A FUZZY GOAL PROGRAMMING METHOD TO SOLVE CONGESTION MANAGEMENT PROBLEM USING GENETIC ALGORITHM

: The objective of this work is to present a priority-based fuzzy goal programming (FGP) method for solving the congestion management (CM) problem in electric power transmission lines by employing genetic algorithm (GA). To formulate the model for this problem, membership functions which are associated with the fuzzy model goals are converted into membership goals by assigning highest membership value (unity) as goal level and adding under-and over-deviational variables to each of them. In solution process, a GA computational scheme is addressed within the framework of FGP model to achieve aspired goal levels of goals according to their priorities in imprecise environment. The standard IEEE 30-Bus 6-Generator test system is taken as a case example to show the effectiveness of the approach. A comparison of model solution is also compared with solution of another approach studied previously.


Introduction
Congestion in thermal power supply system in Bhattacharya et al. (2001) refers to overloading situation in transmission lines when thermal bounds and line capacities of the power supply system are violated in Chung et al. (2015).Congestion actually occurs when power flow in a transmission line is higher than the flow allowed by operating reliability limits in Bachtiar Nappu & Arief (2016).As such, congestion in power system would have to be rectified as and when needed to ensure system security.Further, a lack of paying proper attention to congestion of the system may lead to widespread blackouts which give birth to negative impact to social and economic perspectives.Therefore, congestion management in Emami & Sadri (2012) appears as one the key issues to maintain security and reliability of transmission network.
The mathematical programming method for estimating voltage dropping and line loading for out of service of each network element was first introduced in Abiad & Stagg (1963) in 1963.Then, different classical optimization methods based on load flow were studied for CM in Mamandur & Berg (1978), and Medicherla et al. (1979) in the past century.The decomposition of spot prices to reveal congestion cost component in a pool model was presented in Finney et al. (1997).The DC-optimal power flow (DC-OPF) based approach to compute congestion cost was also propounded by Singh et al. (1998).The real-time operational environment based CM was studied in Fang & David (1999) and Wang & Song (2000) in last two decades.An optimal dispatch with the consideration of dynamic security constraints for CM was discussed in Singh & David (2000).Rau (2000) presented the AC-OPF driven approach to CM along with congestion cost allocation.Then, an effective model to location of unified power flow controller (UPFC) for CM was deeply studied in Verma et al. (2001).The use of Thyristor-Controlled Series Compensation (TCSC) to reduce congestion cost is also presented in Lee (2002).To manage congestion, a minimum load curtailment problem was proposed in Rodrigues & DaSilva (2003).An OPF model with multiplicity of objectives and a set of voltage security constraints was also discussed in Milano et al. (2003) with regard to avoiding congestion through the use of location marginal price.The use of rescheduling of generation and load with voltage security constraints for CM was also discussed in Yamin & Shahidepour (2003).An efficient CM approach using real and reactive power rescheduling via optimal allocation of reactive power resources was proposed in Kumar et al. (2004).A simple cost effective model for generation rescheduling and load shedding was also studied in Talukdar et al. (2005) in the past.
The heuristic methods in Hazra & Sinha (2007); Dutta & Singh (2008); Balaraman & Kamaraj (2010) for global optimizations have been made successfully to solve CM problems in the recent past.Hazra & Sinha (2009) put forth an efficient approach based on fuzzy estimation for identifying collapse sequences to reach the optimal solution of a CM problem.Fuzzily described adaptive bacterial foraging algorithm and gravitational search method have also been studied in Venkaiah & Vinod Kumar (2011) and Vijaya Kumar et al. (2013) previously.
To overcome the various drawbacks associated with the previous approaches concerning CM in thermal power supply system, a priority-based FGP method for multiobjective decision making (MODM) is addressed in this paper to model CM problem and a GA computational scheme is adapted to reach decision in imprecise premises.In model formulation, fuzzy representations of different objectives are considered for minimization of overload alleviation and operation cost subject to various constraints associated with the problem.The experimental test on standard IEEE 6-Generator 30-bus system is made to expound the effective use of the method.The solution is also compared with solution achieved by using Particle Swarm Optimization (PSO) technique in Hazra & Sinha (2007) is performed to present superiority of the proposed method.Now, FGP model formulation of a MODM problem is discussed in the section 2.

FGP problem formulation
In fuzzy environment, objectives are generally described fuzzily, whereas structural resource constraints may be fuzzy or crisp and that depends on how the model parameters are involved there in the decision situation.
In line with the work of Dubois (1987), the generic form of a Fuzzy Programming (FP) problem can be exhibited as follows.
Find X for: where X is a vector of decision variables, gk be the imprecise goal level of kth objective () k FX , k = 1,2,...., K, ≳ and ≲ indicate fuzziness of ≥ and ≤ restrictions, respectively, and where A is a real matrix and b is a constant vector and T means transposition, L X and U X denote the vectors of lower-and upper-limits, respectively, of the vector X , and where L and U indicate lower and upper, respectively.Also, it is assumed that the feasible region () S   is bounded.Now, characterization of fuzzy goals is by associated membership functions concerned with measuring degree of achievement of each of them in a decision making horizon.

