Solving the Economic Dispatch by New Hybrid Algorithm

: The problem of economic dispatch is the minimization of the total cost of production by satisfying the demand of the load. The resolution of this problem is a way of managing an electricity production system taking into account the constraints of equalities and inequalities, in other words it is to find the optimal production for a given combination of units in operation. The appearance of meta-heuristic methods which are part of artificial intelligence, has effectively contributed to solving this problem. Bee colony optimization is a very recent family of meta-heuristics. Its principle is based on the behavior of real bees in life. Bees have properties that are quite different from those of other insect species. They live in colonies, building their nests in tree trunks or other similar enclosed spaces. In this paper, we will apply the optimization by colony of bees in test systems of different sizes with the aim of minimizing the cost of production of electrical energy by taking into account the effect of the valve points of the power plants. In order to see the effectiveness of the proposed algorithm, it has been compared with other algorithms in the literature.


Introduction
Optimal power distribution is therefore the basic computer tool allowing the network manager to determine the conditions for safe and economical operation of the electro-energy system.The Optimal Power Distribution procedure uses mathematical programming-based methods to determine the optimum setting of system control variables while satisfying a specified set of operational and safety requirements.The problem of economic dispatching (ED) or the distribution of the economic load is a particular case of the optimal distribution of powers.Economic dispatching (ED) is a static optimization problem.The objective is to minimize generator fuel consumption and overall system operational cost by determining the optimum output of each generator under the system load constraint conditions within a number of system operational constraints.The fundamental problem of economic dispatching is related to the set of input and output characteristics of power plants.Several classical optimization techniques such as gradient method, lambda iteration method, Newton's method, linear programming, interior point method and dynamic programming have been used to solve the economic power distribution problem.New methods have been developed to solve the (ED) problem such as genetic algorithm (GA) [1][2][3] , tabu search (TS) [4], simulated annealing (SA) [5,6], evolutionary programming (EP) [7,8], particle swarm optimization (PSO) [9,10] and differential evolution (DE) [11,12], cuckoo search algorithm (CSA) [13], the bat algorithm (BA) [14,15].In this paper, a bee colony algorithm is proposed to obtain improved results in the ED problem, taking into account valve point effects.

Problem Formulation
The objective of ED problem is to minimize the total cost of production while respecting the constraints of equality and inequality.The fuel cost curve for any unit is assumed to be approximated by segments of quadratic functions of the generator output active power.

A. Economic Dispatch (ED) formulation
For a given power grid, the problem can be described as an optimization of the total fuel cost function as defined by equation (1) under a set of operating constraints. (1) Where: FT is total fuel cost of generation in the system ($/hr), ai, bi, and ci are the cost coefficient of the i th generator, Pi is the power generated by the i th unit and N is the number of generators.The cost is minimized subjected to the following constraints: Generation capacity constraint, (2) Power balance constraint, (3) Where and are the minimum and maximum power output of the i th unit, respectively.PD is the total load demand and PL is total transmission loss.The transmission loss PL can be calculated by using B matrix technique and is defined by (4) as, (4) Where, Bij, B0i and B00 are transmission loss coefficients.

B. ED problem with valve point effect
Large thermal power plants have several steam inlet valves, which are used to control the power delivered by the unit.Every time you start to open an inlet valve, there is a sudden increase in losses.The valve point effect is considered to be a practical constraint to the operation of generators.The valve point effect brings a ripple in the heat rate function and makes the fuel cost function highly nonlinear, discontinuous and having multiple local optimal.A second-order quadratic cost function is added with the rectified sinusoidal equation for accurate modeling of the generator cost function taking into account valve point effect as follows: (5) Where FT is total fuel cost of generation in ($/hr) including valve point loading, ei, fi are fuel cost coefficients of the i th generating unit reflecting valve-point effects.

Artificial Bee Colony Algorithm for Economic Dispatch Problem
The ABC (Artificial Bee Colony) algorithm is developed by Karaboga and Basturk in 2005, by inspecting the behaviors of real bees to find the source of food, which is called nectar, and share information of sources of food for other bees in the nest.In this algorithm, artificial bees are defined and classified into three groups: employing bees (bees that search for food), spectators (bees observation) and scouts (girl scouts) are in charge of finding new foods, (the new source nectar) [16].For each food source, there is only an employing bee.That is to say, the number of employing bees is equal to the number of food sources [17].If the employing bee of a site fails to find the food source, it must be necessarily become a scout to randomly search for new food sources.The worker bees share information with onlooker bees in a hive so that onlooker bees can choose a food source to explore.The brief working principle of ABC as follows [18]: Step 1 Initialization Determine the number of food source (SN) Calculate vector of possible solution Xi=X1, X2… XSN; Xi is represent by the location of food source.
The fitness of each possible solution can be calculate using the following formula: (6) Step 2 Employed Bees Employed Bees find the new food source position Vij using: Where is a random number between [-1, 1], and k ϵ {1, 2,…,NS} and j ϵ {1,2,…,D} are index randomly chosen.D is number of problem variables.If the new position is found better than the old position, a new position is memorized and otherwise it is removed.The greedy selection method is used to determine the best solution.

Step 3 Onlooker Bees
In this phase, onlooker bees will search the best results according to the probability (Pi) as follows: (8) The solution with better fitness value has high probability of being selected by an onlooker bee in order to exploit the solution near to global optimal value.

