Solution of Combined Economic Emission Dispatch Problem Using Improved and Chaotic Population-Based Polar Bear Optimization Algorithm

This paper proposes a novel improved polar bear optimization (IPBO) algorithm and employs it along with polar bear optimization (PBO) and chaotic population-based variants of polar bear optimization algorithm to solve combined economic emission dispatch (CEED) problem. PBO is a meta-heuristic technique inspired by the hunting mechanisms of polar bears in harsh arctic regions based only on their sense of sight. Polar bears in nature exhibits hunting of prey not only on their sight but also on their keen sense of smell. Hence, a novel improved variant of PBO which enhances its operation by equipping it with tracking capabilities utilizing polar bears sense of smell has been proposed in this study. The validity of novel IPBO is tested through 5 benchmark functions and 140 units Korean ED problem. Furthermore, the impact of different population initialization methods is also observed on the capabilities of conventional PBO. The proposed chaotic population based PBO, improved PBO (IPBO) and PBO are employed to solve IEEE 3 unit and 6-unit CEED problem. CEED is a multi-objective power system optimization problem with conflicting objectives of cost and emission. The simulations performed undertake each objective individually as well as collectively. The results achieved by each technique are analyzed statistically through Wilcoxon rank sum test (WRST), probability density function and cumulative density function. Both the statistical and numerical analysis of results showcase the strength of each solution technique as well as their ability to improve cost and emissions in the solution of CEED problem.


I. INTRODUCTION
All global energy trends indicate monumental increase in electric energy demand in coming years. According to IEA [1], global electrical energy demand increased in 2017 by 2%. Despite the environmental awareness and focus of The associate editor coordinating the review of this manuscript and approving it for publication was Zhouyang Ren . governments to integrate renewable into the grid, this increased demand was largely fulfilled by traditional thermal generators. The renewable sources contribution was increased by a percentage of 15.1% for wind and 21.9% for PV in the year 2017 but still the total contribution of renewable in global energy mix stands at 26.1%. The thermal sources contribute a whopping 59.4% of global electricity demand. Renewable sources are prime candidate in distributed generation (DG) and several renewable issues in DG are under research [2]. While thermal generators Thermal generators have proven their mantle for many decades and our current grids are designed to work around them. Despite their efficiency, resilience, and reliability they come with the added baggage of environmentally hazardous emissions. The SO 2 , CO 2 , and NO X emissions from thermal units have adverse environmental impact and contribute to global warming. As the renewable energy future is still in progress, the emerging challenge nowadays is to handle thermal generation sources such that we can obtain maximum power from them at reduced cost and emissions [3]. The action of controlling thermal generators for a fixed load demand under several physical and operational constraints is a renowned problem of power system operation characterized as economic dispatch (ED) [4]. ED is a specific objective optimization problem with the intention to attain power dispatch at least probable cost with no violation of any constraints. The addition of emission objective turns ED into multi-objective problem aiming minimization of both cost and emission for a power demand without violating any constraints. This multi-objective problem is termed as combined economic emission dispatch (CEED) problem [5], [6].
All these algorithms are population-based strategies having fixed population and they locate the optimum solution within a search space using two distinct stages of search namely local and global search [46]. All these algorithms were successful in achieving solution of the desired problem with varying degree of accuracy and time [47], [48]. Despite the achievement of a successful solution from these techniques the research for a better solution is continuous because of availability of new solution techniques being developed and the opportunity in optimized solution outlined by no free lunch theorem (NFL) [49], [50]. Initially PBO was used by author to solve ED problem [51] and it showed remarkable results. But as stated by NFL theorem the prospect to enhance mechanism of PBO and the possibility to achieve better solution of ED and CEED problem through this proposed novel IPBO were main motivating factors behind this research. In this paper we present solution of CEED problem using polar bear optimization (PBO) [52] algorithm, chaotic population PBO and a novel Improved Polar Bear Optimization (IPBO) algorithm. PBO is a nature instigated population-based metaheuristic approach that simulates the hunting abilities of polar bears in nature. PBO has three couplet stages of search such as global search, local search, and dynamic population. Different to other population methods PBO has capacity to change its population hence decreased number of calculations per iteration causes reduction in time required for execution. The proposed novel IPBO augments the working of PBO and is initially validated by applying it to 5 benchmark functions and large scale 140-unit Korean grid ED problem. The proposed techniques are utilized to work out CEED 3 unit and 6-unit system and the results attained are compared with outcomes in literature.
In this paper, the sections are organized as follows, first section presents the outline of CEED problem, second section gives summary and mathematical formulations of PBO, novel improved PBO (IPBO) and chaotic PBO variants, third section shows the case studies comprising simulation results along with statistical analysis among PBO variants, finally fourth section conveys the conclusion.

