Modified Student Psychology Based Optimization Algorithm for Economic Dispatch Problems

ABSTRACT

This manuscript develops and suggests modified student psychology-based optimization (MSPBO) algorithm to the convoluted economic dispatch problem. The psychology of the students who provide more attempts to perk up their recital in the exam equal to the echelon for turn out to be the best student in the class forms the basis of student psychology-based optimization (SPBO) algorithm. In this manuscript, MSPBO has been suggested to improve convergence speed and better-quality solutions. The efficacy of MSPBO algorithm has been divulged on five different economic dispatch problems incorporating valve-point consequence, proscribed feasible area, ramp rate borders, and manifold fuels. Simulation outcomes of the recommended method are matched up toward those attained as of other affirmed artificial intelligence methods. It is viewed from the assessment that suggested MSPBO is capable of giving better results.

Introduction

Economic dispatch (ED) provides generation among all committed generating units mainly economically simultaneously rewarding various restrictions. Valve-point consequence [3] is epitomized by summing up sinusoidal and quadratic functions.Figure 1

Figure 1. Depicts the flowchart of MSPBO algorithm.

Because of the opening of the steam admission valve, shaft posture shiver produces a proscribed workable area [6] in some sections of the generation region. The optimum cost is attained by evading this feasible area.

Traditional methods are not able to resolve ED problems with such types of complexity. Dynamic programming is able to resolve such types of complications, however, it experiences dimensionality crisis as well as confined optimality.

Preface of artificial intelligence techniques has given choices to resolve these types of complicated ED for example particle swarm optimization (PSO) by Gaing (2003) and Park et al. (2005), Tabu search (TS) by Whei-Min, Fu-Sheng, and Ming-Tong (2002), genetic algorithm (GA) by Walter and Sheble (1993), Cheng and Chang (1995) and Chiang (2005), evolutionary strategy by Pereira-Neto, Unsihuary, and Saavedra (2005), evolutionary programming (EP) by Yang, Yang, and Huang (1996) and Sinha, Chakrabarti, and Chattopadhyay (2003), chaotic differential evolution (CDE) by Coelho Leandro dos and Mariani (2006), new particle swarm optimization by Selvakumar and Thanushkodi (2007), civilized swarm optimization by Selvakumar and Thanushkodi (2009), improved particle swarm optimization by Park et al. (2010), biogeography-based optimization (BBO) by Bhattacharya and Chattopadhyay 2010a), hybrid differential evolution with biogeography-based optimization Bhattacharya and Chattopadhyay 2010a), ant colony optimization (ACO) by Pothiya, Ngamroo, and Kongprawechnon (2010), cultural self-organizing migrating strategy by Coelho Leandro dos and Mariani (2010), artificial bee colony algorithm (ABC) by Hemamalini and Simon (2010), continuous quick group search optimizer by Moradi-Dalvand et al. (2012), differential evolution based on truncated Levy-type flights and population diversity measure by Coelho Leandro dos, Bora, and Mariani (2014), oppositional invasive weed optimization by Braisal and Prusty (2015), crisscross optimization by Meng, Li, and Yin (2016), backtracking search algorithm by Modiri-Delshad et al. (2016), hybrid gray wolf optimize (HGWO) by Jayabarathi et al. (2016), modified symbiotic Organisms search (SOS) algorithm by Calin (2016), fast convergence evolutionary programming (FCEP) by Basu (2017), hybrid artificial algae (HAA) algorithm by Kumar and Dhillon (2018), modified crow search algorithm (MCSA) by Mohammadi and Abdi (2018), ameliorated gray wolf optimization (AGWO) by Singh and Dhillon (2019), conglomerated optimizer based on ion-motion and crisscross search optimizers (C-MIMO–CSO) by Kumar and Dhillon (2019), crow search algorithm (CSA) by Spea Shaimaa (2020), squirrel search optimizer (SSO) by Sumanl, Sakthivel, and Sathya (2020), emended salp swarm algorithm (ESSA) by Kansal and Dhillon (2020), dual-population adaptive differential evolution (DPADE) algorithm by Chen (2020), squirrel search algorithm (SSA) by Sakthivel, Suman, and Sathya (2020), etc.

In very recent times, Das, Mukherjee, and Das (2020) have developed a student psychology-based optimization (SPBO) algorithm, which is rooted in the psychology of the student who tries to acquire good marks in the exam.

In this manuscript, modified student psychology-based optimization (MSPBO) algorithm has been suggested for improving convergence speed and better-quality solutions.

In this manuscript, MSPBO and SPBO have been used for solving convoluted ED problems. Simulation outcomes of the suggested MSPBO algorithm are compared with those attained from other avowed artificial intelligence methods. It is viewed that the suggested MSPBO algorithm provides improved results.

Problem formulation

ED minimizes the fuel cost of power generation plants at the same time fulfilling several constraints. ED gets valve-point consequence, proscribed feasible area, ramp rate borders, as well as manifold fuels together with power demand, transmission losses, plus operational capability frontiers.

Economic dispatch Incorporating quadratic cost function, proscribed feasible area, ramp rate borders, and transmission losses

Economic dispatch is avowed as:

Min

(1) $\sum _{i=1}^{{\mathrm{N}}_g}{F}_i\left({\mathrm{P}}_{gi}\right)=\sum _{i=1}^{{\mathrm{N}}_g}{a}_i+b_i{\mathrm{P}}_{gi}+c_i{\mathrm{P}}_{gi}^2$(1)

where $a_i$, $b_i$, and $c_i$ are the cost coefficients of the $i$th generator, ${\mathrm{P}}_{gi}$ is the power output of the $i$th generator, and ${\mathrm{N}}_g$ is the total number of generators.