Step 4 Scout Bees
After several trials, unimproved food location (solution) will explore other possible location in order to improve the current solution using the following equation: The good position replaced the unimproved solution.
Repeat Steps 2 -4 until satisfied the stopping criteria

Genetic Algorithm
GA [4,19] is usually used in the solutions of the problems, which are hard or even impossible to be solved with conventional methods.Algorithm begins with a solution set which is referred to as population and represented by chromosomes at the beginning.The results which are obtained from this population are used to create a new population which is expected to include better solutions than the previous one.The solutions are chosen to create the new population according to their compatibility.This is due to the fact that the compatible ones will produce better results.This process is continued until a certain condition (for example, development of certain number of societies or the best solution) is maintained.
The process that GA undergoes until it comes to a solution can be described as coding the solution set, creating the initial population, assessing the compatibility of the solutions in the population, choosing the progenitor individuals according to the compatibility and creating new individuals through crossover and mutation processes.As for the control parameters of the GA, the crossover rate and the mutation rate are selected between 0.5-1.0 and 0.0001-0.05,respectively [20][21][22][23].

Applying genetic algorithm to the problem
At first, numbers as much as one less than the number of elements in random NG set (remaining one being reference bus) which enables the constraint in (10) are designated for PG,n values that are output powers of the generation units in order to show bn number of bits (solution sensitivity). (10) Since these designated numbers can get a value out of current constraints of the generation units in the system, they are tailored towards constraints by mapping according to the following equation: (11) There by, the inequality constraints given in Eq. ( 2) have been automatically provided.In this case, solutions that satisfy the condition below are taken as the solution. (12) Therefore, each created individual becomes a solution of the current problem.

Hybrid Algorithm GA/ABC
This section presents hybridization between two meta-heuristic algorithms, the genetic algorithm and the artificial bee colony algorithm, whose goal is to escape local minima and build a more efficient algorithm for the global optimization of non-convex functions.The mechanism of the hybridization between the two algorithms (GA and ABC) is given by Figure 1.

Results and analysis
In this section, we will test and simulate the proposed algorithm on three standard networks of different sizes, the first of three generating units, the second of six generating units and the last of ten generating units.Our objective is to test the validity and efficiency of the proposed algorithm.

A. Test system 1
The first test system consists of three generator units; the characteristic data of this test system is shown in appendix A.1.The following tables show the simulation results of three-unit generator.The first table presents the simulation by an AG, the second table gives the simulation by a bee colony algorithm while the hybridization of the two algorithms mentioned above is presented in the third table.
Table1.Optimum results for three-unit-generator using AG PD Unit 400(MW) 500(MW) 600(MW) 700(MW) The following figures show the variation of the active powers, the active losses as well as the cost of fuel for different powers requested.
Figure2.Variation of the optimal powers obtained by AG Algorithm Figure3.Variation of the optimal powers obtained by ABC Algorithm The variation of the active powers, the transmission losses and the fuel cost are presented as follows: Figure7.Variation of the optimal powers obtained by ABC Algorithm   To test and simulate the convergence of the hybrid algorithm, we changed the requested power at different levels for the three system test networks.
The proposed hybrid algorithm gives us a good result compared to the other basic algorithms AG, PSO and ABC, regarding losses and production cost.Taking the test system network of three generator units (Table 4), for a power demand of 400 MW, a difference of 28.1 ($/h), 26.3 ($/h) and 26.1 ($/h) between (AG /ABC) and the GA reference [24], PSO [24] and the FPA reference [25] respectively.For the test system network of six generator units (comparison table 8), for a requested power of 900 MW, a slight difference of 280.8 ($/h) between the proposed algorithm and references [26] and [27].At the end the table of comparison of the production costs of a test network of ten generator units for a requested power of 2000 MW, a difference of 950 ($/h), 1000 ($/h) and 1119 ($/h) h) between (AG/ABC) and the FPA reference [25], ABC_PSO [28] and the NSGAII reference [29] respectively.The results found by the hybrid algorithm affirm the convergence of the method in the field of electrical energy.Hybridization of meta-heuristic methods is now opening wide application in several fields.

Conclusion
This article proposes a new approach to solve the ED problem with valve point effect using a hybrid optimization algorithm of the two meta-heuristic methods such as the genetic algorithm and the bee colony.As a first step we only applied the genetic algorithm on test systems of three, six and ten generator units.In the second step, another algorithm is applied to solve the economic dispatch problem such as the bee colony.The last algorithm is a hybrid algorithm that brings together the two algorithms mentioned above, the purpose of which is to minimize the cost of fuel and the transmission losses.The comparison of the results with other methods reported in the literature shows the superiority of the proposed method and its potential to solve ED problems taking into account valve point effect.The AG/ABC algorithm can generate an efficient solution with high quality and more stable convergence characteristics than the genetic algorithm and the bee colony algorithm.

Acknowledgment
This work was partially supported by the sustainable development laboratory of electrical energy (LDDEE), USTO-MB, Oran, Algeria.

Appendix
Table A

Figure8Figure9.
Figure8.Variation of the optimal powers obtained by AG/ABC hybridization

Figure11.
Figure11.Convergence characteristic of fuel cost-PD=2000MWThe comparison of the fuel costs obtained by the three algorithms is as follows: This test system consists of six thermal units whose characteristics are shown in TableA2.The simulation results are well classified in the following tables.
Figure6.Comparison of the fuel costs obtained by the three proposed algorithms Table4.ED Comparison of the results for test system 1 (PD=400MW) (AG/ABC) GA [ Ten-unit generator units characterize the last test system.The inputs data for this system is shown in TableA3.