Fuzzy Multi-Objective OPF considering voltage security and fuel emission minimization

Mostafa. Asghari Department of electrical engineering , Miyaneh branch , islamic azad university, Miyaneh,Iran mostafaa.asghari@gmail.com

Jamshid.Mohammadi Department of electrical engineering , Miyaneh branch , islamic azad university, Miyaneh,Iran ma_jamshid@yahoo.com

Abstract— In this paper a fuzzy multi-objective framework is introduced for the optimal power flow optimization problem. The objective functions that are considered in this work are minimizing the total fuel cost, voltage stability index and the total voltage deviation of the system. In the first case each of these items are optimized individually and in the next section using the fuzzy multi objective approach these objective functions are transformed to a single objective function. The fuzzy adaptive PSO approach has been used to solve the optimization problem, also the solution has been tested on the IEEE 30 bus system. Keywords- Optimal Power Flow (OPF), Fuzzy multi-objective optimization, voltage security, fuzzy adaptive PSO, fuel emission minimization

I. INTRODUCTION (HEADING 1) Electric power systems are one of the most complex

human made systems. For the secure and economic operation of these systems many tools have been developed based on the equations that dominate the system. One of these tools is the Optimal power flow (OPF) approach which was introduced by Carpentier for the first time [1]. The goals that are achieved by OPF are different and can include economic or security aspects or both of them.

In OPF the main goal is reached by determining the optimal setting of the control values which are considered in solving the optimization problem. For example the power generated by the generating units, voltage of PV buses, tap settings of transformers and capacitors and reactors are the control variables that there amount should be set optimally. It should be mentioned that in addition to optimizing objective functions a set of constraints exist that have to be satisfied.

In this paper the goals that are considered are the fuel cost minimization, fuel emission minimization voltage deviation minimization and voltage stability improvement. The OPF problem is a nonlinear nonconvex optimization problem which can also have discrete variables. This optimization problem includes many local minimum points which complicate the searching process. Many mathematical methods have been used to solve the OPF optimization problem but due to the problems that these methods have with nonlinear and nonconvex spaces they are often stock in local optimal points [2-3]. Therefore, these optimization

methods that are base on derivatives and gradients are not able to obtain to global optimum.

In the recent years many optimization techniques have been developed based on evolutionary algorithms. In [4] an improved genetic algorithm approach has been used to solve to OPF problem. Also in [5] and [6] tabu search and simulated annealing have been used for optimal power flow. Also other heuristic algoritms such as harmony search algorithm and adaptive harmony search algorithm [7-8] have been used to overcome nonlinear and nonconvex optimization problems.

The other optimization tool that has been used to solve optimization problems is the particle swarm optimization (PSO) approach which was first proposed by Kennedy and Eberhart. In [9] and [10] PSO is used to solve OPF. Also in [11] and [12] the formulation has changed to account the discrete variables.

In this paper a fuzzy adaptive PSO approach is utilized to find the optimal setting of control variables while satisfying the operation constraints. After investigating the optimum operation point of each the objective functions individually the optimization problem is solved simultaneously by using a fuzzy multi-objective approach. In [13] and [14] a fuzzy multi-objective method is used to transform the objective functions in to a single objective function.

II. PROBLEM FORMULATION The OPF problem aims optimizing an objective function

by finding the sufficient setting of the control variables. The problem can be formulated as the following: Minimize F( x, u) S.t (1) H( x, u)=0 G (x, u)≤0

Where in the above equation F is the objective function and H and G are the set of equality and inequality constraints. Also x indicates the control variables and u shows the state variables.

In the following each of these variables are explained.

I.1. Objective Functions In this paper different objective functions are studied. The

objective functions are total fuel cost, total fuel emission,

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voltage stability and voltage deviation. Each of these objective functions are formulated below:

1. Fuel Cost: the goal of this function is to minimize the cost paid for fuel used in the generating units and is exposed as a quadratic function as below:

2. Fuel Emission: Fossil based thermal plants are one of the main sources of pollution. The emission of these generating units has a negative impact on the environment therefore has to be controlled. One way to overcome this problem is to consider the emission of generators is OPF optimization problem. The fuel emission of generating units can exposed by a quadratic function as below:

3. Voltage stability index: One of the necessities for power system operation and control is the reliable assessment of voltage stability. For evaluating voltage stability number of methods have been proposed. One of these approaches represents a L index which varies between zero (system without load) and one (voltage collapse) and evaluates voltage stability of the system [15]. The voltage stability index is based on the hybrid matrix of circuit theory. The transmission system is written as:

Where: H: hybrid matrix VL (IL) :Voltage(Current) at load node VG(IG) :Voltage(Current) at generator node The voltage stability index at load node j may be written as:

Where

i indicates the generator buses. Therefore, the voltage stability index for the whole network may be expressed as: L =Max Lj (6) The introduced index helps the operator to assess the margin to voltage collapse. 4. Total Voltage Deviations: The other security object that is regarded in this work is improving the voltage profile.

