Adaptive differential evolution algorithm for efficient reactive power management

Adaptive differential evolution algorithm for efficient reactive power management

Accepted Manuscript Title: Adaptive Differential Evolution Algorithm for Efficient Reactive Power Management Author: Walaa S. Sakr Ragab A. EL-Sehiem...
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Accepted Manuscript Title: Adaptive Differential Evolution Algorithm for Efficient Reactive Power Management Author: Walaa S. Sakr Ragab A. EL-Sehiemy Ahmed M. Azmy PII: DOI: Reference:

S1568-4946(17)30008-X http://dx.doi.org/doi:10.1016/j.asoc.2017.01.004 ASOC 3998

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

23-5-2014 26-12-2016 3-1-2017

Please cite this article as: Walaa S.Sakr, Ragab A.EL-Sehiemy, Ahmed M.Azmy, Adaptive Differential Evolution Algorithm for Efficient Reactive Power Management, Applied Soft Computing Journal http://dx.doi.org/10.1016/j.asoc.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive Differential Evolution Algorithm for Efficient Reactive Power Management Walaa S. Sakr1,a 1)

Ragab A. EL-Sehiemy1,b,*

Ahmed M. Azmy2

Electrical Engineering Department, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh, Egypt, Email: [email protected]

2)

Elec. Power and Machines Eng. Depart., Faculty of Eng., Tanta University, Tanta, Egypt, [email protected] * Intelligent Systems Research Group (ISRG), Kafrelsheikh University, Egypt

1

Graphical Abstract

Start Read input data for tested systems and the adaptive evaluation differential parameters Perform Newton Raphason load flow calculation for base case to get the voltage profile and power losses

Create the initial population, then compute the objective function of initial population and determine the best fit vector

Perform mutation process

Set parameter values at max or min limits

Yes

Check if Max/Min limits of parameters are violated

No Perform crossover process

Perform selection process

Determine the best, average and worst objective function value of the generation

No

Stop criteria is satisfied

Yes Run the final power flow using the parameters of the best vector to obtain the final voltage profile and power losses

End

2

Highlights

A procedure to solve ORPD based on a multi-objective function by using an adaptive differential evolution algorithm is presented; the convergence characteristic of the differential evolution algorithm is improved; an investigated strategy for adaptive penalty factor to alleviate the effects of dependent variable violation; the voltage profile as well as power losses are improved; Numerical applications for different studied cases are carried out on two standard IEEE test systems; The flexibility of synchronous machines as reactive power sources is proven compared to switchable devices.

3

Abstract— This paper introduces a proposed procedure to solve the optimal reactive power management (ORPM) problem based on a multi-objective function using a modified differential evolution algorithm (MDEA). The proposed MDEA is investigated in order to enhance the voltage profile as well as to reduce the active power losses by solving the ORPM problem. The ORPM objective function aims to minimize transmission power losses and voltage deviation considering the system constraints. The MDEA aims to enhance the convergence characteristic of the differential evolution algorithm through updating the self-adaptive scaling factor, which can exchange information dynamically every generation. The scaling factor dynamically adopts the global and local searches to efficiently eliminate trapping in local optima. In addition, a strategy is developed to update the penalty factor for alleviating the effects of various system constraints. Numerical applications of different case studies are carried out on three standard IEEE systems, i.e., 14-bus, 30-bus and 57-sbus test systems. Also, the proposed procedure is applied on Western Delta Network, which is a real part of the Egyptian main grid system. The flexibility of synchronous machines to provide controllable reactive power is proven with less dependency on the discrete reactive power controllers, such as installing the switchable devices and variations of tap changers. The obtained results show the effectiveness of the proposed enhanced optimization algorithm as an advanced optimization technique that was successively implemented with good performance characteristics. Index Terms: Adaptive penalty factor, Modified differential evolution algorithm, Multi-objective function, Optimal reactive power management.

I.

INTRODUCTION

The optimal reactive power management (ORPM) problem is a non-linear, nonconvex, optimization problem. It is extensively implemented for enhancing economics and secure operation of modern power system. The ORPM problem is a particular form of the optimal power flow (OPF) in which, the control variables are those related to reactive power control. Examples include the control of the generators, synchronous condensers, transformers tapings, shunt reactors, FACTS devices and other reactive power sources. Solving ORPM problem aims to determine the optimal settings of control variables to minimize objective functions, involving power losses and voltage deviation, subject to system equality and inequality constraints [1]-[3]. Because of the complexity and importance of ORPM problem, many researchers developed solution strategies using different optimization techniques. These techniques can be classified into two main categories. The first category is the classical optimization methods [4]-[9] such as linear programming [LP], non-linear programming [NLP], interior point method, quadratic method, gradient method and Lagrangian decomposition-based method [8], [9]. Zhu and Xiong [7] proposed an approach to study the ORPM problem using a modified interior point (MIP) method to minimize the system real power losses and to penalize any reactive power source utilization. The authors in [8] developed a decentralized approach, which was based on Lagrangian decomposition method for solving ORPM problem in multi-area power systems. From the literature survey, it was observed that these classical methods suffer from many drawbacks, such as insecure convergence properties and excessive numerical iterations. These methods suffer also from the incapability of handling nonlinear objective functions and constraints with discontinuous nature. Another problem is the possibility of local optimality, which leads to multiple local minimum points, which prevents their accurate implementation. Thus, these methods have high possibility for trapping in local optimum and they need to linearize their input and output functions. Furthermore, these classical methods exhibit difficulty in handling nonlinearity, discontinuous functions and multimodal characteristic of the problem in addition to the difficulties associated with inequality constraints. Therefore, these solutions have the disadvantage of low accuracy in many situations. 4