Subject to the constraints

(i) Power equilibrium constraint

(2) $\sum _{i=1}^{{\mathrm{N}}_g}{\mathrm{P}}_{gi}-{\mathrm{P}}_D-{\mathrm{P}}_L=0$(2)

The transmission loss ${\mathrm{P}}_L$ is avowed as

(3) ${\mathrm{P}}_L=\sum _{l=1}^{{\mathrm{N}}_g}\sum _{i=1}^{{\mathrm{N}}_g}{\mathrm{P}}_{gl}{\mathrm{B}}_{li}{\mathrm{P}}_{gi}+\sum _{i=1}^{{\mathrm{N}}_g}{\mathrm{B}}_{0i}{\mathrm{P}}_{gi}+{\mathrm{B}}_{00}$(3)

where ${\mathrm{P}}_D$ is load demand and ${\mathrm{B}}_{li}$, ${\mathrm{B}}_{0i}$, and ${\mathrm{B}}_{00}$ are the $\mathrm{B}$-coefficients.

(ii) Generation capacity boundary constraints

The output power of the generator should lie inside its minimum and maximum boundaries.

(4) ${\mathrm{P}}_{gi}^{min}\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi}^{max},i\in {\mathrm{N}}_g$(4)

where ${\mathrm{P}}_{gi}^{min}$ is the minimum boundary and ${\mathrm{P}}_{gi}^{max}$is the maximum boundary of the $i$th generator.

(iii) Proscribed feasible county

Workable region of a generator with the forbidden feasible county is stated as:

(5) $\begin{array}{rl}& {\mathrm{P}}_{gi}^{min}\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi,1}^l\\ & {\mathrm{P}}_{gi,,r-1}^u\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi,,r}^l,r=2,3,\dots ,n_i\\ & {\mathrm{P}}_{gi,,r_i}^u\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi}^{max},i\in {\mathrm{N}}_g\end{array}$(5)

where $r$ denotes the number of the forbidden feasible county of the $i$th generator. ${\mathrm{P}}_{gi,r-1}^u$ is the maximum boundary of $\left(r-1\right)$th the forbidden feasible county, and ${\mathrm{P}}_{gi,r}^l$ is the minimum boundary of the $r$th forbidden feasible county. The total number of forbidden feasible county of the $i$th generator is $n_i$.

(Iv) Ramp rate frontier constraints

Power ${\mathrm{P}}_{gi}$, generated by $i$th generator in a specific period must neither go further than that of the previous period ${\mathrm{P}}_{gi0}$ by above a specific amount $U{R}_i$, ramp-up boundary nor lower than that of the previous period by above a definite amount $D{R}_i$, the ramp-down boundary of the $i$th generator. These produce the following constraints.

As generation enhances

(6) ${\mathrm{P}}_{gi}-{\mathrm{P}}_{gi0}\le U{R}_i$(6)

As generation reduces

(7) ${\mathrm{P}}_{gi0}-{\mathrm{P}}_{gi}\le D{R}_i$(7)

and

(8) $max\left({\mathrm{P}}_{gi}^{min},{\mathrm{P}}_{gi0}-D{R}_i\right)\le {\mathrm{P}}_{gi}\le min\left({\mathrm{P}}_{gi}^{max},{\mathrm{P}}_{gi0}+U{R}_i\right)$(8)

Economic dispatch incorporating valve-point consequence

Economic dispatch subject to constraints bestowed by Equations (2)(2) $\sum _{i=1}^{{\mathrm{N}}_g}{\mathrm{P}}_{gi}-{\mathrm{P}}_D-{\mathrm{P}}_L=0$(2) and (4(4) ${\mathrm{P}}_{gi}^{min}\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi}^{max},i\in {\mathrm{N}}_g$(4) ) is avowed as:

Min

(9) $\sum _{i=1}^{{\mathrm{N}}_g}{F}_i\left({\mathrm{P}}_{gi}\right)=\sum _{i=1}^{{\mathrm{N}}_g}{a}_i+b_i{\mathrm{P}}_{gi}+c_i{\mathrm{P}}_{gi}^2+\left|d_i\times sin\left\{e_i\times \left({\mathrm{P}}_{gi}^{min}-{\mathrm{P}}_{gi}\right)\right\}\right|$(9)

where $d_i$ and $e_i$ are the cost coefficients of the $i$th generator due to valve point consequence.

Economic dispatch incorporating valve-point consequence and manifold fuels

A generator is normally proffered by manifold fuels, epitomized by several piecewise quadratic functions permit for valve point consequence. Cost function of the $i$th generator with ${\mathrm{N}}_F$ fuel categories are avowed as

(10) $F_i\left({\mathrm{P}}_{gi}\right)={\alpha }_{ir}+{\beta }_{ir}{\mathrm{P}}_{gi}+{\gamma }_{ir}{\mathrm{P}}_{gi}^2+\left|{\eta }_{ir}\times sin\left\{{\delta }_{ir}\times \left({\mathrm{P}}_{gir}^{min}-{\mathrm{P}}_{gi}\right)\right\}\right|$(10)

if ${\mathrm{P}}_{gir}^{min}\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gir}^{max}$ for fuel type $r$ and $r=1,2,\dots ,{\mathrm{N}}_F$

where ${\mathrm{P}}_{gir}^{min}$and ${\mathrm{P}}_{gir}^{max}$are the lower and upper power borders of $i$th generator for $r$th fuel category, respectively. ${\alpha }_{ir}$, ${\beta }_{ir}$, ${\gamma }_{ir}$, ${\eta }_{ir}$, and ${\delta }_{ir}$ are cost coefficients of the $i$th generator for fuel category $r$.

