Fuzzy Multi-Objective Optimal Power Flow Using Genetic Algorithms Applied to Algerian Electrical Network

This paper presents a mathematical model for solving Multi-Objective Optimal Power Flow problem considering uncertainties modeled by fuzzy numbers affecting three objective functions given by total generation cost, total gas emission and voltage profile index. The presented resolution approach is based on Genetic Algorithm (GA), where the parameters of this algorithm are determined and optimized after many tests of execution. A model for analyzing trade-off between profit and security constraint is developed. The probabilities of crossover and mutation optimized for GA parameters, dedicated to the presented approach are used to demonstrate a performance and effectiveness of the algorithm compared to other approaches mentioned in this paper. The mathematic model is applied in the Algerian electrical network for 59-bus test system.


Introduction
Modern power grids are facing many paradigms that directly affect their state of planning and operations, while being affected by several uncertain variables.Under the influence of market processes and operating constraints, the decision-making for power utilities is influenced mainly by the volatility of fuel prices, fluctuating pollutant gas emission and instability of the voltage caused by load changes from time to another.For such reasons, the Optimal Power Flow (OPF) problem must deal with three problems men-tioned above as an optimization problem under uncertainties, where objective functions are handled as fuzzy numbers.We are faced a Multi-Objective Optimal Power Flow (MOOPF) issue for three objectives, the fuel cost minimization (economic generation), less polluted environment with minimization of total emission of environmental pollutants (can be caused by nitrogen oxides NOx and sulphur oxides emitted from generation units based on fuel fossil) and best security of power system described by flatter voltage profile (safety operation of equipments in power system is considered for preferred voltage magnitude at buses closer to 1 p.u).
The ordinary applications of the traditional OPF optimize only a single objective function and the rest of objectives functions must be dealt as constraints [1].Nevertheless, when such objective functions describe trade-off statements with each other, the development of an efficient algorithm of MOOPF becomes a necessity to solve the problem [2].
Many deterministic optimization techniques in the literature have been applied for solving OPF problems [3], [4], [5], [6], they depend to the convex form of the objective function to find the global optimal solution.Nevertheless, the most OPF problems are nonlinear, non-convex and non-differential, where the previous techniques do not ensure a global optimal solution.Furthermore, those traditional methods used to optimize specific aspects of power system operations are not efficient, because they are not appropriate to deal with many practical aspects encountered in recent electrical networks, including the uncertainties of the objective functions and operational constraints [7].According to these issues, it is necessary to extend the problem to Fuzzy MOOPF (FMOOPF) problem using Artificial Intelligence (AI) [8].The application of AI in the recent years has undergone a significant evolution [9], they include for example Differential Evolution [10], Particle Swarm Optimization (PSO) [11] and Genetic Algorithms (GA) [12].
In this paper, the study is based on the optimization of three objective functions previously cited, while taking into the account the adjustment devices in the electrical network.We describe three cases for the optimization problem.In the first case, each objective function is optimized independently to another one, with determination of maximal and minimal values in order to define the membership function for each one.In the second case, FMOOPF problem is solved to get the best compromise solution for all membership functions using maximization of Degree of Satisfaction (DS).In the third case, the optimization problem is extended for various scenarios by increasing the load to 20 % and 30 % compared to the base case.The problem resolution is based on the optimal choice of GA parameters (crossover and mutation probability), considered as key factors for efficiency and accuracy of the algorithm.The GA parameters adjustment is determined with respect to our own experimental tests (crossover probability with two points).The mathematical optimization model proves its efficiency to overcome other metaheuristic methods.The mathematical algorithm application is interested in the Algerian electrical network for 59 bus test system.For this aim, our paper is structured as follows: After the introduction, section 2 presents the problem formulation that focuses on the FMOOPF Problem describing the approach and the objective functions.Section 3 is devoted to GA dedicated to the resolution of the FMOOPF problem.Section 4 illustrates the optimization strategy and its application on the mathematical model.Simulation, results and discussions are reserved to the Section 5 with comparison of the algorithm to some other methods.Finally, we finished with conclusion.

