Wind Energy Uncertainties in Multi-objective Environmental / Economic Dispatch Based on Multi-objective Evolutionary Algorithm

This paper a Multi-objective Honey Bee Mating Optimization (MOHBMO) is proposed for Environmental/ Economic Power Dispatch (EED) problem. This paper proposes a new environmental/economic load dispatch model that considers cost and emission function coefficients with uncertainties and the constraints of ramp rate. Due to the environmental concerns that arise from the emissions produced via fossil-fueled electric power plants, the classical economic dispatch, which operates electric power systems so as to minimize only the total fuel cost, can no longer be considered alone. Actually, EED problem is the scheduling of generators which fulfill the load demand of the power plants using fossil fuel and also making combined production, in order for them to perform with minimum cost and emission. Therefore, by EED, emissions can be reduced by dispatch of power generation to minimize emissions. Which is affect on power generated, system loads, fuel cost and emission coefficients in realworld situations. The MOHBMO technique has been carried out on the IEEE 30and 118-bus test system. This technique is compared with other techniques which reveals the superiority of the proposed approach and confirms its potential for solving other power systems problems.


Introduction
The conventional economic dispatch problem mainly concerns minimization of operating cost subject to diverse unit and system constraints.However, the environmental pollution problem caused by generation has been presented in recent years.Therefore, people think more and more of how to decrease the emission of maleficent gas, and have proposed many feasible strategies.The different strategies [1][2] have been proposed to reduce the atmospheric emissions.These include installation of pollutant cleaning equipment, switching to low emission fuels, replacement of the aged fuel-burners and generator units, and emission dispatching.The literature [3] pointed out that the first three options should be as long-term options.The emission dispatching option is an attractive short-term alternative.In fact, the first three options should be determined by generation companies, but not by regulation department, especially in the environment of power market.Secondly, the target we should pursue in a long run is to reduce the emission, in other words, we should reduce the emission of the generation companies with high emission by the rule, which not only makes the generation companies do their best to reduce emission, but also embodies the impracticality principle.So, the environmental/economic load dispatch problem considering emission of maleficent gas is a kernel issue in power market.
The EED problem is formulated as a nonlinear constrained multi-objective problem with competing and non-commensurable objectives of fuel cost, emission and system loss.Consequently, single objective and conventional optimization methods that make use of derivatives and gradients, in general, are not able to locate or identify the global optimum.The considered problem in this paper is a multiobjective problem with conflicting objectives because pollution is conflicting with minimum cost of generation.Several strategies and techniques are proposed for solving the EED problem.Accordingly, multi-objective Genetic Algorithm (GA) is presented in [6-7], hierarchical system approach [1], fuzzified multi-objective particle swarm optimization algorithm [8], fuzzy linear programming [9], fast Newton-Raphson algorithm [10] and linear programming [11][12].It is clear that for this kind of optimization problem in power system, the final cost is really important.Also, saving the cost and decreasing it using several techniques leads to bulk thrift for power system in long time.
Honey Bee Mating Optimization (HBMO) consist of the high ability, great potential and good perspective for solving optimization problems.Its main advantage is the fact that it uses mainly real random numbers, and it is based on the global communication among the swarming particles, and as a result, it seems more effective in optimization of EED problem.In this paper, a MOHBMO is proposed to solve the environmental/ economic power dispatch problem.The proposed algorithm runs on the IEEE 30and 118-bus test systems and the results are compared with techniques which are presented in [14].The achieved numerical results of the proposed technique demonstrate the feasibility of the proposed technique to solve the multi-objective EED problem.