II. PROBLEM FORMULATION
CEED is a multi-objective constrained optimization problem with the objective of arranging electrical power outputs from varied generation units such that the entire operational cost and emission is minimalized with no violation of the respective constraints like generation limits, power balance and valve point effect. CEED problem may also involve the calculation of transmission loses acquired by every single generating unit at its corresponding power output. Arithmetically, the main purpose of CEED problem is reduction of VOLUME 9, 2021 operational cost and emission of generation entities that can be presented in equation (1) as (1) where, N x is the total number of generation units, i represents the i th generator under consideration and W is the weightage factor which determines contribution of fuel cost or emission in total objective value, its value is in range (0,1). F P i and E P i indicate total fuel cost ($/h) and emissions (ton/h) for i th unit respectively and are explained in mathematical form below.
Here equation (2) signifies the quadratic estimate of thermal units fuel cost curves without valve point effect while equation (3) characterizes the detailed cost equation comprising the valve point impact. In above equations a, b, c, e and f are cost coefficients, N x is the maximum number of generation units available for scheduling, P i is the i th generating unit and P il is the least power limit of i th generating unit.
Similarly, from [34] and [53] the emission from each thermal generation can be defined as summed quadratic and exponential functions as: where α, β, η, ξ and λ are emission coefficients. The CEED must comply with the following equality and inequality constraints.

A. EQUALITY CONSTRAINTS
Equality constraints include power generation balance that the load demand is met by considering the transmission line losses shown in equation (5).
where P generated is the total power scheduled, P required is the power demand and P loss is the transmission loss incurred at respective level of power scheduled and can be computed from loss coefficient matrix B formed by Kron's transmission loss formula shown in Eq. (6).
where; B ik , B i0 and B 00 are transmission loss coefficients.

B. INEQUALITY CONSTRAINTS
In CEED, Inequality constraint is mainly named as generation limits on each generator shown in equation (7).
where; P il and P ih are the lower and upper limits of i th generation unit and P i is the power scheduled on the i th generation unit.

C. PENALTY FUNCTION
The overall fitness function including equality constraints and objective can be obtained by penalty function formed as equation (8).

III. OVERVIEW OF PROPOSED METHODOLOGY
A general overview and mathematical description of each technique under consideration is presented below.

A. POLAR BEAR OPTIMIZATION ALGORITHM (PBO)
Polar bear optimization [51] is a population based metaheuristic optimization algorithm that simulates the hunting abilities of polar bear in severe arctic territories. PBO algorithm has three distinctive phases of search in search space namely local search by encircling and catching prey, global search by gliding ice floats and dynamic population. Each of these stages represents some vital characteristic of Polar Bear's hunting method in arctic zones and is described below. PBO algorithm begins its search by arbitrarily adjusting each polar bear having n coordinates as characterized bȳ x = (x 0 , x 1 , . . . . . . , x n−1 ) and then propels itself to find optimum solution in search space using global and local search strategies.
Global search process imitates Polar Bears nature to glide on arctic ice bergs in exploration of food, this behavior is modeled using following equation.
where; (x j t ) i is movement of i th polar bear having j coordinates in t th iteration towards the optimum, α is random number in range (0, 1), ω is distance between the present bear and optimum bear and γ is random number in the range (0, ω). The distance is dealt in Euclidian metrics and is given as During local search, the bears surround the prey and shot it with their teeth. This performance is efficiently modeled employing trifolium equations. To transmute polar bears behavior into these equations two parameters are characterized known as distance of vision 'a' chosen at random in range (0, 0.3) and angle of tumbling o chosen at random in range (0, π 2 ). From these limits, radius of vision is computed as This radius is utilized to calculate movement in local search space for each spatial coordinate correspondingly as where 1 , 2 and 3 are chosen randomly in the range (0, π).
Ultimately, to model the impact of severe arctic climatic conditions and bring in uncertainty to the optimization strategy, PBO algorithm initializes with 75% of population while the left over 25% depends on population growth controlled by reproduction of best or malnourishment of worst. To execute this approach a new constant k is introduced having value in range (0, 1). Dependent on k, creation or destruction of individuals will be performed according to following ruling.
The individuals are destroyed reliant on k until population in more than 50% while the reproduced individual is provided as where;x t (best) j the best solution is up to current iteration and x t (i) j is selected arbitrarily from among top 10% of best individuals up to current iteration.