The objective function subject to constraints bestowed by Equations (2)(2) $\sum _{i=1}^{{\mathrm{N}}_g}{\mathrm{P}}_{gi}-{\mathrm{P}}_D-{\mathrm{P}}_L=0$(2) and (4(4) ${\mathrm{P}}_{gi}^{min}\le {\mathrm{P}}_{gi}\le {\mathrm{P}}_{gi}^{max},i\in {\mathrm{N}}_g$(4) ) is avowed as:

Min

(11) $\sum _{i=1}^{{\mathrm{N}}_g}{F}_i\left({\mathrm{P}}_{gi}\right)$(11)

Calculation of relaxed generator

${\mathrm{N}}_g$ dedicated generators allocate their output power subject to power equilibrium constraint (2) as well as power capability constraints (4). If power loading of first (${\mathrm{N}}_g-1$) generators have been identified, power output of ${\mathrm{N}}_g$th generator (i.e. relaxed generator) is provided by

Devoid of transmission loss

(12) ${\mathrm{P}}_{g{\mathrm{N}}_g}={\mathrm{P}}_D-\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_{gi}$(12)

Pondering transmission loss

(13) ${{\mathrm{P}}_{g\mathrm{N}}}_g={\mathrm{P}}_D+{\mathrm{P}}_L-\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_{gi}$(13)

Transmission loss ${\mathrm{P}}_L$ is provided by

(14) ${\mathrm{P}}_L=\sum _{i=1}^{{\mathrm{N}}_g-1}\sum _{j=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_i{\mathrm{B}}_{ij}{\mathrm{P}}_j+2{\mathrm{P}}_{{\mathrm{N}}_g}\left(\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{B}}_{{\mathrm{N}}_{t}i}{\mathrm{P}}_i\right)+{\mathrm{B}}_{{\mathrm{N}}_g{\mathrm{N}}_g}{\mathrm{P}}_{{\mathrm{N}}_g}^2+\sum _{i=1}^{{\mathrm{N}}_{tg}-1}{\mathrm{B}}_{0i}{\mathrm{P}}_i+{\mathrm{B}}_{0{\mathrm{N}}_g}{\mathrm{P}}_{{\mathrm{N}}_g}+{\mathrm{B}}_{00}$(14)

Augmenting and reshuffling, Equation (13)(13) ${{\mathrm{P}}_{g\mathrm{N}}}_g={\mathrm{P}}_D+{\mathrm{P}}_L-\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_{gi}$(13) turns into

(15) ${\mathrm{B}}_{{\mathrm{N}}_g{\mathrm{N}}_g}{\mathrm{P}}_{{\mathrm{N}}_g}^2+\left(2\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{B}}_{{\mathrm{N}}_{g}i}{\mathrm{P}}_i+{\mathrm{B}}_{0{\mathrm{N}}_g}-1\right){\mathrm{P}}_{{\mathrm{N}}_g}+\left({\mathrm{P}}_D+\sum _{i=1}^{{\mathrm{N}}_g-1}\sum _{j=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_i{\mathrm{B}}_{ij}{\mathrm{P}}_j+\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{B}}_{0i}{\mathrm{P}}_i\right.-\left.\sum _{i=1}^{{\mathrm{N}}_{tg}-1}{\mathrm{P}}_i+{\mathrm{B}}_{00}\right)=0$(15)

Loading of the relaxed generator (i.e. ${\mathrm{N}}_g$th) is established by resolving Equation (15)(15) ${\mathrm{B}}_{{\mathrm{N}}_g{\mathrm{N}}_g}{\mathrm{P}}_{{\mathrm{N}}_g}^2+\left(2\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{B}}_{{\mathrm{N}}_{g}i}{\mathrm{P}}_i+{\mathrm{B}}_{0{\mathrm{N}}_g}-1\right){\mathrm{P}}_{{\mathrm{N}}_g}+\left({\mathrm{P}}_D+\sum _{i=1}^{{\mathrm{N}}_g-1}\sum _{j=1}^{{\mathrm{N}}_g-1}{\mathrm{P}}_i{\mathrm{B}}_{ij}{\mathrm{P}}_j+\sum _{i=1}^{{\mathrm{N}}_g-1}{\mathrm{B}}_{0i}{\mathrm{P}}_i\right.-\left.\sum _{i=1}^{{\mathrm{N}}_{tg}-1}{\mathrm{P}}_i+{\mathrm{B}}_{00}\right)=0$(15) utilizing the classical algebraic technique.

For fulfilling equality constraint (13), the positive root of (15) has been selected as the power output of ${\mathrm{N}}_g$th generating unit.

If positive root breaches operational boundary constraint (4) at initialization procedure, then the power output of first $\left({\mathrm{N}}_g-1\right)$ generators has been reinitialized until positive root fulfills operational boundary constraint (4).

If positive root breaches operational boundary constraint (4) at a later phase of the method, then that customized generation is eliminated, and different steps of the method have been reapplied on its previous value until constraint (4) is fulfilled.

Student psychology based optimization algorithm

Student psychology-based optimization algorithm developed by Das, Mukherjee, and Das (2020) is founded on the psychology of the student who is endeavoring to acquire the best marks in the exam. The recital of a student is determined by considering the marks acquired in the exam. The student acquiring the highest marks in the exam is regarded as the best student in the class. Generally, the student endeavors to improve the recital so that the student can become the best student in the class. For achieving this goal, students must provide more attempts to all subjects proffered to them. However, the attempt provided by any student to any subject hinges on the student’s ability, adeptness in addition to interest in that subject. So, it is observed that the enhancement of recital in the exam of all students is not identical and can be different from student to student. A few students endeavor to perk up recital by providing identical or superior sort of attempts provided by the best student. Simultaneously, a number of students endeavor to provide attempts taking into consideration the attempt provided by the best student and endeavor to provide more attempts than the attempts provided by the average students in the class. Therefore, the students of a class can be classified into four categories. These are the best students, good students, average students, and students who endeavor to perk up at random.

Classification of students

(i) Best student: The student acquiring the highest overall marks in the exam is regarded as the best student in the class who all the time endeavors for providing more attempt to all subjects than the attempts provided by the rest of the students to keep up the best position by acquiring the highest marks in the class. Enhancement of the best student is stated as follows.

