This paper addresses the optimal power flow (OPF) issue by using the CONOPT solver for nonlinear programming embedded in the Generalized Algebraic Modeling System (GAMS) software package. The research is performed on both standard IEEE 30-bus test systems and their modified version. The system modification has been done in order to assess the impact of the integrated renewable energy sources, primarily wind energy sources, on the OPF. The obtained results strongly confirm the GAMS/CONOPT efficiency for solving the OPF problem. GAMS/CONOPT always converges to the same optimal solution contrary to many well-known optimization techniques. Additionally, the GAMS/CONOPT requested computation time considerably outperforms that of the other techniques in the field (in all analyzed cases, in the worst situation, more than 30 times). These performances promote GAMS/CONOPT as a very successive tool for solving the real-time OPF problem.

The energy sector has undergone significant changes over the last few decades. Technological innovations, economic reasons, and political decisions inspired by environmental protection have led to the emergence of modern power systems. Large scale integration of renewable energy sources (RES) and their growing share in electrical energy production have become the main features of modern and market-oriented power systems. In light of new circumstances, the optimal power flow (OPF) problem is gaining importance. The main idea behind the OPF is the optimization of the selected objective functions (OFs), such as minimization of the fuel cost (MFC), minimization of CO2 emission, and minimization of the power losses, by adjusting the system control variables and by taking into account various system constrains. The OPF problem is highly nonlinear and multiobjective, and it has more than one local optimum solution.1 

The OPF problem can be solved by using traditional optimization methods2–6 or by using some of the various population-based optimization techniques.7–53 The traditional optimization methods, such as the Newton method (NM),2 gradient projection method (GPM),3 linear programming method (LPM),4 interior point method (IPM),5 and bisection method (BM),6 require linearization of the objective function or neglecting constrains, and the methods from this group do not guarantee finding the global optimum. Additionally, they are based on complex calculations and therefore are not convenient for implementation. In recent years, many evolutionary programming (EP) algorithms, improved evolutionary programming (IEP) algorithms, and meta-heuristic algorithms (MHA) have been developed aiming at solving the OPF problem (see  Appendix ATable IX).7–54 As general remarks for the mentioned algorithms, their convergence depends on the initial conditions, and they can fall into a local optimum during the optimization process. The requested computation time (RCT) of the optimization methods is profoundly important for solving the OPF problem effectively, especially in the present-day power systems based on the RES.55 However, the above-mentioned papers do not provide comprehensive information related to the issue. In some papers, the RCT is calculated without the described calculation procedure. Also, the importance of solving the OPF problem at high speed and the impact of the system dimension on the RCT are not discussed in the analyzed papers. Note that in the literature, papers dealing with optimal power flow with renewable energy resources,55–58 optimal power flow with renewable energy resources and storage systems,59 and optimal reactive power scheduling60 can also be found.

A large number of the existing optimization methods and their drawbacks point to necessity for further research in this scientific field. To that end, in this paper, we propose the CONOPT solver for nonlinear programming (NLP) imbedded in the Generalized Algebraic Modeling System (GAMS) software package for the OPF problem solving. The GAMS/CONOPT performances are high, enabling one to perform the OPF with high accuracy and extremely low computational time.61–65 This issue will be specially analyzed in this paper. In order to confirm the GAMS/CONOPT superiority with respect to other optimization methods, this paper includes a detailed comparison between GAMS/CONOPT and other methods in terms of achieved accuracy, the RCT, and the system dimension impact on the RCT. The comparison will be carried out for different objective functions.

The remainder of this paper is organized as follows: the OPF problem formulation is shortly described in Sec. II. Section III presents a comparison of different optimization techniques for the OPF problem solving. Fundamentals of the GAMS/CONOPT and its application on the IEEE 30 test bus system aimed at solving the OPF are given in Sec. IV. Finally, Sec. V provides the conclusion and outlines the future work.

The OPF problem can be considered as a general minimization problem with constraints.1 Also, as it is stressed in Introduction, the OPF represents a multiobjective problem, which can be expressed in the following mathematical form:

Minimize f of x , y , Subject to g ec x , y = 0 , h sc x , y 0 ,
(1)

where fof (x,y) is the objective function, gec(x,y) represents the equality constraints (e.g., load flow equations), hsc(x,y) are the system operating constraints, x denotes the vector of dependent variables (load bus voltage, generator reactive power output, transmission line loading, etc.), and y is the vector of control variables such as generator active power output except at slack bus, generator bus voltage, transformer tap setting, and shunt VAR compensation.

The most commonly used objective functions (OFs) in the literature have the following mathematical descriptions:

  • Minimization of total fuel cost (MFC)
    MFC = i = 1 Ng ( a i P gi 2 + b i P gi + c i ) ,
    (2)

    where Ng is the number of thermoaggregates, Pgi is the active power output of each generator i, and ai, bi, and ci are the corresponding cost coefficients.

  • Minimization of total power losses (MTPL)
    MTPL = i , j N b G ij ( V i 2 + V j 2 2 V i V j cos ( δ ij ) ) ,
    (3)

    where Nb is the number of busses, i, j =1, …, Nb, Gij is the conductance between buses i and j, Vi and Vj are the voltages of the ith and the jth bus, respectively, and δij is the voltage angle between buses i and j.

  • Total emission cost minimization (TECM)
    TECM = i = 1 N g ( α i + β i P gi + γ i P gi 2 + ξ i e λ i P gi ) ,
    (4)

    where αi, βi, γi, ξi, and λi are the emission coefficients of the ith generator unit.

Besides the previously listed OFs, voltage profile improvement (VPI), voltage stability enhancement (VSE), and the combinations of different objective functions can also be found in the literature.

The equality constraints gec(x,u) represent the typical example of nonlinear power flow equations

P Gi P Di V i j = 1 N b V j ( G ij cos ( δ ij ) + B ij sin ( δ ij ) ) = 0 , Q Gi Q Di V i j = 1 N b V j ( G ij sin ( δ ij ) B ij cos ( δ ij ) ) = 0 ,
(5)

where PD is the load active power, QD is the load reactive power, and Bij is susceptance between buses i and j, respectively.

Finally, the system operating constraints hsc(x,u) are the sets of constraints which represent the system operational and security limits. These include generation constraints (generator voltages, real power outputs, and reactive power outputs which are restricted by the lower and upper limits), transformer constraints (transformer tap settings are restricted by the minimum and maximum limits), shunt VAR compensator constraints (define minimum and maximum VAR injection limits of the ith shunt compensator), and security constraints (incorporate the constraints of voltage magnitudes of load buses and transmission line loadings).