Problem Statement
There is no doubt that, the EED problem finds the optimal combination of load dispatch of generating units and minimizes both fuel cost and emission while satisfying the total power demand.Hence, the proposed problem is including of two objective functions as economic and emission dispatches [4].The EED problem can be formulated as follows: Objective Function Fuel cost minimization: The cost curves of generators are presented by quadratic functions [5].Also the total fuel cost F(PG) ($/h) is presented as: Where,N= the number of generators ai, bi, ci= the cost coefficients of the ith generator PGi = the real power output of the ith generator


(2) PG = the vector of real power output generator

Emission Minimization
The emission function can be presented as the sum of all types of emission considered, and thermal emission, with suitable pricing or weighting on each pollutant emitted.In this paper, only one type of emission (NOx) is taken into account without loss of generality [9].The amount of NOx emission is given as a function of generator output, that is, the sum of a quadratic and exponential function.
The total amount of emission such as SO2 or NOx depends on the amount of power generated by unit [10].The NOx emission amount which is, the sum of a quadratic and exponential function is given as: Where, αi, βi, γi, ζi and λi are the coefficients of ith generator emission characteristics.Total real power loss's minimization: The objective of the reactive power dispatch is to minimize the real power loss in the transmission network.Also it can be determined by means of a power flow solution exactly and can be presented as: Where, K= the network branches that connects bus i to j (i=1,2,…,ND/ j=1,2,…,Nj) ND= the set of numbers of power demand bus Nj = the set of numbers of buses adjacent to bus j NL = the set of numbers of network branches (transmission lines) Vi, Vj = the voltage magnitudes at bus i and j gk = the transfer conductance between bus i and j θi , θj = the voltage angles at bus i and j, respectively

Problem Constrains
Generation constraints: The upper and lower constrains of generator outputs and bus voltage magnitudes are presented as: Where, Also the power balance constraint is expressed as: The line loading constrain is explain as: Where, Sli max is maximum power flow through the ith transmission line.

Proposed Algorithm
The honey bee is a social insect that can survive only as a member of a community, or colony.This means that they tend to live in colonies while all the individuals are the same family.In the more highly organized societies there is a division of labor in which individuals carry out particular duties.In fact, a colony consists of a queen and several hundred drones, 30,000 to 80,000 workers and broods in the active season.Each bee undertakes sequences of actions which unfold according to genetic, ecological and social condition of the colony [15].The queen is the most important member of the hive because she is the one that keeps the hive going by producing new queen and worker bees and any colony maybe contain one or much queen in it lifes.Drones' role is to mate with the queen.In the marriage process, the queen(s) mate during their mating flights far from the nest [16].In each mating, sperm reaches the spermatheca and accumulates there to form the genetic pool of the colony.The queen's size of spermatheca number equals to the maximum number of mating of the queen in a single mating flight is determined.When the mate be successful, the genotype of the drone is stored.In start the flight, the queen is initialized with some energy content and returns to her nest when her energy is within some threshold from zero or when her spermatheca is full.A drone's mate probabilistically is [17]: Prob(Q,D) = e -(∆f)/(S(t)) (8) Where, Prob (Q, D) = The probability of adding the sperm of drone D to the spermatheca of queen Q ∆(f) = The absolute difference between the fitness of D and the fitness of Q (i.e., f (Q)) S(t) = The speed of the queen at time t After each transition in space, the queen's speed, and energy, decay using the following equations: The amount of energy reduction after each transition.The flowchart of Classic HBMO is presented in "Fig.4", [14].
Thus, HBMO algorithm may be constructed with the following five main stages [13]: The algorithm starts with the mating-flight, where a queen (best solution) selects drones probabilistically to form the spermatheca (list of drones).A drone is then selected from the list at random for the creation of broods.Creation of new broods by crossoverring the drones' genotypes with the queen's.Use of workers (heuristics) to conduct local search on broods (trial solutions).Adaptation of workers' fitness based on the amount of improvement achieved on broods.Replacement of weaker queens by fitter broods.
Where fi min and fi max are the maximum and minimum values of the i th objective function, respectively.For each non-dominated solution k, the normalized membership function FDM k is calculated as: Where M is the number of non-dominated solutions, and Nobj is the number of objective functions.