(16) $x_{best,new}=x_{best}+{\left(-1\right)}^k\times rand\times \left(x_{best}-x_j\right)$(16)

where $x_{best}$ and $x_j$ are the marks acquired by the best student and randomly selected $j$th student, respectively, in a particular subject, $rand$ is a random number in between 0 and 1, and k is a randomly chosen parameter either 1 or 2.

(ii) Good student: The student endeavors to provide increasingly attempt at any subject in which he gets interest for enhancing the recital in general. This type of student is selected randomly as the difference of psychology of the students and is regarded as the subject wise good student. To be the best student in the class by acquiring the highest marks in the exam, few students endeavor to provide identical or superior sort of attempt provided by the best student. This class of students is signified using Equation (17)(17) $x_{i,new}=x_{best}+rand\times \left(x_{best}-x_i\right)$(17) . Simultaneously, a number of students endeavor to provide more attempts in the study than the attempt provided by the average students of the class and endeavor to pursue the attempt provided by the best student. This class of students is signified using Equation (18)(18) $x_{i,new}=x_i+rand\times \left[\left(x_{best}-x_i\right)+\left(x_i-x_{mean}\right)\right]$(18) .

(17) $x_{i,new}=x_{best}+rand\times \left(x_{best}-x_i\right)$(17)

(18) $x_{i,new}=x_i+rand\times \left[\left(x_{best}-x_i\right)+\left(x_i-x_{mean}\right)\right]$(18)

where $x_i$ is the marks acquired by the $i$th student in a particular subject, $x_{mean}$ is the average recital of the class in that particular subject, and $rand$ is a random number in between 0 and 1.

(iii) Average student: Since the attempt provided by a student hinges on the interest of the student in the offered subjects, the students proffer an average attempt to the subject which is less interested to him. Whilst providing an average attempt to that subject, the students endeavor to provide more attempts to other subjects so that their overall marks are perked up. This class of students is referred to as the subject-wise average student. Due to different students’ psychology, this class of students is chosen randomly. Recital of this class of students is signified using Equation (19)(19) $x_{i,new}=x_i+rand\times \left(x_{mean}-x_i\right)$(19) .

(19) $x_{i,new}=x_i+rand\times \left(x_{mean}-x_i\right)$(19)

where $x_i$ and $x_{mean}$ are the marks acquired by the $i$th student and the average marks acquired by the class in that particular subject, respectively, and $rand$ is a random number in between 0 and 1.

(iv) Students who endeavor to perk up at random: Apart from these abovementioned classes of students, a number of students endeavor to perk up their recital by themselves. These students endeavor to provide attempt to the subject arbitrarily hinging on the subject. This class of students endeavors to provide attempt arbitrarily to the subject so that overall recital in the exam perks up. Recital of this class of student is signified using Equation (20)(20) $x_{i,new}=x_{min}+rand\times \left(x_{max}-x_{min}\right)$(20) .

(20) $x_{i,new}=x_{min}+rand\times \left(x_{max}-x_{min}\right)$(20)

where $x_{min}$ and $x_{max}$ are the minimum and the maximum limit of marks of the subject, respectively.

Selection

Compute recital of the class. If the new recital is better than older, then replace older by new one otherwise, keep older recital.

Modified student psychology-based optimization algorithm

In the modified student psychology-based optimization (MSPBO) algorithm, some modification is suggested in student psychology-based optimization (SPBO) algorithm to improve convergence speed and better-quality solution.

Classification of students

(i) Best student: The enhancement of the best student is modified as follows.

(21) $x_{best,new}=x_{best}+N\left(0,1\right)\times \left(x_{best}-x_j\right)$(21)

where $x_{best}$ and $x_j$ are the marks acquired by the best student and randomly selected $j$th student, respectively, in a particular subject, $N\left(0,1\right)$ signifies Gaussian random variable with mean 0 and standard deviation 1.

(ii) Good student: A few students want to be the best student in the class by acquiring the highest marks in the exam and endeavor to provide identical or superior sort of attempt provided by the best student. This class of students is modified as follows:

(22) $x_{i,new}=x_i+N\left(0,1\right)\times \left(x_{best}-x_i\right)$(22)

Simultaneously, a number of students who endeavor to provide more attempts in study than the attempt provided by the average students of the class and endeavor to pursue the attempt provided by the best student are modified as follows:

(23) $x_{i,new}=x_i+N\left(0,1\right)\times \left[\left(x_{best}-x_i\right)+\left(x_i-x_{mean}\right)\right]$(23)

where $x_i$ is the marks acquired by the $i$th student in a particular subject, $x_{mean}$ is the average recital of the class in that particular subject, $N\left(0,1\right)$ signifies Gaussian random variable with mean 0 and standard deviation 1.

(iii) Average student: Recital of the average student is modified as follows:

(24) $x_{i,new}=x_i+N\left(0,1\right)\times \left(x_{mean}-x_i\right)$(24)

where $x_i$ and $x_{mean}$ are the marks acquired by the $i$th student and the average marks acquired by the class in that particular subject, respectively, and $N\left(0,1\right)$ signifies Gaussian random variable with mean 0 and standard deviation 1.

(iv) Students who endeavor to perk up at random: Apart from these aforesaid classes of students, a number of students who endeavor to perk up their recital by themselves and to provide attempt to the subject arbitrarily hinging on the subject so that overall recital in the exam perks up are modified as follows:

(25) $x_{i,new}=x_i+N\left(0,1\right)\times \left(x_{max}-x_{min}\right)$(25)

where $x_{min}$ and $x_{max}$ are the minimum and the maximum limit of marks of the subject, respectively.

Selection

Compute recital of the class. If the new recital is better than older, then replace older by new one otherwise, keep older recital.