From power system operation point of view, there is a necessity for continuous improvement of optimization methodologies to efficiently solve different power systems problems. Several search-based optimization methods have evolved in the last decades [10]-[25]. Population-based optimization methods inspired by nature may be classified into two important categories, which are evolutionary algorithms and swarm intelligence. In recent decades, many heuristic optimization techniques have obtained high interest due to their flexibility, versatility and robustness in seeking global optimal solutions. On the other hand, the second category is the search-based optimization techniques such as ant colony optimization algorithm [1], genetic algorithm [2], self-adaptive real-coded genetic algorithm [10], [11], improved hybrid evolutionary programming technique [12], biogeography-based optimization (BBO) [13], [14], gravitational search algorithm (GSA) [15], seeker optimization algorithm (SOA) [16] and artificial bee colony optimization (ABC) [17]. Khazali et al. proposed harmony search algorithm (HSA) [18] to solve ORPM problem and produced better simulation results compared to other algorithms. A multi-objective fuzzy based procedure for enhancing the reactive power management in power systems is presented in [19]. Applications of several versions of particle swarm optimization (PSO) algorithm are presented in [3], [20]-[25]. Higher-quality solutions for the ORPM problems are obtained using an enhanced version known as comprehensive learning PSO algorithm [20]. A significant reduction in transmission power losses is achieved using fully informed PSO [21] and dynamic PSO version as proposed in [22]. In [23], a particle swarm technique is implemented for optimizing reactive-power and voltage control considering voltage security issues. A multi-agent-based PSO approach for solving the optimal reactive power dispatch problem is presented in [24]. A solution is obtained for ORPM in the existence of wind farms using PSO algorithm in [25]. These methods have extreme superiority in obtaining the global optimum and in handling discontinuous and non-convex objectives. Differential evolution is an advanced version of GA that was introduced by Storn and Price [26], [27]. Initially, differential evolution (DE) is used to deal with real-coded and continuous functions, but it is extended to handle integer, discrete and mixedinteger problems. The main advantages of DE are its simple structure, the need for lower parameters, the ease of use and robustness in addition to its effectiveness to deal with optimization problems having nonlinear constraints. The principle of DE is based on using the difference between two parent vectors, which are selected randomly, to guide the mutation operation. On the other hand, other evolutionary algorithms use probability distribution functions. Hence, the difference between parent vectors is used to facilitate the optimization procedures to reach the global minimum [28], [29]. Due to the advantages of DE, it was developed for many applications in power system engineering such as generation expansion, capacitor placement, reactive power dispatch, economic dispatch, FACTS placement and power flow optimization [30], [34]-[42]. Many efforts are applied to improve DE characteristics and convergence capability such as adapting control parameters, e.g., scaling factor, crossover factor and number of populations [43]-[54]. Variant multi-objective frameworks were developed in [55]-[58]. This paper proposes an improved version of DE algorithm that is implemented for solving ORPM problem. This methodology is intended to enhance the convergence characteristic of the differential evolution algorithm through updating scaling factor and to reduce the possibility of trapping in local optimal. In addition, an updating strategy to the penalty factor for every population is proposed to ensure satisfying all system equality and inequality constraints. The proposed MDEA is investigated in order to enhance the voltage profile as well as to reduce active power losses by solving the ORPM problem. Implementation of the proposed MDEA is successively carried out on three standard IEEE networks, i.e. IEEE 14-bus, 30-bus and 57-bus test systems in addition to a real Egyptian system (Western Delta network).

II.

PROBLEM FORMULATION 5

The ORPM problem is modeled as a nonlinear mixed-integer problem, where it is required to determine the optimal settings of control variables of the power system to fulfill certain objective functions, while satisfying unit and system constraints. The commonly-used objectives in power system are the voltage deviations reduction at load busses, transmission losses minimization, voltage stability and security enhancement. These objectives conflict with each other and cannot be handled by conventional single-objective optimization techniques. Generally, the ORPM model can be mathematically described as follows: a)

Problem objective function nd



min( f )   wPi  1  w vi  v ref i 1



(1)

The first term in the objective function aims to reduce the transmission losses in the network. The magnitudes of these losses need to be accurately estimated. The transmission losses can be expressed as the sum of injected power at all buses as: nb

P L   pi

(2)

i 1

where, P L is the power losses in the system, nb is the number of buses, pi is the injected power at bus i, which is given by: [ cos  ij  Bij sin  ij ] | v j | pi  vi2 Gii  | vi | inb 1 Gij

(3)

j i

where vi , v j are voltage magnitudes at buses i and j, G ij , B ij are mutual conductance and susceptance between bus i and j, G ii is the self-conductance at bus i and  ij is angle between voltages at buses i and j. nb

2

nb P L ( X ,U )  [vi Gii  | vi | i 1 [Gij cos  ij  Bij sin  ij ] | v j |] i 1

(4)

j i

where, X is a vector of control variables, which contain generation-bus voltages and reactive power sources and is expressed as X T  [v g1, v g 2 ,...,v gn , Qsw1, Qsw2 ,...,Qswn ] , and U represents the set of dependent variables, which involve load voltages, and reactive power sources at generation buses. It is expressed as: U T  [vl1, vl 2 ,..., vln d , Q g1, Q g 2 ,..., Qng ] . where, v g , vl are voltages at generation bus and load bus, respectively, Q g , Q sw are reactive power at generation bus and switchable bus, respectively, ng, nd and swn are number of generation buses, number of load buses and number of switchable buses, respectively. The second term in the objective function, (Eq. 1), aims to reduce the voltage deviation and it can be expressed as: nd

V D   |vi  v ref |

(5)

i 1

where, V D is the voltage deviation, vi is voltage for load bus and vref is the reference voltage value. The objective functions of the ORPM problem can be mathematically formulated in a compact form as a nonlinear constrained multi-objective optimization problem as follows:

min F  w P L ( x, u)  (1  w)V D

(6)

where, w is a weighting factor.