When the bus voltages are considered as constraints after the optimization process there is a high probability that many bus voltages move to near there maximum or minimum limits and will violate the voltage constraints in the case of a contingency. So, one of the object functions can be keeping the bus voltages near to a desired amount (1 p.u in this work). The voltage deviation is obtained by the following relation:

I.2. Constraints In this section the constraints of the OPF problem is exposed. The first set of constraints are the equality constraints which are indeed the active and reactive power flow equations:

(8)

The other sets of constraints are the inequality constraints which are: Active and reactive power limits of generating units: Pi,min ≤Pi ≤Pi,max (9) Qi,min ≤Qi ≤Qi,max (10) Where i is the number of the generating unit.

Also the voltages of PQ buses, PV buses and tap settings of transformers are limited to: VPQ,Min ≤VPQ ≤VPQ,Max (11) VPV,Min ≤VPV ≤VPV,Max (12) Ti,Min ≤Ti ≤Ti,Max (13) Where: VPQ is the voltage for PQ buses. VPV is the voltage for PV buses. Ti is the tap setting of the ith transformer.

III. FUZZY BASED MULTI-OBJECTIVE OPTIMIZATION

Two main approaches are usually used for solving multiobjective optimization problems including Pareto sets and fuzzy evaluation techniques. In the second method using membership functions the multi-objective optimization problem is transformed to a single-objective optimization problem[14].

Fuzzy evaluation is a proper way for compromising conflicting object functions in a convex or non-convex multiobjective optimization problem and is used to extract a

single object (known as the fuzzy performance index) from several objects of the optimization problem. Assuming a number of object functions ( f1 , f2 , …, fn ) first each of the object functions are transformed to a fuzzy variable µfi according to the membership function that has been defined for it. After the object functions are fuzzified the single membership function is obtained by the aggregation of all of the fuzzy variables. Finally among all of the solution vectors the solution with the maximum membership function (which is obtained by the aggregation of all fuzzified object functions) is the optimal value. The mathematical formulation can be exposed as:

F =min (µ1 , µ2 ,...µn ) (15) Max F (16) Where in (31) F is the overall fuzzy performance index. Also the membership functions for each object function are exposed in Fig. 1. For the membership functions is the optimal value of the ith object function when it is optimized as a single-objective function. Also is the initial value of the ith object function.

IV. FUZZY ADAPTIVE PSO IV.1. PSO

In the PSO optimization approach first a set of random answers are created. Each of these answers are a vector including settings for each variable and in the form of xi (t) =(xi,1(t), xi,2 (t), ..., xi,k (t), ..., x i,n (t)) also each particle has a velocity vector vi (t) =(vi,1 (t), vi,2 (t), ..., vi,k (t), ..., vi,n (t)) which is updated at each iteration.

Fig. 1. Fuzzy membership functions for different object functions.

The best answer among all of the vectors is pbest i (t) =(pi,1(t), pi,2 (t), ..., pi,k (t), ..., pi,n (t)) at a specific iteration and the best answer among all vectors in all iterations is shown by gbest(t) =(g1 (t), g2(t),..., gk (t), ...,gn (t))

At each iteration following the below relation the velocity vector and position vector of each particle is modified:

v i,k (t +1) =w.v i, k (t)

+c 1 .rand(0,1) .(pbest i, k (t) -x i, k (t)) (17)

+c 2 .rand 1 (0,1).(gbest k (t) -x i, k (t))

In (33) w is the inertia weight. c1 is the cognitive parameter and c2 is the social parameter (both of these parameters are also known as learning factors). rand (0,1) and rand1 (0,1) are random generated values between 0 and 1 for each iteration, each particle and each variable of the particle. The position of each particle is modified as following: xi,k (t +1) =xi,k (t) +vi,k (t +1) (18) This iterative process is continued until reaching the stopping criteria.

IV. 2. Fuzzy adaptive PSO The parameters in (17) have an important impact on the

search process. These parameters should be chosen in a way to provide a sufficient balance between global and local search at each iteration considering the situation and parameters of that iteration. In the search process the inertia weight is the parameter that determines the searching space. High values of this parameter leads to a global search while low amounts of this parameter results to a local search.

Since the search space is nonlinear and very complicated it is very difficult to find the exact relationship between the variations of w and the search procedure. In [27] a linear decreasing inertia weight is proposed to set a global search at the initial iterations and a local search in the last iterations.