B. CHAOTIC POPULATION PBO
In chaotic population-based version of PBO we simply initialize the population of polar bears based on chaotic tent map. Chaotic tent map [54] is mathematically defined as where x (0) is randomly selected from range (0, 1) such that x (0) is not equal to 1/2, 1/4, 2/3 and 3/4. Value represents the scaling factor to which the generated chaotic population will be scaled to. In our case we have used to scaling values.
C. IMPROVED PBO PBO algorithm was designed to mimic the hunting capabilities of polar bears based on their sense of sight completely ignoring polar bears scavenging capabilities. Polar bears have very sharp sense of smell and they make use of it during extreme conditions to find food. To incorporate this feature into already existing PBO algorithm we devised a unique two-tier global search stage in which 1 bear among 30% of least fit bears is selected to undergo global search based on their sense of smell mimicking its scavenging capabilities in extreme food shortage. This behavior is modeled using Levi flight equation (18) taken from [55] as shown below.
where L is the levy factor that maps the random flight behavior of birds. Here it is used here to map wind movement which carries the smell. So, at a global search stage two bears take two different trajectories, most fit bear undergoes ice float global search whereas least fit bears resort to scavenging. In this paper the proposed techniques will be used to solve CEED problem. The overall solution strategy for solution of CEED problem followed by each technique is outlined in flowchart shown in Fig. 1.

IV. SIMULATION RESULTS
Before tackling the CEED system the validity of proposed novel IPBO was tested by applying it to unimodal and multi-modal benchmark functions and compared with the state of art approaches. IPBO was also implemented to take on large scale 140-unit Korean grid ED problem at a demand VOLUME 9, 2021 of 49342 MW under two distinct cases. Finally, IPBO along with chaotic population based PBO variants were applied to solve.
• IEEE 3-unit CEED test system at a load demand of 850 MW • IEEE 6-unit CEED test system at a load demand of 283.4 MW. Simulations were performed on MATLAB 2016 software on Intel Core M-5Y10c@0.8GHz (4 CPU), 4GB RAM system. 20 runs were performed for each case of CEED problem having 100 bears and 100 iterations. For 140-unit ED problem iterations were kept at 1000.

A. VALIDATION FOR BENCHMARK FUNCTIONS
In this subsection, the validity of proposed IPBO is tested by applying it to five standard test functions presented by equation (19) to (23). The equation (19) and (20) represent unimodal functions whereas equations (21) to (23) represent multi-modal functions.    The simulations were performed for 30 independent runs keeping dimension of each function at 20 and the iterations were kept at 500. IPBO was able to achieve solution of each test function as represented by Fig.2.
From Table 1, it can be seen IPBO is able to achieve better average results in almost all cases. Its solution. strength is also highlighted in solution of multi-modal functions where it outclasses most of its competitors.

B. IEEE-140 UNIT TEST SYSTEM
Previously, PBO has been applied by the author to tackle small scale economic dispatch problem [52]. The knowledge gained from that research helped fine tune IPBO to take large scale ED problem. IPBO was employed to solve 140-unit Korean grid ED problem for two cases. The data was taken from [39]. In first case, IPBO is solved for only convex cost solution at a load demand of 49342 MW. Whereas in case 2, 12 units are subjected to valve point effect and 4 units have POZ constraints. The result achieved are presented in Table 2 and Table 3 along with other similar solutions available in literature. Furthermore, the convergence characteristics for both case 1 and case 2 by IPBO is presented in Fig. 3 as follows.
From Table 2 and Table 3, IPBO was able to achieve better solution as compared to QGSO, CCPSO, HHE, FPA, MFPA, CTPSO for case 1 and QGSO, CCPSO, HHE, FPA, MFPA, CTPSO, PSO C-GRASP-SaDE, C-GRASP-MDE, L-SHADE, IL-SHADE for case 2 respectively. The improvement in cost was observed to be as high as 6% both for case 1 and case 2.