6

b) Problem constraints The minimization of the multi-objective function is subjected to operational equality and inequality constraints as follows: 1) Power flow constraints The power flow constraints can be expressed as:

p gi  p di  f p ( X ,U )  0

(7)

Q gi  Qdi  f q ( X ,U )  0

(8)

where: p gi and Q gi are, respectively, the active and reactive power generation at bus i, P di and Q di are the load active and reactive power, respectively, and f p ( X ,U ) and f q ( X ,U ) are injected active and reactive power, respectively. The power flow solution is obtained using Newton Raphson technique. 2) Generation constraints At the generation buses, it is important to preserve the voltages and active and reactive powers within their operational limitations as follows: max vmin gi  v gi  v gi

: i=1, 2,…, ng

(9)

max Qmin gi  Q gi  Q gi

: i=1, 2,…, ng

(10)

max p min gi  p gi  p gi

: i=1, 2,…, ng

(11)

3) Switchable VAR constraints: Switchable VAR compensators ( Q sw ) must be maintained within their minimum and maximum limits as follows: min max  Qswi  Qswi Qswi

(12)

i= 1, 2,..., nsw

4) Load voltage constraints: The voltages at load buses must be within acceptable limits as follows:

vlimin  vli  vlimax ,

i=1, 2,…, nd

(13)

III. SENSITIVITY COEFFICIENTS The fast decoupled power flow method considers that the change in active power is correlated to the change in voltage phase angles, while the change in reactive power is related to the change in voltage magnitudes. Hence, it can be written as:

P v  B'  Q v  B' 'v

(14) (15)

where, P, Q are the changes in active and reactive power, respectively,  and v are the changes in voltage phase angles and magnitudes, respectively, and B' and B" are susceptance matrices. They are calculated as follows: nb

Bii''   1 xiji, j  1,2,..,nb

(16)

B''ji  Bij''  1 xij i, j=1,2,..nb

(17)

j 1 j i

7

where xij is the reactance between buses i and j. The ORPM problem is based on sensitivity coefficients based on the second equation of fast decoupled power model as follows:  Q g v g   BGG BGL  v g        Q L v L   B LG B LL   v L 

(18)

The sensitivity coefficients between control and dependent variables can be introduced as follows: 1. The change in load bus voltages due to changes in control variables: Change in generation voltages only (  Q sw  0 )

 v L   B LL

1

 

B LG  vG

(19)

v L  S GL  vG

(20)

L  where, S G B LL

1

B LG 

Change in reactive power source only ( vG  0 )

 

L Qsw v L  S sw

(21)

L  where, S sw B LL (vsw1) , vsw is the initial bus voltage that is connected to reactive power source. 1

2. The change in reactive power generation due to changes in control variables: The change in reactive power generation due to changes in generation voltage: QG  BGG VG  BGLVL VG

(22)

QG  BGG VG  BGL SGL VG VG

 

(23)

G QG  SG vG

(24)

  where: S   V B G G

G

GG 

BGLSGL .

The change in reactive power generation due to changes in reactive power source:

 

1 QG  BGG  SGL VL  BGLVL VG

 

(25)

 

1 QG   L   BGG  SGL  BGL S sw Qsw vG  

(26)

G QG  Ssw Qsw

(27)

 

S   V  B S  G sw

G

GG

L 1  G

L  BGLSsw

(28)



The relationship between dependent variables and control variables in a matrix form is given as follows:

 VL   SGL V    G  G  SG

L   V  S sw G G  Q  S sw   sw 

(29)

8

IV. TRADITIONAL DIFFERENTIAL EVOLUTION ALGORITHM Differential evolution is an important and effective evolutionary computation algorithm due to its fast convergence characteristic, few required control parameters, simple structure and small computational time. It is an improved version of GA, where GA and DE perform crossover, selection and mutation processes. The main differences between GA algorithm and DE algorithm appear in the selection process, mutation scheme and crossover mechanism, which converts DE to a self-adaptive algorithm. DE solves real-coded problems and can be extended to solve mixed integer problems. It depends on the idea of natural evolution and it uses a population “P” that has “NP” floating-point-encoded individuals. The evolving process is accomplished over G generations to reach an optimal solution. Each population of NP individuals represents a candidate solution or vector that contains a number of parameters, which are called decision variables “D” or control variables. In DE, the size of population “NP” remains constant through the optimization process. DE starts by initial population, then it is improved by using mutation, crossover and selection that are repeated through generation until reaching the convergence criteria. The Initial population contains randomly-generated vectors Uio , i= 1, 2,… NP. After generating initial population, the mutation and crossover processes are applied to population and then offspring vectors compete against their parent to select the best for the next generation. The parallel version of DE contains two arrays: the first contains the current population and the second contains the vectors that are selected for the second generation. The dimension of each array is (NP, D). In brief, DE procedures can be summarized as follows: 1. Initial population and parameters selection: The first step of DE is to create initial population of candidate solutions. The value of parameters of each vector in the population must be within certain limits according to:

U i  U imin  rand  (U imax  U imin)

(30)

where: U imin ,U imax are the minimum and maximum values of decision variables, respectively, and rand is uniformly distributed number between [0 and 1], i=1, 2,..., NP. The selection of NP affects the optimization process and it depends on the size of the problem. Increasing the number of populations means high computational time but decreases the risk to be trapped in local minimum. Storn and Price [26] observed that, for real-world engineering problems with “D” control variables, a value of 5D or 20D may be a proper choice for NP but a value less than 2D for NP is a refused choice. In [27], Storn and Price choose a population size less than 20D in many situations in their investigations. In [47], it is suggested to use NP greater than or equal to 4D. In [48] on the other hand, 5D is preferred as a first choice and then, the value of NP is modified until reaching a good solution. 2. Mutation process: After creating initial population, the operations of mutation, crossover and selection are applied. There are different strategies of mutation, crossover and selection that affect the optimization process. In this work, the code of the mutation type process is DE/rand/1. The mutation process depends on perturbing the selected vector U 3 randomly using the difference between two other randomly-selected vectors ( U2 , U1 ). All of these vectors must be dissimilar so that the population contains at least four individuals. The mutation process is applied according to:

U 4  U 3  F  (U 1 U 2).

