Abstract: 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.
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 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 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 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:
| (1) |
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 as:
| (2) |
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
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:
| (3) |
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 Kron’s loss formula known as B-matrix coefficients. In this study, B- matrix coefficients method is used to calculate system loss as follows:
| (4) |
Generation limits of units:
| (5) |
where,
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:
| (6) |
| (7) |
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:
| (8) |
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:
| (9) |
where,
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:
| Step 1: | Generate an initial set of colonies with a size of NC |
| 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: |
| (10) |
This means the less OF is, the more stronger IPi is.
| Step 4: | Keep the best Nimp colonies as the imperialists and set the power of each imperialist as follows: |
| (11) |
| 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., IPi |
| 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 (CPc > IPi) |
| Step 8: | Compute the empire’s power, i.e., EPi for all empires as follows: |
| (12) |
where, w1 and w2 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 (EPi) |
| 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: 6-unit system: The first test system is a 6-unit 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 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.
| Table 1: | Comparison of simulation results for the 6-unit system |
| TG: Total generation, TL: Total loss, TC: Total cost | |
| Table 2: | 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 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 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.