C. IEEE 3-UNIT TEST SYSTEM
The data for IEEE 3-unit test system including cost coefficients, NO x coefficients and SO 2 coefficients was taken 56158 VOLUME 9, 2021 from [56]. The scaling factor for NO x and SO 2 were taken from [57] having value147582.78814 ($/ton) and 970.031569 ($/ton) respectively. Table 4 shows results for 3-unit system. From the Table 4 all techniques were successful in achieving solution of CEED problem for minimization of fuel cost, NO x emission and SO 2 emission, respectively. When VOLUME 9, 2021     compared among each other IPBO outclassed all its companions achieving an improvement as high as 0.027452% in cost, 0.1525% in NO x emissions and 0.002464% in SO 2 emission, respectively.

D. IEEE 6-UNIT TEST SYSTEM
The data for 6-unit system including cost coefficients, NOx coefficients and B matrix was taken from [34]. The scaling factor of NOx values was 1000 ($/ton). The results for 6-unit system for case 1 without considering loses and case 2 including loses are tabulated in Table 5 for best cost solution. From Table 5 IPBO was able to achieve an improvement as high as 0.0272% and 0.0766% in cost for case 1 and case 2, respectively.
Similarly, Table 6 and 7 represent best emission and best compromise solutions. From Table 6 IPBO was able to achieve an improvement as high as 0.0325% and 0.056% in emission for case 1 and 2, respectively. Whereas data from Table 7 indicates that IPBO achieves best compromise solution at minimum cost at a comparable level of emission. Table 8 and 9 show comparison of case 1 and case 2 respectively with other techniques available in literature. From Table 8 it can be seen that IPBO was able to achieve best cost and best compromise solution as compared to MSA [15], FFA [21], PSOGSA [38], MBFA [35], SOA [24], PSO [17], MOPSO [41], DE [31], and MODE/PSO [42], respectively. Other PBO variants PBO, PBO-CM and PBO-CU were also successful in achieving comparable cost and compromise solutions. The improvement in cost was in the range 0.88$ to 0.0105$ when compared to literature and in the range 0.0163$ to 0.092$ when compared to other PBO variants for best cost solution. For best compromise solution IPBO was able to achieve a cost improvement in the range 19.69$ to 1.88$ when compared to literature at a comparable emission level. For IPBO achieved comparable emission levels at an improved cost as compared to literature whereas PBO, PBO-CM  and PBO-CU showed comparable cost and emissions reviewer.
From Table 9, it can be seen than IPBO was able to achieve best cost and best compromise solution as compared to MSA [15], FFA [21], PSOGSA [38], MBFA [35], PSO [17], MOPSO [41], DE [31], MODE/PSO [42], IABC, FSO [33], and NGPSO [40], respectively. In case of IABC [34], IPBO achieved better emission level at comparable cost for best cost solution. The overall improvement in cost was in the range 2.23$ to 0.057$ whereas for best compromise      solution the improvement in cost was in the range 16.09$ to 0.29$ at a comparable emission level. Other PBO variants PBO, PBO-CM and PBO-CU were also successful in solving CEED problem at comparable cost and emissions. In case of best compromise solution all PBO variant achieved better cost at comparable emission level. For best emission solution all PBO variants achieved comparable emission levels at comparable cost as compared to literature. Fig. 4 and 3 show convergence characteristics for best cost solution of both cases. In Fig. 4 all PBO variants are plotted, chaotic PBO variants start search from higher fitness values because of compulsion on initial population according to chaotic level employed. Fig. 5