Problem Formulation
The problem formulation is based on fuzzy quantification of objective functions in the aim to define the Multi-Objective Problem.

Fuzzy Multi-Objective Problem
Regarding human ambiguity, it is more convenient for the Decision Maker (DM) to have such fuzzy targets as, "it is desirable to make each objective function below a certain value f o " and "it is necessary for each objective function to have a desirable limit value f m " Fig. 1.
Then the quantitative implementation can be made by defining a membership function for each objective function.For the multi-objective problem formulated as a vector minimization, it becomes a vector of maximization of all membership functions and evaluation of membership function can be expressed as formulating the DS of DM by changing the problem to conventional OPF with the aim to maximize the DS [13].The work in this paper is based on the above formulation using GA.We adopt a fuzzy linear distribution for each objective function as shown in the following expression. .

Objective Functions
The three most important problems considered in the power system operation are economy, environment, and security conflicts selected as the evaluation of three functions described as below:

1) Total Cost Generation Function
The most commonly used objective in the OPF problem formulation is the minimization of the total operating cost of the fuel consumed in producing electric power.The generation cost related to each thermal unit is considered as an economic function with respect to real power, describing a curve in second order and expressing a quadratic function.The total cost of power generation can be calculated as: P gi : active power generation at unit i. n g : number of total units for active power generation.a i , b i and c i : are the cost coefficients of the ith generator.

2) Total Gas Emission Function
The Kyoto Protocol, adopted in 1997, has forced electricity power companies to reduce their pollutant gases emitted by power plants to keep a clean environment.The objective function of total emission gases is expressed as a function of active power generation and can be summarized as follows: α i , β i , γ i , δ i and ε i are coefficients of the i th generator emission characteristics.

3) Function of Voltage Profile Index
The voltage magnitude of load buses must be in proximity of 1 p.u (nominal values), which means a flattened voltage profile for a security issue.The function of Voltage Profile Index (VPI) is as below: V i -Voltage magnitude in p.u at PQ bus i, N pg -Number of PQ buses.

Fuzzy Multi-Objective Optimization
The generalized problem optimization can be formulated in the following manner: where g(x) and h(x) are respectively the set of equality and inequality constraints described respectively in Eq. ( 6) and Eq. ( 7).
We have two types of equality constraints; the first is the necessity to keep at each moment, the total generation power equal to the total load demand plus transmission line losses.The second and third equality constraints are described by the power flow equations reflected by the injection of active and reactive powers at each bus.
P gi and P di , are the active powers of generation and demand respectively of the i-th bus, Q gi and Q di are the reactive powers of generation and demand respectively of the i-th bus.The voltage magnitudes of the i-th and j-th buses are V i and V j respectively.The angle θ ij signify the difference of voltage angles for i-th and j-th buses, while the transfer admittance is modeled by real and imaginary parts G ij and B ij for the same buses.The parameters nb is the number of buses and nld is the number of loads in power system, Ploss is the total active losses.
The inequality constraints are reflected by the operating limits of electrical network components which are considered for secure power system operation.The most adopted inequality constraints in OPF problem are the limited active and reactive powers at generation buses, voltage limits corresponding to voltage generators at PV-buses and the limited interval of transformer tap setting.
where: p and r are the number of transformer tap settings and the number of generators with controlled voltages respectively.x is the vector of control and state variables.The control variables are: • Unit active power output.
• Generation bus voltage magnitude.
The state variables are voltage and angle of load buses.
For the optimization in fuzzy set theory, objectives and constraints are modeled as membership functions.The decision making of Bellman and Zadeh [14] in fuzzy environment is adopted to get the overall degree of membership using the minimum operator with conjunction of all membership functions.The intersection of all fuzzy sets must lead to the optimal solution (optimal decision for the DM) [13].If µ f (x) is the membership function of all fuzzy objective functions and µ h(x) is the membership function of all fuzzy constraints, the overall degree of membership is given as follows : m: number of inequality constraints.
The maximization of the overall degree of membership µ DS leads to the optimal solution defined with the highest degree of membership, subject to the crisp and fuzzy constraints.The multi-objective optimization problem Eq. ( 5) can be transformed as: with i = 1,...,N obj ; k = 1,...,m and X denotes the feasible region satisfying all goals and constraints of the problem.N obj is the number of objective functions.