Although the OPF problem has been recognized many years ago, it still represents an actual research problem in power systems. To that end, we have put a lot of efforts to collect and analyze the literature dealing with optimization techniques for OPF. The results of this activity are summarized in Table I, which provides the reference details, IEEE test system, applied optimization method, and objective function. More than 45 papers published in prominent journals were analyzed. Two main conclusions are derived from the analysis:

TABLE I.

List of the papers which deal with the OPF problem, together with the used optimization method and objective function. * denotes modified parameters/function.

Literature IEEE test system Optimization method Objective function
MFC ($/h) MTPL (MW) TECM (ton/h) VPI VSE
Surender6   30  EP  …  … 
Yuryevich7   30  EP  …  …  … 
Somasundaram8   30  EP  …  …  …  … 
Ongsakul10   30  IEP  …  …  …  … 
Surender Reddy11   30, 300  MH  …  …  … 
Osman12   …  GA  …  …  …  … 
Lai13   30  IGA  …  …  …  … 
Bakirtzis14   3-area RTS96, 30  EGA  …  …  …  … 
Sailaja Kunari15   30  EGA  …  … 
Attia16   30  AGA  …  …  … 
Abido17   30  TSA  …  …  …  … 
Roa-Sepulveda18   30  SAA  …  …  …  … 
Abido19   30  PSO  …  … 
Hazra20   30, 118  PSO  …  …  … 
Kim21   30, 118  PSO and Parallel PSO  …  …  …  … 
Liang22   30, 118  PSO, PSO–LRS  …  … 
Radosavljevic23   30, 118  PSO–GSA  …  …  … 
Niknam24   30  IPSO  … 
Varadarajan25   30  DE  …  … 
Abou El Ela26   30  DEA  …  … 
El-Sehiemy27   30, 57  DEA  … 
Sayah28   30  MDEA  …  …  …  … 
Bouchekara-DSA29   30, 118  DSA  …  …  … 
Duman30   30, 57  GSA  …  … 
Bhattacharya31   118  GSA  …  … 
Bhowmik32   30  GSA 
Jahan33   30, 57, 118, 300, 2746  GSA  …  …  …  … 
Niknam34   30  MSFLA  …  …  … 
Sivasubramani35   I30  HSA  …  … 
Arul36   30, 300  CSADHSA  …  …  …  … 
Bhattacharya37   30  BBO  …  … 
Rezaei Adaryani38   9, 30, 57  ABCA  …  … 
El-Sehiemy39   30, 57  MPSOA, MPSO, HSOA, HPSOA  …  …  …  … 
Bouchekara40   30  ICBOA  …  …  … 
Bouchekara41   30, 57, 118  BHOA  …  …  …  … 
El-Fergany42   30, 118  DE, GWO  …  … 
Abdel-Fattah43   30, 118  SCA, MSC  …  … 
Bouchekara44   …  LCA  … 
Bouchekara45   30  TLBO  …  … 
Chaib46   30, 57, 118  BTSA  …  …  …  … 
Ghasemi47   57  ICA  …  …  …  …  … 
Daryani48   30, 57  AGSO  …  …  … 
Tripathy49   …  BFA  …  …  … 
Reddy50   30  GSOT, PSO  …  …  …  … 
Christy51   30  ABBPPOT  … 
Warid52   30, 118  Jaya  …  … 
Mishra53   30, 57  MCS  …  …  …  … 
Surender Reddy54   30  CATLBO  …  … 
Surender Reddy56   …  SA  …  …  … 
Surender Reddy58   30  SPEA  …  …  …  … 
Literature IEEE test system Optimization method Objective function
MFC ($/h) MTPL (MW) TECM (ton/h) VPI VSE
Surender6   30  EP  …  … 
Yuryevich7   30  EP  …  …  … 
Somasundaram8   30  EP  …  …  …  … 
Ongsakul10   30  IEP  …  …  …  … 
Surender Reddy11   30, 300  MH  …  …  … 
Osman12   …  GA  …  …  …  … 
Lai13   30  IGA  …  …  …  … 
Bakirtzis14   3-area RTS96, 30  EGA  …  …  …  … 
Sailaja Kunari15   30  EGA  …  … 
Attia16   30  AGA  …  …  … 
Abido17   30  TSA  …  …  …  … 
Roa-Sepulveda18   30  SAA  …  …  …  … 
Abido19   30  PSO  …  … 
Hazra20   30, 118  PSO  …  …  … 
Kim21   30, 118  PSO and Parallel PSO  …  …  …  … 
Liang22   30, 118  PSO, PSO–LRS  …  … 
Radosavljevic23   30, 118  PSO–GSA  …  …  … 
Niknam24   30  IPSO  … 
Varadarajan25   30  DE  …  … 
Abou El Ela26   30  DEA  …  … 
El-Sehiemy27   30, 57  DEA  … 
Sayah28   30  MDEA  …  …  …  … 
Bouchekara-DSA29   30, 118  DSA  …  …  … 
Duman30   30, 57  GSA  …  … 
Bhattacharya31   118  GSA  …  … 
Bhowmik32   30  GSA 
Jahan33   30, 57, 118, 300, 2746  GSA  …  …  …  … 
Niknam34   30  MSFLA  …  …  … 
Sivasubramani35   I30  HSA  …  … 
Arul36   30, 300  CSADHSA  …  …  …  … 
Bhattacharya37   30  BBO  …  … 
Rezaei Adaryani38   9, 30, 57  ABCA  …  … 
El-Sehiemy39   30, 57  MPSOA, MPSO, HSOA, HPSOA  …  …  …  … 
Bouchekara40   30  ICBOA  …  …  … 
Bouchekara41   30, 57, 118  BHOA  …  …  …  … 
El-Fergany42   30, 118  DE, GWO  …  … 
Abdel-Fattah43   30, 118  SCA, MSC  …  … 
Bouchekara44   …  LCA  … 
Bouchekara45   30  TLBO  …  … 
Chaib46   30, 57, 118  BTSA  …  …  …  … 
Ghasemi47   57  ICA  …  …  …  …  … 
Daryani48   30, 57  AGSO  …  …  … 
Tripathy49   …  BFA  …  …  … 
Reddy50   30  GSOT, PSO  …  …  …  … 
Christy51   30  ABBPPOT  … 
Warid52   30, 118  Jaya  …  … 
Mishra53   30, 57  MCS  …  …  …  … 
Surender Reddy54   30  CATLBO  …  … 
Surender Reddy56   …  SA  …  …  … 
Surender Reddy58   30  SPEA  …  …  …  … 
  • The IEEE 30 bus system, containing 6 generators and 24 load nodes, is the most commonly used test system for the OPF and

  • The most commonly used objective function is the fuel cost minimization.

In many of these papers, the comparison of different optimization algorithms can be found. Table II gives the results of comparison for the IEEE 30 test bus system.

