Improved particle swarm optimisation for multi-objective optimal power ...

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 19th September 2011 doi: 10.1049/iet-gtd.2011.0851

ISSN 1751-8687

Improved particle swarm optimisation for multi-objective optimal power flow considering the cost, loss, emission and voltage stability index T. Niknam M.R. Narimani J. Aghaei R. Azizipanah-Abarghooee Department of Electronic and Electrical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran E-mail: [email protected]; [email protected]

Abstract: The study presents an improved particle swarm optimisation (IPSO) method for the multi-objective optimal power flow (OPF) problem. The proposed multi-objective OPF considers the cost, loss, voltage stability and emission impacts as the objective functions. A fuzzy decision-based mechanism is used to select the best compromise solution of Pareto set obtained by the proposed algorithm. Furthermore, to improve the quality of the solution, particularly to avoid being trapped in local optima, this study presents an IPSO that profits from chaos queues and self-adaptive concepts to adjust the particle swarm optimisation (PSO) parameters. Also, a new mutation is applied to increase the search ability of the proposed algorithm. The 30-bus IEEE test system is presented to illustrate the application of the proposed problem. The obtained results are compared with those in the literatures and the superiority of the proposed approach over other methods is demonstrated.

1

Introduction

Optimal power flow (OPF) is a static non-linear programming problem which optimises a certain objective function while satisfying a set of physical and operational constraints imposed by equipment and network limitations [1, 2]. It is also a large-scale static optimisation problem with both continuous and discrete control variables [3]. Many mathematical techniques such as quadratic programming [4], linear programming [5], non-linear programming [6] and the interior point method [7] have been applied to solve the OPF problem. All the above mathematical techniques have some drawbacks such as being trapped in local optima or they are suitable for considering a specific objective function in the OPF problem [8]. These shortcomings can be overcome if evolutionary methods are utilised to solve the OPF problem. Particle swarm optimisation (PSO) is one of the known optimisation algorithms that has been used to solve complicated problems [9 – 12]. Also, it is a strong and accurate algorithm that can find high-quality solutions for complicated problems such as the OPF. However, the traditional PSO is often trapped in local optima and converges to the optimal value in a long time. In order to avoid these problems and increase the efficiency of the PSO algorithm, this study uses a chaos concept to tune the inertia weight (v) and a self-adaptive approach for adjusting the learning factors (C1 and C2) of the PSO algorithm. These parameters (v, C1 and C2) play an important role in the PSO convergence property. In other words, the performance of the PSO algorithm greatly depends on these parameters. Also for increasing the search ability, a new mutation is applied in the proposed algorithm. The used IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

mutation increases the diversity of the generated population and causes to escape from local optima during the optimisation process. The thermal power houses release sulphur oxides (SOx), nitrogen oxides (NOx) and carbon dioxides into the atmosphere. The passage of the US Clean Air Act amendments of 1990 forces the utilities to modify their operation strategies for generating electrical power not only at a minimum generation cost but also with minimum pollution level [13]. There are a lot of solutions proposed in the literature for reducing the emission. The short-term and applicable solution for the emission problem is an environmental dispatch that needs no additional equipments, therefore this method is applied for emission reduction. Since electrical business enters a deregulated environment, power companies try their best to operate with economic efficiency. Loss reduction is an effective method to decrease the generation cost, also active power transmission loss is considered as an objective function. By decreasing the loss in power systems, the total generation and consequently generation cost are reduced which increase social welfare. The stability of power systems is an important task that the power system operator should keep at an acceptable level. One of the important branches related to the stability in power systems is voltage stability. So, during previous years a great number of approaches have been considered for satisfying the voltage stability. Therefore beside economic and environmental issues in this study, the voltage stability is considered as an objective function, in this regard the Voltage Stability Index (VSI) is optimised to increase the secure operation of the power system. 515

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www.ietdl.org There are several techniques that have been considered in the literature to solve multi-objective problems. One of these methods is reducing the multi-objective problem into a single objective problem by considering one objective as a target and others as a constraint. Another strategy is combining all objective functions into one objective function. The above strategies have some weak points such as the limitation of the available choices and their priori selection need of weights for each objective function. Besides the above drawbacks, finding just one solution for the multi-objective problem is known as the most important weak point of these strategies. Over the past few years, the studies on evolutionary algorithms have revealed that these methods can be efficiently used for solving the multiobjective optimisation problem, some of these algorithms are multi-objective evolutionary algorithm [14], strength Pareto evolutionary algorithm (SPEA) [15], non-dominating sorting genetic algorithm (NSGA) [16] and multi-objective PSO algorithm [17]. Since these algorithms are populationbased techniques, multiple Pareto-optimal solutions can be found in one program run. Since the objectives are in conflict with each other in multiobjectives problems, it is usual to obtain a set of solutions instead of one. In this study, the proposed improved PSO (IPSO) algorithm is implemented in order to extract the non-dominated solutions. In this regard, this study utilises an external repository to save all non-dominated solutions during the evolutionary process, and a fuzzy decisionmaking method is applied to sort these solutions according to their importance. Power system decision makers can select the desired solution between them by applying the fuzzy decision-making method to the Pareto-optimal solution. Finally, to authenticate the obtained Pareto-optimal solutions, three criterions involving generational distance (GD) [18], spacing (SP) [18] and diversity metric [19] are used. The 30-bus IEEE test system is presented to illustrate the efficiency of the proposed method.

2 2.1

The generation cost function can be mathematically stated as follows Ngen

(ai P 2gi + bi P gi + ci ) $/h

The emission function can be presented as the sum of all types of emissions considered, such as NOx , SOx , thermal emission, etc. In the present study, two important types of emission gases are taken into account. The amount of NOx and SOx emission is given as a function of generator output that is the sum of a quadratic and exponential function as follows F2 (X ) =

Ngen 

2 (gi Pgi + bi Pgi + ai + ji exp (li Pgi ))

ton/h

i=1

(8) where F2(X ) is the total emission (ton/h), gi , bi , ai , zi and li are the emission coefficients of the ith unit. 2.3

Transmission loss

The power flow solution gives all bus voltage magnitudes and angles. Then, the active power loss in transmission line can be computed as follows F3 (X) =

Nline 

gk [Vi2 − Vj2 − 2Vi Vj cos(ui − uj )]