Numerical results

Modified student psychology-based optimization (MSPBO) algorithm and student psychology-based optimization (SPBO) algorithm are employed for solving ED. The suggested MSPBO and SPBO have been employed by utilizing MATLAB (Version: (R2013a)). MSPBO and SPBO have been employed to solve five different economic dispatch problems such as 15-unit system with proscribed feasible area, ramp rate borders, and transmission losses, a 40-unit system with valve point effect, a 10-unit system with manifold fuels, and valve point effect, a 140-unit convoluted Korean system, and 280-unit system. A total of 100 runs are executed for all systems to compare the resolution quality. Simulation outcomes have been used to compare the concert of MSPBO and SPBO with that of other avowed evolutionary techniques.

Test system 1

This test system includes 15 generating units with proscribed feasible area, ramp rate borders, and transmission losses. Generating units 2, 5, and 6 have three proscribed feasible areas and generating unit 12 has two proscribed feasible areas. The power demand is 2630 MW. The data is espoused from Gaing (2003) and revealed in Table 1. Transmission loss coefficients are not mentioned here on account of word limitations.

Table 1. Generation constraints and best power output attained from MSPBO for test system 1 with power demand of 2630 MW

Unit	${\mathrm{P}}^{min}$	${\mathrm{P}}^{max}$	$a$	$b$	$c$	Proscribed workable area	${\mathrm{P}}^0$	UR	DR	Generation
	(MW)	(MW)	($)	($/MW)	($./MW^2)		(MW)	(MW)	(MW)	(MW)
1	150	455	671	10.1	0.000299		400	80	120	455.0000
2	150	455	574	10.2	0.000183	[185,225],[305,335], [420, 450]	300	80	120	380.0000
3	20	130	374	8.8	0.001126		105	130	130	130.0000
4	20	130	374	8.8	0.001126		100	130	130	130.0000
5	150	470	461	10.4	0.000205	[180,200], [305,335], [390,420]	90	80	120	170.0000
6	135	460	630	10.1	0.000301	[230,255], [365,395], [430,455]	400	80	120	460.0000
7	135	465	548	9.8	0.000364		350	80	120	430.0000
8	60	300	227	11.2	0.000338		95	65	100	110.4140
9	25	162	173	11.2	0.000807		105	60	100	25.0000
10	25	160	175	10.7	0.001203		110	60	100	154.0518
11	20	80	186	10.2	0.003586		60	80	80	80.0000
12	20	80	230	9.9	0.005513	[30, 40], [55,65]	40	80	80	80.0000
13	25	85	225	13.1	0.000371		30	80	80	25.0000
14	15	55	309	12.1	0.001929		20	55	55	15.0000
15	15	55	323	12.4	0.004447		20	55	55	15.0000
Total generation (MW) 2659.4658
Power loss (MW) 29.4658
Total generation cost ($) 32,691.87

Problem is resolved by utilizing both MSPBO and SPBO. Parameters have been chosen as ${\mathrm{N}}_{\mathrm{P}}=50$ and $(Itr{)}_{max}=100$.

Power generation, power loss, and cost related to the preeminent cost among 100 runs attained from recommended MSPBO have been summed up in Table 1. Preeminent, mean, and worst costs among 100 runs attained from recommended MSPBO have been summed up in Table 2. Cost acquired from SPBO, particle swarm optimization (PSO), improved particle swarm optimization (IPSO), backtracking search algorithm (BSA), fast convergence evolutionary programming (FCEP), conglomerated optimizer based on ion-motion and crisscross search optimizers (C-MIMO–CSO), and emended salp swarm algorithm (ESSA) is also summed up in Table 2. Cost convergence characteristic achieved from MSPBO and SPBO is revealed in Figure 2. It is viewed from Table 2 that cost attained by utilizing MSPBO is lower than all other avowed methods.

Table 2. Cost and CPU time assessment of test system 1

Techniques	Preeminent cost ($)	Mean cost ($)	Worst cost ($)
MSPBO	32,691.87	32,691.89	32,691.93
SPBO	32,692.78	32,692.88	32,692.99
FCEP	32,691.91	32,691.94	32,692.14
C-MIMO–CSO	32,701.21	32,701.21	32,701.22
ESSA	32,701.21	32,701.22	32,701.22
BSA	32,704.45	32,704.47	32,704.58
IPSO	32,704.45	32,704.45	32,704.45
PSO	32,858.00	33,039.00	33,331.00

Figure 2. Cost convergence characteristic for test system 1.

Test system 2

This test system includes 40 generating units and the data is espoused from Sinha, Chakrabarti, and Chattopadhyay (2003) and revealed in Table 3. Power demand is 10,500 MW.

Table 3. Generation constraints and best power output attained from MSPBO for test system 2 with power demand of 10,500 MW