(31)

where, U 1, U 2 , U 3 and U 4 are randomly selected vectors, U4  U3  U2  U1 and F is the scaling factor, whose value is (0 ≤ F 9

≤ 1.2). This factor controls the robustness and speed of the search. Low values of the scaling factor increase the rate of convergence but also increase the risk of being trapped in local optima. 3. Crossover process: To increase the diversity of the population, a crossover operator is applied. The crossover operator produces a trial vector, which is used in the selection process. The trial vector is the combination of parent vector ( Ui ) and mutant vector ( U 4 ). It depends on different distributions such as uniform, binomial and exponential distributions. DE uses two types of crossover, where the first one is the exponential crossover that was used in the original work of Storn and price [26], while the binomial distribution was used in many recent applications [47]. According to [50]-[54], comparative studies between both types of crossover process showed better performance for binomial crossover and hence, it is applied in this paper. In addition, the exponential-type crossover is performed with the D variables in one loop until its value is maintained within the crossover constant (CR) bound. When a picked number between 0 and 1 is beyond the CR value, the crossover process will stop and the remaining D variables are left intact. The crossover is performed in binomial type on all D variables as far as the randomly picked number, between 0 and 1, is within the CR value. Generally, exponential and binomial crossovers provide similar results for high values of CR. Furthermore, there is a small range of CR values (typically [0.9, 1]) to which the DE is sensitive in case of exponential crossover. On the other hand, for the same value of CR, the exponential variant needs a larger value for the scaling parameter F in order to avoid premature convergence [30], [37] and [53]. The random distribution is chosen in the range of [0, 1] and it is compared to CR. The parameters of trial vector are selected to be equal to parameters of mutant vector if the value of random distribution is lower than or equal to the CR. On the other hand, the parameters of the trial vector are selected equal to parameters of parent vector if random is greater than CR according to (32). U 4 if r(0,1)  CR or j  q Ut   U i if r(0,1)  CR and j  q

(32)

where, j=1, 2,…, D and q is a randomly chosen index  [1, 2,…, D]. The value of CR is chosen in the range of [0, 1], where the value of 1 means selecting parameters from mutant vector, while the zero value means selecting most of parameters from parent vector. The value of crossover is selected to ensure that the trial vector must have parameters from mutant vector.

4. Selection process: In this process, the vectors that compose the next generation will be selected. The fitness of parent vector is compared with the fitness of trial vectors and the vectors of better fitness are selected according to (33): U t

Ut 1  

 Ui

if f(x)  U t   f Ui 

(33)

otherwise

V.

PROPOSED MODIFICATION OF DIFFERENTIAL EVALUATION

A) Modified Mutation-Factor Differential Evaluation In this section, modified mutation-factor differential evaluation (MFDE) is applied to improve the traditional DE. The scaling factor (F) is modified according to (34): F  S r (0, 1)2 d  b

(34) 10

where: d is a linear decreasing factor in the range between [0.2 and 1.2], r is a random variable, S is an acceleration factor, b is a deceleration factor against acceleration factor. The factor “d” cannot equal zero to avoid setting “F” to 0. This factor increases the capacity to explore the search space and the ability of exploration. On the other hand, when combining the random variable “r” in the optimization process, the value of scaling factor “F” will fluctuate. The scaling factor gives a probability to decrease the risk of trapping in local minima and increases the regional improvement capacity. The acceleration factor (S) balances between global exploration and local exploitation. When improved fitness function reduces the value of F, the speed of convergence increases. The importance of deceleration factor appears when the fitness function is not improved. In this case, the value of F will be increased and hence, prevents dropping in local optima.

B) Modifying Penalty Factors Due to the importance of penalty factors in the optimization algorithm, the choice of proper penalty factors affects the efficiency of optimization. Few papers presented methods for dealing with penalty factors in case of DE algorithm. In the proposed optimization technique, three rules are considered as follows: 1) Any feasible solution is preferred over infeasible solution 2) For two feasible solutions, the individual that has better fitness is preferred 3) For two infeasible solutions, the individual with lower violation is preferred The first two rules are applied for feasible solutions, where the dependent variables do not violate the constraints. Thus, these rules are applied for acceptable solutions. Applying the third rule in the study is important to alleviate the violation effects of the system dependent variables. From this view point, the proposed MDEA is enhanced with an adaptive penalty term, which is added to the fitness function. This penalty term aims to enhance the system convergence. The violation of dependent variables can be modeled as follows: Violation term= [max (0, (current dependent variables-maximum limits of dependent variables)), max (0, (minimum limits of dependent variables – current dependent variables))]

(35)

In terms of ORPM problem, the added violation term is written as: min max min viol ( DV )  [max(0, Q gi  Qmax gi ), max(0, v Li  v Li ), max(0, Q gi  Q gi ), max(0, v Li  v Li )]

(36)

where, DV is an abbreviation of dependent variables. The violation term is learned by an adaptive penalty coefficient α. The penalty factor is updated according to:

 n1   n  

(37)

where,  n 1 is a new penalty factor for iteration (n+1),  n is the previous penalty factor for iteration (n) and  is the updating formula of the penalty factor. The updating formula of penalty factor is calculated as follows:

 

| fit n  fit n 1   |

(38)

Depn 1  Depn  

where, fit n , fit n 1 are old and new fitness functions, Depn 1, Depn are new and old dependent variables, respectively, and  and  are constants that are obtained empirically.