MW (9)

k=1

where F3(X ) is the total transmission loss (MW), Nline is the number of transmission lines, di and dj are the bus voltage angles at the two ends of the kth line, Vi and Vj are bus voltage amplitudes at the two ends of the kth line and gk is the conductance of the kth line. Voltage stability enhancement index

    Ngen    V Lj = 1 − Fji i  Vj   i=1

j = Ngen + 1, . . . , n

(10)

(1) Matrix (F ) is computed by

i=1

X = [P g , V g , TAP, Qc]1×n

(2)

P g = [P g1 , P g2 , . . . , P g(Ngen −1) ]1×(Ngen −1)

(3)

V g = [V g1 , V g2 , . . . , V g Ngen ]1×Ngen

(4)

TAP = [TAP1 , TAP2 , . . . , TAPNtran ]1×Ntran

(5)

Qc = [Qc1 , Qc2 , . . . , QcNcap ]1×Ncap

(6)

n = (Ntran + Ncap + Ngen + (Ngen − 1))

(7)

where F1(X ) is the total fuel cost ($/h), ai , bi , ci are fuel cost coefficients of the ith unit, Pgi is the real power generation of the ith unit, Vgi is the voltage magnitude of the ith generator, TAPi is the tap of the ith transformer, Qci is the reactive power of the ith compensator capacitor, Ngen 516 & The Institution of Engineering and Technology 2012

Emission objective

The static voltage stability margin can be measured by the minimal L index which is described as follows [20]

Generation cost objective



2.2

2.4

Problem formulation and constraints

F1 (X ) =

is the total number of generation units, Ntran is the number of tap transformer and Ncap is the number of the compensation capacitor.

[F] = −[Y LL ]−1 [Y LG ]

(11)

where [YLL] and [YLG] are submatrices of the Y bus matrix. The network equations in terms of the node admittance matrix can be simply written as I bus = Y bus V bus

(12)

For computing the VSI value, it is necessary to cluster all nodes into two categories that involve load buses (aL) and generator buses (aG) as follows 

IL IG





Y LL = Y GL

Y LG Y GG



VL Vg



I L = Y LL × V L + Y LG × V g

(13) (14)

IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515 –527 doi: 10.1049/iet-gtd.2011.0851

www.ietdl.org The above equation can be written after adding the diagonal elements of IL matrix into the YLL matrix as follows Y ′ LL × V L + Y LG × V G = 0

(15)

From (15), it is clear that all load bus voltages can be calculated by using the generator bus voltages. According to the superposition principle, the voltage VLk in a load bus k can be calculated by

constraints are expressed in the following equations [20]

Pi = Pgi − Pdi =

nbus 

Vi Vj (Gij cos uij + Bij sin uij )

(19)

j=1

Qi = Qgi − Qdi =

nbus 

Vi Vj (Gij sin uij − Bij cos uij )

(20)

j=1

VLk =

Ngen 

((Y LL )−1 Y LG )k,i × Vgk

(16)

(18)

where i ¼ 1, 2, . . . , nbus and uij ¼ ui 2 uj that ui and uj are the voltage angle of two ending buses of an arbitrary branch and nbus is expressed as the number of the buses. It is worthwhile to note that all generator outputs except slack generator are generated randomly in their limits. Furthermore output of slack generator puts in its limit according to Fig. 1. The mechanism of handling the equality constraint related to the equality of generation level with load level plus loss is shown in Fig. 1. It is noticeable that whenever each output of generator is set to its maximum or minimum level the related velocity of the control vector for the next iteration is declined. In this regard, a negative value is added to the current velocity in order to change the direction of aforementioned element that is output power of the generator.

2.5.1 Equality constraints: The OPF equality constraints reflect the physics of the power systems. Equality

2.5.2 Inequality constraints: The inequality constraints of the OPF reflect the limits on physical devices in the power system as well as the limits created to ensure system

i=1

Lj =

Ngen 

((Y ′ LL )−1 Y LG )k,i

(17)

i=1

The L indices for the given load condition are computed for all load buses and the maximum of L indices shows that the system tends towards the voltage collapse. For stable situations, the condition 0 ≤ Lj ≤ 1 must not be violated for any of the nodes j. Hence, a global indicator L that describes the stability of the whole system is given by F4 (X ) = L = max (Lj ) 2.5

j [ aL

Constraints

Fig. 1 Diagram of satisfying equality constraint IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

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www.ietdl.org security. They are presented in the following inequalities Pgi min ≤ Pgi ≤ Pgi max

i = 1, 2, . . . , Ngen

(21)

Qgi min ≤ Qgi ≤ Qgi max

i = 1, 2, . . . , Ngen

(22)

|Pij | ≤ Pij max Vi min ≤ Vi ≤ Vi max Qci min ≤ Qci ≤ Qci max Ti min ≤ Ti ≤ Ti max

i = 1, 2, . . . , NL i = 1, 2, . . . , Ncap i = 1, 2, . . . , Ntran

(23) (24) (25) (26)

where NL is the number of load bus and Pij is the power that flows between bus i and bus j. Vimax and Vimin voltages are, respectively, the maximum and minimum valid voltages for each bus. Pijmax is the maximum power flow through the branch. Pgimax and Pgimin are the maximum and minimum active power values of the ith bus, respectively. Qgimax and Qgimin are the maximum and minimum reactive power values of the ith bus. In this study, the penalty factor method is utilised for handling the inequality constraints. In this regard, each control vector which violates constraints will be fined by these penalty factors therefore in the next step this control vector will be deleted automatically.

3 Fuzzy adaptive particle swarm optimisation 3.1

3.2 Self-adaptive method for computing the learning factors The learning factors C1 and C2 determine the influence of the k personal best pki and the global best gbest , respectively. They have a significant effect on the PSO convergence. When the global fitness is large, the particles are far away from the optimum point. Therefore a large velocity is needed for global search in the search space of the problem and so C1 and C2 must be larger values. Also, for the local search, a small movement is needed, therefore C1 and C2 must set to small values. According to above statement, learning factors in the proposed method are adjusted as a function of the global optimum fitness. Therefore a self-adaptive method is developed to adjust the learning factors, which is expressed as follows Cj = 1 +

PSO is an evolutionary optimisation method modelled by the social behaviour of birds flocking or fish schooling. This algorithm consists of a population continuously updating the searching space knowledge. This population is formed by individuals, where each one represents a possible solution and can be modelled as a particle that moves through hyperspace. The position and velocity of each particle are updated as follows [21] Velocityik+1 = vk × Velocityki + C1 × rand()1 k × (pki − Xik ) + C2 × rand()2 × (gbest − Xik )

j = 1, 2

(29)

k ) is the fitness of the global optimum solution. The where F(gbest value of a can be selected as the inverse of the best compromise objective value in the first iteration which is shown as follows [11] 1 a = 1/(F(gbest ))

(30)

k By this manner, C1 and C2 change according to the gbest rate. k Since the proposed problem is a multi-objective problem, gbest cannot be selected according to the fitness of the objective k function. The proposed method for computing the gbest is explained in Section 5.