TABLE II.

Comparison of OPF results for five objective functions obtained by using different optimization methods.

Literature Optimization method Objective function
MFC ($/h) MTPL (MW) TECM (ton/h) VPI VSE
Surender Reddy6   BM  799.445  3.0554  …  …  0.0892 
Somasundaram8   EP  802.4  …  …  …  … 
Bakirtzis14   EGA  802.06  …  …  …  … 
Sailaja Kunari15   EGA  799.56  3.2008  …  …  0.10402 
Attia16   AGA  799.8441  …  …  0.1207  … 
Abido17   TSA  802.29  …  …  …  … 
Roa-Sepulveda18   SA  799.45  …  …  …  … 
Abido19   PSO  800.41  …  …  0.0891  0.1246 
Hazra20   PSO  …  …  0.1942  …  … 
Kim21   PSO  800.68  …  …  …  … 
  Parallel PSO  800.64  …  …  …  … 
Liang22   PSO    3.48  …  …  … 
PSO–LRS    3.41  …  …  … 
Radosavljevic23   PSOGSA  799.7055  …  …  0.09638  … 
Niknam24   IPSO  801.978  5.0732  0.2058  …  0.1037 
Abou El Ela26   DE  799.2891  …  …  0.1357  0.1219 
El-Sehiemy27   FIDEA  799.0827  2.8535  …  0.0939  0.1243 
Sayah28   MDE  802.376  …  …  …  … 
Bouchekara29   DSA  799.0943  …  …  0.0977  … 
Duman30   GSA  798.67514  …  …  0.093269  0.116247 
Bhowmik32   NSMOOGSA  796.124  1.495  0.2236  0.1019  0.0647 
Niknam34   MSLFA  802.287  …  0.2056  …  … 
Sivasubramani35   HSA  798.8  2.9678  …  …  0.1006 
Arul36   CSADHSA  801.5888  …  …  …  … 
Bhattacharya37   BBO  799.1116  …  …  0.0951  0.09803 
Rezaei Adaryani38   ABC  800.66  3.1078  0.204826  …  … 
El-Sehiemy39   MPSOA  798.198  …  …  …  … 
MPSO  799.93  …  …  …  … 
HSOA  810.68  …  …  …  … 
HPSOA  812.48  …  …  …  … 
Bouchekara40   BHBO  799.9217  …  …  0.1262  0.11671 
Bouchekara41   ICBO  799.053  …  …  …  … 
El-Fergany42   DE  801.23  3.38  …  …  0.047 
  GWO  801.41  3.41  …    0.048 
Abdel-Fattah43   SCA  800.102  2.9425  …  0.1082  … 
  MSCA  799.31  2.9334  …  0.103  … 
Bouchekara45   TLA  799.0715  …  …  0.0945  0.11311 
Chaib46   BSO  799.0706  …  …  …  … 
Daryani48   AGSOA  801.75  …  0.2059  …  … 
Reddy50   GSO  799.05  …  …  …  … 
PSO  800.05  …  …  …  … 
Christy51   ABBPPO  799.0566  2.867  …  0.0887  0.11388 
Warid52   Jaya  800.4794  3.1035  …  …  0.1243 
Surender Reddy54   CATLBO  799.0522  3.0554  …  …  … 
Literature Optimization method Objective function
MFC ($/h) MTPL (MW) TECM (ton/h) VPI VSE
Surender Reddy6   BM  799.445  3.0554  …  …  0.0892 
Somasundaram8   EP  802.4  …  …  …  … 
Bakirtzis14   EGA  802.06  …  …  …  … 
Sailaja Kunari15   EGA  799.56  3.2008  …  …  0.10402 
Attia16   AGA  799.8441  …  …  0.1207  … 
Abido17   TSA  802.29  …  …  …  … 
Roa-Sepulveda18   SA  799.45  …  …  …  … 
Abido19   PSO  800.41  …  …  0.0891  0.1246 
Hazra20   PSO  …  …  0.1942  …  … 
Kim21   PSO  800.68  …  …  …  … 
  Parallel PSO  800.64  …  …  …  … 
Liang22   PSO    3.48  …  …  … 
PSO–LRS    3.41  …  …  … 
Radosavljevic23   PSOGSA  799.7055  …  …  0.09638  … 
Niknam24   IPSO  801.978  5.0732  0.2058  …  0.1037 
Abou El Ela26   DE  799.2891  …  …  0.1357  0.1219 
El-Sehiemy27   FIDEA  799.0827  2.8535  …  0.0939  0.1243 
Sayah28   MDE  802.376  …  …  …  … 
Bouchekara29   DSA  799.0943  …  …  0.0977  … 
Duman30   GSA  798.67514  …  …  0.093269  0.116247 
Bhowmik32   NSMOOGSA  796.124  1.495  0.2236  0.1019  0.0647 
Niknam34   MSLFA  802.287  …  0.2056  …  … 
Sivasubramani35   HSA  798.8  2.9678  …  …  0.1006 
Arul36   CSADHSA  801.5888  …  …  …  … 
Bhattacharya37   BBO  799.1116  …  …  0.0951  0.09803 
Rezaei Adaryani38   ABC  800.66  3.1078  0.204826  …  … 
El-Sehiemy39   MPSOA  798.198  …  …  …  … 
MPSO  799.93  …  …  …  … 
HSOA  810.68  …  …  …  … 
HPSOA  812.48  …  …  …  … 
Bouchekara40   BHBO  799.9217  …  …  0.1262  0.11671 
Bouchekara41   ICBO  799.053  …  …  …  … 
El-Fergany42   DE  801.23  3.38  …  …  0.047 
  GWO  801.41  3.41  …    0.048 
Abdel-Fattah43   SCA  800.102  2.9425  …  0.1082  … 
  MSCA  799.31  2.9334  …  0.103  … 
Bouchekara45   TLA  799.0715  …  …  0.0945  0.11311 
Chaib46   BSO  799.0706  …  …  …  … 
Daryani48   AGSOA  801.75  …  0.2059  …  … 
Reddy50   GSO  799.05  …  …  …  … 
PSO  800.05  …  …  …  … 
Christy51   ABBPPO  799.0566  2.867  …  0.0887  0.11388 
Warid52   Jaya  800.4794  3.1035  …  …  0.1243 
Surender Reddy54   CATLBO  799.0522  3.0554  …  …  … 

By analyzing the results presented in Table II, as well as the results from the previously listed papers, some methods obtain more accurate results although the differences between the optimal solutions are almost negligible. For example, the OPF solutions for the fuel cost minimization are in the range of 798.1 $/h to 812.5 $/h.

