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Multi-Objective Stochastic Distribution Feeder Reconfiguration in Systems With Wind Power Generators and Fuel Cells Using the Point Estimate Method Ahmad Reza Malekpour, Graduate Student Member, IEEE, Taher Niknam, Anil Pahwa, Fellow, IEEE, and Abdollah Kavousi Fard

Abstract—This paper presents a multi-objective algorithm to solve stochastic distribution feeder reconfiguration (SDFR) problem for systems with distributed wind power generation (WPG) and fuel cells (FC). The four objective functions investigated are 1) the total electrical energy losses, 2) the cost of electrical energy generated, 3) the total emissions produced, and 4) the bus voltage deviation. A probabilistic power flow based on the point estimate method (PEM) is employed to include uncertainty in the WPG output and load demand, concurrently. Different wind penetration strategies are examined to capture all economical, operational and environmental aspects of the problem. An interactive fuzzy satisfying optimization algorithm based on adaptive particle swarm optimization (APSO) is employed to determine the optimal plan under different conditions. The proposed method is applied to Taiwan Power system and the results are validated in terms of efficiency and accuracy.

Inertia weight of PSO. Mean of th solution. Standard deviation of the th solution. Lower limit of th objective function. Upper limit of the th objective function. th objective function. ,

Random numbers between 0 and 1. Number of swarms in each iteration. ,

Index Terms—Distribution system reconfiguration, fuel cells, fuzzy set theory, particle swarm optimization, point estimate methods, wind power generation.

Weighting acceleration factors. Velocity of the th particle in th iteration. Best previous experience of th particle. Best particle among the entire population. th standard location of Initial value of

NOMENCLATURE ,

.

Chaotic parameter at iteration .

Mean value of the random variable ,

.

Control parameter.

.

Estimated locations of random variable

.

th individual in the population. th element in th individual at th iteration.

,

,

Weighting factors for estimated locations of random variable . Total number of input variables. th moment of th output random variable. Standard deviation of Skewness coefficient of

. .

Number of WPGs. th membership function. Random function generator. Expected value of the current for the th branch. Number of main loops. Number of branches.

Manuscript received March 26, 2012; revised July 13, 2012; accepted August 26, 2012. Date of publication October 23, 2012; date of current version April 18, 2013. Paper no. TPWRS-00227-2012. A. R. Malekpour and A. Pahwa are with the Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS 66503 USA (e-mail: [email protected]; [email protected]). T. Niknam and A. Kavousi Fard are with the Department of Electronics and Electrical Engineering, Shiraz University of Technology (SUTech), Shiraz 11456-7856, Iran (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2012.2218261 0885-8950/$31.00 © 2012 IEEE

Resistance of the th branch. Current amplitude of the th transformer. Expected value of emission produced by WPGs. Current amplitude of the th feeder. Maximum current of the th feeder. Cost of power supplied by the grid. Generated cost of the th WPG.

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Nitrogen oxide pollutants of FC.

Mathematical round operator.

Expected value of power supplied by the grid.

Random numbers in

Expected value of power generated by the th WPG.

Mathematical mean operator.

.

Mean value of the th element in the population.

Expected value of power generated by the th FC.

PSO population.

Amplitude of the voltage at the th bus.

Probability of th modification method.

Angle of the voltage at the th bus.

Accumulator of the th modification method.

Amplitude of the branch admittance through buses , .

Vector of the mean value of the population.

Angle of branch through buses , .

I. INTRODUCTION

Minimum active power of the th FC. Expected value of active power flow in line Maximum active power of the th FC. Maximum reactive power of the th FC. th probabilistic output variable. Number of self-adapting parameters. Probability density function. Total number of decision variables. Position of the th variable in iteration . Number of FCs. Number of buses. Emission factor for FCs. Emission factor for grid. Expected value of emission produced by FCs. Maximum current of the th transformer. Expected value of emission produced by grid. Generated cost of the th FC. Sulfur oxide pollutants of FC. Carbon dioxide pollutants of FC. Nitrogen oxide pollutants of grid. Sulfur oxide pollutants of grid. Carbon dioxide pollutants of grid. Net injected active power at the th bus. Net injected reactive power at the th bus. Maximum active power flow in line

.

Minimum reactive power of the th FC. Expected value of voltage magnitude of the th bus. Expected value of the maximum voltage. Expected value of the minimum voltage. Probabilistic solution set of output variables. Set of deterministic power flow equations. Input set of uncertain variables. Uncertain input random variable.

.

W

ITHIN the last decade, there has been significant interest in distributed energy sources from both utilities and end users. Wind and solar are popular types of renewable generation resources being considered for connection to distribution networks [1]. Governmental policies for promotion of clean technologies have also encouraged utilization of low emission fuel cell power generation [2]. However, intermittent power generation associated with wind and solar generation poses new challenges for operation of distribution systems [3]. Although distribution system reconfiguration is a well-researched topic [4]–[15], integration of distributed energy resources as well as randomness associated with renewable generation has made it one of the most significant challenges for distribution system operators. Traditionally, the reconfiguration problem has been studied based on deterministic approaches. In [4], a method based on artificial neural network (ANN) was proposed to determine configuration of the network with different load levels. The ANN determined the appropriate system topology such that the total power loss reduced under different load patterns. In [5], Gomes et al. have presented a method based on heuristic search to find the best structure of the network. The algorithm started with the system in a meshed status with all maneuverable switches in closed position, and tried to find the topology with the least power losses. Similar to [4], the problem is solved for the single objective of loss minimization. Kashem et al. [6] have suggested a method based on distance measurement to locate the loops in the system and then improve the load balancing in the corresponding loop. The objective function was load balancing in feeder branches. In [7], Das has proposed a new approach based on fuzzy theory to solve the multi-objective reconfiguration problem including minimization of real power loss, deviation of nodes voltage, and branch current constraint. Baran and Wu [8] have suggested a network reconfiguration method based on mixed integer programming (MIP) to solve the load balancing problem. In [9], Zhou et al. have used an algorithm based on heuristic rules and fuzzy logic approach to support load balancing and service restoration simultaneously. Ahuja et al. [10] have proposed a hybrid algorithm based on artificial immune systems and ant colony optimization for multi-objective distribution system reconfiguration. Tsai et al. [11] proposed an evolutionary programming and used the grey correlation analysis to solve the multi-objective reconfiguration problem. In all

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

these papers, distributed generators (DGs) and their impact on reconfiguration problem has not been considered. In [12], Khodr et al. presented a methodology for reconfiguration problem including DGs using Benders decomposition approach. The authors have considered real loss as well as load balancing of feeder branches as the objective functions. However, some important objective functions such as the cost of energy were neglected. Wu et al. [13] presented an ant colony algorithm to solve the multi-objective reconfiguration problem with DGs to achieve the optimal power loss and load balance of radial networks. In [14], Niknam has proposed a hybrid method to solve the multi-objective reconfiguration problem considering real power loss, voltage deviation, number of switching, and load balancing as the objectives. Since this method only can find a single optimal solution, the several satisfying solutions which could be considered as good preferences were neglected. Despite some deterministic approaches which have been proposed to solve the multi-objective reconfiguration problem in the presence of DGs, there is still no technical literature available to tackle the problem under uncertainties such as those created by intermittent nature of renewable DGs. Monte Carlo simulation (MCS) is an option for analysis including uncertainties, but it can be computationally intensive. This difficulty is more evident in problems in which the main optimization algorithm is solved based on evolutionary methods. This paper aims to study the multi-objective reconfiguration problem with uncertainties in both load and wind power generation using the point estimate method (PEM) [15]. Unlike the MCS method, the PEM is generally simpler and more flexible to deal with complex models. Solar generation is not included, but it can be considered using an approach similar to that used for WPGs. In order to achieve realistic system performance, probabilistic approaches [16] are employed to model the random variation of wind speed and load demand concurrently. The uncertainties involved in the multi-objective reconfiguration problem are handled through an effective probabilistic power flow based on the PEM [17]. Different wind penetration strategies are considered to solve the multi-objective reconfiguration problem. The discrete nature of the tie and sectionalizing switches make the multi-objective stochastic reconfiguration problem a complex nonlinear optimization problem with discrete variables. Therefore, a new adaptive particle swarm optimization (APSO) coupled with an interactive fuzzy satisfying method is proposed to solve the problem. The reminder of this paper is organized as follows. In Section II, the mathematical formulation of multi-objective SDFR is presented. In Section III, the PEM and its background are explained. The introduction of APSO and its application in SDFR problem is described in Section IV. In Section V, the proposed method is applied on a test system and the results are discussed. Finally, in Section VI, the relevant conclusions are described.