Characterization of membership function
Let k t and tuk be lower-and upper-tolerance ranges, respectively, regarding achievement of aspired level gk of kth fuzzy goal.
Then, membership function, say () k X


, associated with () k FX can be characterized as follows. For appear as in Zimmermann (1987): where ( ) kk gt  denotes the lower-tolerance limit to achieve the stated fuzzy goal.
Further, for  type of constraint,  can be presented as: where (gk + tuk) denotes the upper-tolerance limit to achieve the stated fuzzy goal.The membership functions in (2) and (3) can be graphically depicted as in Figure 1 and Figure 2, respectively.
The formulation of an FGP model under a pre-emptive priority structure by defining membership goals is described in section 2.2.

FGP model
Since in a MODM context, various conflicting goals are dealt for achieving the aspired levels, priority-based FGP is adopted by Pal & Chakraborti (2013) for formulating the model of the problem.In priority-based FGP, priorities are assigned to goals according to importance of achieving goal levels, where a set of goals which seems equally important for their goal achievements are included at a same priority level and numerical weights are introduced there according to relative weights of importance to achieve goal levels.
The generic form of a priority-based FGP model can be presented as follows., are under and over-deviational variables introduced to kth goal, and where Z represents the vector of R priority achievement function. () Pd  is a linear function of vector of weighted under-deviational variables, and ., . where rk and it is the weight of importance of achieving kth goal level relative to others which are grouped together at rth priority level and where w  rk values are determined in Pal et al. (2003) as: , where ">>>" implies "much greater than".
In the formulated model, the notion of using pre-emptive priorities is that the goals which are at rth priority level r P are preferred most to achieve the corresponding aspired levels before taking the achievement problem of goals included at next lower priority level 1 r P  .Now, to design the model of a CM problem, it is worth noting that objectives and some system constraints are with nonlinear characteristics.To avoid computational complexity in Awerbach et al. (1976) with nonlinearity in model goals and constraints as well as to overcome the burden of hand calculations for linearization of them using approximation technique in Pal et al. (2009), GA as a goal satisfier in Deb (2002) for multiobjective decision analysis is considered for searching solution of the problem.The GA computational scheme is presented in the section 3.

GA Computational scheme for CM problem
The three probabilistically defined operators in Goldberg (1989): selection, crossover and mutation are used to generate new population (i.e., new solution candidates) in the GA scheme to search solution.The real-value coded chromosomes are considered to perform operations with GA in random fashion.To evaluate a function, say () v Eval E , the fitness score of a chromosome, say v, according to maximization or minimization of an objective function defined by decision maker (DM) in the decision making context.In the proposed MODM model, since () v Eval E is a single-objective linear program, roulette-wheel selection, arithmetic crossover and uniform mutation are adapted to search decision of the problem.
The algorithmic steps of GA computational process are described in the following section 3.1.

GA algorithm
Step 1. Representation and initialization.Let E denote the double vector representation of chromosome in a population as 12 ( , ,..., ) . The population size is defined by pop_size, and pop_size chromosomes are randomly initialized in the domain of searching solution. .
The fitness value of each chromosome is judged by the value of an objective function.The fitness function is defined as: 4) for measuring the fitness value of vth chromosome, when attainments of goals included at rth priority level P r is considered.
The best value of a chromosome is determined as in course of searching minimum value of achievement function.
The simple roulette-wheel scheme is employed for selection of two parents for mating purpose in solution search process.
The probability of crossover is defined by parameter pc.The single-point crossover in Goldberg (1989) is applied here with a view to obtaining offspring that always satisfy linear constraints set.dom number for generating two offspring Step 5. Mutation.A parameter pm is defined as the probability of mutation.The mutation operation is made uniformly, where for a random number [0,1] r  , a chromosome is selected for mutation provided that .
m rp  Step 6. Termination.The solution search process terminates when best decision for a chromosome is received at a certain generation number in decision making premises.
The pseudo code of the GA is as follows:

Initialize population of chromosomes E x
Evaluate the initialized population by computing its fitness measure

CM problem Formulation
The various objectives that are inherently associated with a CM problem are defined as follows.

Defining the objective functions
(a) "Overload alleviation" function.
In decision premises, the alleviation of overload on a transmission line is essentially needed to ensure security and stability of system, and thereby taking preventing measure against happening of system outage.Here, transmission line overload can be alleviated by line switching, generation rescheduling and load shedding.
The alleviation of overload in the system takes the form: where, F1 represents cumulative overload, NL is number of overloaded lines, and where max ii

S
and S be the MVA flow and MVA capacity of line i in power supply system, respectively.Also, square form of objective is made to avoid masking effect.
(b) Operational cost function.In this context, the total incurring cost for thermal power plant operation and which is associated with CM problem can be expressed as sum of the fuel cost and cost of load shedding.The total operational cost function is expressed as: where F2 denotes total operating cost, NG be the number of participating generators, PL is used to represent number of associated loads, PGi is generation of power from ith generator, Lshd,k is amount of load shedding at bus k, and where (c) Power-loss function.
A certain function called real power-loss function which is inherent to a power transmission line and directly affect the ability to transfer power.The mathematical expression of real power-loss function, F3 (MW) can be defined as in Talukdar et al. (2005): where TL represents total transmission lines, gl be the conductance of lth line, Vi and Vj are voltage magnitudes, i and j are voltage phase angles at the end buses i and j of lth line, respectively, of the system, where 'cos' designates cosine function.