Numerical Results IEEE 30-bus Test System
The IEEE 6-generator 30-bus test system is used for the first case study for solving the EED problem using the proposed MOHBMO technique.The values of the fuel and emission coefficients of the IEEE 30-bus and 118-bus system are given in "Table 1" and "Table 2" , respectively [18-19].The line data and bus data of the system are presented in [18].Also, the load of the IEEE 30-bus and 118-bus system was set to 2.834 p.u. on a 100MVA and 950MW, respectively.
To demonstrate the effectiveness of the proposed MOHBMO, the multi-objective EED problem with two objective functions of fuel cost is considered in case one.Case two is the emission objective function.Case 3 is the fuel cost and emission together.Also three objective functions of fuel cost, emission and system loss are considered which is called case four.Actually the fuel cost, emission and system loss objectives are optimized individually to explore the extreme points of the tradeoff surface in all cases.The minimum and maximum objective values of case studies when optimized individually for all cases are presented in "Table 3" and "Table 4 5 and 6.According to the presented results, there is no doubt that the applied technique is superior to the other techniques.Also the trend of objective function variation of cost function and variation of emission function are presented in Fig. 1-2, respectively.Also the achieved results for case 3 of the best compromise solution are presented in Table .7. The typical Pareto front of case 3 and 4 obtained by MOHBMO which is shown in Fig. 3 and 4, respectively [24].For case 4, the solutions of MODE, MOPSO and MOHBMO are presented in Table. 8. The Fig. 4, presents the Pareto front of this case.For second case study, the standard IEEE 14-generator 118-bus test system [18-19] is considered.Also, the transmission loss for this system is calculated using the Kron's loss formula [19].
For testing the proposed case study, two options are considered as a test functions.For Case 1, the bi-objective optimization problem with cost and emission objectives is considered.And for Case 2, the transmission losses PL is regard as the third objective.The numerical results of case 1 and 2 are presented in Table 9 and 10,respectively [14].For proposed cases the Pareto fronts are presented in Fig. 5 and 6 respectively.proposed approach to handle multi-objective optimization problems as the best solution of EED problem for each objective in case studies.

Conclusion
This paper presented the EED optimization problem formulated as multi-objective optimization problem with competing objectives of fuel cost, emission and system loss using the MOHBMO technique.According to the presented results, the proposed technique demonstrates the feasibility to solve the multiobjective EED problem.The IEEE 30-and 118-bus test systems were used to investigate the effectiveness of the proposed MOHBMO approach.The proposed technique is compared with other MOEAs, such as NPGA, NSGA, SPEA, MOPSO and MODE.It is obvious that, the proposed technique achieve appropriate results is power systems.Hence, the MOHBMO gives lower cost for several cases in two test systems.
minimum and maximum real power output of the ith generator, respectively min Gi Q , max Gi Q = the minimum and maximum active power output of the ith generator, respectively min Gi V , max Gi V = the minimum and maximum voltage magnitude of the ith transmission line, respectively.

Figure 1 .
Figure 1.The Classic HBMO technique Fuzzy Decision in Multi Objective HBMO Usually, a membership function for each of the objective functions is defined by the experiences and intuitive knowledge of the decision maker.In this work, a simple linear membership function was considered for each of the objective functions.The membership function is defined as:

Figure 2 .
Figure 2. Objective function variation of cost function

Table 1 .
Generator And Emission Coefficents Of The Ieee 30-Bus Power System

Table 2 .
Generator and emission coefficients of the IEEE 118-bus system

Table 4 .
The minimum and maximum objective values of IEEE 118-bus system The proposed technique is compared with the MODE [14], NSGA [20], NPGA [21], SPEA [22] and MOPSO [23] through solving the EED problem.The achieved numerical results of best cost and best emission solutions are presented in Tables

Table 5 .
IEEE 30-bus system best solutions out of ten runs for cost of MOHBMO, Case 1

Table 6 .
IEEE 30-bus System Best Solutions out of Ten Runs for Emission of MOHBMO, Case 2