A New Heuristic Algorithm for Solving Non-convex Economic Power Dispatch
Valve-points effects and prohibited operation zones (POZ) make the generating units cost functions non-linear and non-smooth. Hence, the Economic Dispatch (ED) problem is a highly non-convex optimization problem. The consideration of the transmission losses makes the ED problem even more complicated. This study presents a novel and heuristic algorithm for solving ED problems, by employing a new heuristic method, called Imperialist Competition Algorithm (ICA). The effectiveness of the proposed method is examined and validated by carrying out extensive tests on two different test systems. The valve-point effects, POZs, ramp-rate constraints and transmission losses are considered in the analysis. The numerical results show that the ICA method has good convergence property. Furthermore, the generation costs of the ICA approach are lower than other optimization algorithms reported in recent literature.
August 02, 2011; Accepted: November 19, 2011;
Published: January 02, 2012
The Economic Dispatch (ED) problem is one of the important problems in the operation of power systems. The purpose of ED problem is to determine the economical combination of generators production to satisfy the constraints while supplying the demand.
ED problem is a non-convex and nonlinear optimization problem. Due to the high
economic benefit of better solutions, many researches have been carried out
to present solution methods over the years. Many optimization methods including
classical and stochastic search approaches applied to ED problem. Applying the
conventional methods has their own shortcomings. The most popular conventional
method to solve ED problem is lambda-iteration method (Wood
and Wollenberg, 1996) which needs that the cost function be continuous and
However, the actual cost function is not continuous due to prohibited operation
zones. Many stochastic search based methods have been used to solve the ED problem
more efficiently. Genetic algorithm (Chiang, 2007; Walters
and Sheble, 1993) and its variants like as improved genetic algorithm (Chiang,
2005; Ling and Leung, 2007), Atavistic genetic algorithm
(Kim et al., 2002), Hybrid genetic algorithm
(He et al., 2008), self adaptive real-coded genetic
algorithm (Subbaraj et al., 2009) and real coded
genetic algorithm (Amjady and Nasiri-Rad, 2010a) proposed
to solve different types of ED problem. Hybrid particle swarm optimization presented
(Niknam et al., 2011a) for solving ED problem
considering valve-point effects. A hybrid multi-agent based particle swarm optimization
algorithm proposed by Kumar et al. (2011) to
solve valve-point constrained ED problem. Other modifications of Particle Swarm
Optimization method like as Parallel particle swarm optimization (Subbaraj
et al., 2010), quantum-behaved particle swarm optimization (Sun
et al., 2009), fuzzy adaptive hybrid particle swarm optimization
algorithm (Niknam, 2010), Anti predatory particle swarm
optimization (Selvakumar and Thanushkodi, 2008) applied
to solve Ed problem, also. Differential Evolution is used (Nomana
and Iba, 2010) to solve ED problem. Application of Modified DE and Hybrid
DE in ED presented (Amjady and Sharifzadeh, 2010b; Wang
et al., 2007).
In this study, an Imperialist Competition Algorithm (ICA) is proposed to solve non-convex dynamic economic dispatch problem with constraints.
ED PROBLEM FORMULATION
The objective function of ED problem is to minimize the total production cost over the operating horizon, which can be written as:
where, Ci is the production cost of unit i at time t, N is the number
of dispatchable power generation units and Pi is the power output
of I-th unit at time t. T is the total number of hours in the operating horizon.
The production cost of generation unit considering valve-point effects is defined
where, ai, bi, ci are the fuel cost coefficients
of the i-th unit, ei and fi are the valve-point coefficients
of the i-th unit. is the minimum capacity limit of unit .
It should be noted that the added sinusoidal term in the production cost function
reflects the effect of valve-points. The ED problem will be non-convex and non-differentiable
considering valve-point effects.
The objective function of the ED problem (1) should be minimized subject to the following equality and inequality constraints:
Real power balance: Hourly power balance considering network transmission losses is written as:
where, Ploss (t) and PD (t) are the total transmission loss and total load demand of the system at time t, respectively. The system loss is a function of units power production and can be calculated using the results of load flow problem Krons loss formula known as B-matrix coefficients. In this study, B- matrix coefficients method is used to calculate system loss as follows:
Generation limits of units:
is the maximum power outputs of i-th unit.
Ramp up and ramp down constraints: The output power change rate of the thermal unit must be in an acceptable range to avoid undue stresses on the boiler and combustion equipments. The ramp rate limits of generation units can be mathematically stated as follows:
where, URi is the ramp up limit of the i-th generator (MW/hr) and DRi is the ramp down limit of the i-th generator (MW/hr). Considering the ramp rate limits of unit, generator capacity limit (5) can be rewritten as follows:
Prohibited Operation Zones limits (POZs): Generating units may have certain restricted operation zone due to limitations of machine components or instability concerns. The allowable operation zones of generation unit can be defined as:
are the lower and upper limits of the jth prohibited zone of unit
i, respectively. Mi is the number of prohibited operation zones of
IMPERIALIST COMPETITION ALGORITHM (ICA)
The Imperialist Competition Algorithm (ICA) was first proposed by Atashpaz-Gargari
and Lucas (2007). It is inspired by the imperialistic competition. It starts
with an initial population called colonies. The colonies are then categorized
into two groups namely, imperialists (best solutions) and colonies (rest of
the solutions). The imperialists try to absorb more colonies to their empire.