Unit	${\mathrm{P}}^{min}$	${\mathrm{P}}^{max}$	$a$	$b$	$c$	$d$	$e$	Generation
	(MW)	(MW)	($)	($/MW)	($./MW^2)	($)	($/rad)	(MW)
1	36	114	94.705	6.73	0.00690	100	0.084	110.8978
2	36	114	94.705	6.73	0.00690	100	0.084	110.8906
3	60	120	309.540	7.07	0.02028	100	0.084	97.4034
4	80	190	369.030	8.18	0.00942	150	0.063	179.7349
5	47	97	148.890	5.35	0.01140	120	0.077	88.2235
6	68	140	222.330	8.05	0.01142	100	0.084	140.0000
7	110	300	287.710	8.03	0.00357	200	0.042	259.6222
8	135	300	391.98	6.99	0.00492	200	0.042	284.6003
9	135	300	455.76	6.60	0.00573	200	0.042	284.6007
10	130	300	722.82	12.9	0.00605	200	0.042	130.0000
11	94	375	635.20	12.9	0.00515	200	0.042	168.8027
12	94	375	654.69	12.8	0.00569	200	0.042	94.0000
13	125	500	913.40	12.5	0.00421	300	0.035	214.7644
14	125	500	1760.40	8.84	0.00752	300	0.035	394.2827
15	125	500	1760.40	8.84	0.00752	300	0.035	394.2791
16	125	500	1760.40	8.84	0.00752	300	0.035	304.5224
17	220	500	647.85	7.97	0.00313	300	0.035	489.2826
18	220	500	649.69	7.95	0.00313	300	0.035	489.2798
19	242	550	647.83	7.97	0.00313	300	0.035	511.2837
20	242	550	647.81	7.97	0.00313	300	0.035	511.2829
21	254	550	785.96	6.63	0.00298	300	0.035	523.2837
22	254	550	785.96	6.63	0.00298	300	0.035	523.2907
23	254	550	794.53	6.66	0.00284	300	0.035	523.2908
24	254	550	794.53	6.66	0.00284	300	0.035	523.2887
25	254	550	801.32	7.10	0.00277	300	0.035	523.2879
26	254	550	801.32	7.10	0.00277	300	0.035	523.2800
27	10	150	1055.10	3.33	0.52124	120	0.077	10.0000
28	10	150	1055.10	3.33	0.52124	120	0.077	10.0000
29	10	150	1055.10	3.33	0.52124	120	0.077	10.0000
30	47	97	148.89	5.35	0.01140	120	0.077	96.4156
31	60	190	222.92	6.43	0.00160	150	0.063	190.0000
32	60	190	222.92	6.43	0.00160	150	0.063	190.0000
33	60	190	222.92	6.43	0.00160	150	0.063	190.0000
34	90	200	107.87	8.95	0.00010	200	0.042	164.8243
35	90	200	116.58	8.62	0.00010	200	0.042	200.0000
36	90	200	116.58	8.62	0.00010	200	0.042	200.0000
37	25	110	307.45	5.88	0.01610	80	0.098	110.0000
38	25	110	307.45	5.88	0.01610	80	0.098	110.0000
39	25	110	307.45	5.88	0.01610	80	0.098	110.0000
40	242	550	647.83	7.97	0.00313	300	0.035	511.2846
Total generation (MW) 10,500
Total generation cost ($) 1,21,379.22

Problem is resolved by utilizing both MSPBO and SPBO. Parameters have been chosen as ${\mathrm{N}}_{\mathrm{P}}=100$ and $(Itr{)}_{max}=300$.

Power generation related to the preeminent cost among 100 runs attained from MSPBO has been summed up in Table 3. Preeminent, mean, and worst costs among 100 runs attained from MSPBO have been summed up in Table 4. Cost attained from SPBO, NPSO-LRS, DEC-SQP, CSO, IPSO, BBO, DE/BBO, ACO, CSOMA, ABC, CQGSO, OIWO, HGWO, MSOS, FCEP, HAAA, MCSA, AGWO, C-MIMO–CSO, SSO, and ESSA has been also summed up in Table 4. Cost convergence characteristic achieved from MSPBO and SPBO has been revealed in Figure 3. It is viewed from Table 4 that cost attained by using MSPBO is lower than all other avowed methods.

Table 4. Cost and CPU time assessment of test system 2

Techniques	Preeminent cost ($)	Mean cost ($)	Worst cost ($)
MSPBO	1,21,379	1,21,380	1,21,381
SPBO	1,21,401	1,21,403	1,21,406
FCEP	1,21,393	1,21,394	1,21,395
AGWO	1,21,404	1,21,412	1,21,446
HAAA]	1,21,403	1,21,425	1,21,428
MSOS]	1,21,412	1,21,412	1,21,412
HGWO	1,21,412	1,21,419	NA
OIWO	1,21,412	NA	NA
CQGSO	1,21,412	1,21,423	1,21,438
MCSA	1,21,412	1,21,413	1,21,414
SSO	1,21,412	1,21,413	-
C-MIMO–CSO	1,21,412	1,21,451	1,21,536.7
ESSA	1,21,412	1,2,1450	1,21,517
ABC]	1,21,441	1,21,995	1,22,123
CSOMA	1,21,414	1,21,415	1,21,417
ACO	1,21,532	1,21,606	1,21,679
DE/BBO]	1,21,420	1,21,420	1,21,420
BBO	1,21,426	1,21,503	1,21,688
IPSO	1,21,403	121,445	1,21,525
CSO	1,21,461	1,,21,936	1,26,902
DEC-SQP	1,21,741	1,22,295	1,22,981
NPSO-LRS	1,21,664	1,22,209	1,22,981

Figure 3. Cost convergence characteristic for test system 2.

Test system 3

This test system includes 10 generators with valve point effect and manifold fuels. Power demand is 2700 MW and data is espoused from Chiang (20005) and not mentioned here because of the word limitations.

Problem is resolved by utilizing both MSPBO and SPBO. Parameters have been chosen as ${\mathrm{N}}_{\mathrm{P}}=50$ and $(Itr{)}_{max}=100$.

Power generation related to the preeminent cost among 100 runs attained from recommended MSPBO has been summed up in Table 5. Preeminent, mean, and worst costs among 100 runs attained from recommended MSPBO have been summed up in Table 6. Cost attained from SPBO, NPSO-LRS, IPSO, CQGSO, CSO, BSA, FCEP, MCSA, C-MIMO–CSO, CSA, SSO, and DPADE has been also summed up in Table 6. Cost convergence characteristic achieved from MSPBO and SPBO has been revealed in Figure 4. It is viewed from Table 6 that cost attained by using MSPBO is lower than all other avowed methods.