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is solved for the peak hour. The related objective functions are expressed as follows: 1) Energy losses: (1) 2) Cost of electricity generation:

(2) 3) Emission produced:

(3) 4) Voltage deviation: (4) subject to a) Distribution power flow equations:

(5) b) Distribution line limits: (6) c) Radial structure of the network: (7) d) Limit on the currents of the transformers: (8) e) Limit on the current of the feeders: (9) f) Active power constraints of FCs: (10) g) Reactive power constraints of FCs:

II. PROBLEM FORMULATION The stochastic distribution feeder reconfiguration (SDFR) problem includes minimization of four objective functions and

(11) The decision variables in the SDFR are the status of normal open/close (tie/sectionalizing) switches and the active/reactive

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powers of FCs. In the above formulas, the superscript the expectation of the random variables.

denotes

III. PROBABILISTIC POWER FLOW A. Background algorithms are required to calIn the original PEM [15], culate the statistical moments of a random quantity which is a function of random variables. In 1998, Hong [18] attempted to improve the original PEM method and introduced the and schemes by reducing the number of computations from to and ( is a parameter determining the type of Hong’s PEM scheme). This property makes PEM an efficient tool to handle engineering problems with numerous random variables. Su [19] was the first to approach the power flow problem as a probabilistic problem using Hong’s scheme. Later, Morales et al. [17] and Caramia et al. [20] used , and PEM schemes to solve the probabilistic power flow problem. They concluded that the scheme provides the best solution in terms of accuracy and computational efforts and is the most efficient scheme in dealing with Gaussian distribution. B.

Fig. 1. Concept of the

where

PEM scheme.

is the standard location expressed as

PEM Scheme

(15)

Mathematically, the deterministic power flow can be expressed as (12) where is the set of input variables representing the network configuration, load and distributed power generation; is the set of output variables (bus voltage or line flows) which are calculated through the power flow function . The idea behind the PEM is to calculate the moments of the output variables of interest through solution of only a few deterministic power flows. In this regard, only few statistical moments of input random variables are required. Considering input random variables in the system, (12) can be written as (13) According to scheme, (13) has to be calculated times. Let be a random variable with probability density function . As , the scheme uses the first four central moments of each random variable to approximate its distribution by three points. Fig. 1 depicts the concept of PEM scheme. The variables , and are transformed to the variables , , and through the functional relationship of . Also, the three weighting factors , , and are utilized to scale the three estimates of variants ( , , and ). Each point or concentration consists of three pairs , , 2, 3 where and are the location and the weighting factor, respectively. For each random variable , the three locations are calculated using mean and variance of : (14)

The parameters and are the third and the fourth central moments of which are defined as coefficients of skewness and kurtosis as follows: (16) where is the expectation operator. The weighting factor associated with the location can be computed as

(17)

Using (13)–(17), two pairs of locations and weights are calculated for each point. The power flow is calculated for two locations assuming that other variables are set at their mean value:

(18) has zero standard location The third location and thus according to (14) . Hence, it is required to run power flow using the input variables set at their mean value: (19)

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

Once solutions of the power flows are obtained using (18) and (19), the th order raw moment of can be formulated as

of objective functions. The reference value is a real number in the range of which represents the importance of each obgives more importance jective function. A higher value of to the th objective function such that the algorithm would be encouraged to explore solutions with smaller value for the specific objective function. A unique optimal solution based on the preference of the decision maker is obtained by solving the following min-max problem:

(20) According to (19), point

number of locations have the same . Thus, (20) can be simplified as

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(23) is the th memwhere is the set of non-inferior solutions, is the th reference membership value. bership function and The reference value for each objective function is estimated from the decision maker’s experience or by trial and error. C. Adaptive Particle Swarm Optimization (APSO)

(21) initial runs and only one further run of power Therefore, flow are required. Hence, the number of times to calculate power flow in a system with input random variables is reduced to . Upon calculation of the mean and the standard deviation of output random variables through (21), the cumulative probability density function can be approximated by Gram-Charlier series approach [21]. IV. SOLUTION METHODOLOGY Firstly, the objective functions of the SDFR problem are represented by fuzzy models. Next, the interactive fuzzy satisfying method is used to convert the objective functions into a min-max problem. Finally, the proposed APSO method is implemented to solve the multi-objective SDFR problem. A. Fuzzy Models of Objective Functions Fuzzy models are able to handle the multi-objective SDFR problem with non-commensurable objective functions. In order to optimize the objective functions simultaneously, each objective function is modeled by a linear membership function [22]: (22)

, where separately.

are extracted by optimizing each objective

B. Interactive Fuzzy Satisfying Method In this section an interactive fuzzy satisfying method [23], [24] is presented to obtain the satisfying solution from the set of non-inferior solutions. In the proposed algorithm, the membership functions are defined separately for each objective. Later, the decision maker is asked to specify the desired preference level for each of the membership functions, called the reference membership values , , where is the number

1) Standard PSO: Particle swarm optimization (PSO) is one of the evolutionary optimization algorithms developed by Eberhart and Kennedy [25]. This algorithm was inspired by the movement of a flock of birds searching for food. In the PSO algorithm, position and velocity of each particle are updated according to the following formula:

(24) (25) The inertia weight factor is used to control the impact of the previous history of velocities on the current velocity. A large leads to a global search whereas a small tends to facilitate local search. This factor usually decreases linearly from 0.9 to 0.4 [26]. Also, the acceleration factors and , named cognitive and social parameters, determine influence of the personal best and the global best in determining new solutions. and represent functions that generate independent random numbers which are uniformly distributed between 0 and 1; is the number of swarms in each iteration. 2) Adaptive PSO: Despite numerous advantages of PSO such as simple concept, easy implementation, minimal storage requirement, and straightforward incorporation with other optimization tools, it may experience inappropriate convergence and fall in local minima. Therefore, in this paper new self-adaptive modifications are presented to overcome the above-mentioned deficiency of PSO algorithm while improving its ability for global exploration. The proposed self adaptive modifications employ three different approaches with specific probability. Each particle will choose the approach with the highest probability during the optimization process. Modification Approach 1: This approach updates the inertia weight and accelerating factors self adaptively. The inertia weight factor is dynamically tuned in each iteration as follows: (26) The value of is determined by an iterative chaotic method called the logistic map [27] is (27)

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The control parameter, , is selected randomly within 0 to 4 range. Variation of determines whether becomes stable at a constant size, oscillates between limited sequences of sizes, or behaves chaotically in an unpredictable pattern. Since (27) is an iterative equation, its behavior is greatly sensitive to the initial value of . Equation (27) without any stochastic pattern in the value of indicates chaotic dynamic when and [26]. The acceleration coefficients and are incorporated into the optimization problem by augmenting the position vector, the velocity vector, and the best position vector. In general, if is the dimension of the problem is the number of self adapting parameters, the new and vectors for the particle will be (28)

(35) (36) , , will reThe best individual among the in the population. place The initial probability of each modification approach is 0.33 ( and , 2, 3). Also, an accumulator named is defined for each modification approach. In each iteration, the particles are sorted according to their objective function such that and indicate the best and the worst individual in the sorted population. Afterwards, each particle would gain a weighting factor according to its position in the population as follows:

(29)

(37)

(30) Obviously, the first variables correspond to the particle in varithe search space (decision variables), while the last ables account for the personal acceleration constants (cognitive and social parameters). These new variables do not enter the fitness function but are manipulated using the same mixed individual-social learning paradigm as the one used in PSO. Modification Approach 2: This approach is utilized to encourage movement of the swarm population toward the best individual found to date. Using this modification, convergence velocity of the algorithm is enhanced effectively. In this regard, the mean value of the population is calculated column-wise as follows:

(31) Now the th particle will move toward the best particle as follows:

(32) will be accepted if it is better than . Modification Approach 3: This approach is utilized to increase diversity of the swarm population and reduce the possibility of non-mature convergence. For each particle three are selected from the population randomly particles , , such that . Next, a new test individual is generated as follows:

(33) Later, two mutant particles are achieved using the following scheme: (34)

The accumulator of each method is updated as follows:

(38) shows the number of solutions which have where selected the th modification approach. Then, the probability of each modification approach is updated as follows: (39) Each particle would select the th modification approach by the roulette mechanism based on its probability . D. Implementation of APSO in SDFR Problem To apply the proposed APSO algorithm to solve the multiobjective SDFR problem, the following steps are performed: Step1: Read the input data. Step2: Generate an initial population and velocity . Step3: Evaluate the objective function for each individual using probabilistic power flow based on PEM scheme. Step4: Select and . Step5: Update the APSO parameters as described in IV. Step6: Update the velocity and position using (24) and (25). Step7: Check the constraints and modify the position. Step8: Update and . Step9: Check the termination criterion. If the iteration number reaches the predetermined maximum value, the search is stopped, else go to step 5. V. SIMULATION RESULTS The case study is the 86 buses Taiwan Power Company (TPC) system [28], which consists of 2 substations, 11 feeders, and 96 switches (83 sectionalizing switches and 13 tie switches). The

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TABLE I CHARACTERISTIC OF INSTALLED FCS

TABLE II COMPARISON OF EXPECTED VALUES FOR AVERAGE, BEST, AND WORST SOLUTION USING THE GA, PSO, AND APSO METHODS

Fig. 2. Single line diagram of the test system.

studies have been implemented in MATLAB 7.4 using a Pentium P4, Core 2 Duo 2.4-GHz personal computer with 1 GB of RAM. The simulation results will be described in the following three sections: A. Assumptions Table I shows the location and capacities of FCs in the TPC. The WPGs connected to nodes #7, #72, #80 are considered to have a fixed power factor of 0.9 lagging. The random output power of each WPG is modeled as a five-step discrete distribution function corresponding to a Weibull distribution with mean speed of 7 meters and the shape parameter of 2 [29]. Since the three WPGs are considered to be located in different locations, spatial correlations between them are neglected. However, such correlations can be included using the approach provided in [30], which will be considered in the future research. The mean and the standard deviation of power production for the WPGs are computed based on the corresponding discrete distribution through a standard statistical formula which gives the final mean power of 500 kW for the WPGs. The energy cost is considered to be 0.07 ($/kWh) for FCs, 0.044 ($/kWh) for the grid [31] and zero for the WPGs. The loads on buses #10, #11, #15, #19, #28, #42, #46, #52, #59, #84 have the normal distribution function with a constant standard deviation of 5%. The other loads are assumed to be deterministic (zero standard deviation). The emission factor of FCs and grid are 1078.036 and 2043.96 (lb/MWh) [32], respectively. B. Comparison of APSO With Other Methods In order to compare performance of the proposed APSO algorithm with other well-known algorithms, the SDFR problem is also solved using GA and PSO, and the results are shown in Table II. For all the methods, the initial population equals four times the total number of decision variables and the best solution

is determined by running the algorithm for 200 iterations. The mutation probability and crossover rate for GA are set to 0.09 and 0.3, respectively. The cognitive parameter, social parameter, initial and final inertia weight factors for PSO and APSO are set to 2, 2, 0.9 and 0.4, respectively. The problem is solved for 25 trials with each method and the mean values of the corresponding best solution, the worst solution, the average of the mean values, the standard deviation of the mean values, and the CPU time obtained are shown in Table II. It reveals that APSO provides better results in comparison to GA and PSO methods in terms of robustness and CPU Time. C. Application of APSO to TPS Since the APSO gave the best performance, it is used to study the test system under different conditions including changing outputs of the WPGs and changing the reference membership of the objective functions to implement decision maker’s preferences. In the first two cases the reference membership for the objectives are considered to be unity. It is assumed that each objective function follows the Gaussian distribution and the values obtained from scheme are used to generate the cumulative probability distribution of the objective functions. 1) Effect of Reconfiguration: Fig. 3 depicts the cumulative probability distributions of the objective functions before and after reconfiguration considering 500 kW average output for the WPGs and reference membership value of unity for all the objectives. The proposed method resulted in statistically lower total electric loss, total emission, operating costs and voltage deviation with mean values of 0.448447 MW, $1254.56, 55 816.16 lb., and 0.5295 p.u., respectively. Not only the mean value, but also the standard deviation of the objective functions reduced due to reconfiguration. Table III shows expected values of the optimal dispatch of the FCs and switch positions before and after reconfiguration. 2) Effect of Wind Generator Capacity: Two different strategies with average output of 250 and 750 kW for the WPGs are considered to investigate their effect on the optimal results. Fig. 4 shows the objective functions’ values as the WPGs output is changed. The strategy with the total average output of 2250

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Fig. 3. Cumulative probability distributions of the four objective functions before and after reconfiguration.

Fig. 4. Cumulative probability distributions of the four objective functions for different wind power penetration strategies.

TABLE III EXPECTED VALUES FOR OPTIMAL DISPATCH OF THE FCS AND SWITCH POSITIONS BEFORE AND AFTER RECONFIGURATION

TABLE IV EXPECTED VALUES FOR OPTIMAL DISPATCHES OF THE FCS AND SWITCH POSITIONS FOR THREE WPG CAPACITY STRATEGIES

kW (third strategy) provides the lowest expected value for loss, emission, cost, and voltage deviation. This is because the total combined output of the WPGs and the FCs is the highest in the third strategy (3166.408 kW) in comparison to the first (2158.73 kW) and second (2632.9 kW) strategies. Also, the standard deviation of the objective functions increased with higher output of the WPGs. Table IV shows the detailed results of the strategies. It can be observed that utilization of the FCs decreased as the WPGs output increased. 3) Interactive Fuzzy Satisfying Method: So far, the reference membership values were considered to be unity to give equal weight to all the objective functions (Interaction 0). However, in problems with non-commensurable objectives, the decision maker can evaluate the operational requirements and the associated cost to change his/her decision through an interactive procedure. To illustrate this concept, the base case (second strategy) with average output of 500 kW for the WPGs is considered for analysis. To give lower importance to the cost, its reference membership value is reduced to 0.8 (Interaction 1). This resulted in 0.431146 MW losses, $1265.67 cost, 55 389.16 lb. emissions and 0.5324 p.u. voltage deviation resulting in a decrease of 3.858% in losses, increase of 0.886% in cost, decrease of 0.765% in emissions and increase of 0.554% in voltage deviation in comparison to values given in Section V-C1 for the base case (Interaction 0). Further, to give lower importance to

emission in addition to cost, in the second step is reduced from 1 to 0.8 (Interaction 2). This resulted in 0.4126 MW losses, $1260.23 cost, 55 738.11 lb. emissions and 0.5359 p.u. voltage deviation. In comparison to Interaction 1, losses and cost decreased by 4.29% and 0.43% while emissions and voltage deviation increased by 0.63% and 0.66%, respectively. Fig. 5 shows the cumulative probability distributions of the four objective functions for these interactions. Table V shows the expected value for the optimal dispatch of the FCs and switch positions for different interactions. The proposed tradeoff method provides the decision maker with an opportunity to select a compromised solution or the most satisfactory plan based on preference. VI. CONCLUSION This paper offers a new method based on an APSO for investigating the multi-objective SDFR problem. Various sources of uncertainties including output of the WPGs and load demands are handled through an effective probabilistic power flow method based on PEM scheme. The simulation results show that the proposed APSO method performs better than the standard PSO and GA and is more robust. In order to handle conflicting objectives of the SDFR problem, the non-inferior optimal solution is derived through an interactive fuzzy satisfying method. Application of the method is demonstrated through various examples in which the output of the WPGs is changed

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

Fig. 5. Cumulative probability distributions of the four objective functions for different interactive processes.