VI. DIFFERENTIAL EVALUATION-BASED OPTIMAL REACTIVE POWER MANAGEMENT 11

In the ORPM problem, the solution represents the setting of control variables. In this paper, the control variables are the generation-bus voltage ( vg ) and sources reactive power ( Qsw ). The objective is to minimize a function comprising active power losses and voltage deviation considering equality and inequality constraints according to (6) through (12). Equality constraints are satisfied by Newton-Raphson Algorithm. The inequality constraints are taken into consideration by adding penalty functions to the objective function for dependent variables. On the other hand, control variables are self-constrained through the algorithm. The objective function is calculated considering a penalty term as follows: min fitness  wPL  (1  w)VD    viol  DV . 

(39)

where,  is the penalty factor of reactive power generation and load bus voltage. The previous objective is subjected to system constraints presented by equations (7)-(13). Figure 1 shows the flowchart of the proposed MDEA for solving ORPM problem, while Figures 2 through 5 show the single line diagrams of the IEEE 14-bus, the IEEE 30-bus, the IEEE 57-bus power systems and the Western Delta network, respectively.

VII. IMPLEMENTATION OF THE METHODOLOGY A) Test systems Three standard IEEE systems are used to show the effectiveness and capability of differential evolution algorithm after modifying scaling factor and penalty factor for solving ORPM problem. The test networks are the IEEE 14-bus, the IEEE 30-bus, the IEEE 57-bus power systems, with their data are obtained from [49]. The results obtained for different case studies are compared with those obtained using LP, particle swarm optimization technique, genetic algorithm, ant colony and two versions of DE. These comparable results are reported in [1], [34]. The IEEE 14-bus test system contains five generation buses, i.e., buses 1, 2, 3, 6 and 8, three on-load tap changers, i.e., in lines 8, 9 and 10. In addition, it has reactive power sources at buses 9 and 14. The control variables in this paper are (v g1, v g 2 , Qsw9 and Qsw14) , while the dependent variables are (Q g1, Q g 2) in addition to load voltages. The second test system is the IEEE 30-bus test system that comprises 41-lines, six generators allocated at buses 1, 2, 5, 8, 11 and 13, four on-load tap changers allocated in lines 11, 12, 15 and 36 and reactive power sources at buses 10, 17 and 24. The allowable range of the voltages is (1.0  5%) of the specified voltages for both systems. The third system, i.e., IEEE 57-bus system, contains 80 lines, 7 generators allocated at buses 1, 2, 3, 6, 8, 9 and 12. In this system, no reactive power sources are used. In addition to the previous test systems, a real system, i.e., the Western Delta network is used to prove the validity of the algorithm. This system contains 52 buses, 108 lines and 8 generators allocated at buses 1-8. Table 1 summarizes the parameters of the modified DE algorithm for the four test systems. From this table, it is observed that parameters of MDEA are constant for the test power systems. The modified algorithm is used for solving the optimization problems considering three case studies as follows: 

The first case study (Case 1, w1=1, w2=0) aims to minimize the transmission power loss as a single objective function



The second case study (Case 2, w1=0, w2=1) aims to minimize the voltage deviation at load buses as a single objective function



The third case study (Case 3, w1=w2=0.5) is to minimize a multi-objective function of both power loss and voltage deviation. The control variables are the generation voltage and setting of the reactive power sources, while dependent variables are load voltages and reactive power of generators. The modified differential evolution procedure is implemented by using 12

MATLAB 7.14-2012a. The base case involves the initial values of control variable and dependent variable. This case is customized using NR power flow solution. The following two subsections describe the ORPM solution using the proposed approach for the four test systems. B) Results and discussion 1) Results of IEEE 14-bus system The three case studies are discussed considering two operating conditions. The first refers to the existence of generator and switchable devices as control variables, while the other operating condition considers only the generators as controllable devices. Table 2 summarizes the ORPM results obtained using the MDEA for the three case studies. This table shows the settings of control variable, dependent variables and the corresponding objective functions (power losses and voltage profile) for all case studies. From this table, it is observed that high saving in power losses and reduction in voltage deviation are achieved. In Case 1, the power losses are reduced from 13.4 MW in the base case to 12 MW with a total reduction of about 10.45%, which is the highest reduction level compared to the case studies (Cases 2 and 3). In Cases 2 and 3, on the other hand, the power losses are reduced by 6.94% and 8.4%, respectively. The voltage deviation is minimized to the lowest value for Case 2 (0.0165 pu) when the voltage deviation is considered as a single objective. In the third case, the multi-objective function is minimized to improve the voltage and power loss at the same time. Both variables achieve acceptable levels compared to the base case with a total saving in the power losses of 8.4% and enhanced voltage profile by reducing the voltage deviation from 0.0479 to 0.0173 pu. When voltages at generation buses are selected as the only control variables without switchable devices, more flexible reactive power management is achieved compared to the corresponding case study with switchable devices and two voltages at generation buses. Table 3 summarizes the setting values of reactive power generation and violated buses with considering reactive power sources and the possible control variables related to the voltages at generation buses. In Case 1, the reduction in power losses is increased to 13.43%, while an increase in power losses occurs when the voltage profile is optimized as a single objective (13.47 MW). The multi-objective solution improves both real power losses and the corresponding voltage deviation to 11.76 MW and 0.0139 pu, respectively. The lowest level of voltage deviation is occurred in case 2 (0.0053 pu). Table 4 illustrates a comparison between results obtained using different optimization techniques in the literature such as LP, GA, DE, PSO and ant colony optimization (ACO). The obtained results in this paper are competitive with those reported in the literature. Figure 6 shows the convergence curves for optimizing the 14-bus system for Cases 1 through 3 when considering multiple reactive power sources. In addition, Figure 7 shows the convergence characteristics of the fitness function for the three case studies for the IEEE 14-bus system when the voltages at generation buses are considered as the only controllable reactive power sources. These figures show good convergence characteristics for the case studies. The statistical evaluation factors of the proposed adaptive differential evolution algorithm for IEEE 14-bus test system are given in Table 5. It is clear that the highest reduction in system losses is obtained in Case 1 (11.596 MW) compared to other two cases (2 and 3). The lowest voltage deviation in the three case studies is obtained in Case 2 (0.00527) with a standard deviation of 0.0000268 pu. The multi-objective case study (Case 3) results in 11.651 MW power losses and 0.01321 pu voltage deviation. 2) Results of IEEE 30-bus system Similar to the previous network, Table 6 shows power loss, voltage deviation, maximum and minimum limits for all cases considering the IEEE 30-bus test system. The obtained ORPM solution using the MDEA is also shown for the three case studies. This table shows the settings of control variable, dependent variables and the corresponding objective functions (power losses and 13