It is important to point out the inconsistency of the presented results, i.e., the existence of infeasible solutions for the OPF problem in some papers. For example, one or several inequality constraints of the problem violate their limitations during the optimization process. Furthermore, the authors compare these solutions with feasible solutions and derive unfair and inaccurate conclusions. The observed problem is also discussed in Refs. 38 and 52. The list of papers with infeasible solutions can be found in Ref. 38.

Let us conclude that the evolutionary computation techniques are very sensitive to appropriate selection of control parameters. Additionally, within the limited number of iterations, many of the analyzed optimization techniques do not converge to the global optimal solution at all the time of computation, so their performance cannot be judged by the results of a single run.

This section is divided into three subsections. Subsection IV A contains the fundamentals of the GAMS (GAMS/CONOPT) programming. The comparison of the OPF results obtained by using GAMS and literature known optimization methods is presented in Subsection IV B. The possibilities of the application of the GAMS/CONOPT on the systems with integrated wind generators are analyzed in Subsection IV C.

GAMS is one of the most powerful optimizer packages available for industrial applications or academic research in the fields of applied sciences or mathematics.61 

The most striking feature of the GAMS application for solving certain mathematical optimization model states is the capability to solve both linear and nonlinear models and to take into account continuous, discrete, or binary variables.

The basic structure of the mathematical model coded in GAMS has the following components: “sets” (assignments of members), “data” (parameters, tables, and scalars), “variable” (bounds and/or initial values), “equation” (components that hold the definition of the objective function), “model” (component that selects constraints that GAMS uses in objective function optimization), and “output.”61 The GAMS solution procedure is depicted in Fig. 1.

FIG. 1.

GAMS solution procedure.

FIG. 1.

GAMS solution procedure.

Close modal

A large number of solvers for mathematical programming models (deterministic global optimization, stochastic programming solver, linear programming, linear regression solver, etc.) are embedded in GAMS. However, nonlinear models, such as OPF problem, created with GAMS must be solved with a nonlinear programming (NLP) algorithm.

GAMS has two classes of nonlinear models—nonlinear programming (NLP) and differentiable nonlinear programming (DNLP). NLP models are defined as models in which all functions that appear with endogenous arguments, i.e., arguments that depend on model variables, are smooth with smooth derivatives. DNLP models can in addition use functions that are smooth but have discontinuous derivatives. For that reason, in this paper, we have used NLP models.

Currently, there are a large number of different solvers available for solving NLP in GAMS. We decide to use the CONOPT solver for the following reasons:

  • GAMS/CONOPT is well suited for models with very nonlinear constraints,

  • GAMS/CONOPT can use second derivatives,

  • GAMS/CONOPT has been designed for large models, and

  • GAMS/CONOPT is designed for models with smooth functions, but it can also be applied to models that do not have differentiable functions.

The CONOPT optimization algorithm could be formulated in the following form:63,64

min i = 1 E x ji
(6)

subject to

c i x i , x i 1 , , x i t = b i , i = 1 , , E ,
(7)
l i x i u i , i = 1 , , E ,
(8)

where x i is an m dimensional vector of optimization variables in period i , c i is an n dimensional function of constraint values in period i , b i is an n dimensional vector of right-hand sides in period i , l i and u i are the m dimensional vectors of upper and lower bounds in period i , x ji represents the j -th component of vector x i , and E is the time horizon. The flowchart of the CONOPT algorithm (see Fig. 2) can be found in Ref. 65, and a detailed description of all flow chart steps.

FIG. 2.

Flowchart of the CONOPT algorithm.

FIG. 2.

Flowchart of the CONOPT algorithm.

Close modal

In this paper, we will point out also two main characteristics of CONOPT. Namely, the widely used algorithms embedded in GAMS for solving large-scale nonlinear optimization problems are as follows: in addition to CONOPT, MINOS, KNITRO/ACTIVE, and SNOPT. However, in Ref. 65, it is shown that the CONOPT solver is the unequaled GAMS solver in terms of RCT. On the other hand, CONOPT has implemented three active-set methods. The first one is a gradient projection method, the second one a sequential linear programming algorithm, and the third one a sequential quadratic programming algorithm. CONOPT includes algorithmic switches that automatically detect the best method.65 For this reason, CONOPT is a high speed solver.

Finally, the usage of the CONOPT solver for solving different scientific problems is presented in Refs. 66–68.

The feasibility and effectiveness of the GAMS/CONOPT for solving OPF are tested on an IEEE 30-bus test system. The main parts of the developed GAMS/CONOPT code for the IEEE 30-bus system solving are presented in  Appendix B (Figs. 3–7). Note, we used GAMS version 24.7.1.

FIG. 3.

Example of declaring variables and equations in GAMS.

FIG. 3.

Example of declaring variables and equations in GAMS.

Close modal
FIG. 4.

Example of equation realization in GAMS.

FIG. 4.

Example of equation realization in GAMS.

Close modal
FIG. 5.

Example of setting bounds in GAMS.

FIG. 5.

Example of setting bounds in GAMS.

Close modal
FIG. 6.

Example of displaying results in GAMS.

FIG. 6.

Example of displaying results in GAMS.

Close modal
FIG. 7.

Example of loading data in GAMS.

FIG. 7.

Example of loading data in GAMS.

Close modal

In order to compare the GAMS/CONOPT results with those obtained by using other optimization algorithms such as PSO, GSA, ABC, and WDO, we have developed these algorithms using MATLAB (Version 2015). To that end, we have carried out numerical experiments on several test systems including IEEE test systems ranging in size from 9 to 30 buses. In all experiments, two different objective functions (fuel cost and power loss minimization) were taken into account and experiments were realized on a PC with a 2.5 GHz Intel Core i7 and 8 GB RAM.

The observed drawbacks of the evolutionary computation techniques in terms of their sensitivity to appropriate selection of control parameters and convergence problem within the limited number of iterations have pointed us to perform 30 test runs for all these algorithms. The best results are presented in Tables III and IV. For all optimization algorithms, the population size is set to 25, while the maximum number of iterations is set to 100. It should be pointed out that the maximal deviation between the obtained results in all performed tests was lower than 0.2%.

TABLE III.

Comparison of OPF solutions obtained by using different algorithms and solver CONOPT embedded in GAMS.