Definitions of system constraints
The constraints on the power generation system r are as follows: a) Power balance constraints.
The power balance constraints appear as: where H be the number of buses, PGi and QGi are real-and reactive-power of the generator connected to ith bus, respectively, and where PDi and QDi be real-and reactive-power of the load connected to ith bus, respectively, gij and bij indicate transfer conductance and susceptance between bus i and bus j, respectively, δi and δj are bus voltage angles of buses i and j, respectively.b) Determining the Generation capacity & voltage constraint.Similar to conventional power generation and dispatch system, constraints on power generation and voltage appear as: Now, to show the effective use of the proposed approach, an example is considered in the section 5.

Case example
The IEEE 30-bus 6-generator test system Talukdar et al. (2005) is addressed to present the effectiveness of the method.The diagram of the system depicted in Figure 3 below.
The diagram shows that the system is with 6 generators, 41 lines and 30 buses.The total demand on 21 load buses is 283.4MW.

Figure 3. Diagram of IEEE 30-bus test system
The model data were collected from the studies (Talukdar et al., 2005;Hazra & Sinha, 2007) made previously.The cost-coefficients of power generation and that of load shedding are presented in the Table 1 and Table 2, respectively  The data associated with transmission lines and loads at buses are presented in the Table 3 and Table 4, respectively.

3
Overload simulation for outage of unit 3 at bus 5 and with reduction of capacity of line 2-5 from 130 MW to 50 MW.
In this case, the Optimization Toolbox under MATLAB (MATLAB R2010a) has been employed to conduct the experiments by employing GA at different stages for program evaluation.The computational environment is Intel Pentium IV with 2.66 GHz.Clockpulse and 3 GB RAM.In the solution search process, initial population= 50; Roulette-Wheel selection; Single-point crossover with probability= 0.8; Mutation probability= 0.07 and Maximum generation number= 100 are taken into account for exploration and exploitation of search space in the domain of interest.
Then, following the procedure and fitting the data presented in Tables 1 -Table 5, the membership goals can be obtained by addressing the second goal expression in (4).
The executable FGP models for individual three simulation runs under a priority structure considered for the system are presented as follows.
Run-3: Simulation of system under overload with outage of unit 3 at bus 5 and by reducing capacity of line 2-5 from 130 MW to 50 MW The executable model is obtained as follows.Find ( , ) i Gi SP so as to: 1, subject to the problem constraints in ( 14) -( 27).The goal achievement function ( Z ) defined for the three runs actually describes the evaluation function in GA search process for solving the problem.
The evaluation function for determining the fitness of a chromosome is given as: The solutions obtained from the three runs of the test system are presented in the Table 6.

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It is clear from the results that the decision is a satisfactory one from the view point of proper management of MVA flow with incurring of minimum operational cost of the power plant in imprecise environment.
To show the effective use of the approach, a performance comparison is made in the section 6.

Performance comparison
The PSO technique in Hazra & Sinha (2007) is considered for a solution comparison.The resulting decision is presented in the Table 7.

Conclusion
The main merit of the method presented here is that the fuzzy characteristics regarding attainment of objectives values are preserved there in all possible instances of executing the model of the CM problem.Again, computational complexity arising out of the nonlinearity in the goals and constraints associated with the model can easily be avoided here with the use of GA based solution search approach for solving problems in imprecise environment.The proposed method is also advantageous in the sense that here a multi-objective optimization problem can be converted into a goal oriented single objective optimization problem for achieving a compromise solution in the decision making horizon.Further, the proposed approach is flexible enough to accommodate different other restrictions as and when needed for CM in electric power transmission system.However, the use of interval data in Pal (2018), instead of considering fuzziness of model parameters, towards promoting CM performances and thereby improving quality of solution is an interesting alley of research for optimization of a power supply problem.
cost coefficients of objective associated with load shedding at bus k.
and total incurring cost of the CM problem under the proposed model and PSO technique are diagrammatically presented in Figure4and Figure5, respectively.

Figure 4 .
Figure 4. Graphical representation of MVA flow comparison

Figure 5 .
Figure 5. Graphical representation of cost comparisonThe result comparisons show that the proposed approach is superior over the PSO to arrive at appropriate decision in imprecise environment.
Now, formulation of FGP model of CM problem is discussed in the section 4.

Table 1 .
. Power generation cost -coefficient data

Table 2 .
Load shedding cost-coefficient data

Table 3 .
Transmission-line data

Table 4 .
Bus-load data

Table 5
exhibits various simulation runs which were carried out in the test system.

Table 6 .
Solution achievements under different runs

Table 7 .
Results of three simulation cases under PSO technique