Genetic Algorithms
The GA is proposed as a computational model inspired from the natural evolution of survivals by Holand [15] in 1970.It is an optimization procedure to find the globally optimal solution using natural selection process, in which stronger individuals of the population have the highest probability to reproduce based on their level of goodness.Each individual represents a potential solution reflecting a chromosome structure.
In the beginning of GA algorithm, an initial population (encoded in chromosome) satisfying all constraints of the problem is selected.The evaluation of fitness function based on the objective function value is accomplished to rank all members of the population in the goodness order and to perform the selection process of chromosomes that must be reproduced.The combination of the current solutions of the population (pairs from subsets) is achieved based on a crossover process to create a new population of children (called offspring) and mutation mechanism describing occasional interchanges on the genes of chromosomes (particular variables of the problem).The process continues with the same reasoning described previously in every generation until the number of generations is reached.The genetic operators; number of generations, crossover and mutation forms the key factors to find the optimum solution efficiently and accurately [16].GA increases the probability to achieve an absolute optimum point of the optimization problem without trapping in local optimum points [17].The parameters of GA for this paper are given as follows: Chromosome Type Each variable of the optimization problem plays as a gene and a set of genes form a chromosome [18].In the presented paper approach, the chromosome is an answer to the optimization problem as it is represented in Fig. 2.

Crossover Type
We must specify a crossover type to produce children from one pair of parents.Among crossover type used in GAs, we choose crossover two points cleared as below: for example, choose randomly two integers pi and qi from 1 to variables number.The process must select: • Sub-vector from the parent number 1, for entries numbered less than or equal to pi.
• Sub-vector from the parent number 2, for entries numbered from pi +1 to qi.
• Sub-vector from the parent number 1, numbered greater than qi.
The Fig. 3 illustrates two point crossover process with pi = 2 and qi = 5.

Mutation Type
The mutation is an essential process of the genetic algorithm, by applying random changes to a single in- dividual in the current generation to create a child.It is a mechanism which ensures the diversity of the population and thereby increases the likelihood that the algorithm will generate individuals with better fitness function values.In this paper, a uniform mutation is used and defined in a two-step process.First, the algorithm selects a fraction of the vector entries of an individual for mutation, where each entry has a probability rate of being mutated.In the second step, the algorithm replaces each selected entry by a random number selected uniformly from the range for that entry [19].

Fitness Function
GAs are usually designed so as to maximize the Fitness Function (FF), which is a measure of the quality of each candidate solution.In this paper the fitness function is defined as follows: F obj : Objective Function.P f un : Penalty function for violating constraint in the first equation of system (6).W : Penalty factor.

Optimization Strategy
The strategy of optimization aims at the maximization of the DS for the problem formulated in Eq. ( 9) by GA while being based on the couple (f o , f m ) for each objective function f, resulting from DM.For this objective, three cases are presented: • Case 2: The couple (f o , f m ) is introduced for each objective function in problem Eq. ( 9), the DS is maximized by GA and the objective function in Eq. ( 10) becomes • Case 3: The optimization is focused on various load levels with different scenarios given by: where, P DO and Q DO are active and reactive power demands at base case.λ: is the load factor.(λ=1 for the base case).