3.3

Overview of the PSO

1 k (1 + exp(−a × F(gbest ))k )

Chaotic formula for inertia weight

The inertia weight factor is used to control the impact of the previous experience of velocities on the current velocity. A large value of v leads to a global search whereas a small value of v tends to facilitate local search. This factor usually decreases linearly from 0.9 down to 0.4 [22]. Despite many major advantages of the PSO such as its simple concept, easy implementation, minimal storage requirement and straightforward incorporation with other optimisation methods, it may experience inappropriate convergence and fall into the local optima. Therefore a proper tuning of the PSO parameters is very important for finding the global optimum solution. In this study, the inertia weight factor is tuned dynamically in each iteration as follows

(27) Xik+1

=

Xik

+

Velocityik+1

Velocityki

cvk = vk Dk (28)

is the velocity of the ith particle at the where kth iteration, Velocityik+1 is the velocity of the ith particle at (k + 1)th iteration, vk is an inertia weight in the kth iteration and rand()1 and rand()2 are random numbers selected between 0 and 1. C1 and C2 are positive coefficients between 0 and 2 that is C1 + C2 ≤ 4. Original PSO often suffers from the problem of being trapped in local optima so as to prematurely converge. The performance of the original PSO greatly depends on C1 , C2 and vk. In order to avoid this disadvantage, this study proposes a chaos algorithm to compute the inertia weight (v) and a self-adaptive method to compute learning factors (C1 and C2). 518 & The Institution of Engineering and Technology 2012

(31)

where cvk is the chaotic inertia weight factor at iteration k, vk is the inertia weight factor which is decreased linearly from 0.9 down to 0.4 and D k is the chaotic parameter at iteration k. The value of D k is determined by an iterator chaotic system, namely the logistic map [23] Dk = mDk−1 (1 − Dk−1 )

(32)

where m displays a control parameters which is selected between [0, 4]. The variation of m is greatly influenced on D k that is able to evaluate the D k in a constant size, oscillates between a limited sequence of size or oscillates randomly. Equation (32) is without any stochastic pattern in the value of m and indicating chaotic dynamic when m ¼ 4 and IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515 –527 doi: 10.1049/iet-gtd.2011.0851

www.ietdl.org D 0  {0, 0.25, 0.50, 0.75, 1} [22]. The searching capability and efficiency of the proposed algorithm increase greatly with the chaotic v as illustrated in the numerical result. Fig. 2 shows the chaotic sequence, the conventional weight factor and presented chaotic weight factor. From this figure it is clear that the change in the proposed weight factor is erratic. The aforementioned characteristic of the proposed weight factor leads to improvement of global search ability and prevent from trapping in local optima. 3.4

Mutation

To avoid being trapped in local optima, a new mutation is proposed in this study. For building the mutant vectors in each iterate, firstly four vectors Xr1 , Xr2 , Xr3 and Xr4 are selected randomly among the population. It is worthwhile to note that for increasing the mutation effect each mutant vector is generated by different mutant rules according to (33) – (37) [24]. This feature causes searching of the entire search space in different directions to be improved. Therefore the probability of being trapped in the local optima decreased drastically and consequently the ability of finding the global optima increased intensely. Equations (33) and (34) pave the way for the proposed optimisation problem to have an appropriate search in different directions around the gbest . Also (35) and (36) generate two random vectors in order to increase the diversity of population in the condition that the algorithm loses its diversity in its population. Therefore this procedure compensates the progressing of the algorithm in local optima direction or weak it at least. X imutant 1 = X ir1 + K1 × (X igbest − X ir2 ) + K1 × (X ir3 − X ir4 ) (33) X imutant 2 = X igbest + K2 × (X ir1 − X ir2 )

(34)

X imutant 3 = X ir1 + K3 × (X ir2 − X ir3 ) + K3 × (X ir1 − X ir4 ) (35) X imutant 4

=

(X ir1

+

X ir2

+ (h3 −

+

X ir3 )/3

h2 )(X ir2

−

+ (h 2 −

X ir3 )

h1 )(X ir1

−

X ir2 )

+ (h1 − h3 )(X ir3 − X ir1 ) (36)

h1 =

|f (X ir1 )| , h∗

h2 =

|f (X ir2 )| , h∗

h3 =

|f (X ir3 )| h∗

(37)

where h∗ = |F(X ir1 ) + F(X ir2 ) + F(X ir3 )| and F(X ) is the function to be optimised, K1 to K3 are the coefficients between 0 and 1. After generating the mutant vectors, we should check the limits for all elements in each mutant vector to ensure that there are not any breaks the limits, if any element of each vector break its constraint (xmax or xmin), it should be replaced by its own limit, which are the elements of upper and lower boundaries (Xmin , Xmax) of control vectors. Furthermore the inequality and equality constraints are checked for these new vectors in the way mentioned before. After all mutant vectors are compared with the X gbest , if each mutant vector dominates the X gbest , then X gbest is replaced by the mutant vector.

4

Multi-objective strategy

4.1

Fuzzy model for the multi-objective problem

A fuzzy system can handle the multi-objective problems with objective functions in contradiction. In this study, a fuzzy optimisation approach is used to normalise the objectives of the multi-objective problem. The objective functions are modelled by membership functions to achieve the optimal point. The higher the values of the membership function are, the greater the solution satisfaction is. The membership function includes lower and upper boundary values together with a strictly monotonically decreasing and continuous function. Fmin and Fmax are lower and upper bounds of ith objective i i function. The membership function is calculated under given constraints for each objective function, Fi(X ). The ith membership function is defined as follows ⎧ ⎪ 1 if Fi (X ) ≤ Fimin ⎪ ⎪ max ⎨ Fi − Fi (X ) mi (X) = if Fimin , Fi (X ) , Fimax max min ⎪ F − F ⎪ i ⎪ ⎩ i 0 if Fi (X ) ≥ Fimax

(38)

max Fmin are extracted from optimisation of each objective i , Fi separately.

