
Research Article


A New Heuristic Algorithm for Solving Nonconvex Economic Power Dispatch


A. Rabii,
S. Mobaieen,
B. Mohamady
and
A. Suroody



ABSTRACT

Valvepoints effects and prohibited operation zones (POZ) make the generating units’ cost functions nonlinear and nonsmooth. Hence, the Economic Dispatch (ED) problem is a highly nonconvex 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 valvepoint effects, POZs, ramprate 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.





Received: August 02, 2011;
Accepted: November 19, 2011;
Published: January 02, 2012


INTRODUCTION 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 nonconvex 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 lambdaiteration method (Wood
and Wollenberg, 1996) which needs that the cost function be continuous and
monotonically increasing.
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 realcoded genetic
algorithm (Subbaraj et al., 2009) and real coded
genetic algorithm (Amjady and NasiriRad, 2010a) proposed
to solve different types of ED problem. Hybrid particle swarm optimization presented
(Niknam et al., 2011a) for solving ED problem
considering valvepoint effects. A hybrid multiagent based particle swarm optimization
algorithm proposed by Kumar et al. (2011) to
solve valvepoint constrained ED problem. Other modifications of Particle Swarm
Optimization method like as Parallel particle swarm optimization (Subbaraj
et al., 2010), quantumbehaved 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 nonconvex 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, C_{i} is the production cost of unit i at time t, N is the number
of dispatchable power generation units and P_{i} is the power output
of Ith unit at time t. T is the total number of hours in the operating horizon.
The production cost of generation unit considering valvepoint effects is defined
as:
where, a_{i}, b_{i}, c_{i} are the fuel cost coefficients
of the ith unit, e_{i} and f_{i} are the valvepoint coefficients
of the ith 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 valvepoints. The ED problem will be nonconvex and nondifferentiable
considering valvepoint 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, P_{loss} (t) and P_{D} (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 Kron’s loss formula known as Bmatrix coefficients. In this study, B matrix coefficients method is used to calculate system loss as follows: Generation limits of units:
where,
is the maximum power outputs of ith 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, UR_{i} is the ramp up limit of the ith generator (MW/hr) and DR_{i} is the ramp down limit of the ith 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:
where,
and
are the lower and upper limits of the j^{th} prohibited zone of unit
i, respectively. M_{i} is the number of prohibited operation zones of
unit i.
IMPERIALIST COMPETITION ALGORITHM (ICA)
The Imperialist Competition Algorithm (ICA) was first proposed by AtashpazGargari
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:
Step 1: 
Generate an initial set of colonies with a size of N_{C} 
Step 2: 
Set iteration = 1 
Step 3: 
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 IP_{i} is.
Step 4: 
Keep the best N_{imp} colonies as the imperialists
and set the power of each imperialist as follows: 
Step 5: 
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., IP_{i} 
Step 6: 
Move the colonies toward their relevant imperialist using crossover and
mutation operators 
Step 7: 
Exchange the position of a colony and the imperialist if it is stronger
(CP_{c} > IP_{i}) 
Step 8: 
Compute the empire’s power, i.e., EP_{i} for all empires
as follows: 
where, w_{1} and w_{2} are weighting factors which are adaptively
selected.
Step 9: 
Pick the weakest colony and give it to one of the best empires
(select the destination empire probabilistically based on its power (EP_{i}) 
Step 10: 
Eliminate the empire that has no colony 
Step 11: 
If more than one empire remained then go to Step. 6 
Step 12: 
End 
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.

Fig. 1: 
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: 6unit system: The first test system is a 6unit system. System’s
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 ramprate 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 nonconvex ED problems.
Case 2: 40unit system: This test case consists of 40 generating units
with valvepoint 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.
Table 1: 
Comparison of simulation results for the 6unit system 

TG: Total generation, TL: Total loss, TC: Total cost 
Table 2: 
Comparison of simulation results for the 40unit system 

TG: Total generation, TC: Total cost 
The obtained results are compared with the results of hybrid DE with biogeographybased
optimization (DE/BBO) algorithm (Bhattacharya and Chattopadhyay,
2010a), variable DE with the fuzzy adaptive PSO (FAPSOVDE) (Niknam
et al., 2011a), quantuminspired PSO (QPSO) (Meng
et al., 2010), multiagent based hybrid PSO technique (HMAPSO) (Kumar
et al., 2011), biogeographybased optimization (BBO) algorithm (Bhattacharya
and Chattopadhyay, 2010b) and hybrid GA (HGA) (He et
al., 2008).
These results imply the efficiency of the proposed ICA method for dealing with the ED problem. CONCLUSION In this study, a new approach to solve power systems economic dispatch problem, called imperialist competition algorithm (ICA), is proposed. The valvepoint effect, prohibited operation zones, ramprate constraints and transmission losses are modeled and the resulting nonlinear and nonconvex 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.

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