voltage profile) for all case studies. The power losses for the base-case is 17.5 MW, while in case 1, the power loss is reduced to 15.64 MW by a percentage of 10.83% of the base case. In case 2, the power loss is reduced to 17.38MW, which represents a small improvement because the objective function is to minimize voltage deviation only. In case 3, the power loss is 16.54 MW, which represents a significant improvement, i.e., about 5.75% saving in power losses. On the other hand, voltage deviation for base case is about 0.0261 pu, while it becomes 0. 0243 pu in case 1. Thus, a reduction of 6.89% is achieved regarding the voltage deviation. In case 2, the voltage deviation is improved to be 0.0105 pu, which is a high reduction since the priority is for improving the voltage profile. In the third case, the voltage deviation reaches 0.0154 pu, which is an acceptable value because the fitness function considers both power losses and voltage deviation. Figure 8 shows the convergence curves for IEEE 30-bus system for Cases 1 through 3. This figure shows good convergence characteristics for all case studies. Table 7 compares the performance of the proposed algorithm with other algorithms. From this table, it is obvious that power loss obtained by MDEA is less than that from other algorithms, where it is 15.6 MW according to the proposed algorithm, while it is about 16.49 MW according to ACO, which represents the best results obtained before. The statistical evaluation factors of proposed MDEA for IEEE 30-bus test system are presented in Table 8. In this table, the highest reduction in system losses are obtained in Case 1 (15.644 MW) compared to Cases 2 and 3. The lowest voltage deviation in the three case studies is obtained in Case 2 (0.01051) with a standard deviation of 0.000077 pu. The multi-objective case study (Case 3) results in 16.496 MW power losses and 0.01438 pu voltage deviation. 3) Results of IEEE 57-bus system Table 9 summarizes the values of control variables, reactive power generation, active power losses, reactive power losses and voltage deviation for the four cases, i.e. the base case and the three optimized cases. In this section, voltages at six generation buses are considered as controlled variables, while the slack-bus voltage is kept constant. In the base case, the power losses are 0.2787 pu, average voltage deviation is 0.0246 pu, while reactive power losses are 1.47 pu. After applying optimization algorithm considering voltages at generation buses as the only control variables, power losses become 0.2690 pu, average voltage deviation becomes 0.0262 pu and reactive power losses are decreased to 1.42 pu. In case of minimizing the voltage deviation only, the power losses become 0.2718 pu, voltage deviation becomes 0.0255 pu and the reactive power losses are decreased to 1.4217 pu. For minimizing power losses only, the power losses are decreased to 0.2687 pu, the average voltage deviation becomes 0.0257 pu and reactive power losses are decreased to 1.4162 pu. It is obvious that there is a little improvement in the active power loss for this system. Figure 9 shows the convergence curves for optimizing the IEEE 57-bus system for Cases 1 through 3. This figure shows good convergence characteristics for all case studies. 4) Results of Western Delta network Table 10 shows the values of control variables, reactive power generation, power losses and voltage deviation for base case in addition to the three optimized cases of the Western delta network. Here, the control variables are voltages at generation buses only expect slack bus to reduce the number of switching processes in addition to ensure the validity of the algorithm in the worst case. The power losses for the initial case are 0.4046 pu and average voltage deviation is 0.0265 pu. For the case of minimizing power losses only, the power losses are reduced to 0.3706 pu with a percentage improvement of 8.4%, while the voltage deviation is about 0.0214 pu with a percentage reduction of 19.24%. For case 2, i.e., minimizing voltage deviation only, the power losses are reduced to 0.3853 pu with a percentage improvement of 4.77%, while the average voltage deviation is 0.0150 pu with a 43.39% improvement. For minimizing the summation of power losses and voltage deviation, significant improvements are achieved, where the power losses are minimized to 0.3818 pu, while the voltage deviation is about 0.0152 pu. 14

Figure 10 shows the convergence curves for the Western Delta system for Cases 1 through 3. This figure shows good convergence characteristics for all case studies. 5) Results of multi-objective model The weights are varied to get the Pareto-dominated solutions for all test systems. Figure 13 gives the Pareto solutions for the tested power systems. The best compromise solutions for the tested system are reported in Table 11. It is clear that the obtained compromise solution is worse than the obtained results using the proposed MDEA.

VIII. INFLUENCE OF PROPOSED OPTIMIZATION PARAMETERS In this section, the influence of various optimization parameters is discussed. The studied tests for parameters influences are carried out on the IEEE 30-bus test system. The influence of the parameter settings of the proposed MDEA are reported in Table 12 through Table 14. Figure 11 shows the influence of adaptive parameters of the proposed MDEA on four test conditions when the power loss is considered as an objective function. It has found more stable solutions over the 50 run using the proposed adaptive setting parameters regarding least standard variation as shown in Table 12. Figure 12 shows the penalty factor variation, which is dynamically updated for each population for different values of β and γ factors. In Figure 12, the smallest penalty settings at different buses need lower levels of β and γ factors. Therefore, the influence of changing factors like β and γ is greatly affected. Table 13 and Table 14 show the parameters variation and their influence on the fitness function for different runs and weighting factors. IX.