Fig. 2 Comparison of conventional weight factor and proposed chaotic weight factor IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

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www.ietdl.org For fuzzy multiple objectives, the fuzzy solution can be calculated as F(X ) = min [m1 (X), m2 (X ), m3 (X ), m4 (X )]

(39)

The maximum value of F(X ) is considered as the optimal solution. 4.2

Pareto-optimal solution

Since the target of the proposed approach is to obtain a set of solutions for the multi-objective OPF problem it is necessary to apply a method which can obtain a set of optimal solutions. The Pareto-optimal method is a suitable approach for the multi-objective problem which can obtain a set of solutions instead of one. This method works based on the dominance concept, the vector X1 dominates X2 , when the following conditions are satisfied [2] ∀ i = {1, 2, . . . , M }, Fi (X 1 ) ≤ Fi (X 2 ) ∃ j [ {1, 2, . . . , M }, Fj (X 1 ) , Fj (X 2 ) 4.3

Step 1: The input data involving the generator real powers, generator bus voltages, fuel cost coefficients of generators, emission coefficients of generators, the tap position of the transformer, the reactive power of switchable VAR, Fimin and Fimax are defined. Step 2: Transfer the constraint multi-objective problem to an unconstraint one as follows (see (42)) where Neq and Nueq are the number of equality and inequality constraints, respectively. hj (X ) and sj (X ) are the equality and inequality constraints, respectively, and Z1 and Z2 are the penalty factors. Step 3: Produce the initial population. An initial population based on state variables is produced randomly as follows ⎡

X1 X2 .. .

⎢ ⎢ Population = ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎦

(43)

X Nswarm (40)

X i = [xi,1 , xi,2 , . . . , xi,n ]

(44)

xi,;j = rand(.) ∗ (xj, max − xj, min ) + xj, min , j = 1, 2, 3, . . . , n;

Fuzzy decision maker

In the proposed approach, a repository is defined to save non-dominated solutions in each iteration. The solutions that are saved in this repository in all iterations are sorted by a type of decision-making function. Therefore it is possible to select the best solution by selecting the top solutions in this collection. A decision-making function that is utilised in this study is defined as follows [25]

Nobj b × mk (j)

n k Nmj = m k=1 j=1 k=1 bk × mk (j)

(41)

where bk is the weight factor for the kth objective functions, Nobj is the number of the objective function and m is the number of non-dominated solutions. The weight values bk can be selected by the operator based on the importance of the objective function. The solution with the maximum membership function Nm is the most preferred compromise solution based on the adopted weight factors.

5 Application of the IPSO to the multi-objective OPF

i = 1, 2, 3, . . . , Nswarm

where xj is the position of the jth state variable, n and Nswarm are the number of control variables and the number of initial population, respectively. Step 4: Calculate the objective functions value and normalise them by the fuzzy decision-making which is proposed in (41). Also for each individual (Xi) the membership values of all the different objectives are computed. Step 5: Apply the Pareto method and save non-dominated solutions in the repository. k Step 6: Select (pki ) and gbest in each iterate, as follows: In the first iteration, the initial population is considered as pki but in the second to final iteration, if the pk+1 could i dominate the pki then, the pki is replaced by pk+1 and if i could not dominate pki then pki is remained as the best solution in next iteration. If none of them dominates each other, then pki is updated by max – min method in the next iteration which is described as follows: After computing all objective functions and normalising them by (38) in each iterate, by calculating the following equation for all objectives, mDi is computed for each individual as follows

To apply the IPSO algorithm to solve the proposed problem, the following steps should be taken and repeated. ⎡

N eq 

mDi = max(min(m1 , m2 , m3 , m4 )) 

N ueq 

⎢ F1 (X ) + Z1 (hj (X)) + Z2 (Max[0, ⎢ j=1 j=1 ⎢ ⎢ N  N eq ueq ⎢   ⎡ ⎤ ⎢ 2 J1 (X ) (hj (X)) + Z2 (Max[0, ⎢ F2 (X ) + Z1 ⎢ ⎢ J2 (X ) ⎥ j=1 j=1 ⎢ ⎥ J (X ) = ⎢ N  N ⎣ J3 (X ) ⎦ = ⎢ eq ueq ⎢   ⎢ 2 J4 (X ) 4×1 ⎢ F3 (X ) + Z1 (hj (X)) + Z2 (Max[0, ⎢ j=1 j=1 ⎢ N  N ⎢ eq ueq ⎢   ⎣ 2 F4 (X ) + Z1 (hj (X)) + Z2 (Max[0, j=1

520 & The Institution of Engineering and Technology 2012

(45)

2

(46)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −sj (X )])2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −sj (X )])2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −sj (X )])2 −sj (X )])

2

(42)

j=1

IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515 –527 doi: 10.1049/iet-gtd.2011.0851

www.ietdl.org k pik+1 = max(mD (Xik+1 ), mD (Xmut ), mD (pki ))

6

(47)

To validate the performance of the proposed method, this method is tested on the IEEE 30-bus test system. The detailed data of the test system are given in [26]. The 30-bus IEEE test system has 41 transmission lines, six generators and four transformers (T6 – 9 , T6 – 10 , T4 – 12 and T27 – 28). The lower and upper voltage magnitudes and transformer tap limits are considered between 0.9 and 1.1 pu, the emission and generation cost coefficients are given in Table 1. The parameters required for implementation of the proposed IPSO algorithm are adjusted by 100 times running of this algorithm. These parameters involve maximum iteration and number of population which are set on 200 and 100, respectively. The importance of this act is having compromise between accuracy and run time which is a necessity feature of the evolutionary algorithms. For the purpose of comparison, the simulations are carried out for two different cases categorised as cases I and II. Also, the obtained results are compared with those reported by other approaches. The rest of this study is divided into two categories, as follows:

The aim of the proposed approach is to find the welldistributed Pareto front that covers all Pareto trade-off front, in this regard a new method that considers all objective values in obtaining Pareto fronts is presented as follows. In the first iterations of the algorithm, all weight factors related to each objective are equal (e.g. in two-dimensional Pareto front both of them are equal to 0.5), after specific iteration numbers (usually three-tenth of the maximum iteration) one objective function is selected randomly and its weight factors are considered as one for specific iteration numbers (usually one-tenth of the rest of the iteration). This process is repeated until the number of the iteration reaches its maximum value. The above approach causes each objective function to obtain an opportunity to conduct the Pareto front trajectory towards its minimum value point. Step 7: Update the velocity, the position and the IPSO parameters (C1 , C2 and v). The position and the velocity of each individual are modified by (28) and (27), respectively. Step 8: If any element of each individual breaks its limit then the position of the individual is fixed to its maximum/ minimum operating point. Step 9: Compute the objective functions for the new generated individuals and normalised them. Step 10: Apply the Pareto method and add the non-dominated solutions to the repository. Step 11: Determine the non-dominated solutions between the repository members using the Pareto method. In each iteration, some non-dominated solutions are added to the repository and they might dominate solutions which had existed in the repository. Therefore it is necessary to apply the Pareto method again to determine the real non-dominated solutions. Step 12: If the current iteration number obtains the preordained maximum iteration number, the algorithm is stopped, otherwise go to step 6. Table 1

G2

Fuel cost coefficient a 100 b 200 c 10 Emission coefficient g 0.06490 b 20.05554 a 0.04091 j 0.0002 l 2.857

Pg1 , MW Pg2 , MW Pg3, MW Pg4 , MW Pg5 , MW Pg6 , MW cost, $/h

Case I: All objective functions are optimised individually. Case II: Objectives are optimised simultaneously. Case I: Initially, each objective function is considered individually in order to explore the extreme points of the trade-off curve and assess Fimin and Fimax (which are the best and worst result of the ith objective function while it is optimised as a single objective), these values are needed to normalise objective functions in multi-objective optimisation process. The obtained results from IPSO are compared with evolutionary programming (EP) [27], tabu search (TS) [28], improved evolutionary programming (IEP) [29], modified differential evolution optimal power flow (MDE-OPF) [30], stochastic genetic algorithm (SGA) [31],

Cost and emission coefficients for 30-bus IEEE test system G1

Table 2

Simulation and numerical results

G3

G4

G5

G6

120 150 10

40 180 30

60 100 10

40 180 20

100 150 10

0.05638 20.06047 0.02543 0.0005 3.333

0.04586 20.05094 0.04258 0.000001 8.000

0.03380 20.03550 0.05326 0.002 2.000

0.04586 20.05094 0.04258 0.000001 8.000

0.05151 20.05555 0.06131 0.00001 6.667

Best generation cost for different algorithms EP [27]

TS [28]

IEP [29]

MDE-OPF [30]

SGA [31]

EGA [32]

ACO [33]

FGA [34]

PSO

IPSO

173.848 49.998 21.386 22.63 12.928 12 802.62

176.04 48.76 21.56 22.05 12.44 12 802.29

176.2358 49.0093 21.5023 21.8115 12.3387 12.0129 802.465

176.009 48.801 21.334 22.262 12.46 12 802.376

175.974 48.884 21.51 22.24 12.251 12 803.699

176.2 48.75 21.44 21.95 12.42 12.02 802.06

181.945 47.001 20.553 21.146 10.433 12.173 802.578

175.137 50.353 21.451 21.176 12.667 12.11 802

178.4646 46.274 21.4596 21.446 13.207 12.0134 802.205

177.0431 49.209 21.5135 22.648 10.4146 12 801.978

The bold numbers show that the proposed IPSO algorithm resulted in obtaining better global solutions with respect to the other approaches in the area for single-objectives cost, transmission loss, VSI and emission IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

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www.ietdl.org Table 3

Table 4

Best transmission loss for different algorithms Initial [35] SPEA [35] GA [35]

Vg (1) Vg (2) Vg (5) Vg (8) Vg (11) Vg (13) T6 – 9 , pu T6 – 10 , pu T4 – 12 , pu T27 – 28 , pu power loss, MW

– 1.045 1.01 1.01 1.05 1.05 0.97 0.96 0.93 0.96 5.4356

– 1.044 1.023 1.022 1.042 1.043 1.09 0.90 1.02 0.96 5.1995

– 1.03 1.00 1.00 1.02 1.04 1.00 1.01 1.00 1.04 5.3513

PSO

IPSO

1.045 1.043 0.998 1.009 1.014 1.047 1.012 0.971 1.023 1.014 5.2105

1.047 1.044 0.976 1.035 0.984 1.042 1.029 0.98 1.01 0.97 5.0732

Initial [36] EGA-DQLF [36] PSO [37] FAPSO [37] IPSO Vg (1) Vg (2) Vg (5) Vg (8) Vg (11) Vg (13) T6 – 9 , pu T6 – 10 , pu T4 – 12 , pu T27 – 28 , pu VSI

Table 5

enhanced genetic algorithm (EGA) [32], ant colony optimisation (ACO) [33], fuzzy genetic algorithm (FGA) [34], and PSO for the cost objective function, SPEA [35], GA [35] and PSO for loss objective function, enhanced genetic algorithm-decoupled quadratic load flow (EGADQLF) [20], PSO [36] and fuzzy adaptive particle swarm optimisation (FAPSO) [36] for VSI objective function which are summarised in Tables 2 – 4. Unfortunately, there is no reference for the emission objective function to compare with our results, therefore this objective is optimised by GA, PSO and IPSO methods which are all implemented in MATLAB computational environment and the results are shown in Table 5. It is clear that the proposed method can achieve better results with respect to other algorithms and it is evident from Tables 2 – 5. These values are not negligible because of the continuous operations of power dispatch throughout the years as well as the numerous power plants in the entire world. Also the comparison of the best, worst and average solutions for the emission objective function is shown in Fig. 3 for the Table 6

1 1 1 1 1 1 1 1 1 1 0.23

1.0618 1.053 1.053 1.014 1.025 1.046 0.9125 0.9 0.9 0.925 0.10402

– – – – – – – – – – 0.1307

– – – – – – – – – – 0.1238

1.05 1.047 1.049 1.021 1.023 1.043 0.92 0.91 0.95 0.91 0.1037

Best emission for different algorithms

Pg1 , MW

Pg2 , MW

Pg3 , MW

Pg4 , MW

Pg5 , MW

Pg6 , MW

Emission, ton/h

IPSO 67.04 PSO 67.13 GA 69.73

68.14 68.94 67.84

50 49.86 49.73

35 34.89 34.42

30 29.67 29.15

40 39.94 39.29

0.2058 0.2063 0.20723

Fig. 3 Best, worst and average of emission objective function for 50 trials

Control variables related to multi-objective problem for PSO and IPSO algorithms Cost, $/h