V. STATISTICAL ANALYSIS
To demonstrate the supremacy of one method a statistical analysis is executed demonstrating best, worst, mean, standard deviation and rank of each state is performed [58]- [60]. This statistical analysis is achieved by taking the data of 20 runs individually for all methods as examined earlier and results are exhibited in Table 10.
Wilcoxon rank sum test was introduced by Wilcoxon [61], [62].This is non-parametric test that can reflect the relationship between two different data sets in both cases. The Wilcoxon rank sum test is based on the hypothesis. We made a hypothesis that most of the results of IPBO as shown in both cases are less than other techniques. The probability-value of Wilcoxon rank-sum test shows the probability that how many times the results of PBO, PBO-CM and PBO-CU are less than IPBO results. The probability-value in Table 10 shows that there is very low probability that the other cases have cost values less than IPBO for case 1. In case 2 IPBO outclasses PBO significantly whereas PBO-CM or PBO-CU show significant probability for better results but at a higher standard deviation and variance. Table 11 proves our hypothesis is true.
The results of all four techniques are taken by independent trial runs for both cases. To show the distribution of data for each case Probability Density Function (PDF) and Cumulative Density Function (CDF) are plotted as shown in Fig.6 and Fig. 7.
It can be seen from Fig. 6 that highest peak is obtained for IPBO in both cases and widest data distribution is in PBO-CU. So, statistically IPBO is best, and PBO-CU is worst. It can also be seen that slope is highest for IPBO in both the cases and it reaches to 1 first than other techniques.

VI. CONCLUSION
In this paper novel improved PBO (IPBO), PBO and chaotic population PBO were employed to solve CEED problem for the first time in literature. Also, IPBO was validated by applying it to solve 140-unit Korean grid ED problem and 5 standard benchmark functions. All the proposed algorithms were successful in achieving solution of 3 unit and 6-unit CEED problem. Statistical analysis performed established that IPBO is superior to other PBO variants when it comes to solution of CEED problem showing an improvement as high as 0.027452% in cost, 0.1525% in NO x emissions and 0.002464% in SO 2 emission respectively for 3-unit system and an improvement as high as 0.0272% and 0.0766% in cost, and 0.0325% & 0.056% in emission for 6-unit system case 1 and case 2, respectively. From convergence behavior we can see that IPBO converges to optimum value in a smaller number of iterations as compared to other PBO variants with the difference in the range of 6 to 1 iteration. When compared to literature IPBO showed best cost and best compromise solutions for both cases. IPBO achieved an improvement in cost as high as 0.1475% and 3.255% for best cost and best compromise solutions of 6-unit system case 1, whereas an improvement in cost as high as 0.3686% and 2.65% for best cost and best compromise solutions of 6-unit system case 2 was observed. For best emission solution IPBO achieved better cost at a comparable emission level. The success of all PBO variants in achieving better solution of CEED problem is a motivating factor for further research applying IPBO and other PBO variants to engineering problems. Different demand response programs with integrated distributed energy resources will be explored for energy management in different energy consumption sectors. AFTAB AHMAD received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Engineering and Technology at Lahore, in 1985 and 1994, respectively, and the Ph.D. degree from the University of Engineering and Technology (UET) at Taxila, in 2010. He is currently working as a Professor of Electrical Engineering with the Department of Electrical Engineering, UET Taxila, and acting as the Chairman of the Electrical Engineering Department. He has supervised more than 50 Masters, one Ph.D. student, and contributed in many national/international journals and conference proceedings. His research interests include power system optimization, unit commitment, power system operation and control, and energy management systems. He is a Life Member of the Pakistan Engineering Council (PEC).
HAFIZ ASHIQ HUSSAIN (Member, IEEE) received the Ph.D. degree in telecom engineering from BUPT, China. He is currently working as an Associate Professor of Electrical Engineering with the Department of Electrical Engineering, HITEC University at Taxila, and acting as the Chairman of the Electrical Engineering Department. He has supervised more than 50 Masters, two Ph.D. students and contributed in many national and international journals and conference proceedings. His research interests include power systems, optical fiber communication, and silicon photonics. He is a Life Member of the Pakistan Engineering Council (PEC).
KHALID ALSUBHI received the B.Sc. degree in computer science from King Abdulaziz University (KAU), Jeddah, Saudi Arabia, in 2003, and the M.Math. and Ph.D. degrees in computer science from the University of Waterloo, Waterloo, ON, Canada, in 2009 and 2016, respectively. He is currently an Assistant Professor of Computer Science with KAU. His research interests include network security and management, cloud computing, and security and privacy of healthcare applications. VOLUME 9, 2021