Optimization algorithm/solver IEEE 9 test system IEEE 14 test system IEEE 30 test system
MFC ($/h)  PSO  5293.335  8191.084  804.343 
GSA  5293.429  8182.425  802.0909 
ABC  5282.725  8094.125  802.6145 
WDO  5295.208  8097.412  801.4839 
GAMS/CONOPT  5296.686  8082.595  800.148 
MTPL (MW)  PSO  2.433808  1.369987  3.817803 
GSA  2.331828  1.511266  4.376247 
ABC  2.317738  0.804722  3.294266 
WDO  2.31776  0.61412  3.58926 
GAMS/CONOPT  2.317  0.612  3.136 
Optimization algorithm/solver IEEE 9 test system IEEE 14 test system IEEE 30 test system
MFC ($/h)  PSO  5293.335  8191.084  804.343 
GSA  5293.429  8182.425  802.0909 
ABC  5282.725  8094.125  802.6145 
WDO  5295.208  8097.412  801.4839 
GAMS/CONOPT  5296.686  8082.595  800.148 
MTPL (MW)  PSO  2.433808  1.369987  3.817803 
GSA  2.331828  1.511266  4.376247 
ABC  2.317738  0.804722  3.294266 
WDO  2.31776  0.61412  3.58926 
GAMS/CONOPT  2.317  0.612  3.136 
TABLE IV.

Comparison of the computation time for the OPF.

Optimization algorithm/solver IEEE 9 test system IEEE 14 test system IEEE 30 test system
Computation time in (s) for MFC  PSO  2.126728  5.314809  17.82697 
GSA  2.244032  6.164463  18.86064 
ABC  4.474348  12.99995  38.64705 
WDO  1.689555  6.232187  18.63043 
GAMS/CONOPT  0.078  0.085  0.093 
Computation time in (s) for MTPL  PSO  2.59023  6.069107  16.93375 
GSA  2.273988  6.343528  18.92881 
ABC  4.57543  13.23358  38.05651 
WDO  2.289727  6.504237  19.09615 
GAMS/CONOPT  0.073  0.075  0.092 
Optimization algorithm/solver IEEE 9 test system IEEE 14 test system IEEE 30 test system
Computation time in (s) for MFC  PSO  2.126728  5.314809  17.82697 
GSA  2.244032  6.164463  18.86064 
ABC  4.474348  12.99995  38.64705 
WDO  1.689555  6.232187  18.63043 
GAMS/CONOPT  0.078  0.085  0.093 
Computation time in (s) for MTPL  PSO  2.59023  6.069107  16.93375 
GSA  2.273988  6.343528  18.92881 
ABC  4.57543  13.23358  38.05651 
WDO  2.289727  6.504237  19.09615 
GAMS/CONOPT  0.073  0.075  0.092 

Table III reveals that all analyzed optimization methods yield very close results in terms of accuracy (optimal solution). In this regard, it can be concluded that GAMS/CONOPT enables at least the same level of accuracy.

However, Table IV presents the comparison between the methods regarding total computation time for obtaining the optimal solution. The following two main conclusions can be derived:

  1. The requested computation time grows with increasing dimensions of the tested systems (number of buses) and

  2. The requested computation time is extremely low when GAMS/CONOPT is used for the OPF problem solving.

Besides these conclusions, it is very important to stress that GAMS/CONOPT always find the same optimal solution, which is opposite to other optimization algorithms.

It should be noted that developed MATLAB algorithms (PSO, GSA, ABC, and GDO) have not been optimized regarding the computation time. However, GAMS/CONOPT and all used MATLAB algorithms had the same operating constraints. On the other hand, in Table III, the results obtained on the IEEE 9 test bus system show that GAMS/CONOPT has the lowest accuracy when a minimum fuel cost objective function is used. However, GAMS/CONOPT is superior for the IEEE 14 and 30-test bus systems. The reason for this situation lies in the fact that GAMS/CONOPT in general is superior for large system solving, as noted in Ref. 65.

The GAMS/CONOPT superiority over other literature known optimization techniques is given in Table V. Although different PC configurations (with respect to CPU speed and RAM capacity—computing hardware) have been used in the analyzed papers, it can be concluded that all current approaches yield very close optimal solution, but they require very high computation time for OPF solving.

TABLE V.

RTC of optimization algorithms for the OPF problem solving.

Optimization algorithm RTC Computing hardware
Fuel cost Power loss
Surender Reddy6   BM  3.054 s  3.149 s  … 
Lai13   IGA  5.25 min  …  50 MHz computer 
Kim21   PSO  8.03 s  …  CPU-Intel 2.0 GHz; RAM-DDR SD 256 MB; 8PC's 
Parallel PSO  1.73 s  … 
Liang22   PSO  …  218 s  Personal computer P4-2.8 GHz 
PSO–LRS  …  128 s 
Sayah28   MDE  23.07 s  …  Personal computer with Intel Pentium IV 3.0 GHz processor and 512 MB total memory 
Arul36   CSADHSA  2.16 min  …  1.6-GHz, 2-GB RAM, Pentium-IV, IBM personal computer (IBM, India) 
Bhattacharya37   BBO  11.02 s  …  2.3 GHz Pentium IV personal computer with 512-MB RAM 
El-Fergany42   DE  16.2 s  16.7 s  Intel(R) Core (TM)i7-3630QM CPU @ 2.4 GHz Processor, 8 GB RAM, 64-bitoperating system, PC 
GWO  15.8 s  16.1 s 
Reddy50   GSO  4.904 s  …  … 
PSO  7.98 s  … 
Surender Reddy11   BSM  3.054 s  3.149 s  … 
Optimization algorithm RTC Computing hardware
Fuel cost Power loss
Surender Reddy6   BM  3.054 s  3.149 s  … 
Lai13   IGA  5.25 min  …  50 MHz computer 
Kim21   PSO  8.03 s  …  CPU-Intel 2.0 GHz; RAM-DDR SD 256 MB; 8PC's 
Parallel PSO  1.73 s  … 
Liang22   PSO  …  218 s  Personal computer P4-2.8 GHz 
PSO–LRS  …  128 s 
Sayah28   MDE  23.07 s  …  Personal computer with Intel Pentium IV 3.0 GHz processor and 512 MB total memory 
Arul36   CSADHSA  2.16 min  …  1.6-GHz, 2-GB RAM, Pentium-IV, IBM personal computer (IBM, India) 
Bhattacharya37   BBO  11.02 s  …  2.3 GHz Pentium IV personal computer with 512-MB RAM 
El-Fergany42   DE  16.2 s  16.7 s  Intel(R) Core (TM)i7-3630QM CPU @ 2.4 GHz Processor, 8 GB RAM, 64-bitoperating system, PC 
GWO  15.8 s  16.1 s 
Reddy50   GSO  4.904 s  …  … 
PSO  7.98 s  … 
Surender Reddy11   BSM  3.054 s  3.149 s  … 

As previously stated, the GAMS/CONOPT is proved to be a very powerful optimization tool, enabling efficient optimization.

The large scale integration of the RES into modern power systems leads to new conditions for OPF. We have found it challenging to check the GAMS applicability and validity for these conditions.