Pg1 , MW Pg2 , MW Pg3 , MW Pg4 , MW Pg5 , MW Pg6 , MW Vg1 Vg2 Vg3 Vg4 Vg5 Vg6 T6 – 9 , pu T6 – 10 , pu T4 – 12 , pu T27 – 28 , pu Qc10 Qc24 Cost emission loss VSI

Best VSI for different algorithms

Emission, ton/h

Loss, MW

VSI

PSO

IPSO

PSO

IPSO

PSO

IPSO

PSO

IPSO

177.126 49.172 21.413 22.538 10.621 12 1.05 1.0442 1.446 1.0408 0.9601 1.05 1.01 0.99 1.01 1.02 25.62 23.51 802.105 0.3665 13.58 0.1197

177.0431 49.209 21.5135 22.648 10.4146 12 1.05 1.0462 1.0459 1.04165 0.95231 1.05 1.01 0.98 1.01 1.02 27.27 22.43 801.97 0.3513 13.39 0.1192

67.13 68.94 49.86 34.89 29.67 39.94 1.0287 1.0341 1.0387 1.0254 1.0317 1.0395 1.03 1.02 1.05 0.99 11.85 8.251 954.475 0.2063 5.385 0.1152

67.04 68.14 50 35 30 40 1.0351 1.0273 1.0417 1.0248 1.0321 1.041 1.03 1.01 1.06 0.99 12.53 8.741 954.248 0.2058 5.362 0.1131

77.18 60.36 49.55 34.33 30 39.84 1.05 1.0426 0.985 1.0372 0.973 1.0436 1.02 0.98 1.02 0.98 15.03 12.42 942.188 0.20834 5.1204 0.1307

76.94 60.14 49.532 34.743 30 39.74 1.047 1.044 0.976 1.035 0.984 1.042 1.02 0.98 1.01 0.97 15.81 11.63 941.672 0.20807 5.0732 0.1294

105.916 73.26 48.92 34.53 10 12.19 1.0493 1.0485 1.049 1.026 1.025 1.031 0.98 0.92 0.97 1.01 27.84 23.36 869.47 0.2471 14.158 0.1042

105.742 73.32 48.32 35 10 12 1.05 1.047 1.049 1.021 1.023 1.033 0.98 0.91 0.97 1.02 28.62 23.79 866.11393 0.247089 14.001 0.1037

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IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515 –527 doi: 10.1049/iet-gtd.2011.0851

www.ietdl.org proposed approach, PSO and GA algorithms. It can be noticed from this figure that the proposed algorithm can obtain better results with respect to other mentioned algorithms. The emission difference between the highest and lowest emission is only 0.08 ton/h, which confirms the effectiveness of the proposed method. From Table 6, it is clear that the control settings corresponding to the cost-based OPF result in a reduction in fuel cost but cause the system loss to increase by 146% of its base case value and VSI to decrease by 48% of its base case value. From the above mentioned statement, it is clear that the cost and VSI are in line with each other, but cost and loss are not in line with each other. Minimisation of emission causes the system loss and VSI to decrease by 1.25 and 50.8% of their base case values, respectively. The

minimisation of the loss objective results in the reduction of 6.5% in loss and 43% reduction in the VSI. Moreover, the minimisation of the VSI objective results in a 55% reduction in the VSI over its base case. Therefore it is clear that considering a specific objective function in the OPF problem cannot obtain a solution which satisfies other objective functions, and for achieving a result that satisfies some objectives, it is necessary to solve the multi-objective OPF problem. Case II: In this case, the proposed problem is handled as a multi-objective optimisation problem, whereas each two and three objective functions are optimised simultaneously. All obtained Pareto fronts with the IPSO algorithm are shown in Figs. 4 and 5 for two-dimensional Pareto fronts and Fig. 6 for three-dimensional Pareto fronts.

Fig. 4 Two-dimensional Pareto-optimal fronts

Fig. 5 Pareto-optimal front for

a Cost loss b Emission–cost c Emission–loss

a Cost voltage stability index b Emission–voltage stability index c Voltage stability index loss

IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

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Fig. 6 Three-dimensional Pareto fronts for different objective functions a b c d

Voltage stability index – emission loss Voltage stability index – emission–cost Loss–emission–cost Voltage stability index – cost– loss

It is worthwhile to say that each Pareto-optimal solution is an optional choice for a system operator. As a matter of fact, after acquiring the Pareto-optimal solutions, the decision maker needs to choose one best compromised solution according to the specific preference for different applications. In this regard, (41) is computed for all non-dominated solutions, after that they are sorted according to their Nm value, the best solution among them is considered as a best compromised solution. It is noted that in (40), the importance of each objective function should be determined (in this study b1 , b2 , b3 and b4 are importance of cost, emission, loss and VSI

Nobj objective functions, respectively, also i=1 bi = 1). The results of selecting the best compromise over Pareto-optimal solutions are proposed in Table 7. By comparing the results, the following observations can be inferred: in case I, when each objective is minimised individually, its value is the best one over cases II–IV. Also, it can be seen that when each objective reaches its minimum value, the other objectives’ value will increase with respect to their minimum (when they are minimised as single objective process). Also in cases II– IV the values of the objective functions are shown for different weight factors. In other words, the importance of the objective functions is considered by weight factors. For instance, the cost function and emission are 860.421 $/h and 0.2383 ton/h, respectively, when b2 ¼ 0.8 and b1 ¼ 0.2 in case II, in contrast when b1 ¼ 0.8 and b2 ¼ 0.2, the cost and emission are 823.134 $/h and 0.2751 ton/h, respectively. It is obvious that by raising the weight factor the significance of objective function is raised and consequently its value is diminished. 524 & The Institution of Engineering and Technology 2012