The colonies will change according to the policies of imperialists. The colonies
may take the place of their imperialist if they become stronger than it (propose
a better solution). This algorithm has been successfully applied to PSS design
(Jalilvand et al., 2010) and data clustering
(Niknam et al., 2011b). The flowchart of proposed
algorithm is depicted in Fig. 1. The steps of the proposed
ICA are described as follows:
||Generate an initial set of colonies with a size of NC
||Set iteration = 1
||Calculate the objective function for each colony using (2) and set the
power of each colony as Follows:
This means the less OF is, the more stronger IPi is.
|| Keep the best Nimp colonies as the imperialists
and set the power of each imperialist as follows:
|| Assign the colonies to each imperialist according to calculated.
This means the number of colonies owned by each imperialist is proportional
to its power, i.e., IPi
||Move the colonies toward their relevant imperialist using crossover and
||Exchange the position of a colony and the imperialist if it is stronger
(CPc > IPi)
||Compute the empires power, i.e., EPi for all empires
where, w1 and w2 are weighting factors which are adaptively
|| Pick the weakest colony and give it to one of the best empires
(select the destination empire probabilistically based on its power (EPi)
||Eliminate the empire that has no colony
||If more than one empire remained then go to Step. 6
The flowchart of the proposed Algorithm is depicted in Fig. 1.
CASE STUDIES AND NUMERICAL RESULTS
Here, the proposed ICA is applied on two test systems with different number of generating units. After a number of careful experimentation, following optimum values of ICA parameters have finally been settled: = 100; crossover probability = 0.6, mutation probability = 0.2.
|| The flowchart of the proposed ICA algorithm
The stopping criteria is defined as reaching to the maximum number of iterations (here 600 iterations) or when no significant changes observed in the objective function.
Case 1: 6-unit system: The first test system is a 6-unit system. Systems
total demand is 1263 MW. The data for this system is provided by Pothiya
et al. (2008). In this test system, the transmission losses, POZs
and ramp-rate constraints are considered.
Table 1 shows the obtained results for this system. These
results are compared with multiple Tabu search (MTS) algorithm (Pothiya
et al., 2008), new PSO with local random search (NPSOLRS) (Selvakumar
and Thanushkodi, 2007), bacterial foraging optimization (BFO) (Panigrahi
and Pandi, 2008), new adaptive PSO (NAPSO) algorithm (Niknam
et al., 2011c), GA (Selvakumar and Thanushkodi,
2007) and PSO (Selvakumar and Thanushkodi, 2007).
The obtained results outperform the existing methods and hence the proposed method is so effective for solving such non-convex ED problems.
Case 2: 40-unit system: This test case consists of 40 generating units
with valve-point effects (Sinha et al., 2003).
Total load demand of the system is 10500 MW. The obtained results by the proposed
ICA are presented in Table 2.
|| Comparison of simulation results for the 6-unit system
|TG: Total generation, TL: Total loss, TC: Total cost
|| Comparison of simulation results for the 40-unit system
|TG: Total generation, TC: Total cost
The obtained results are compared with the results of hybrid DE with biogeography-based
optimization (DE/BBO) algorithm (Bhattacharya and Chattopadhyay,
2010a), variable DE with the fuzzy adaptive PSO (FAPSO-VDE) (Niknam
et al., 2011a), quantum-inspired PSO (QPSO) (Meng
et al., 2010), multi-agent based hybrid PSO technique (HMAPSO) (Kumar
et al., 2011), biogeography-based optimization (BBO) algorithm (Bhattacharya
and Chattopadhyay, 2010b) and hybrid GA (HGA) (He et
These results imply the efficiency of the proposed ICA method for dealing with the ED problem.
In this study, a new approach to solve power systems economic dispatch problem, called imperialist competition algorithm (ICA), is proposed. The valve-point effect, prohibited operation zones, ramp-rate constraints and transmission losses are modeled and the resulting non-linear and non-convex optimization problem is solved by ICA. The proposed method is applied on two test systems. The analysis results have demonstrated that for such a complicated problem, ICA founds solutions better than so far best known results by any other method in terms of cost and power losses.
Amjady, N. and H. Nasiri-Rad, 2010. Solution of nonconvex and nonsmooth economic dispatch by a new adaptive real coded genetic algorithm. Expert Syst. Appl., 37: 5239-5245.
Amjady, N. and H. Sharifzadeh, 2010. Solution of non-convex economic dispatch problem considering valve loading effect by a new modified differential evolution algorithm. Int. J. Electr. Power Energy Syst., 32: 893-903.
Atashpaz-Gargari, E. and C. Lucas, 2007. Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. Proceedings of the IEEE Congress on Evolutionary Computation, September 25-28, 2007, Singapore, pp: 4661-4667.