CONCLUSIONS

This paper has investigated the implementation of a modified differential evolution algorithm (MDEA) for solving the optimal reactive power dispatch applied to three standard systems, namely the IEEE 14-bus, the IEEE 30-bus and the IEEE 57-bus power systems in addition to a large real power system in Egypt, i.e., Western Delta network. The results from this paper ensure that the modified DE algorithm is an efficient tool in minimizing power losses and voltage deviation considering equality and inequality constraints. In terms of the ORPM problem, the highest level of power loss reduction is obtained using the proposed MDEA with adaptive penalty factor added to the fitness function. Also, the voltage profile for all case studies is improved with more reduction in the voltage deviation. Adaptive settings of the violation elimination strategy prevent the violation with adaptive penalty factor applied to dependent variables. Updating the penalty factor each generation for each population improves the performance of MDEA as an optimization algorithm and minimizes constraints violation. The obtained results prove also that using generation bus voltages as control variable is more effective in solving ORPD problem. The results prove the flexibility of synchronous generators as efficient reactive power sources compared to other switchable devices in terms of the power losses reduction and voltage profile enhancement at load buses. This strategy has the advantages of reducing the number of switching processes and control actions in addition to increasing the life time of equipment. Modifying the scaling factor and updating penalty factor improved the performance and convergence capability of DE algorithm as an optimization tool. The proposed MDEA is efficient and has the capability to solve single/multi-objective problems with reasonable solutions compared to other optimization techniques. The results from this paper ensure that the MDEA with adaptive penalty factor is an efficient alternative competitive tool to solve other optimization problems in power systems.

REFERENCES 15

[1]

A. A. Abou El-Ela, A. M. Kinawy, R. A. El-Sehiemy and M. T. Mouwafi, Optimal reactive power dispatch using ant colony optimization algorithm, Electrical Engineering, 93 (2): 103-116, 2011.

[2]

D. Devaraj and J. Preetha Roselyn, Genetic algorithm based reactive power dispatch for voltage stability improvement, International Journal of Electrical Power & Energy Systems 32 (10): 1151-1156, 2010.

[3]

N. Mancer, B. Madad, K. Sariri and M. Hamed, Multi objective for optimal reactive flow using modified PSO considering TCSC, International Journal of Energy Engineering, 2(4):165-170, 2012.

[4]

D. C.Yu, J. E. Fagan, B. Foote and A. A. Aly, An optimal load flow study by the generalized reduced gradient approach, Electric Power Systems Research, 10 (1):47-53, 1986.

[5]

K. Aoki, M. Fan and A. Nishikori. Optimal VAR planning by approximation method for recursive mixed integer linear programming, IEEE Transactions on Power Systems, 3 (4):1741–7, 1988.

[6]

N. I. Deeb and S. M. Shahidehpour, An efficient technique reactive power dispatch using a revised linear programming approach, Electric Power Systems Research, 15(2):121–34, 1988.

[7]

J. Z. Zhu and X. F. Xiong, Optimal reactive power control using modified interior point method, Electric Power Systems Research, 66 (2):187–192, 2003.

[8]

V.A. Sousa, E.C. Baptista and G.R.M. Costa, Optimal reactive power flow via the modified barrier Lagrangian function approach. Electric Power Systems Research, 84 (1):159–64, 2012.

[9]

M. Granada, M. J. Rider, J. R.S. Mantovani and M. Shahidehpour, A decentralized approach for optimal reactive power dispatch using a Lagrangian decomposition method, Electric Power Systems Research, 89 (1): 148-156, 2012.

[10]

P. Subbaraj and P. N. Rajnaryanan, Optimal reactive power dispatch using self-adaptive real coded genetic algorithm, Electric Power Systems Research, 79 (2):374–381, 2009.

[11]

Q.H. Wu, Y.J. Cao and J.Y. Wen, Optimal reactive power dispatch using an adaptive genetic algorithm, International Journal of Electrical Power & Energy Systems, 20 (8):563-569, 1998.

[12]

W. Yan, S. Lu and D.C. Yu, A novel optimal reactive power dispatch method based on an improved hybrid evolutionary programming technique, IEEE Transactions on Power Systems, 19 (2):913–918, 2004.

[13]

P. K. Roy, S. P. Ghoshal and S. S. Thakur, Optimal reactive power dispatch considering FACTS devices using biogeography based optimization, Electric Power Component & Systems, 39 (8):733–750, 2011.

[14]

P. K. Roy, S. P. Ghoshal and S. S. Thakur, Optimal VAR control for improvements in voltage profiles and for real power loss minimization using biogeography based optimization, International Journal of Electrical Power & Energy Systems, 43(2):830–838, 2012.

[15]

P. K. Roy, B. Mandal and K. Bhattacharya, Gravitational search algorithm based optimal reactive power dispatch for voltage stability enhancement, Electric Power Component Systems, 40 (9):956–976, 2012.

[16]

D. Chaohua, C. Weirong, Z. Yunfang and Z. Xuexia, Seeker optimization algorithm for optimal reactive power dispatch. IEEE Transactions on Power Systems, 24(3):1218–1231, 2009.

[17]

K. Ayan and U. Kılıc, Artificial bee colony algorithm solution for optimal reactive power flow, Applied Soft Computing, 12 (5):1477–1482, 2012.

[18]

A.H. Khazali and M. Kalantar, Optimal reactive power dispatch based on harmony search algorithm. International Journal of Electrical Power & Energy Systems, 33(3):684–692, 2011.

[19]

R. El Sehiemy, A. Abou El-Ela and A. Shaheen, Multi-objective fuzzy-based procedure for enhancing reactive power management, IET Generation, Transmission & Distribution 7 (12): 1453-1460, 2013. 16

[20]

K. Mahadevan and P. S. Kannan, Comprehensive learning particle swarm optimization for reactive power dispatch, Applied Soft Computing, 10 (2):641–652, 2010.