Simulation and Results
Simulation is applied on the Algerian electrical network for 59-bus test system represented in Fig. 4. The model has 9 generators (except 13th generator bus, no generation power at output), 83 branches (lines and transformers) and 50 load buses.Number of control variables is 22 with n g = 9, p = 5 and r = 8.The first type of control variables is reserved to the power generation output at buses 1, 2, 3, 4, 27, 37, 41, 42 and 53 the second type is devoted to tap setting transformers at branches 20-22, 29-30, 33-34, 52-53 and 58-59, while the third type is addressed to voltage of generating buses (except the slack bus).Upper and lower active power generating limits, reactive power limits and unit cost coefficients are given in Tab. 1. Emission characteristics of generators for the test system are presented in [20].Lower and upper limits of voltage magnitude for generator buses are 0,95 and 1,05 p.u respectively.Permissible range related to tap setting transformers is considered from 0,9 to 1,1 p.u.
The proposed approach is developed by the use of MATLAB 7.9 tested with Pentium (R) -Dual-Core CPU 3 GHz, 1 GHz DDR RAM, consistently acceptable results were observed.For base case λ=1, the total active and reactive load are P dt =684,1 [

First Case
The optimal results of several objective functions independently of the one to another are illustrated in Tab. 2 in the base case, specifying the optimal values of the parameters of the adjusting devices.Generation's number, population's size, crossover probability (two points) and mutation probability are respectively 250, 200, 0,7 and 0,061.These parameters are optimized based on our own experimental results (after several tests of execution).
For the fuel cost minimization, our approach gives a more important profit of 1693,6 $/h compared to three other approaches Genetic Algorithm/Fuzzy Rules (FGA) [21], Decomposed Parallel GA (PGA) [22] and Ant Colony Optimization (ACO) [23], given in Tab. 3. The good simulation results show the best choice of the probabilities of crossover and mutation for the presented approach (compared to the choice of the same parameters indicated in [21] and [22]).Ploss have an acceptable value of 29,19 MW and acceptable voltage profile with minimal value of voltage magnitude 0,9166 p.u referred to Fig. 5.The previous figure describes the voltage profile for each case of optimization.
Best situation of electrical power system security is assigned to the voltage profile index minimization with value of 0,1718 and relatively higher active total losses 33,74 MW, but maximal values of the total cost generation 2118,2 $/h and emission level 0,882 ton/h referred to other functions minimization.Such remark shows the trade-off between security and cost-benefits problems.
In the case of emission level minimization, 0,3970 ton/h is the better results assigned to our approach compared to optimal emission values for PGA method in [22] with 0,4213 ton/h and for Fast Successive Linear Programming method (FSLP) with 0,4329 ton/h in [24] based on Tab. 4. Medium values of generation cost 1840,8 $/h and Ploss of 23,3859 MW with violation of voltage magnitude (0,8891 p.u at 35th bus noticed in Fig. 5).All control variables are in their allowable limits referred to declared values in the beginning of this section.It is clearly remarked the tradeoff between market benefits, gas emission and voltage security issues.

Second Case
The couple (f o , f m ) to define each membership function related to each objective function is shown in Tab. 5 referred to Tab. 2.
The DS is optimized using GA (maximizing the fitness function i.e optimization of three functions simultaneously).The crossover probability pc chosen from four values 0,6; 0,7; 0,8 and 0,9 based on simulation results taken at base case with fixed mutation probability at 0,061.The best value of F obj to maximize the DS is observed for simulation results in Fig. 6 and for pc equal to 0,7 (which proves the best choice previously indicated), where it is established in following simulations.The previous figure exposes the change of the objective function at each generation, with a multitude of crossover probability values and mutation probability of 0,061.Control variables are extracted after simulation for the optimization of three functions simultaneously, DS is maximized and optimal values of objective functions are obtained in the base case and results are arranged in Tab. 6.The convergence has been achieved after 226,116 seconds with 250 iterations.
Voltage magnitudes are closer to 1 p.u for the optimization of three functions simultaneously than the optimization of the cost function and emission function Fig. 5 and Fig. 7 (voltage profile index has the best value in optimization of f V P I ).Based on Fig. 7, voltage profile is assigned to the DS maximization without voltage violation (minimal voltage magnitude 0,9005 at 35th bus).
In the base case and for the optimization of three functions simultaneously, we remark based on simulation results in Tab.6 intermediary's values of three conflicting functions.The tap setting transformers and generation bus voltage magnitudes have contributed to the improvement of the objective functions.As seen in Tab. 7, the presented approach is significantly better than PGA [22] and FSLP [24] methods (for simultaneous minimization of fuel cost and gas emission).The optimized choice of GA parameters confirms the efficiency of the proposed approach which makes it promising to solve the FMOOPF.