Case II-A: Two-dimensional optimisation: All twodimensional Pareto fronts for different objective functions are shown in Figs. 4 and 5. It is clear that the proposed IPSO algorithm is able to obtain well-distributed Pareto-optimal fronts. The best compromise solution which is found by the fuzzy decision method according to (41) is shown in each figure. In the proposed approach the trade-off among the competing objectives is obtained by emphasising on non-dominated solutions and getting a well-distributed set of solutions, respectively. These result in finding a set of well-distributed Pareto-optimal solutions. Fig. 5a shows the Pareto-optimal front for fuel cost and VSI objective functions. The best compromise solution is 822.72 $/h and 0.10916. The fuel cost and VSI values in the best compromise solution are very close to their optimised values, whereas these objectives are optimised individually, the truth of the aforementioned statement is clear in all Pareto fronts. All the above features are provided by applying the mutation and self-adaptive method for tuning the learning factors, because these methods increase the search ability of the PSO by adapting its learning factors according to its convergence trajectory and increasing its population diversity. Being the minimum values of both objectives in each two-dimensional Pareto front has approved the statement that the proposed method is able to cover the entire trade-off front. Pareto fronts also almost show that there is no gap in their shapes except some gaps which may appear in some regions because of the nature of the proposed multi-objective OPF problem and objectives. IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515 –527 doi: 10.1049/iet-gtd.2011.0851

www.ietdl.org Table 7

Multi-objective obtained results for different weight factors

Cases

b1

b2

b3

b4

Cost, $/h

Emission, ton/h

Loss, MW

VSI

case I

– – – – 0.8 0.5 0.2 0 0 0 0 0 0 0.8 0.5 0.2 0.8 0.5 0.2 0 0 0 0.3 0.3 0 0 0.3 0.4 0.5 0.2 0.1 0.5 0.4 0.7

– – – – 0.2 0.5 0.8 0 0 0 0.8 0.5 0.2 0 0 0 0 0 0 0.8 0.5 0.2 0.6 0.1 0.2 0.7 0 0 0.3 0.3 0.3 0.2 0.2 0.1

– – – –

– – – – 0 0 0 0.2 0.5 0.8 0 0 0 0.2 0.5 0.8 0 0 0 0.2 0.5 0.8 0 0 0.1 0.1 0.4 0.3 0.2 0.5 0.4 0.1 0.2 0.1

801.978 – – – 823.134 841.052 860.421 – – – – – – 815.109 822.638 832.374 839.843 850.916 869.731 – – – 867.854 855.281 – – 889. 631 872.481 828.632 923.452 910.542 883.514 897.371 869.518

– 0.2058 – – 0.2751 0.2585 0.2383 – – – 0.2061 0.2063 0.2066 – – – – – – 0.2146 0.2226 0.2286 0.2489 0.3169 0.2496 0.2136 – – 0.3613 0.27084 0.2441 0.3659 0.3341 0. 3715

– – 5.0732 – – – – 6.821 7.219 7.913 5.213 5.179 5.162 – – – 8.976 7.893 6.775 – – – 8.793 6.184 7.818 9.632 11.357 11.152 – – 10.4521 13.892 12.764 15.518

– – – 0.1037 – – – 0.1121 0.1087 0.1073 – – – 0.1105 0.1092 0.1085 – – – 0.1073 0.10586 0.1051 – – 0.1537 0.1243 0.1137 0.1185 0.1286 0.1136 0.1124 0.1238 0.1213 0.1208

case II

case III

case IV

0 0 0 0.8 0.5 0.2 0.2 0.5 0.8 0 0 0 0.2 0.5 0.08 0 0 0 0.1 0.6 0.7 0.2 0.3 0.3 0 0 0.2 0.2 0.2 0.1

Such important optimal solutions could not have been discovered without the proposed multi-objective Pareto optimisation method. Unlike the single-objective optimisation, there are two targets in a multi-objective optimisation: (i) convergence to the Pareto-optimal set and (ii) maintaining the diversity in Pareto-optimal solution. These two goals are satisfied by the proposed method. The Chaotic formula for the inertia weight and the self-adaptive method for computing the learning factors beside the proposed mutation in the presented IPSO algorithm appear to work well for multi-objective problems. Case II-B: three-dimensional Pareto fronts: In this case, three competing objectives are optimised simultaneously by the proposed algorithm and the obtained results are shown in Fig. 6. It is clear that cost, emission, loss and VSI cannot be further improved without degrading the other two related optimised objectives. Fig. 6 clearly shows the relationships among all presented objective functions. Between the obtained Pareto-optimal solutions, it is necessary to choose one of them as a best compromise for implementation which has been done by the fuzzy decision maker. It is necessary to note that like the two-dimensional Pareto fronts, the best compromise is shown in red colour in this case. It can once again be proved that the proposed method is giving well-distributed Pareto-optimal front. The results confirm that the multi-objective IPSO algorithm is an impressive tool for solving the multi-objective optimisation IET Gener. Transm. Distrib., 2012, Vol. 6, Iss. 6, pp. 515– 527 doi: 10.1049/iet-gtd.2011.0851

problem where multiple Pareto-optimal solutions can be obtained in a single run. Case II-C: authenticate the obtained Pareto-optimal solutions: Since one of the significant targets of the proposed approach is solving the multi-objective problem, so it is very important to use some criterions to authenticate the obtained Pareto-optimal solutions. Moreover this study compares the obtained Pareto solutions with those which are obtained by PSO and NSGA II [37] optimisation algorithms. In this regard, this study utilises GD, SP and diversity metric criterions which are described as follows: Generational distance: GD criterion is proposed by Van Veldhuizen and Lamont [38] to estimate the value of being far of the elements in the set of non-dominated vectors found so far from those in the Pareto-optimal set. This criterion is explained as follows 

m 2 i=1 di GD = m

(48)

where di is the Euclidean distance (measured in objective space) between each of these non-dominated solution vectors and the nearest member of the Pareto-optimal set. It is noticeable to note that a value of GD ¼ 0 indicates that all the elements generated are in the Pareto-optimal set and consequently whatever this criterion be closed to zero it 525

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www.ietdl.org shows that all the generated elements are close to the Paretooptimal set. Therefore if this criterion is less, it is more acceptable and the related Pareto-optimal solutions have better condition respect to those which have greater GD. Spacing: Another important factor of the obtained Pareto solutions is the distribution of these solutions in the Pareto fronts. In this regard, a criterion in which measures the range (distance) variance of neighbouring vectors in the nondominated solutions which are found so far is proposed in [18]. Since the ‘beginning’ and the ‘end’ of the current Pareto front found are known, a suitably delineated metric judges how the solutions in such front are scattered. This criterion is defined as follows  m 1  SP W (d − di )2 m − 1 i=1

and shows the space among Pareto-optimal solutions in a Pareto front are divided equally. Diversity metric: This metric does not require any optimal Pareto front and bases on the Hamming and Euclidean distance between solutions. If there is m number of points on a Pareto front and the space is N-dimensional then centroid Ci for ith dimension is computed as follows m

Ci =

j=1

xij

m

,

for i = 1, . . . , Nobj

(50)

where xij indicates the ith dimension of the jth point. So the diversity measuring D-metric is calculated as follows

(49)

D- metric =

Nobj m  

(xij − Ci )2

(51)

i=1 j=1

where di = minj (|F1i (X ) − F1j (X )| + |F2i (X )− F2j (X )| + · · · + |FNi obj (X ) − FNj obj (X )|), i, j ¼ 1, . . ., m, d is the mean of all di . A value of zero for this metric indicates all members of the Pareto front currently available are equidistantly spaced. Therefore the less value of this parameter is more desirable Table 8

Although this metric is greater, it shows that all the elements generated are close to each other. In other words, the distances between Pareto-optimal solutions in a Pareto front are less. Therefore the higher value of this parameter shows the higher diversity of the obtained Pareto-optimal solutions.