TABLE V EXPECTED VALUE FOR OPTIMAL DISPATCHES OF FCS AND SWITCH POSITIONS IN DIFFERENT INTERACTIVE PROCESSES

and the reference membership values of different objectives are changed to implement the decision maker’s preferences related to the objectives. Since penetration of renewal DGs is expected to increase in the future, the methodology presented in this paper would be valuable for planning and operation of distribution systems. Future research will focus on including correlation between output of DGs. REFERENCES [1] P. R. Lopez, F. Jurado, N. R. Reyes, S. G. Galan, and M. Gomez, “Particle swarm optimization for biomass-fuelled systems with technical constraints,” Eng. Appl. Artif. Intell., vol. 21, pp. 1389–1396, Dec. 2008. [2] A. U. Dufour, “Fuel cells—A new contributor to stationary power,” Power Sources. J, vol. 71, pp. 19–25, 1998. [3] J. M. Morales, “Impact on system economics and security of a high penetration of wind power,” Ph.D. dissertation, Dept. Elect. Eng., Castilla-La Mancha. Univ., Ciudad Real, Spain, 2010. [4] H. Kim, Y. Ko, and K. H. Jung, “Artificial neural-network based feeder reconfiguration for loss reduction in distribution systems,” IEEE Trans. Power Del., vol. 8, no. 3, pp. 1356–1366, Jul. 1993. [5] F. V. Gomes, S. Carneiro, J. L. R. Pereira, M. P. Vinagre, P. A. N. Garcia, and L. R. Araujo, “A new heuristic reconfiguration algorithm for large distribution systems,” IEEE Trans. Power Syst., vol. 20, no. 3, pp. 1373–1378, Aug. 2005. [6] M. A. Kashem, V. Ganapathy, and G. B. Jasmon, “Network reconfiguration for load balancing in distribution networks,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 146, pp. 563–567, 1999.

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[7] D. Das, “A fuzzy multi-objective approach for network reconfiguration of distribution systems,” IEEE Trans. Power Del., vol. 21, no. 1, pp. 202–209, Jan. 2006. [8] M. E. Baran and F. F. Wu, “Network reconfiguration in distribution systems for loss reduction and load balancing,” IEEE Trans. Power Del., vol. 4, no. 2, pp. 1401–1407, Apr. 1989. [9] Q. Zhou, D. Shirmohammadi, and W. H. E. Liu, “Distribution feeder reconfiguration for service restoration and load Balancing,” IEEE Trans. Power Syst., vol. 12, no. 2, pp. 724–729, May 1997. [10] A. Ahuja, S. Das, and A. Pahwa, “An AIS-ACO hybrid approach for multi-objective distribution system reconfiguration,” IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1101–1111, Aug. 2007. [11] M.-S. Tsai and F.-Y. Hsu, “Application of grey correlation analysis in evolutionary programming for distribution system feeder reconfiguration,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1126–1133, May 2010. [12] H. M. Khodr, J. Martinez-Crespo, M. A. Matos, and J. Pereira, “Distribution systems reconfiguration based on OPF using Benders decomposition power,” IEEE Trans. Power Del., vol. 24, no. 4, pp. 2166–2176, Oct. 2009. [13] Y.-K. Wu, C.-Y. Lee, L.-C. Liu, and S.-H. Tsai, “Study of reconfiguration for the distribution system with distributed generators,” IEEE Trans. Power Del., vol. 25, no. 3, pp. 1678–1685, Jul. 2010. [14] T. Niknam, “An efficient hybrid evolutionary based on PSO and ACO algorithms for distribution feeder reconfiguration,” Eur. Trans. Elect. Power, vol. 20, pp. 575–590, 2010. [15] E. Rosenblueth, “Point estimates for probability moments,” Proc. Nat. Acad. Sci., vol. 72, pp. 3812–3814, Oct. 1975. [16] A. R. Malekpour and T. Niknam, “A probabilistic multi-objective daily Volt/Var control at distribution networks including renewable energy sources,” J. Energy, vol. 36, pp. 3477–3488, May 2011. [17] J. M. Morales and J. Perez-Ruiz, “Point estimate schemes to solve the probabilistic power flow,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 1594–1601, Nov. 2007. [18] H. P. Hong, “An efficient point estimate method for probabilistic analysis,” Reliab. Eng. Sys. Saf., vol. 59, pp. 261–267, Mar. 1998. [19] C. Su, “Probabilistic load-flow computation using point estimate method,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1842–1851, Nov. 2005. [20] P. Caramia, G. Carpinelli, and P. Varilone, “Point estimate schemes for probabilistic three phase load flow,” Elect. Power Syst. Res., vol. 80, pp. 168–175, 2010. [21] P. Zhang and S. T. Lee, “Probabilistic load flow computation using the method of combined cumulants and Gram-Charlier expansion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 676–682, Feb. 2004. [22] H. J. Zimmermann, Fuzzy Set Theory—And its Applications, 4th ed. Boston, MA: Kluwer, 2001. [23] Y. L. Chen and C. C. Liu, “Interactive fuzzy satisfying method for optimal multi-objective VAR planning in power systems,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 141, pp. 554–560, Nov. 1994. [24] Y. T. Hsiao, “Multiobjective evolution programming method for feeder reconfiguration,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 594–599, Feb. 2004. [25] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. IEEE Int. Conf. Neural Networks, 1995, vol. 4, pp. 1942–1948. [26] J. B. Park, Y. W. Jeong, H. H. Kim, and J. R. Shin, “An improved particle swarm optimization for economic dispatch with valve-point effect,” Int. J. Innov. Energy Syst. Power, vol. 1, pp. 1–7, Nov. 2006. [27] R. Caponetto, L. Fortuna, S. Fazzino, and M. G. Xibilia, “Chaotic sequences to improve the performance of evolutionary algorithms,” IEEE Trans. Evol. Comput., vol. 7, pp. 289–304, Jun. 2003. [28] Su, Ching-Tzong, Lee, and Chu-sheng, “Network reconfiguration of distribution systems using improved mixed-integer hybrid differential evolution,” IEEE Trans. Power Syst., vol. 22, pp. 1022–1027, 2003. [29] C. Su, “Stochastic evaluation of voltages in distribution networks with distributed generation using detailed distribution operation models,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 786–795, May 2010. [30] J. M. Morales, L. Baringo, A. J. Conejo, and R. Minguez, “Probabilistic power flow with correlated wind sources,” IET Gen., Transm., Distrib., vol. 4, no. 5, pp. 641–651, 2010. [31] D. B. Nelson, M. H. Nehrir, and V. Gerez, “Economic evaluation of grid-connected fuel-cell systems,” IEEE Trans. Energy Convers., vol. 20, no. 2, pp. 452–458, Jun. 2005. [32] M. Pipattanasomporn, M. Willingham, and S. Rahman, “Implications of on-site distributed generation for commercial/industrial facilities,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 206–212, Feb. 2005.

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Ahmad Reza Malekpour (S’06–GS’12) received the B.S. and M.S. degrees in electrical engineering from Shiraz University, Shiraz, Iran, in 2003 and 2006, respectively. Currently, he is pursuing the Ph.D. degree at Kansas State University, Manhattan. Since 2009, he has served as a lecturer at several academic institutions in Fars, Iran. From 2009 to 2011, he was an engineer at Fars Regional Electric Company (FREC), Shiraz, where he managed various distribution and transmission planning studies as well as distributed generation interconnection and integration analysis. His research interests include renewable energy, smart grids, stochastic electric power system planning, power system reliability, and energy market.

Taher Niknam was born in Shiraz, Iran. He received the B.S., M.S., and Ph.D. degrees from Shiraz University, Shiraz, Iran, and Sharif University of Technology, Tehran, Iran. He is a faculty member at the Electrical Engineering Department of Shiraz University of Technology. His research interests include power system restructuring, impact of distributed generations on power systems, optimization methods, and evolutionary algorithms.

Anil Pahwa (F’03) received the B.E. (honors) degree in electrical engineering from Birla Institute of Technology and Science, Pilani, India, in 1975, the M.S. degree in electrical engineering from the University of Maine, Orono, in 1979, and the Ph.D. degree in electrical engineering from Texas A&M University, College Station, in 1983. Since 1983, he has been with Kansas State University, Manhattan, where presently he is a Professor and has the Logan-Fetterhoof Chair in the Electrical and Computer Engineering Department. He worked at ABB, Raleigh, NC, during sabbatical from August 1999 to August 2000. His research interests include distribution automation, distribution system planning and analysis, distribution system reliability, intelligent computational methods for power system applications, and integration of renewable generation into power systems. Dr. Pahwa is a member of Eta Kappa Nu, Tau Beta Pi, and ASEE. He is currently the Chair of the Power and Energy Education Committee (PEEC) and Vice Chair of Power System Planning and Implementation Committee of the Power and Energy Society.

Abdollah Kavousifard was born in Shiraz, Iran, in 1987. He received the B.Sc. and M.Sc. degrees from Shiraz University, Shiraz, Iran, in 2009 and 2011, respectively. His major research interests are distribution systems analysis, renewable energy sources (Wind/FC), stochastic analysis, evolutionary algorithms, reliability of power systems, energy market, and load forecasting.