Also as it can be seen in (17) the learning factors adjust the influence of pbesti and gbest on modifying the velocity vector of each particle for the next iteration. While c1 determines how much the particle should use it's own data in the search procedure, c2 adjusts the impact of the best data obtained by the other particles on the velocity vector [14].

As mentioned, optimization by PSO is a nonlinear and complicated procedure and linearly decreasing the parameters does not provide an accurate adjustment of the parameters in the search process. Also considering the complication process achieving an exact mathematical formulation for the search process seems to be impossible. Although there is no mathematical formulation for the search process but it can be expresses and understood by a linguistic description [14]. So using a fuzzy system can be sufficient for handling the adjustment parameters of the PSO algorithm. In the next section a fuzzy system will be organized based on two initial experiences and linguistic descriptions: 1. When the best answer obtained for the problem is low and the last iterations are elapsed the inertia weight has decrease and learning factors should increase to provide a local search. 2. When the best answer is

unchanged for a number of iterations it means it has been stuck in a local minima so for coming out of this point the inertia weight has to increase and the learning factors should decrease proportionally. The inputs of the fuzzy system are two parameters. 1. Normalized object function (NOF) which is defined as:

NOF =(OF -OFmin ) /(OFmax -OFmin ) (19) 2. Normalized unchanged best object function (NU) which is exposed as: NU =(NU -NUmin ) /(NUmax -NUmin ) (20) The output of the fuzzy system is the variation of the inertia weight (Δw ) and also the variations of learning factors ( Δc1 and Δc2 ). The fuzzy system used in this study is composed of the four following steps: 1. Fuzzification In this stage membership functions are defined for the input and output parameters of the fuzzy system. The membership functions for the input variables are shown in Fig. 6.

Figure.2 Membership functions for the input and output of the fuzzy

system In the above figures S, M, B and VB respectively indicate small, medium, big and very big membership functions. Also the membership function used for output variables is represented in Fig. 7. NB, NS, Z, PS and PB respectively present negative big, negative small, zero, positive small and positive big membership functions. The algorithm control parameters ( w, c1 and c2 ) are obtained from the below relations: WC+1=W0+ C1=C10+ C2=C20+ Where in (38) wc+1 is the inertia weight for the next iteration. Also for this relation we have w0 =0.7 and c10 =c20 =1.5 . 2. Fuzzy rules Conditional statements are formulated using Mammdani fuzzy rules. The fuzzy rules used in this paper are exposed in the appendix.

3. Fuzzy inference Mammdani fuzzy inference method is used in this paper. 4. Defuzzification The method "center of sums" has been used for defuzzification by the following relation: Y=

In (37) y is a number and µBi is a fuzzy set. V. SIMULATION AND RESULTS In this work the IEEE 30 bus system has been chosen to

investigate the fuzzy multi-objective OPF problem. The system has 41 lines, 4 transformers with tap settings and three reactive compensation devices which are installed at the buses 3, 10 and 24. The reactive compensation devices are restricted to [-12, 36] MVar and the tap settings of the generators are limited to [0.95, 1.05] p.u. also the generator voltages are between 0.9p.u and 1.1 p.u. The limitation on the active power generated by the generators and there fuel and emission coefficients are exposed in Table.1 and Table.2 [16]. Table1. Cost and emission coefficients for generators 1,2 and 5

BUS NUMBER 1 2 5

a 0 0 0 b 2 1.75 1 c 0.00375 0.01750 0.06250 α 0.04091 0.02543 0.04258

-0.05554 -0.06047 -0.05094 0.06490 0.05638 0.04586

P max 200 80 50 P max 50 20 15

Tables 3 and 4 respectively represent the results for the FAPSO and PSO approach for each objective function after twenty runs. The best, worse and medum of the answers for the four objective functions are exposed. The results show the better performance of the FAPSO approach. Also the convergence process has been shown in Figure – for each objective function.

In the second case the OPF problem is solved in a multiobjective framework. The objective functions including total fuel cost, voltage deviation and the voltage stability index are optimized simultaneously and the results are presented at Table.5 .