OPF in the presence of distributed generators (DGs) is analyzed in a very limited number of papers. The Jaya algorithm for this purpose is presented in Ref. 52, and the obtained results will be taken into account for performing the comparison with the results obtained by using GAMS/CONOPT (Table VI).

TABLE VI.

GAMS vs Jaya algorithm52 for the OPF before and after DG integration. *The Jaya algorithm is performed in the computational environment of MATLAB R2015b and implemented on a PC with a 2.7 GHz Intel® Core™ (Intel Corporation, 2200 Mission College Blvd.: Santa Clara, CA, USA) i7 CPU and 16 GB RAM.52 

Variable Without wind generator Wind generator at node 3 Wind generator at node 30
Jaya algorithm52* GAMS/CONOPT Jaya algorithm52  GAMS/CONOPT Jaya algorithm52  GAMS/CONOPT
PG1 (MW)  177.744  177.139  169.467  172.488  169.723  172.536 
PG2 (MW)  48.1929  48.729  47.9308  47.656  47.6386  47.608 
PG5 (MW)  21.4679  21.318  21.1194  20.996  20.8386  20.967 
PG8 (MW)  21.1103  21.217  20.8342  18.818  20.6944  18.363 
PG11 (MW)  11.782  11.928  11.8917  11.1  11.8375  11.064 
PG13 (MW)  12.1669  12  12.0307  12  12.0173  12 
VG1 (p.u.)  1.0862  1.1  1.07033  1.1  1.07264  1.1 
VG2 (p.u.)  1.06653  1.088  1.05308  1.089  1.05512  1.088 
VG5 (p.u.)  1.0335  1.062  1.02076  1.063  1.01985  1.062 
VG8 (p.u.)  1.03722  1.071  1.02941  1.072  1.03177  1.071 
VG11 (p.u.)  1.09983  1.1  1.07827  1.1  1.07907  1.1 
VG13 (p.u.)  1.05041  1.09  1.04283  1.088  1.04054  1.093 
Cost ($/h)  800.479  800.148  769.963  768.329  768.039  766.473 
PDG (MW)  9.1169  9.1169  9.1478  9.1478 
QDG (MVAr)  5.6501  5.6501  5.6692  5.6692 
Loss (MW)  9.06481  8.932  8.9912  8.782  8.4983  8.294 
Variable Without wind generator Wind generator at node 3 Wind generator at node 30
Jaya algorithm52* GAMS/CONOPT Jaya algorithm52  GAMS/CONOPT Jaya algorithm52  GAMS/CONOPT
PG1 (MW)  177.744  177.139  169.467  172.488  169.723  172.536 
PG2 (MW)  48.1929  48.729  47.9308  47.656  47.6386  47.608 
PG5 (MW)  21.4679  21.318  21.1194  20.996  20.8386  20.967 
PG8 (MW)  21.1103  21.217  20.8342  18.818  20.6944  18.363 
PG11 (MW)  11.782  11.928  11.8917  11.1  11.8375  11.064 
PG13 (MW)  12.1669  12  12.0307  12  12.0173  12 
VG1 (p.u.)  1.0862  1.1  1.07033  1.1  1.07264  1.1 
VG2 (p.u.)  1.06653  1.088  1.05308  1.089  1.05512  1.088 
VG5 (p.u.)  1.0335  1.062  1.02076  1.063  1.01985  1.062 
VG8 (p.u.)  1.03722  1.071  1.02941  1.072  1.03177  1.071 
VG11 (p.u.)  1.09983  1.1  1.07827  1.1  1.07907  1.1 
VG13 (p.u.)  1.05041  1.09  1.04283  1.088  1.04054  1.093 
Cost ($/h)  800.479  800.148  769.963  768.329  768.039  766.473 
PDG (MW)  9.1169  9.1169  9.1478  9.1478 
QDG (MVAr)  5.6501  5.6501  5.6692  5.6692 
Loss (MW)  9.06481  8.932  8.9912  8.782  8.4983  8.294 

The modified IEEE 30-bus test system has been taken as a base for the GAMS/CONOPT application. Two cases will be analyzed: the system optimization before and after DG integration. Related to the second case, it is assumed that wind generators are connected at nodes 30 and 3.52 Their individual active power does not exceed 10 MW, while the power factor is 0.85. Fuel cost minimization is taken as a goal function during the GAMS/CONOPT implementation. The same goal function was considered in Ref. 52 with the Jaya algorithm.

As can be derived from Table VI, for the same value of active and reactive powers of integrated wind generators, the production costs are lower if the OPF problem is solved using GAMS. Also, although the production of generators obtained by using GAMS/CONOPT is very close to that obtained by using the Jaya algorithm, it can be seen that in this case, GAMS/CONOPT enables obtaining lower system losses. On the other side, it can be concluded that the integration of wind generator reduces the cost of energy production of other power sources. In addition, GAMS/CONOPT outperforms Jaya in terms of CPU time (the computation time for the Jaya algorithm is about 35 s—see Ref. 52, page 10).

We have tested the GAMS/CONOPT also for the modified IEEE 30-bus test with wind farm at node 30. We have assumed that the wind farm gives constant active power, without any reactive power compensation. For the demonstration purpose, we have considered three values of wind active power, namely, 5 MW, 10 MW, and 15 MW, and all optimizations are realized for two objective functions (fuel cost and power loss minimization). The results of calculations are presented in Tables VII and VIII, which confirm the following:

TABLE VII.

Optimal value of control variables obtained by using GAMS/CONOPT without and with the presence of the RES obtained for the fuel cost minimization objective function (modified IEEE 30-bus network).