Best, worst and average values of GD, SP and D-metric for different optimisation algorithms in two-dimensional Pareto fronts

Algorithms

IPSO

Objectives loss –emission

cost –emission

emission –VSI

loss –VSI

cost –loss

cost –VSI

SP GD D-metric SP GD D-metric SP GD D-metric SP GD D-metric SP GD D-metric SP GD D-metric

PSO

NSGA II

Best

Average

Worst

Best

Average

Worst

Best

Average

Worst

0.001360 0.000153 0.812698 1.697219 0.338549 132919.7 0.000369 0.000070 0.017317 0.082554 0.018233 469.2634 0.897170 0.107436 151051.83 1.270777 0.335409 13545.013

0.001732 0.000948 0.739690 2.106907 0.397213 119893.2 0.000486 0.000079 0.015732 0.116354 0.019843 441.7878 1.093265 0.118743 142753.38 1.510842 0.345183 12494.282

0.002573 0.001984 0.631683 3.084262 0.484126 98674.3 0.000674 0.000105 0.012329 0.174527 0.022153 386.9124 1.582948 0.139172 130387.94 1.973527 0.369821 10650.715

0.004146 0.000775 0.275222 2.703175 0.438114 89796.0 0.000410 0.000078 0.013486 0.080661 0.020785 415.7619 1.276503 0.149522 125264.40 1.544961 0.399661 10379.771

0.009988 0.000943 0.222158 4.715341 0.541632 70025.8 0.000925 0.000142 0.010163 0.203413 0.024932 308.4173 2.274323 0.174715 99093.67 2.978063 0.493415 7752.827

0.015405 0.001075 0.144417 6.578263 0.627423 52662.7 0.001352 0.000183 0.007736 0.309906 0.028515 246.7909 3.040720 0.192352 76845.95 4.132463 0.579372 5629.884

0.003976 0.000666 0.312606 1.722061 0.438114 93183.4 0.000397 0.000072 0.013908 0.084822 0.019075 419.9626 0.987462 0.117923 137017.93 1.300492 0.364580 10459.242

0.006742 0.000741 0.283617 2.874325 0.479327 81957.3 0.000726 0.000089 0.009734 0.146212 0.021075 310.9859 1.361819 0.129876 113214.72 1.963890 0.409365 8642.474

0.011860 0.000857 0.241940 4.582061 0.575769 57932.1 0.001233 0.000123 0.008954 0.267528 0.024539 249.5638 2.456410 0.155830 81402.35 3.477516 0.490397 6004.651

Table 9 Best, worst and average values of GD, SP and D-metric for different optimisation algorithms in three-dimensional Pareto fronts Algorithms

IPSO

Objectives loss –emission –VSI

cost –emission –VSI

cost –emission –loss

cost –loss –VSI

SP GD D-metric SP GD D-metric SP GD D-metric SP GD D-metric

PSO

NSGA II

Best

Average

Worst

Best

Average

Worst

Best

Average

Worst

0.0186168 0.0441499 410.006 0.30100 0.04164 348652.8 1.252664 0.179002 160650.4 0.63177 0.03526 391962.4

0.02136 0.04682 386.453 0.37511 0.04348 317773.4 1.41075 0.19075 148863.7 0.74220 0.04792 372188.8

0.02979 0.05176 328.526 0.52137 0.05306 262884.2 1.91458 0.21164 129221.1 0.96737 0.07493 318414.6

0.05112 0.15202 270.733 1.05239 0.16825 210473.5 4.21748 0.51074 92543.8 1.53205 0.09275 273184.5

0.11131 0.20127 202.482 2.31160 0.21631 149029.5 6.95823 0.67310 68442.8 3.37516 0.10844 201629.5

0.15950 0.23329 173.301 3.31081 0.24849 119570.0 8.63428 0.75043 54541.6 4.86042 0.11428 150060.2

0.03463 0.09625 348.752 0.86433 0.11749 276482.8 2.34184 0.32846 127381.5 0.97465 0.05282 321846.9

0.04842 0.10853 291.013 1.29525 0.13174 242188.5 2.95173 0.37361 102321.6 1.54174 0.05208 281241.0

0.06421 0.12632 217.273 1.88078 0.15094 185879.4 4.13837 0.43633 76268.4 2.28166 0.06733 232051.6

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www.ietdl.org Tables 8 and 9 depict the obtained best, worst and average of GD, SP and D-metric for IPSO, PSO and NSGA II in 25 runs. From these tables, it is clear that the proposed algorithm can obtain better Pareto front with respect to other algorithms, because the values of SP and GD related to solutions of IPSO algorithm are lower than those obtained by PSO and NSGA II. Also the value of D-metric related to proposed algorithm is greater than those obtained by other algorithms. These tables reveal the superiority of the proposed algorithm in solving the multi-objective optimisation problem once again.

7

Conclusion

The goal of the proposed multi-objective OPF problem is to compute advised set points for power system controls that satisfy the security, the environment and the economical conditions simultaneously. The most important privilege of the proposed approach for the multi-objective formulation is obtaining several non-dominated solutions allowing the system operator to use his/her personal preference in selecting one solution for implementation. Furthermore the proposed fuzzy decision method helps the power system operator to apply his/her decisions very easily. Also in single objective cases, the proposed approach can obtain better results with respect to other algorithm in the literature. In multi-objective cases, the proposed method proves its ability to obtain well-distributed Pareto fronts. Since the proposed algorithm profits the chaos formula, the self-adaptive method and mutation, its search ability is increased, therefore this method is suggested for solving the complicated optimisation problems.

8

13 14 15 16 17

18 19 20 21 22 23 24 25

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