Third Case
For this case, the load level λ=1,2 and λ=1,3 (load is increased by 20 % and 30 %), we extend the code of simulation to obtain the state of the test system in such cases.For the same reasoning as described in the base case, results are detailed in Tab. 8, Tab.9 and Tab.10.Increased values are assigned to our objective functions due to expansion of demand (active and reactive total load).Results appear in Tab. 8 for each couple (f o , f m ).Tab. 9 and Tab. 10 show simulation results at such cases obtained by the proposed approach for DS maximization (control variables are between lower and upper limits).We obtain the Pareto solutions in wide case, they will be increased by percentage ∆F (%) (for each objective function) related to the base case (for DS maximiza-

Conclusion
An efficient algorithm of Multi-Objective Optimal Power Flow with Fuzzy Logic using Genetic Algorithms  The optimized GA parameters of the proposed approach show its effectiveness to enhance the solution quality in comparison to other Algorithms PGA, FSLP and ACO presented in this paper.The security constraints are satisfied for voltage magnitudes (0, 9 < V < 1, 1 p.u) for the optimization of three functions simultaneously at the base and wide cases.Results his M.Sc.degree in electrical engineering on 2003 from Constantine University, Algeria.He is also a member of LGEB laboratory.His research activities include integration of renewable energy in power system grid, optimization, and power flow and power system stability.
His teaching includes modeling and optimization in power system and power system stability.
Tarek BOUKTIR was born in Ras El-Oued, Algeria in 1971.He received the B.Sc. degree in Electrical Engineering Power system from Setif University (Algeria) in 1994, his M.Sc.degree from Annaba University in 1998, his Ph.D. degree in power system from Batna University (Algeria) in 2004.He is with the Department of Electrical Engineering in Ferhat Abbes University (Setif), ALGERIA.His areas of interest are the application of the meta-heuristic methods in linebreak optimal power flow, FACTS control and improvement in electric power systems, Multi-Objective Optimization for power systems, and Voltage Stability and Security Analysis.He is the Editor-In-Chief of Journal of Electrical Systems (Algeria), He currently serves on the editorial boards of TELKOMNIKA Journal, Indonesia.He serves as reviewer with the Journals: IEEE Transactions on SYSTEMS, MAN, AND CYBERNETICS, IEEE Transactions on Power Systems (USA), ETEP-European Transactions on Electrical Power Engineering.He is also the head of the research team "Algerian Smart Grid".

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Case1: Each objective function is optimized independently to another one and the couple (f o , f m ) is extracted.f o : optimal value of the objective function f given independently to another one.f m : maximum value of the objective function f imposed during the optimization of another objective functions.The membership function for each objective function is defined based on the couple (f o , f m ).The objective function in Eq. (10) becomes f C , f E or f V P I .

Fig. 4 :
Fig. 4: Topology of the Algerian production and transmission network.

Fig. 5 :
Fig. 5: Voltage profile for the optimization of one objective function at base case.

Fig. 6 :
Fig. 6: Evolution of DS with respect to crossover probability at base case λ=1.

Fig. 7 :
Fig. 7: Voltage profile for the optimization of three functions simultaneously at base case.
MW] and Q dt =311,6[MVAR].Lower and upper limits of parameters related to generators with cost coefficients.Optimization of one objective function independently to another one at base case.
c 2013 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING Tab. 3: Comparison of the proposed approach with other techniques for the total cost minimization case.Comparison of the proposed approach with other techniques for total gas emission minimization case.