DG30 P = 0 DG30 P = 5 DG30 P = 10 DG30 P = 15
PG1 (MW)  177.1394592  174.5927826  172.0986875  169.6535107 
PG2 (MW)  48.72873085  48.10931648  47.50565079  46.91641169 
PG5 (MW)  21.31837919  21.12513966  20.93665456  20.75256685 
PG8 (MW)  21.21658372  19.64537235  18.12216665  16.64268374 
PG11 (MW)  11.9279301  11.45513406  10.99513413  10.54685323 
PG13 (MW)  12  12  12  12 
VG1 (p.u.)  1.1  1.1  1.1  1.1 
VG2 (p.u.)  1.088262832  1.088075693  1.088056329  1.088177945 
VG5 (p.u.)  1.062461021  1.062134744  1.062124407  1.06237939 
VG8 (p.u.)  1.071417806  1.071148198  1.07133836  1.07191591 
VG11 (p.u.)  1.1  1.1  1.1  1.1 
VG13 (p.u.)  1.089838119  1.08840122  1.087657318  1.087470988 
Cost ($/h)  800.148445  781.5198487  763.4702329  745.9541081 
PDG (MW)  10  15 
QDG (MVAr) 
Loss (MW)  8.932057018  8.528288636  8.259134476  8.113737997 
RTC (s)  0.093  0.097  0.097  0.097 
DG30 P = 0 DG30 P = 5 DG30 P = 10 DG30 P = 15
PG1 (MW)  177.1394592  174.5927826  172.0986875  169.6535107 
PG2 (MW)  48.72873085  48.10931648  47.50565079  46.91641169 
PG5 (MW)  21.31837919  21.12513966  20.93665456  20.75256685 
PG8 (MW)  21.21658372  19.64537235  18.12216665  16.64268374 
PG11 (MW)  11.9279301  11.45513406  10.99513413  10.54685323 
PG13 (MW)  12  12  12  12 
VG1 (p.u.)  1.1  1.1  1.1  1.1 
VG2 (p.u.)  1.088262832  1.088075693  1.088056329  1.088177945 
VG5 (p.u.)  1.062461021  1.062134744  1.062124407  1.06237939 
VG8 (p.u.)  1.071417806  1.071148198  1.07133836  1.07191591 
VG11 (p.u.)  1.1  1.1  1.1  1.1 
VG13 (p.u.)  1.089838119  1.08840122  1.087657318  1.087470988 
Cost ($/h)  800.148445  781.5198487  763.4702329  745.9541081 
PDG (MW)  10  15 
QDG (MVAr) 
Loss (MW)  8.932057018  8.528288636  8.259134476  8.113737997 
RTC (s)  0.093  0.097  0.097  0.097 
TABLE VIII.

Optimal value of control variables obtained by using GAMS/CONOPT without and with the presence of the RES obtained for the power loss minimization objective function (modified IEEE 30-bus network).

DG30 P = 0 DG30 P = 5 DG30 P = 10 DG30 P = 15
PG1 (MW)  51.536195  50  50  50 
PG2 (MW)  80  76.17959147  70.97042744  65.88515127 
PG5 (MW)  50  50  50  50 
PG8 (MW)  35  35  35  35 
PG11 (MW)  30  30  30  30 
PG13 (MW)  40  40  40  40 
VG1 (p.u.)  1.1  1.1  1.1  1.1 
VG2 (p.u.)  1.098090327  1.097711898  1.097293023  1.097019712 
VG5 (p.u.)  1.080773693  1.080447958  1.080240158  1.080294276 
VG8 (p.u.)  1.089026805  1.089124599  1.089500922  1.090245484 
VG11 (p.u.)  1.1  1.1  1.1  1.1 
VG13 (p.u.)  1.096262009  1.094902366  1.094125547  1.093890713 
Cost ($/h)  967.5670138  946.7900628  924.2597755  903.181445 
PDG (MW)   0  10  15 
QDG (MVAr)   0  0  
Loss (MW)  3.136194999  2.779591471  2.570427443  2.485151268 
RTC (s)  0.092  0.094  0.094  0.094 
DG30 P = 0 DG30 P = 5 DG30 P = 10 DG30 P = 15
PG1 (MW)  51.536195  50  50  50 
PG2 (MW)  80  76.17959147  70.97042744  65.88515127 
PG5 (MW)  50  50  50  50 
PG8 (MW)  35  35  35  35 
PG11 (MW)  30  30  30  30 
PG13 (MW)  40  40  40  40 
VG1 (p.u.)  1.1  1.1  1.1  1.1 
VG2 (p.u.)  1.098090327  1.097711898  1.097293023  1.097019712 
VG5 (p.u.)  1.080773693  1.080447958  1.080240158  1.080294276 
VG8 (p.u.)  1.089026805  1.089124599  1.089500922  1.090245484 
VG11 (p.u.)  1.1  1.1  1.1  1.1 
VG13 (p.u.)  1.096262009  1.094902366  1.094125547  1.093890713 
Cost ($/h)  967.5670138  946.7900628  924.2597755  903.181445 
PDG (MW)   0  10  15 
QDG (MVAr)   0  0  
Loss (MW)  3.136194999  2.779591471  2.570427443  2.485151268 
RTC (s)  0.092  0.094  0.094  0.094 
  1. The GAMS/CONOPT accuracy,

  2. High efficiency in solving the OPF problem, and

  3. Full GAMS/CONOPT applicability in the OPF problem solving in the presence of the RES.

It can also be seen that the integration of a wind generator with higher power reduces the cost of energy production of other power sources (see Table VII) and reduces the total power system losses (see Table VIII). However, Tables VII and VIII show that the difference in generator output power obtained for two different objective functions is huge. Therefore, minimizing the total operation cost will increase losses and vice versa. In both cases, however, technical limits of all electric components must be satisfied.

Finally, it should be noted that in this paper, we tested the usage of GAMS when the system contains renewable power sources. In real-time OPF, the wind capacity of wind farm, and other constraints (such as load scheduling) should be taken into account.

This paper deals with the application of the CONOPT solver for nonlinear programming (NLP) imbedded in the GAMS software package for the OPF problem. The proposed approach has been evaluated on the IEEE 30-bus and modified IEEE 30-bus systems with integrated renewable energy sources. Simulation results show that GAMS/CONOPT provides a very efficient and accurate solution. Moreover, the results obtained using GAMS/CONOPT are on par or better than those obtained using other proposed optimization techniques. The requested computation time of GAMS is significantly lower compared with other techniques for the OPF problem. Therefore, the GAMS program is suitable for real-time OPF.

The future work will be oriented toward using GAMS/CONOPT for solving OPF under more comprehensive real-time conditions. Furthermore, our plan is to test the impact of renewable power sources together with the storage system on the total production cost, system loss, etc.

This work was supported through European Union's Horizon 2020 research and innovation program under project CROSSBOW—CROSS BOrder management of variable renewable energies and storage units enabling a transnational Wholesale market (Grant No. 773430).

