G. R. Venkatakrishnan , R. Rengaraj and S. Salivahanan

1 Introduction

Real power economic dispatch (RPED) is one of the most important non - linear problem to be solved in the modern power system. The objective of the RPED problem is to allocate optimal real power generation to the existing thermal units without violating the constraints in the system. Conventional methods like lambda-iteration method, and so on are used to solve traditional RPED problem with assumptions many assumptions [Park, Lee, Shin et al. (2005); Sayah and Hamouda (2013)].

However, in practical, the nonlinearities and discontinuities like valve point loading, ramp rate limits and so on represent RPED problem as a non-smooth or non-convex optimization problem which makes it difficult for the traditional methods to obtain the global optimum[Park, Lee, Shin et al. (2005); Sayah and Hamouda (2013)]. Moreover considerable number of researchers has shown interest in developing an efficient algorithm in solving the RPED problem with nonlinearities [Mandal, Roy and Mandal (2014)]. Though the conventional methods have advantages like few control parameters and less computational time, it fails to reach global optima for the ELD problems with large dimensional and discrete search space [Nguyen and Vo (2015)].

According to No Free Lunch (NFL) theorem, there exist no meta heuristic optimization algorithm which is applicable in solving all real world optimization problems [Mirjalili,Mirjalili and Lewis (2014); Basu (2014)]. The development of numerous meta heuristic algorithms by various researchers around the world over the past two decades has successfully solved the ELD problem with superior convergence characteristics, high solution quality and robustness, eliminating most of the difficulties of classical methods[Mandal, Roy and Mandal (2014); Basu (2015)].

Grey Wolf Optimization (GWO) algorithm, a recent swarm intelligence algorithm is proposed to solve the non-convex optimization problem [Mirjalili, Mirjalili and Lewis(2014)]. The leadership and hunting behaviors of grey wolves in nature is incorporated in the algorithm and has superior exploration and exploitation ability. In solving real world problems, the GWO algorithm has the capability of providing higher quality solutions and good computational efficiency with few parameters and ease of implementation [Mandal,Roy and Mandal (2014); Mirjalili, Mirjalili and Lewis (2014)]. These properties have motivated few researchers to implement the GWO algorithm in solving problems like combined heat and power dispatch [Mandal, Roy and Mandal (2014)], hyper spectral band selection [Medjaheda, Ait Saadib, Benyettoua et al. (2015)], load frequency control [Guha,Roy and Banerjee (2015)], optimal reactive power dispatch [Sulaimana, Mustaffab,Mohameda et al. (2015)], power system stabilizer design [Shakarami and Faraji Davoudkhani (2015)], MPPT design [Mohanty, Subudhi and Ray (2016)], flow shop scheduling [Komakia and Kayvanfar (2015)], attribute reduction [Emarya, Yamany,Hassaniena et al. (2015)], feature selection [Emary, Zawbaa and Hassaniena (2015)],parameter estimation [Song, Tang, Zhao et al. (2015)] and automatic generation control[Sharma and Saikia (2015)]. In this paper, GWO algorithm is implemented to solve the RPED problem to validate its effectiveness over other meta heuristic algorithms. The simulation results show that this algorithm performs better than the other algorithms in terms of solution quality, convergence efficiency and robustness.

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Section 2 describes the formulation of ELD problem with constraints like ramp rate limits and so on. The detailed description of GWO algorithm is discussed in Section 3. Section 4 describes the implementation of GWO to the complex RPED problem. The numerical results and discussion of the GWO algorithm for different test systems are presented in Section 5 and conclusion is drawn in Section 6.

2 Formulation of the ELD problem

Minimization of the total cost in producing real power in a power system without violating constraints is the main aim of RPED [Sahoo, Dash, Prusty et al. (2015)]. In this paper,RPED problem without valve point loading is considered.

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2.1 RPED problem with smooth cost function

The objective of the RPED problem with smooth cost function is given by

Step 2: Initialization of GWO parameters i.e. population size N and select the stopping criteria.

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whererepresents total fuel cost of all the thermal units present in the system ($/hr),N is the total number of thermal units existing in the system andrepresents fuel cost of thethermal unit ($/hr) andrepresents power generated by thethermal unit (MW).In general, the fuel cost functionthermal unit is expressed in quadratic polynomial as

where aq,bqandare the cost coefficients ofthermal unit.

The different practical constraints to which the above minimization problem is subjected are power balance or demand constraint, generator output limits, prohibited operating zones and ramp rate limits.

2.2 Power balance or demand constraint

The sum of individual power generated from each thermal unit existing in the system must be equal to the sum of transmission loss and total demand of the system which is represented as

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