Table 5. Generation (MW) of test system 3

Unit	GEN	F	Unit	GEN	F
1	214.0142	2	6	247.3183	3
2	230.0000	3	7	342.1730	2
3	290.0243	1	8	265.0000	3
4	137.3776	1	9	318.3436	1
5	333.0599	1	10	322.6893	1
Total generation (MW) 2700
Total generation cost ($) 623.2247

Table 6. Cost and CPU time assessment of test system 3

Techniques	Preeminent cost ($)	Mean cost ($)	Worst cost ($)
MSPBO	623.2247	623.2274	623.2295
SPBO	623.6496	623.6745	623.7198
FCEP	623.6385	623.6564	623.7179
SSO	623.7129	624.2592	-
CSA	623.8092	623.8650	679.6398
DPADE	623.8226	623.8236	623.8243
C-MIMO–CSO	623.8237	623.8254	623.827
MCSA	623.8280	623.8311	623.8439
BSA	623.9016	623.97	624.08
CSO	623.82	623.82	623.84
CQGSO	623.82	623.83	623.85
IPSO	623.8266	623.8273	623.8291
NPSO-LRS	624.1273	624.9985	-

Figure 4. Cost convergence characteristic for test system 3.

Test system 4

This is 140 unit convoluted Korean system of 12 generators with valve point effect and four generators with proscribed feasible area. Ramp rate borders have been pondered here. Power demand is 49,342 MW. Data has been espoused from Park et al. (2010)] and not mentioned here due to the word limitations.

Problem is resolved by utilizing both MSPBO and SPBO. Parameters have been chosen as ${\mathrm{N}}_{\mathrm{P}}=150$ and $(Itr{)}_{max}=1000$.

Power generation related to the preeminent cost among 100 runs attained from recommended MSPBO has been summed up in Table 7. Preeminent, mean, and worst costs among 100 runs obtained from recommended MSPBO have been summarized in Table 8. Cost attained from SPBO, IPSO, CQGSO, DEL, FCEP, and AGWO have been summarized in Table 8. Cost convergence characteristic achieved from MSPBO and SPBO is revealed in Figure 5. It is viewed from Table 8 that cost attained by using MSPBO is lower than all other avowed methods.

Table 7. Generation (MW) of test system 4

Unit	GEN	Unit	GEN	Unit	GEN	Unit	GEN
1	118.4000	36	500.0000	71	141.5000	106	890.0000
2	164.0000	37	241.0000	72	388.3272	107	873.0000
3	190.0000	38	241.0000	73	195.0000	108	877.0000
4	190.0000	39	774.0000	74	196.2280	109	871.0000
5	168.5399	40	769.0000	75	196.2569	110	866.0000
6	190.0000	41	3.0000	76	255.8500	111	882.0000
7	490.0000	42	3.0000	77	402.9899	112	94.0000
8	490.0000	43	250.0000	78	330.0000	113	94.0000
9	496.0000	44	250.0000	79	531.0000	114	94.0000
10	496.0000	45	250.0000	80	531.0000	115	244.0000
11	496.0000	46	250.0000	81	542.0000	116	244.0000
12	496.0000	47	250.0000	82	56.0000	117	244.0000
13	506.0000	48	250.0000	83	115.0000	118	95.0000
14	509.0000	49	250.0000	84	115.0000	119	95.0000
15	506.0000	50	250.0000	85	116.0000	120	116.0000
16	505.0000	51	165.0000	86	207.0000	121	175.0000
17	506.0000	52	165.0000	87	207.0000	122	2.0000
18	506.0000	53	165.0000	88	175.0000	123	4.0000
19	505.0000	54	165.0000	89	175.0000	124	15.0000
20	505.0000	55	180.0000	90	180.0000	125	9.0000
21	505.0000	56	180.0000	91	175.0000	126	12.0000
22	505.0000	57	103.0000	92	575.4000	127	10.0000
23	505.0000	58	198.0000	93	547.5000	128	112.0000
24	505.0000	59	312.0000	94	836.8000	129	4.0000
25	537.0000	60	309.5965	95	836.9332	130	5.0000
26	537.0000	61	162.5995	96	682.0000	131	5.0000
27	549.0000	62	95.0000	97	720.0000	132	50.0000
28	549.0000	63	511.0000	98	718.0000	133	5.0000
29	501.0000	64	511.0000	99	720.0000	134	42.0000
30	490.0000	65	490.0000	100	964.0000	135	42.0000
31	506.0000	66	256.7389	101	958.0000	136	41.0000
32	506.0000	67	489.9997	102	947.9000	137	17.0000
33	506.0000	68	490.0000	103	934.0000	138	7.0000
34	506.0000	69	130.0000	104	935.0000	139	7.0000
35	500.0000	70	339.4397	105	877.0000	140	26.0000
Total generation (MW) 49,342
Total generation cost ($) 16,57,886

Table 8. Cost and CPU time assessment of test system 4

Techniques	Best cost ($)	Average cost ($)	Worst cost ($)
MSPBO	16,57,886	16,57,889	16,57,893
SPBO	16,57,938	16,57,942	16,57,948
FCEP	16,57,924	16,57,926	16,57,931
IPSO	16,57,962	16,57,962	16,57,962
CQGSO	16,57,962	16,57,962	16,57,962
DEL	16,57,962	16,58,001	-
AGWO	16,57,962	16,57,963	16,57,964

Figure 5. Cost convergence characteristic for test system 4.

Test system 5

This is a 280 unit system of 24 generators with valve point effect and eight generators with proscribed feasible area. Ramp rate borders have been pondered here. Power demand is 98,684 MW. Data of test system 5 is attained by duplicating data of test system 4.

Problem is resolved by using both MSPBO and SPBO. Parameters have been chosen as ${\mathrm{N}}_{\mathrm{P}}=200$ and $(Itr{)}_{max}=1000$.

Power generations of first 140 generators of test system 5 related to the preeminent cost among 100 runs attained from MSPBO are summed up in Table 9. Power generations of last 140 generators of test system 5 related to the preeminent cost among 100 runs attained from MSPBO are summed up in Table 10. Preeminent, mean, and worst costs among 100 runs obtained from suggested MSPBO and SPBO have been summarized in Table 11. Cost convergence characteristic achieved from MSPBO and SPBO is revealed in Figure 6.