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Multi-Objective Stochastic Distribution Feeder Reconfiguration in Systems With Wind Power Generators and Fuel Cells Using the Point Estimate Method Ahmad Reza Malekpour, Graduate Student Member, IEEE, Taher Niknam, Anil Pahwa, Fellow, IEEE, and Abdollah Kavousi Fard

Abstract—This paper presents a multi-objective algorithm to solve stochastic distribution feeder reconfiguration (SDFR) problem for systems with distributed wind power generation (WPG) and fuel cells (FC). The four objective functions investigated are 1) the total electrical energy losses, 2) the cost of electrical energy generated, 3) the total emissions produced, and 4) the bus voltage deviation. A probabilistic power flow based on the point estimate method (PEM) is employed to include uncertainty in the WPG output and load demand, concurrently. Different wind penetration strategies are examined to capture all economical, operational and environmental aspects of the problem. An interactive fuzzy satisfying optimization algorithm based on adaptive particle swarm optimization (APSO) is employed to determine the optimal plan under different conditions. The proposed method is applied to Taiwan Power system and the results are validated in terms of efficiency and accuracy.

Inertia weight of PSO. Mean of th solution. Standard deviation of the th solution. Lower limit of th objective function. Upper limit of the th objective function. th objective function. ,

Random numbers between 0 and 1. Number of swarms in each iteration. ,

Index Terms—Distribution system reconfiguration, fuel cells, fuzzy set theory, particle swarm optimization, point estimate methods, wind power generation.

Weighting acceleration factors. Velocity of the th particle in th iteration. Best previous experience of th particle. Best particle among the entire population. th standard location of Initial value of

NOMENCLATURE ,

.

Chaotic parameter at iteration .

Mean value of the random variable ,

.

Control parameter.

.

Estimated locations of random variable

.

th individual in the population. th element in th individual at th iteration.

,

,

Weighting factors for estimated locations of random variable . Total number of input variables. th moment of th output random variable. Standard deviation of Skewness coefficient of

. .

Number of WPGs. th membership function. Random function generator. Expected value of the current for the th branch. Number of main loops. Number of branches.

Manuscript received March 26, 2012; revised July 13, 2012; accepted August 26, 2012. Date of publication October 23, 2012; date of current version April 18, 2013. Paper no. TPWRS-00227-2012. A. R. Malekpour and A. Pahwa are with the Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS 66503 USA (e-mail: [email protected]; [email protected]). T. Niknam and A. Kavousi Fard are with the Department of Electronics and Electrical Engineering, Shiraz University of Technology (SUTech), Shiraz 11456-7856, Iran (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2012.2218261 0885-8950/$31.00 © 2012 IEEE

Resistance of the th branch. Current amplitude of the th transformer. Expected value of emission produced by WPGs. Current amplitude of the th feeder. Maximum current of the th feeder. Cost of power supplied by the grid. Generated cost of the th WPG.

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Nitrogen oxide pollutants of FC.

Mathematical round operator.

Expected value of power supplied by the grid.

Random numbers in

Expected value of power generated by the th WPG.

Mathematical mean operator.

.

Mean value of the th element in the population.

Expected value of power generated by the th FC.

PSO population.

Amplitude of the voltage at the th bus.

Probability of th modification method.

Angle of the voltage at the th bus.

Accumulator of the th modification method.

Amplitude of the branch admittance through buses , .

Vector of the mean value of the population.

Angle of branch through buses , .

I. INTRODUCTION

Minimum active power of the th FC. Expected value of active power flow in line Maximum active power of the th FC. Maximum reactive power of the th FC. th probabilistic output variable. Number of self-adapting parameters. Probability density function. Total number of decision variables. Position of the th variable in iteration . Number of FCs. Number of buses. Emission factor for FCs. Emission factor for grid. Expected value of emission produced by FCs. Maximum current of the th transformer. Expected value of emission produced by grid. Generated cost of the th FC. Sulfur oxide pollutants of FC. Carbon dioxide pollutants of FC. Nitrogen oxide pollutants of grid. Sulfur oxide pollutants of grid. Carbon dioxide pollutants of grid. Net injected active power at the th bus. Net injected reactive power at the th bus. Maximum active power flow in line

.

Minimum reactive power of the th FC. Expected value of voltage magnitude of the th bus. Expected value of the maximum voltage. Expected value of the minimum voltage. Probabilistic solution set of output variables. Set of deterministic power flow equations. Input set of uncertain variables. Uncertain input random variable.

.

W

ITHIN the last decade, there has been significant interest in distributed energy sources from both utilities and end users. Wind and solar are popular types of renewable generation resources being considered for connection to distribution networks [1]. Governmental policies for promotion of clean technologies have also encouraged utilization of low emission fuel cell power generation [2]. However, intermittent power generation associated with wind and solar generation poses new challenges for operation of distribution systems [3]. Although distribution system reconfiguration is a well-researched topic [4]–[15], integration of distributed energy resources as well as randomness associated with renewable generation has made it one of the most significant challenges for distribution system operators. Traditionally, the reconfiguration problem has been studied based on deterministic approaches. In [4], a method based on artificial neural network (ANN) was proposed to determine configuration of the network with different load levels. The ANN determined the appropriate system topology such that the total power loss reduced under different load patterns. In [5], Gomes et al. have presented a method based on heuristic search to find the best structure of the network. The algorithm started with the system in a meshed status with all maneuverable switches in closed position, and tried to find the topology with the least power losses. Similar to [4], the problem is solved for the single objective of loss minimization. Kashem et al. [6] have suggested a method based on distance measurement to locate the loops in the system and then improve the load balancing in the corresponding loop. The objective function was load balancing in feeder branches. In [7], Das has proposed a new approach based on fuzzy theory to solve the multi-objective reconfiguration problem including minimization of real power loss, deviation of nodes voltage, and branch current constraint. Baran and Wu [8] have suggested a network reconfiguration method based on mixed integer programming (MIP) to solve the load balancing problem. In [9], Zhou et al. have used an algorithm based on heuristic rules and fuzzy logic approach to support load balancing and service restoration simultaneously. Ahuja et al. [10] have proposed a hybrid algorithm based on artificial immune systems and ant colony optimization for multi-objective distribution system reconfiguration. Tsai et al. [11] proposed an evolutionary programming and used the grey correlation analysis to solve the multi-objective reconfiguration problem. In all

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

these papers, distributed generators (DGs) and their impact on reconfiguration problem has not been considered. In [12], Khodr et al. presented a methodology for reconfiguration problem including DGs using Benders decomposition approach. The authors have considered real loss as well as load balancing of feeder branches as the objective functions. However, some important objective functions such as the cost of energy were neglected. Wu et al. [13] presented an ant colony algorithm to solve the multi-objective reconfiguration problem with DGs to achieve the optimal power loss and load balance of radial networks. In [14], Niknam has proposed a hybrid method to solve the multi-objective reconfiguration problem considering real power loss, voltage deviation, number of switching, and load balancing as the objectives. Since this method only can find a single optimal solution, the several satisfying solutions which could be considered as good preferences were neglected. Despite some deterministic approaches which have been proposed to solve the multi-objective reconfiguration problem in the presence of DGs, there is still no technical literature available to tackle the problem under uncertainties such as those created by intermittent nature of renewable DGs. Monte Carlo simulation (MCS) is an option for analysis including uncertainties, but it can be computationally intensive. This difficulty is more evident in problems in which the main optimization algorithm is solved based on evolutionary methods. This paper aims to study the multi-objective reconfiguration problem with uncertainties in both load and wind power generation using the point estimate method (PEM) [15]. Unlike the MCS method, the PEM is generally simpler and more flexible to deal with complex models. Solar generation is not included, but it can be considered using an approach similar to that used for WPGs. In order to achieve realistic system performance, probabilistic approaches [16] are employed to model the random variation of wind speed and load demand concurrently. The uncertainties involved in the multi-objective reconfiguration problem are handled through an effective probabilistic power flow based on the PEM [17]. Different wind penetration strategies are considered to solve the multi-objective reconfiguration problem. The discrete nature of the tie and sectionalizing switches make the multi-objective stochastic reconfiguration problem a complex nonlinear optimization problem with discrete variables. Therefore, a new adaptive particle swarm optimization (APSO) coupled with an interactive fuzzy satisfying method is proposed to solve the problem. The reminder of this paper is organized as follows. In Section II, the mathematical formulation of multi-objective SDFR is presented. In Section III, the PEM and its background are explained. The introduction of APSO and its application in SDFR problem is described in Section IV. In Section V, the proposed method is applied on a test system and the results are discussed. Finally, in Section VI, the relevant conclusions are described.