Bhattacharya, A. and P.K. Chattopadhyay, 2010. Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch. Inst. Electr. Electron. Eng. Trans. Power Syst., 25: 1955-1964.
Bhattacharya, A. and P.K. Chattopadhyay, 2010. Biogeography-based optimization for different economic load dispatch problems. IEEE Trans. Power Syst., 25: 1064-1077.
Chiang, C.L., 2005. Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels. Inst. Electr. Electron. Eng. Trans. Power Syst., 15: 1690-1699.
Chiang, C.L., 2007. Genetic-based algorithm for power economic load dispatch. IET Gen. Transm. Distrib., 1: 261-269.
He, D., F. Wang and Z. Mao, 2008. A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect. Int. J. Electr. Power Energy Syst., 30: 31-38.
Jalilvand, A., S. Behzadpoor and M. Hashemi, 2010. Imperialist competitive algorithmbased design of pss to improve the power system. Proceedings of the International Conference on Power Electronics, Drives and Energy Systems, December 20-23, 2010, IEEE., pp: 1-5.
Kim, J.O., D.J. Shin, J.N. Park and C. Singh, 2002. Atavistic genetic algorithm for economic dispatch with valve point effect. Electr. Power Syst. Res., 62: 201-207.
Direct Link |
Kumar, R., D. Sharma and A. Sadu, 2011. A hybrid multi-agent based particle swarm optimization algorithm for economic power dispatch. Int. J. Electr. Power Energy Syst., 33: 115-123.
Ling, S.H. and F.H.F. Leung, 2007. An improved genetic algorithm with average-bound crossover and wavelet mutation operation. Soft Comput. Fusion Found. Method. Appl., 11: 7-31.
Meng, K., H.G. Wang, Z.Y. Dong and K.P. Wong, 2010. Quantum-inspired particle swarm optimization for valve-point economic load dispatch. IEEE Trans. Power Syst., 25: 215-222.
Niknam, T., 2010. A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem. Applied Energy, 87: 327-339.
Niknam, T., E.T. Fard, N. Pourjafarian and A. Rousta, 2011. An efficient hybrid algorithm based on modified imperialist competitive algorithm and k-means for data clustering. Eng. Appl. Artificial Intell., 24: 306-317.
Niknam, T., H.D. Mojarrad and H.Z. Meymand, 2011. A novel hybrid particle swarm optimization for economic dispatch with valve-point loading effects. Energy Convers. Manage., 52: 1800-1809.
Niknam, T., H.D. Mojarrad and H.Z. Meymand, 2011. A new particle swarm optimization for non-convex economic dispatch. Eur. Trans. Electr. Power, 21: 656-679.
Nomana, N. and H. Iba, 2010. Differential evolution for economic load dispatch problems. Electr. Power Syst. Res., 78: 1322-1331.
Panigrahi, B.K. and V.R. Pandi, 2008. Bacterial foraging optimisation: Nelder-mead hybrid algorithm for economic load dispatch. Gener. Transm. Distrib. Inst. Eng. Technol., 2: 556-565.
Direct Link |
Pothiya, S., I. Ngamroo and W. Kongprawechnon, 2008. Application of multiple tabu search algorithm to solve dynamic economic dispatch considering generator constraints. Energy Convers. Manage., 49: 506-516.
Selvakumar, A.I. and K. Thanushkodi, 2007. A new particle swarm optimization solution to nonconvex economic dispatch problem. IEEE Trans. Power Syst., 22: 42-51.
Selvakumar, A.I. and K. Thanushkodi, 2008. Anti-predatory particle swarm optimization: Solution to nonconvex economic dispatch problems. Electr. Power Syst. Res., 78: 2-10.
Direct Link |
Sinha, N., R. Chakrabarti and P.K. Chattopadhyay, 2003. Evolutionary programming techniques for economic load dispatch. IEEE Trans. Evolutionary Comput., 7: 83-94.
Subbaraj, P., R. Rengaraj and S. Salivahanan, 2009. Enhancement of combined heat and power economic dispatch using self adaptive real-coded genetic algorithm. Applied Energy, 86: 915-921.
CrossRef | Direct Link |
Subbaraj, P., R. Rengaraj, S. Salivahanan and T.R. Senthilkumar, 2010. Parallel particle swarm optimization with modified stochastic acceleration factors for solving large scale economic dispatch problem. Int. J. Electr. Power Energy Syst., 32: 1014-1023.
Direct Link |
Sun, J., W. Fang, D. Wang and W. Xu, 2009. Solving the economic dispatch problem with a modified quantum-behaved particle swarm optimization method. Energy Convers. Manage., 50: 2967-2975.
Walters, D.C. and G.B. Sheble, 1993. Genetic algorithm solution of economic dispatch with valve point loading. IEEE Trans. Power Syst., 8: 1325-1332.
Wang, S.K., J.P. Chiou and C.W. Liu, 2007. Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm. IET Gen. Transm. Distrib., 1: 793-803.
CrossRef | Direct Link |
Wood, A.J. and B.F. Wollenberg, 1996. Power Generation Operation and Control. 2nd Edn., John Wiley and Sons, New York, ISBN: 9780471586999, Pages: 569.