Table 9. Generation (MW) of first 140 generators of test system 5

Unit	GEN	Unit	GEN	Unit	GEN	Unit	GEN
1	71.0000	36	500.0000	71	273.2652	106	873.2686
2	120.0000	37	240.6115	72	137.0000	107 864.8235
3	190.0000	38	241.0000	73	195.0000	108	875.2350
4	190.0000	39	774.0000	74	175.0000	109	855.7327
5	90.0000	40	769.0000	75	175.0005	110	864.7526
6	90.0000	41	3.0000	76	408.3752	111	879.7868
7	490.0000	42	3.0000	77	483.9282	112	94.0291
8	490.0000	43	250.0000	78	338.2052	113	94.0000
9	496.0000	44	250.0000	79	522.8227	114	97.5228
10	496.0000	45	250.0000	80	531.0000	115	244.0180
11	496.0000	46	250.0000	81	530.7277	116	244.0121
12	496.0000	47	160.0000	82	60.7221	117	244.0191
13	506.0000	48	160.0000	83	115.0000	118	143.2085
14	509.0000	49	250.0000	84	115.2557	119	95.0000
15	506.0000	50	160.0000	85	115.0000	120	116.3578
16	505.0000	51	165.0000	86	225.2802	121	175.0057
17	506.0000	52	165.0000	87	207.0000	122	2.6674
18	506.0000	53	504.0000	88	188.7465	123	4.6971
19	505.0000	54	165.0000	89	305.6775	124	15.1126
20	505.0000	55	180.0000	90	199.6031	125	9.0000
21	505.0000	56	180.0000	91	175.0002	126	14.4961
22	505.0000	57	341.0000	92	573.5634	127	12.6441
23	505.0000	58	198.0000	93	544.8726	128	112.0000
24	505.0000	59	312.0000	94	833.6595	129	10.5068
25	537.0000	60	153.0000	95	824.6423	130	5.4104
26	537.0000	61	163.0000	96	674.7542	131	13.9177
27	549.0000	62	95.0000	97	720.0000	132	50.0018
28	549.0000	63	160.0000	98	715.7370	133	5.0144
29	501.0000	64	511.0000	99	720.0000	134	51.6952
30	489.6054	65	490.0000	100	963.0802	135	53.9444
31	506.0000	66	196.0000	101	958.0000	136	41.0000
32	506.0000	67	490.0000	102	933.2756	137	17.0000
33	506.0000	68	460.8615	103	931.3053	138	19.0000
34	506.0000	69	130.0000	104	932.4971	139	7.0000
35	500.0000	70	130.0000	105	876.2858	140	35.2220

Table 10. Generation (MW) of last 140 generators of test system 5

Unit	GEN	Unit	GEN	Unit	GEN	Unit	GEN
141	119.0000	176	500.0000	211	454.3694	246	875.1671
142	120.0000	177	241.0000	212	455.0000	247	870.5072
143	190.0000	178	241.0000	213	195.0000	248	873.2416
144	190.0000	179	774.0000	214	175.0000	249	843.7753
145	90.0000	180	769.0000	215	441.0623	250	860.6351
146	90.0000	181	3.0000	216	535.3810	251	881.9941
147	490.0000	182	3.0000	217	274.6947	252	136.5808
148	490.0000	183	160.0000	218	521.5265	253	94.2266
149	496.0000	184	250.0000	219	531.0000	254	94.9944
150	496.0000	185	160.0000	220	514.9125	255	244.0004
151	496.0000	186	160.0000	221	208.6127	256	245.1332
152	496.0000	187	160.0000	222	82.2956	257	244.1329
153	506.0000	188	160.0000	223	115.1901	258	95.0000
154	509.0000	189	250.0000	224	115.9650	259	95.0000
155	506.0000	190	250.0000	225	115.0000	260	147.8593
156	505.0000	191	165.0000	226	207.0000	261	175.0000
157	506.0000	192	504.0000	227	207.0000	262	2.0000
158	506.0000	193	504.0000	228	175.5713	263	4.0102
159	505.0000	194	504.0000	229	176.6496	264	29.1420
160	505.0000	195	180.0000	230	180.1128	265	9.2712
161	505.0000	196	180.0000	231	193.2238	266	12.7515
162	505.0000	197	103.0000	232	572.8236	267	10.0015
163	505.0000	198	198.0000	233	546.8287	268	115.0218
164	505.0000	199	312.0000	234	831.1953	269	4.3671
165	537.0000	200	153.0000	235	813.8951	270	5.0000
166	537.0000	201	163.0000	236	682.0000	271	5.0749
167	549.0000	202	95.0000	237	720.0000	272	67.6614
168	549.0000	203	511.0000	238	717.8419	273	5.1181
169	501.0000	204	160.0000	239	719.7932	274	42.8641
170	501.0000	205	490.0000	240	953.2564	275	42.0000
171	506.0000	206	490.0000	241	955.2000	276	41.0000
172	506.0000	207	490.0000	242	946.1383	277	17.0000
173	506.0000	208	490.0000	243	927.6376	278	7.0000
174	506.0000	209	130.0000	244	918.4329	279	7.0000
175	500.0000	210	130.0000	245	869.4772	280	29.9169
Total generation (MW) 98,684
Total generation cost ($) 33,62,573

Table 11. Cost and CPU time assessment of test system 5

Techniques	Best cost ($)	Average cost ($)	Worst cost ($)
MSPBO	33,62,573	33,62,576	33,62,581
SPBO	33,62,686	33,62,692	33,62,699

Figure 6. Cost convergence characteristic for test system 5.

Conclusion

MSPBO has been developed and employed to solve five convoluted economic dispatch problems. Simulation outcomes attained from the MSPBO and SPBO are compared with those attained from other artificial intelligence techniques. It is viewed that MSPBO offers better of solution.

Disclosure statement

No potential conflict of interest was reported by the author(s).