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is solved for the peak hour. The related objective functions are expressed as follows: 1) Energy losses: (1) 2) Cost of electricity generation:

(2) 3) Emission produced:

(3) 4) Voltage deviation: (4) subject to a) Distribution power flow equations:

(5) b) Distribution line limits: (6) c) Radial structure of the network: (7) d) Limit on the currents of the transformers: (8) e) Limit on the current of the feeders: (9) f) Active power constraints of FCs: (10) g) Reactive power constraints of FCs:

II. PROBLEM FORMULATION The stochastic distribution feeder reconfiguration (SDFR) problem includes minimization of four objective functions and

(11) The decision variables in the SDFR are the status of normal open/close (tie/sectionalizing) switches and the active/reactive

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powers of FCs. In the above formulas, the superscript the expectation of the random variables.

denotes

III. PROBABILISTIC POWER FLOW A. Background algorithms are required to calIn the original PEM [15], culate the statistical moments of a random quantity which is a function of random variables. In 1998, Hong [18] attempted to improve the original PEM method and introduced the and schemes by reducing the number of computations from to and ( is a parameter determining the type of Hong’s PEM scheme). This property makes PEM an efficient tool to handle engineering problems with numerous random variables. Su [19] was the first to approach the power flow problem as a probabilistic problem using Hong’s scheme. Later, Morales et al. [17] and Caramia et al. [20] used , and PEM schemes to solve the probabilistic power flow problem. They concluded that the scheme provides the best solution in terms of accuracy and computational efforts and is the most efficient scheme in dealing with Gaussian distribution. B.

Fig. 1. Concept of the

where

PEM scheme.

is the standard location expressed as

PEM Scheme

(15)

Mathematically, the deterministic power flow can be expressed as (12) where is the set of input variables representing the network configuration, load and distributed power generation; is the set of output variables (bus voltage or line flows) which are calculated through the power flow function . The idea behind the PEM is to calculate the moments of the output variables of interest through solution of only a few deterministic power flows. In this regard, only few statistical moments of input random variables are required. Considering input random variables in the system, (12) can be written as (13) According to scheme, (13) has to be calculated times. Let be a random variable with probability density function . As , the scheme uses the first four central moments of each random variable to approximate its distribution by three points. Fig. 1 depicts the concept of PEM scheme. The variables , and are transformed to the variables , , and through the functional relationship of . Also, the three weighting factors , , and are utilized to scale the three estimates of variants ( , , and ). Each point or concentration consists of three pairs , , 2, 3 where and are the location and the weighting factor, respectively. For each random variable , the three locations are calculated using mean and variance of : (14)

The parameters and are the third and the fourth central moments of which are defined as coefficients of skewness and kurtosis as follows: (16) where is the expectation operator. The weighting factor associated with the location can be computed as

(17)

Using (13)–(17), two pairs of locations and weights are calculated for each point. The power flow is calculated for two locations assuming that other variables are set at their mean value:

(18) has zero standard location The third location and thus according to (14) . Hence, it is required to run power flow using the input variables set at their mean value: (19)

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

Once solutions of the power flows are obtained using (18) and (19), the th order raw moment of can be formulated as

of objective functions. The reference value is a real number in the range of which represents the importance of each obgives more importance jective function. A higher value of to the th objective function such that the algorithm would be encouraged to explore solutions with smaller value for the specific objective function. A unique optimal solution based on the preference of the decision maker is obtained by solving the following min-max problem:

(20) According to (19), point

number of locations have the same . Thus, (20) can be simplified as

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(23) is the th memwhere is the set of non-inferior solutions, is the th reference membership value. bership function and The reference value for each objective function is estimated from the decision maker’s experience or by trial and error. C. Adaptive Particle Swarm Optimization (APSO)

(21) initial runs and only one further run of power Therefore, flow are required. Hence, the number of times to calculate power flow in a system with input random variables is reduced to . Upon calculation of the mean and the standard deviation of output random variables through (21), the cumulative probability density function can be approximated by Gram-Charlier series approach [21]. IV. SOLUTION METHODOLOGY Firstly, the objective functions of the SDFR problem are represented by fuzzy models. Next, the interactive fuzzy satisfying method is used to convert the objective functions into a min-max problem. Finally, the proposed APSO method is implemented to solve the multi-objective SDFR problem. A. Fuzzy Models of Objective Functions Fuzzy models are able to handle the multi-objective SDFR problem with non-commensurable objective functions. In order to optimize the objective functions simultaneously, each objective function is modeled by a linear membership function [22]: (22)

, where separately.

are extracted by optimizing each objective

B. Interactive Fuzzy Satisfying Method In this section an interactive fuzzy satisfying method [23], [24] is presented to obtain the satisfying solution from the set of non-inferior solutions. In the proposed algorithm, the membership functions are defined separately for each objective. Later, the decision maker is asked to specify the desired preference level for each of the membership functions, called the reference membership values , , where is the number

1) Standard PSO: Particle swarm optimization (PSO) is one of the evolutionary optimization algorithms developed by Eberhart and Kennedy [25]. This algorithm was inspired by the movement of a flock of birds searching for food. In the PSO algorithm, position and velocity of each particle are updated according to the following formula:

(24) (25) The inertia weight factor is used to control the impact of the previous history of velocities on the current velocity. A large leads to a global search whereas a small tends to facilitate local search. This factor usually decreases linearly from 0.9 to 0.4 [26]. Also, the acceleration factors and , named cognitive and social parameters, determine influence of the personal best and the global best in determining new solutions. and represent functions that generate independent random numbers which are uniformly distributed between 0 and 1; is the number of swarms in each iteration. 2) Adaptive PSO: Despite numerous advantages of PSO such as simple concept, easy implementation, minimal storage requirement, and straightforward incorporation with other optimization tools, it may experience inappropriate convergence and fall in local minima. Therefore, in this paper new self-adaptive modifications are presented to overcome the above-mentioned deficiency of PSO algorithm while improving its ability for global exploration. The proposed self adaptive modifications employ three different approaches with specific probability. Each particle will choose the approach with the highest probability during the optimization process. Modification Approach 1: This approach updates the inertia weight and accelerating factors self adaptively. The inertia weight factor is dynamically tuned in each iteration as follows: (26) The value of is determined by an iterative chaotic method called the logistic map [27] is (27)

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The control parameter, , is selected randomly within 0 to 4 range. Variation of determines whether becomes stable at a constant size, oscillates between limited sequences of sizes, or behaves chaotically in an unpredictable pattern. Since (27) is an iterative equation, its behavior is greatly sensitive to the initial value of . Equation (27) without any stochastic pattern in the value of indicates chaotic dynamic when and [26]. The acceleration coefficients and are incorporated into the optimization problem by augmenting the position vector, the velocity vector, and the best position vector. In general, if is the dimension of the problem is the number of self adapting parameters, the new and vectors for the particle will be (28)

(35) (36) , , will reThe best individual among the in the population. place The initial probability of each modification approach is 0.33 ( and , 2, 3). Also, an accumulator named is defined for each modification approach. In each iteration, the particles are sorted according to their objective function such that and indicate the best and the worst individual in the sorted population. Afterwards, each particle would gain a weighting factor according to its position in the population as follows:

(29)

(37)

(30) Obviously, the first variables correspond to the particle in varithe search space (decision variables), while the last ables account for the personal acceleration constants (cognitive and social parameters). These new variables do not enter the fitness function but are manipulated using the same mixed individual-social learning paradigm as the one used in PSO. Modification Approach 2: This approach is utilized to encourage movement of the swarm population toward the best individual found to date. Using this modification, convergence velocity of the algorithm is enhanced effectively. In this regard, the mean value of the population is calculated column-wise as follows:

(31) Now the th particle will move toward the best particle as follows:

(32) will be accepted if it is better than . Modification Approach 3: This approach is utilized to increase diversity of the swarm population and reduce the possibility of non-mature convergence. For each particle three are selected from the population randomly particles , , such that . Next, a new test individual is generated as follows:

(33) Later, two mutant particles are achieved using the following scheme: (34)

The accumulator of each method is updated as follows:

(38) shows the number of solutions which have where selected the th modification approach. Then, the probability of each modification approach is updated as follows: (39) Each particle would select the th modification approach by the roulette mechanism based on its probability . D. Implementation of APSO in SDFR Problem To apply the proposed APSO algorithm to solve the multiobjective SDFR problem, the following steps are performed: Step1: Read the input data. Step2: Generate an initial population and velocity . Step3: Evaluate the objective function for each individual using probabilistic power flow based on PEM scheme. Step4: Select and . Step5: Update the APSO parameters as described in IV. Step6: Update the velocity and position using (24) and (25). Step7: Check the constraints and modify the position. Step8: Update and . Step9: Check the termination criterion. If the iteration number reaches the predetermined maximum value, the search is stopped, else go to step 5. V. SIMULATION RESULTS The case study is the 86 buses Taiwan Power Company (TPC) system [28], which consists of 2 substations, 11 feeders, and 96 switches (83 sectionalizing switches and 13 tie switches). The

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TABLE I CHARACTERISTIC OF INSTALLED FCS

TABLE II COMPARISON OF EXPECTED VALUES FOR AVERAGE, BEST, AND WORST SOLUTION USING THE GA, PSO, AND APSO METHODS

Fig. 2. Single line diagram of the test system.

studies have been implemented in MATLAB 7.4 using a Pentium P4, Core 2 Duo 2.4-GHz personal computer with 1 GB of RAM. The simulation results will be described in the following three sections: A. Assumptions Table I shows the location and capacities of FCs in the TPC. The WPGs connected to nodes #7, #72, #80 are considered to have a fixed power factor of 0.9 lagging. The random output power of each WPG is modeled as a five-step discrete distribution function corresponding to a Weibull distribution with mean speed of 7 meters and the shape parameter of 2 [29]. Since the three WPGs are considered to be located in different locations, spatial correlations between them are neglected. However, such correlations can be included using the approach provided in [30], which will be considered in the future research. The mean and the standard deviation of power production for the WPGs are computed based on the corresponding discrete distribution through a standard statistical formula which gives the final mean power of 500 kW for the WPGs. The energy cost is considered to be 0.07 ($/kWh) for FCs, 0.044 ($/kWh) for the grid [31] and zero for the WPGs. The loads on buses #10, #11, #15, #19, #28, #42, #46, #52, #59, #84 have the normal distribution function with a constant standard deviation of 5%. The other loads are assumed to be deterministic (zero standard deviation). The emission factor of FCs and grid are 1078.036 and 2043.96 (lb/MWh) [32], respectively. B. Comparison of APSO With Other Methods In order to compare performance of the proposed APSO algorithm with other well-known algorithms, the SDFR problem is also solved using GA and PSO, and the results are shown in Table II. For all the methods, the initial population equals four times the total number of decision variables and the best solution

is determined by running the algorithm for 200 iterations. The mutation probability and crossover rate for GA are set to 0.09 and 0.3, respectively. The cognitive parameter, social parameter, initial and final inertia weight factors for PSO and APSO are set to 2, 2, 0.9 and 0.4, respectively. The problem is solved for 25 trials with each method and the mean values of the corresponding best solution, the worst solution, the average of the mean values, the standard deviation of the mean values, and the CPU time obtained are shown in Table II. It reveals that APSO provides better results in comparison to GA and PSO methods in terms of robustness and CPU Time. C. Application of APSO to TPS Since the APSO gave the best performance, it is used to study the test system under different conditions including changing outputs of the WPGs and changing the reference membership of the objective functions to implement decision maker’s preferences. In the first two cases the reference membership for the objectives are considered to be unity. It is assumed that each objective function follows the Gaussian distribution and the values obtained from scheme are used to generate the cumulative probability distribution of the objective functions. 1) Effect of Reconfiguration: Fig. 3 depicts the cumulative probability distributions of the objective functions before and after reconfiguration considering 500 kW average output for the WPGs and reference membership value of unity for all the objectives. The proposed method resulted in statistically lower total electric loss, total emission, operating costs and voltage deviation with mean values of 0.448447 MW, $1254.56, 55 816.16 lb., and 0.5295 p.u., respectively. Not only the mean value, but also the standard deviation of the objective functions reduced due to reconfiguration. Table III shows expected values of the optimal dispatch of the FCs and switch positions before and after reconfiguration. 2) Effect of Wind Generator Capacity: Two different strategies with average output of 250 and 750 kW for the WPGs are considered to investigate their effect on the optimal results. Fig. 4 shows the objective functions’ values as the WPGs output is changed. The strategy with the total average output of 2250

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Fig. 3. Cumulative probability distributions of the four objective functions before and after reconfiguration.

Fig. 4. Cumulative probability distributions of the four objective functions for different wind power penetration strategies.

TABLE III EXPECTED VALUES FOR OPTIMAL DISPATCH OF THE FCS AND SWITCH POSITIONS BEFORE AND AFTER RECONFIGURATION

TABLE IV EXPECTED VALUES FOR OPTIMAL DISPATCHES OF THE FCS AND SWITCH POSITIONS FOR THREE WPG CAPACITY STRATEGIES

kW (third strategy) provides the lowest expected value for loss, emission, cost, and voltage deviation. This is because the total combined output of the WPGs and the FCs is the highest in the third strategy (3166.408 kW) in comparison to the first (2158.73 kW) and second (2632.9 kW) strategies. Also, the standard deviation of the objective functions increased with higher output of the WPGs. Table IV shows the detailed results of the strategies. It can be observed that utilization of the FCs decreased as the WPGs output increased. 3) Interactive Fuzzy Satisfying Method: So far, the reference membership values were considered to be unity to give equal weight to all the objective functions (Interaction 0). However, in problems with non-commensurable objectives, the decision maker can evaluate the operational requirements and the associated cost to change his/her decision through an interactive procedure. To illustrate this concept, the base case (second strategy) with average output of 500 kW for the WPGs is considered for analysis. To give lower importance to the cost, its reference membership value is reduced to 0.8 (Interaction 1). This resulted in 0.431146 MW losses, $1265.67 cost, 55 389.16 lb. emissions and 0.5324 p.u. voltage deviation resulting in a decrease of 3.858% in losses, increase of 0.886% in cost, decrease of 0.765% in emissions and increase of 0.554% in voltage deviation in comparison to values given in Section V-C1 for the base case (Interaction 0). Further, to give lower importance to

emission in addition to cost, in the second step is reduced from 1 to 0.8 (Interaction 2). This resulted in 0.4126 MW losses, $1260.23 cost, 55 738.11 lb. emissions and 0.5359 p.u. voltage deviation. In comparison to Interaction 1, losses and cost decreased by 4.29% and 0.43% while emissions and voltage deviation increased by 0.63% and 0.66%, respectively. Fig. 5 shows the cumulative probability distributions of the four objective functions for these interactions. Table V shows the expected value for the optimal dispatch of the FCs and switch positions for different interactions. The proposed tradeoff method provides the decision maker with an opportunity to select a compromised solution or the most satisfactory plan based on preference. VI. CONCLUSION This paper offers a new method based on an APSO for investigating the multi-objective SDFR problem. Various sources of uncertainties including output of the WPGs and load demands are handled through an effective probabilistic power flow method based on PEM scheme. The simulation results show that the proposed APSO method performs better than the standard PSO and GA and is more robust. In order to handle conflicting objectives of the SDFR problem, the non-inferior optimal solution is derived through an interactive fuzzy satisfying method. Application of the method is demonstrated through various examples in which the output of the WPGs is changed

MALEKPOUR et al.: MULTI-OBJECTIVE STOCHASTIC DISTRIBUTION FEEDER RECONFIGURATION

Fig. 5. Cumulative probability distributions of the four objective functions for different interactive processes.