ABBPPOT

Adaptive Biogeography Based Predator-Pray Optimization Technique

ABCA

Artificial Bee Colony Algorithm

AGA

Adapted Genetic Algorithm

AGSOA

Adaptive Group Search Optimization Algorithm

BBOA

Biogeography Based Optimization Algorithm

BFA

Bacteria Foraging Algorithm

BHOA

Black Hole Optimization Approach

BM

Bisection Method

BTSA

Back Tracking Search Algorithm

CATLBO

Clustered Adaptive Teaching Learning-Based Optimization

CPU

Central Processing Unit

CSA

Cuckoo Search Algorithm

CSADHSA

Chaotic Self-Adaptive Differential Harmony Search Algorithm

DE

Differential evolution

DEA

Differential Evolution Algorithm

DNLP

Differentiable Non-Linear Programming

DSA

Differential Search Algorithm

EGA

Enhanced Genetic Algorithm

EGSA

Enhanced Gravitational Search Algorithm

EP

Evolutionary Programming

GA

Genetic Algorithm

GAMS

Generalized Algebraic Modeling Systems

GPM

Gradient Projection Method

GSA

Gravitational Search Algorithm

GSOT

Glowworm Swarm Optimization Technique

GWO

Gray Wolf optimizer

HPSOA

Hybrid Particle Swarm Optimization Approach

HSA

Harmony Search Algorithm

HSOA

Hybrid Search Optimization Algorithm

ICA

Imperialist Competitive Algorithm

ICBOA

Improved Colliding Bodies Optimization Algorithm

IEP

Improved Evolutionary Programming

IGA

Improved Genetic Algorithm

IPM

Interior Point Method

IPSO

Improved Particle Swarm Optimization

JA

Jaya algorithm

LCA

League Championship Algorithm

LPM

Linear Programming Method

MCS

Modified Cuckoo Search

MDEA

Modified Differential Evolution Algorithm

MFC

Minimization of total fuel cost

MHA

Meta-Heuristic Algorithms

MPSO

Modified Particle Swarm Optimization

MPSOA

Multi-Phase Search Optimization algorithm

MSCA

Modified Sine-Cosine algorithm

MSFLA

Modified Shuffle Frog Leaping Algorithm

MTPL

Minimization of total power losses

NLP

Non-Linear programming

NM

Newton method

OF

Objective function

OPF

Optimal power flow

PC

Personal Computer

PSO

Particle Swarm Optimization

PSO-LRS

Particle swarm optimization – Local random search

RCT

Requested Computation Time

RES

Renewable Energy Sources

SA

Stochastic Approach

SAA

Simulated Annealing Algorithm

SCA

Sine-Cosine Algorithm

SPEA

Strength Pareto Evolutionary Algorithm

TECM

Total emission cost minimization

TLA

Teaching-Learning Algorithm

TSA

Tabu Search Algorithm

VPI

Voltage profile improvement

VSE

Voltage stability enhancement

WDO

Wind Driven Optimization

TABLE IX.

List of evolutionary programming (EP) algorithms, improved evolutionary programming (IEP) algorithms, and meta-heuristic algorithms (MHA) for OPF problem solving.

Algorithm References
Evolutionary programming (EP)  7–9  
Improved evolutionary programming (IEP)  10  
Meta-heuristic algorithms (MHA)  11  
Genetic algorithm (GA)  12  
Improved genetic algorithm (IGA)  13  
Enhanced genetic algorithm (EGA)  14 and 15  
Adapted genetic algorithm (AGA)  16  
Tabu search algorithm (TSA)  17  
Simulated annealing algorithm (SAA)  18  
Particle swarm optimization (PSO)  19–21  
Hybrid particle swarm optimization approach (HPSOA)  22 and 23  
Improved particle swarm optimization (IPSO)  24  
Modified particle swarm optimization (MPSO)  39  
Differential evolution algorithm (DEA)  25–27  
Modified differential evolution algorithm (MDEA)  28  
Differential search algorithm (DSA)  29  
Gravitational search algorithm (GSA)  23 and 30–32  
Enhanced gravitational search algorithm (EGSA)  33  
Modified shuffle frog leaping algorithm (MSFLA)  34  
Harmony search algorithm (HAS)  35  
Chaotic self-adaptive differential harmony search algorithm (CSADHSA)  36  
Biogeography based optimization algorithm (BBOA)  37  
Artificial bee colony algorithm (ABCA)  38  
Multiphase search optimization algorithm (MPSOA)  39  
Hybrid search optimization algorithm (HSOA)  39  
Improved colliding bodies optimization algorithm (ICBOA)  40  
Black hole optimization approach (BHOA)  41  
Gray wolf optimizer (GWO)  42  
Sine-cosine algorithm (SCA)  43  
League championship algorithm (LCA)  44  
Teaching-learning algorithm (TLA)  45  
Back tracking search algorithm (BTSA)  46  
Imperialist competitive algorithm (ICA)  47  
Adaptive group search optimization algorithm (AGSOA)  48  
Bacteria foraging algorithm (BFA)  49  
Glowworm swarm optimization technique (GSOT)  50  
Adaptive biogeography based predator-pray optimization technique (ABBPPOT)  51  
Jaya algorithm (JA)  52  
Modified cuckoo search (MCS)  53  
Clustered adaptive teaching learning-based optimization (CATLBO)  54  
Algorithm References
Evolutionary programming (EP)  7–9  
Improved evolutionary programming (IEP)  10  
Meta-heuristic algorithms (MHA)  11  
Genetic algorithm (GA)  12  
Improved genetic algorithm (IGA)  13  
Enhanced genetic algorithm (EGA)  14 and 15  
Adapted genetic algorithm (AGA)  16  
Tabu search algorithm (TSA)  17  
Simulated annealing algorithm (SAA)  18  
Particle swarm optimization (PSO)  19–21  
Hybrid particle swarm optimization approach (HPSOA)  22 and 23  
Improved particle swarm optimization (IPSO)  24  
Modified particle swarm optimization (MPSO)  39  
Differential evolution algorithm (DEA)  25–27  
Modified differential evolution algorithm (MDEA)  28  
Differential search algorithm (DSA)  29  
Gravitational search algorithm (GSA)  23 and 30–32  
Enhanced gravitational search algorithm (EGSA)  33  
Modified shuffle frog leaping algorithm (MSFLA)  34  
Harmony search algorithm (HAS)  35  
Chaotic self-adaptive differential harmony search algorithm (CSADHSA)  36  
Biogeography based optimization algorithm (BBOA)  37  
Artificial bee colony algorithm (ABCA)  38  
Multiphase search optimization algorithm (MPSOA)  39  
Hybrid search optimization algorithm (HSOA)  39  
Improved colliding bodies optimization algorithm (ICBOA)  40  
Black hole optimization approach (BHOA)  41  
Gray wolf optimizer (GWO)  42  
Sine-cosine algorithm (SCA)  43  
League championship algorithm (LCA)  44  
Teaching-learning algorithm (TLA)  45  
Back tracking search algorithm (BTSA)  46  
Imperialist competitive algorithm (ICA)  47  
Adaptive group search optimization algorithm (AGSOA)  48  
Bacteria foraging algorithm (BFA)  49  
Glowworm swarm optimization technique (GSOT)  50  
Adaptive biogeography based predator-pray optimization technique (ABBPPOT)  51  
Jaya algorithm (JA)  52  
Modified cuckoo search (MCS)  53  
Clustered adaptive teaching learning-based optimization (CATLBO)  54  

In this appendix, in Figs. 3–7, the main parts of realized GAMS/CONOPT for OPF are presented.

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