Table.2 Cost and emission coefficients for generators 8,11 and 13 BUS

NUMBER 8 11 13

a 0 0 0 b 3.25 3 3 c 0.00834 0.025 0.025 α 0.05326 0.04258 0.06131

-0.03550 -0.05094 -0.05555

0.03380 0.04586 0.05151 P max 35 30 40 P max 10 10 12

Table3. Results after twenty runs for the PSO approach

BEST ANSWER

BEST ANSWER

BEST ANSWER

TOTAL FUEL

COST ($/h) 800.5232 801.2590 800.8343

TOTAL EMISSION

(ton/h) 900.6049 1081.2 1017.74

LMAX 0.1198 0.1211 0.1204

VOLTAGE DEVIATION

(p.u) 0.1220 0.14 0.1274

Table.4 Results after twenty runs for the FAPSO approach

BEST ANSWER

BEST ANSWER

BEST ANSWER

TOTAL FUEL

COST ($/h) 800.5240 801.8890 801.472

TOTAL EMISSION

(ton/h) 905.31 1153.4 1071.922

LMAX 0.1203 0.1220 0.1209 VOLTAGE

DEVIATION (p.u)

0.1231 0.144 0.131

Table5. Results for the multi-objective OPF optimization problem

TOTAL FUEL

COST ($/h) Lmax

VOLTAGE DEVIATION

(p.u) MULTI

OBJECTIVE OPTIMIZING

800.8876 0.1281 0.6596

In the second case the OPF problem is solved in a multiobjective framework. The objective functions including total fuel cost, voltage deviation and the voltage stability index are optimized simultaneously and the results are presented at Table.5 .

FIGURE.3. TOTAL FUEL COST VARIATION

FIGURE.4 VOLTAGE STABILITY INDEX VARIATION

VI. CONCLUSION In this paper the OPF optimization problem has been solved by a fuzzy adaptive PSO approach. The objective functions that have been considered are the total fuel cost, total emission, voltage stability index and the total voltage deviation. Controlling the PSO parameters by a fuzzy approach results to a proper balance between local and global searching and leads to better answers with a faster convergence rate. Finally all of the objective functions are transformed to a single objective function by a fuzzy approach and are optimized simultaneously.

FIGURE.5 TOTAL VOLTAGE DEVIATION VARIATION I. References [1] J. Carpentier, “Contribution e létude do Dispatching Economique,” Bull. Soc. Franc. Elect., pp. 431–447, 1962. [2] J. Momoh, E. El-Hawary, R. Adapa, A review of selected optimal power flow literature to 1993, Part I: Non-Linear and quadratic programming approaches, IEEE Transactions on Power Systems, 14 February 1999 96-104. [3] J. Momoh, E. El-Hawary, R. Adapa, A review of selected optimal power flow literature to 1993, Part II: Newton linear programming and interior point methods, IEEE Transactions on Power Systems, 14 February 1999 105-111. [4] L. Lai, J. Ma, R. Yokohoma, M. Zhao, Improved genetic algorithm for optimal power flow under both normal and contingent operation states, Electrical Power Energy System, 19 (1997) 287-29. Modality of Chapters in Books’ quote: [5] V. Miranda, D. Srinivasan, L. M. Proenca, Evolutionary computation in power systems, Electrical Power Energy System, 20 (1998) 89-98. [6] M. A. Abido, Optimal power flow using tabu search algorithm . Electric Power Components and Systems 30 May 2002 469-483 [7] A. H. Khazali, M. Kalantar, Optimal reactive power dispatch based on a harmony search algorithm, [8] Optimal Reactive /Voltage control by an improved harmony search algorithm, International Review of Electrical Engineering [9] M. A. Abido Optimal power flow using particle swarm optimization, Elect power energy systems no.24, pp563-571, 2002. [10] B. Zhao, C. X. Gao , Y. J. Cao, Improved particle swarm optimization algorithm for OPF problems, IEEE/PES Power systems conf. 2004 233-238. [11] C. K. Mohan B. Al-Kazemi, Discrete particle swarm optimization, Inproc. Workshop particle swarm optimization, Indianapolis, IN, 2001. [12] J. Kennedy, R. C. Eberhart, A discrete binary version of the particle swarm algorithm, In Proc. IEEE Int. Conf. Syst. Man, Cyber, Piscataway, NJ 1997, pp. 4104-4108. [13] M. Gitizadeh and M. Kalantar, “Genetic algorithm-based fuzzy multiobjective approach to congestion management using FACTS devices”, Electrical Engineering Journal, Springer, vol. 90, no. 8, pp. 539-549, February 2009. [14] W. Zhang, Y. Liu, " Multi-objective reactive power and voltage control based on fuzzy optimization strategy and fuzzy adaptive particle swarm", Electrical power and energy systems 30, 528-532 2008. [15] P. Kessel and H. Glavitsch, "Estimating the Voltage Stability of Power System," IEEE Trans. Power Delivery, Vol. PWRD-1, No. 3, pp. 346-354, July 1986. [16] M. R. AlRashidi, M. E. El hawary, Hybrid particle swarm optimization approach for solving the discrete OPF problem considering the valve loading effects, IEEE Trans. On Power Syst., Vol 22, No. 4November 2007