TABLE V EXPECTED VALUE FOR OPTIMAL DISPATCHES OF FCS AND SWITCH POSITIONS IN DIFFERENT INTERACTIVE PROCESSES

and the reference membership values of different objectives are changed to implement the decision maker’s preferences related to the objectives. Since penetration of renewal DGs is expected to increase in the future, the methodology presented in this paper would be valuable for planning and operation of distribution systems. Future research will focus on including correlation between output of DGs. REFERENCES [1] P. R. Lopez, F. Jurado, N. R. Reyes, S. G. Galan, and M. Gomez, “Particle swarm optimization for biomass-fuelled systems with technical constraints,” Eng. Appl. Artif. Intell., vol. 21, pp. 1389–1396, Dec. 2008. [2] A. U. Dufour, “Fuel cells—A new contributor to stationary power,” Power Sources. J, vol. 71, pp. 19–25, 1998. [3] J. M. Morales, “Impact on system economics and security of a high penetration of wind power,” Ph.D. dissertation, Dept. Elect. Eng., Castilla-La Mancha. Univ., Ciudad Real, Spain, 2010. [4] H. Kim, Y. Ko, and K. H. Jung, “Artificial neural-network based feeder reconfiguration for loss reduction in distribution systems,” IEEE Trans. Power Del., vol. 8, no. 3, pp. 1356–1366, Jul. 1993. [5] F. V. Gomes, S. Carneiro, J. L. R. Pereira, M. P. Vinagre, P. A. N. Garcia, and L. R. Araujo, “A new heuristic reconfiguration algorithm for large distribution systems,” IEEE Trans. Power Syst., vol. 20, no. 3, pp. 1373–1378, Aug. 2005. [6] M. A. Kashem, V. Ganapathy, and G. B. Jasmon, “Network reconfiguration for load balancing in distribution networks,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 146, pp. 563–567, 1999.

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[7] D. Das, “A fuzzy multi-objective approach for network reconfiguration of distribution systems,” IEEE Trans. Power Del., vol. 21, no. 1, pp. 202–209, Jan. 2006. [8] M. E. Baran and F. F. Wu, “Network reconfiguration in distribution systems for loss reduction and load balancing,” IEEE Trans. Power Del., vol. 4, no. 2, pp. 1401–1407, Apr. 1989. [9] Q. Zhou, D. Shirmohammadi, and W. H. E. Liu, “Distribution feeder reconfiguration for service restoration and load Balancing,” IEEE Trans. Power Syst., vol. 12, no. 2, pp. 724–729, May 1997. [10] A. Ahuja, S. Das, and A. Pahwa, “An AIS-ACO hybrid approach for multi-objective distribution system reconfiguration,” IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1101–1111, Aug. 2007. [11] M.-S. Tsai and F.-Y. Hsu, “Application of grey correlation analysis in evolutionary programming for distribution system feeder reconfiguration,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1126–1133, May 2010. [12] H. M. Khodr, J. Martinez-Crespo, M. A. Matos, and J. Pereira, “Distribution systems reconfiguration based on OPF using Benders decomposition power,” IEEE Trans. Power Del., vol. 24, no. 4, pp. 2166–2176, Oct. 2009. [13] Y.-K. Wu, C.-Y. Lee, L.-C. Liu, and S.-H. Tsai, “Study of reconfiguration for the distribution system with distributed generators,” IEEE Trans. Power Del., vol. 25, no. 3, pp. 1678–1685, Jul. 2010. [14] T. Niknam, “An efficient hybrid evolutionary based on PSO and ACO algorithms for distribution feeder reconfiguration,” Eur. Trans. Elect. Power, vol. 20, pp. 575–590, 2010. [15] E. Rosenblueth, “Point estimates for probability moments,” Proc. Nat. Acad. Sci., vol. 72, pp. 3812–3814, Oct. 1975. [16] A. R. Malekpour and T. Niknam, “A probabilistic multi-objective daily Volt/Var control at distribution networks including renewable energy sources,” J. Energy, vol. 36, pp. 3477–3488, May 2011. [17] J. M. Morales and J. Perez-Ruiz, “Point estimate schemes to solve the probabilistic power flow,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 1594–1601, Nov. 2007. [18] H. P. Hong, “An efficient point estimate method for probabilistic analysis,” Reliab. Eng. Sys. Saf., vol. 59, pp. 261–267, Mar. 1998. [19] C. Su, “Probabilistic load-flow computation using point estimate method,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1842–1851, Nov. 2005. [20] P. Caramia, G. Carpinelli, and P. Varilone, “Point estimate schemes for probabilistic three phase load flow,” Elect. Power Syst. Res., vol. 80, pp. 168–175, 2010. [21] P. Zhang and S. T. Lee, “Probabilistic load flow computation using the method of combined cumulants and Gram-Charlier expansion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 676–682, Feb. 2004. [22] H. J. Zimmermann, Fuzzy Set Theory—And its Applications, 4th ed. Boston, MA: Kluwer, 2001. [23] Y. L. Chen and C. C. Liu, “Interactive fuzzy satisfying method for optimal multi-objective VAR planning in power systems,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 141, pp. 554–560, Nov. 1994. [24] Y. T. Hsiao, “Multiobjective evolution programming method for feeder reconfiguration,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 594–599, Feb. 2004. [25] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. IEEE Int. Conf. Neural Networks, 1995, vol. 4, pp. 1942–1948. [26] J. B. Park, Y. W. Jeong, H. H. Kim, and J. R. Shin, “An improved particle swarm optimization for economic dispatch with valve-point effect,” Int. J. Innov. Energy Syst. Power, vol. 1, pp. 1–7, Nov. 2006. [27] R. Caponetto, L. Fortuna, S. Fazzino, and M. G. Xibilia, “Chaotic sequences to improve the performance of evolutionary algorithms,” IEEE Trans. Evol. Comput., vol. 7, pp. 289–304, Jun. 2003. [28] Su, Ching-Tzong, Lee, and Chu-sheng, “Network reconfiguration of distribution systems using improved mixed-integer hybrid differential evolution,” IEEE Trans. Power Syst., vol. 22, pp. 1022–1027, 2003. [29] C. Su, “Stochastic evaluation of voltages in distribution networks with distributed generation using detailed distribution operation models,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 786–795, May 2010. [30] J. M. Morales, L. Baringo, A. J. Conejo, and R. Minguez, “Probabilistic power flow with correlated wind sources,” IET Gen., Transm., Distrib., vol. 4, no. 5, pp. 641–651, 2010. [31] D. B. Nelson, M. H. Nehrir, and V. Gerez, “Economic evaluation of grid-connected fuel-cell systems,” IEEE Trans. Energy Convers., vol. 20, no. 2, pp. 452–458, Jun. 2005. [32] M. Pipattanasomporn, M. Willingham, and S. Rahman, “Implications of on-site distributed generation for commercial/industrial facilities,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 206–212, Feb. 2005.

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Ahmad Reza Malekpour (S’06–GS’12) received the B.S. and M.S. degrees in electrical engineering from Shiraz University, Shiraz, Iran, in 2003 and 2006, respectively. Currently, he is pursuing the Ph.D. degree at Kansas State University, Manhattan. Since 2009, he has served as a lecturer at several academic institutions in Fars, Iran. From 2009 to 2011, he was an engineer at Fars Regional Electric Company (FREC), Shiraz, where he managed various distribution and transmission planning studies as well as distributed generation interconnection and integration analysis. His research interests include renewable energy, smart grids, stochastic electric power system planning, power system reliability, and energy market.

Taher Niknam was born in Shiraz, Iran. He received the B.S., M.S., and Ph.D. degrees from Shiraz University, Shiraz, Iran, and Sharif University of Technology, Tehran, Iran. He is a faculty member at the Electrical Engineering Department of Shiraz University of Technology. His research interests include power system restructuring, impact of distributed generations on power systems, optimization methods, and evolutionary algorithms.

Anil Pahwa (F’03) received the B.E. (honors) degree in electrical engineering from Birla Institute of Technology and Science, Pilani, India, in 1975, the M.S. degree in electrical engineering from the University of Maine, Orono, in 1979, and the Ph.D. degree in electrical engineering from Texas A&M University, College Station, in 1983. Since 1983, he has been with Kansas State University, Manhattan, where presently he is a Professor and has the Logan-Fetterhoof Chair in the Electrical and Computer Engineering Department. He worked at ABB, Raleigh, NC, during sabbatical from August 1999 to August 2000. His research interests include distribution automation, distribution system planning and analysis, distribution system reliability, intelligent computational methods for power system applications, and integration of renewable generation into power systems. Dr. Pahwa is a member of Eta Kappa Nu, Tau Beta Pi, and ASEE. He is currently the Chair of the Power and Energy Education Committee (PEEC) and Vice Chair of Power System Planning and Implementation Committee of the Power and Energy Society.

Abdollah Kavousifard was born in Shiraz, Iran, in 1987. He received the B.Sc. and M.Sc. degrees from Shiraz University, Shiraz, Iran, in 2009 and 2011, respectively. His major research interests are distribution systems analysis, renewable energy sources (Wind/FC), stochastic analysis, evolutionary algorithms, reliability of power systems, energy market, and load forecasting.