.9969

73.6467

36.7714

0.0036

39.5781

13668.8326

II

250

175

CPSO [19]

135.0000

40.3446

10.0506

64.6060

70.9318

39.9918

4.0773

60.0000

12132.8579

TVAC-PSO [19]

135.0000

40.0118

10.0391

64.9491

74.8263

39.8443

16.1867

44.1428

12117.3895

GSA [31]

135.0000

39.9998

10.0000

64.9807

74.9844

40.0000

17.8939

42.1095

12,117.3700

EMA [32]

135.0000

40.0000

10.0002

64.9997

74.9980

40.0001

14.0624

45.9394

12,117.0785

BD [27]

135.0000

40.0000

10.0000

65.0000

75.0000

40.0000

14.4029

45.5971

12116.60

ICA

129.7710

40.4355

14.0021

65.7911

75.1881

27.3526

22.3190

50.1401

12253.1006

MICA

135.0000

39.9994

9.9997

64.9008

75.0003

40.0003

14.4391

45.5601

12113.5990

III

160

220

CPSO [19]

35.5972

57.3554

10.0070

57.0587

89.9767

40.0025

30.0232

60.0000

11781.3690

TVAC-PSO [19]

42.1433

64.6271

10.0001

43.2295

96.2593

40.0001

23.7407

60.0000

11758.0625

GSA [31]

39.2183

60.1454^{a}

10.0000

50.6296

92.8700^{a}

40.0000

27.1044

60.0000

11745.5546

EMA [32]

42.1433

64.6378

10.0000

43.2188

96.2653

40.0000

23.7338

60.0000

11757.9124

BD [27]

42.1454

64.6296

10.0000

43.2250

96.2614

40.0000

23.7386

60.0000

11758.06

ICA

35.5789

57.3554

10.0070

57.0587

89.9767

40.0025

30.0232

59.9976

11781.2024

MICA

41.6965

63.8884

10.1123

44.3026

95.6224

40.0485

24.2289

60

11754.9219

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 minima, is one of the challenging ELD test cases [19, 33]. This test system proposed by Mohammadi et. al [19], consists of 13 power-only units, 6 cogeneration units and 5 heat-only units. Power and heat demands are 2350MW and 1250MWth, respectively. All units’ data 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 those of other algorithms. It can be observed that the total cost obtained by this method (57823.1426 $) 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, respectively, 0.27%, 0.012%, 3.26%, 0.47% and 2.74% is 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. It is noteworthy that the worst result obtained from the proposed method is also better than the best results of the TLBO [5], CPSO [32] and TVAC-PSO [29].

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

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 number of units in the previous section, is 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 units’ characteristics of this system are similar to the case study 2 section A and obtains 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 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 those of other approaches. So, it is clear that the proposed method don’t suffers 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.

5.4. Case study 3: Large scale systems (72 unit-system considering valve point effects and 96 unit-system considering valve point effects)

In the interest of better assessment of MICA 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.

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 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 $ 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 solutions 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 considering the non-smooth and non-convex version. Therefore, the overall performance of each algorithm decreases by increasing the size of the problems. However, the proposed algorithm has acceptable performance and achieves to good optimal solutions at handling the 72 units with 90 variables problem. The presented results in Table 8 confirm these issues. The initial population and maximum iterations for this problem are assumed 100 and 2000, respectively.

Table (4): Comparison of the 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

TVAC-PSO [19]

58122.7460

58198.3106

58359.5520

ICA

59431.8103

59780.9874

60188.2073

MICA

57823.1426

57845.9767

589421.0543

Table (5): Optimal dispatch results of ICA and MICA 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

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. 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 highly better solution than ICA. So, it can be said that the proposed algorithm has high resolution quality, especially at dealing with the large-scale problems. In order to demonstrate how 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.

Fig. 5. Convergence cost curve of ICA and MICA for the case study 3 Section B.

Table (6): Comparison of the 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

TVAC-PSO [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.

Table (7): Optimal dispatch results of ICA and MICA 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 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

6. Conclusion

Combining the 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 imperialists besides the effect of the other imperialists. Experimental systems with various units (4, 5, 24, 48, 72 and 96-unit) 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 near-global (maybe global) optimal solutions of the proposed method is obviously better than the classical ICA and the other state-of-the-art algorithms. Besides, the results demonstrate that the high potential of MICA in solving non-convex with different-scale CHPED problems. Satisfactory performance of MICA in this study, acknowledges that this method can be used as a suitable tool for solving many practical problems of a power system in the future.