[21]

H. Tehzeeb-Ul-Hassan, R. Zafar, S.A. Mohsin and O. Lateef, Reduction in power transmission loss using fully informed particle swarm optimization, International Journal of Electrical Power & Energy Systems 43 (1): 364-368, 2012

[22]

A. Q. Badar, B. S. Umre and A. S. Junghare, Reactive power control using dynamic Particle Swarm Optimization for real power loss minimization, International Journal of Electrical Power & Energy Systems, 41 (1) :133–136, 2012

[23]

H. Yoshida, Y. Fukuyama, K. Kawata, S. Takayama and Y. Nakanishi Y, ―A particle swarm optimization for reactive power and voltage control considering voltage security assessment,‖ IEEE Transactions on Power Systems 15 (4):1232-9, 2001

[24]

B. Zhao, C. X. Guo and Y. J. Cao, ―A multi-agent based particle swarm optimization approach for reactive power dispatch,‖ IEEE Transactions on Power Systems 20 (2):1070-8, 2005.

[25]

M. M. Rojas A. Sumper, O. G. Bellmunt and A. S. Andreu, ―Reactive power dispatch in wind farms using particle swarm optimization technique and feasible solutions search,‖ Applied Energy; 88 (12):4678–4686, 2011

[26]

R. Storn, and P. Kenneth, Differential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces. Berkeley: ICSI, 1995.

[27]

R. Storn, and P. Kenneth, ―Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,‖ Journal of global optimization 11 (4): 341-359, 1997.

[28]

J. Vesterstrøm and R. Thomsen, ―A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems,‖ in: IEEE Congress on Evolutionary Computation, 2004, pp. 980–987.

[29]

Z. Yang, K. Tang and X. Yao, ―Differential evolution for high-dimensional function optimization,‖ IEEE Congress on Evolutionary Computation (CEC2007), 2007: 3523–3530.

[30]

J. M. Ramirez, J. M. Gonzalez and T. O. Ruben, ―An investigation about the impact of the optimal reactive power dispatch solved by DE,‖ International Journal of Electrical Power & Energy Systems, 33 (2):236–244, 2011.

[31]

M. Varadarajan and K. S. Swarup, ―Network loss minimization with voltage security using differential evolution,‖ Electric Power Systems Research 78 (5):815-823, 2008.

[32]

C.H. Liang, C.Y. Chung, K.P. Wong, X.Z. Duan and C.T. Tse, ―Study of differential evolution for optimal reactive power flow,‖ IET Generation, Transmission & Distribution, 1 (2): 253-260, 2007.

[33]

S. Kannan, S. M. R. Slochanal and N. P. Pandhy, ―Application and comparison of metaheuristic techniques to generation expansion planning problem,‖ IEEE Transactions on Power Systems, 20 (1): 466-475, 2003.

[34]

J. Li, Yuancheng, Yiliang Wang, and Bin Li. "A hybrid artificial bee colony assisted differential evolution algorithm for optimal reactive power flow." International Journal of Electrical Power & Energy Systems 52 (1): 25-33, 2013.

[35]

P. Chiou, C. F. Chang and C. T. Su, ―Ant direct hybrid differential evolution for solving large capacitor placement problems,‖ IEEE Trans. on Power Systems, 19 (4): 1794-1800, 2004.

[36]

L.H. Wu, Y.N. Wang, X.F. Yuan and S.W. Zhou, Environmental/economic power dispatch problem using multi-objective differential evolution algorithm, Electric Power Systems Research 80 (9): 1171-118, 2010.

[37]

A. A. Abou El Ela, M. A. Abido, and S. R. Spea, ―Optimal power flow using differential evolution algorithm,‖ Electric Power Systems Research 80 (7): 878-885, 2010.

17

[38]

Shaheen, A. M., El-Sehiemy, R. A., & Farrag, S. M. (2016). Solving multi-objective optimal power flow problem via forced initialised differential evolution algorithm. IET Generation, Transmission & Distribution, 10(7), 1634-1647.

[39]

Shaheen, A. M., El Sehiemy R. A., and Farrag, S. M. (2015). Adequate Planning of Shunt Power Capacitors Involving Transformer Capacity Release Benefit‖, IEEE systems Journal, doi: 10.1109/JSYST.2015.2491966.

[40]

Sakr, W. S., El-Sehiemy, R. A., & Azmy, A. M. (2016). ―Optimal allocation of TCSCs by adaptive DE algorithm‖. IET Generation, Transmission & Distribution, 10(15), 3844-3854.

[41]

Shaheen, A. M., Farrag, S. M., & El-Sehiemy, R. A. (2016), ―MOPF solution methodology,‖ IET Generation, Transmission & Distribution,pp. 1-12, doi: 10.1049/iet-gtd.2016.1379.

[42]

S.M. Islam, S. Das, S. Ghosh, S. Roy and P.N. Suganthan, An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42 (2): 482-500, 2012.

[43]

C. Wei, Z. Rong and W. Ping, Improving the performance of differential evolution algorithm with modified mutation factor, in the proceeding 2009 international conference machine learning and computing volume 3(2011).

[44]

O. Yeniay, Penalty function methods for constrained optimization with genetic algorithms, Mathematical and Computational Applications, 10 (1): 45-56, 2005.

[45]

M. Varadarajan, and K. S. Swarup, Differential evolutionary algorithm for optimal reactive power dispatch, International Journal of Electrical Power & Energy Systems 30 (8): 435-441, 2008.

[46]

M. Varadarajan, and K. S. Swarup, Differential evolution approach for optimal reactive power dispatch, Applied Soft Computing 8 (4): 1549-1561, 2008.

[47]

J. Vesterstrøm and R. Thomsen, A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems, in: IEEE Congress on Evolutionary Computation, 2004, pp. 980–987.

[48]

Zhenyu Yang, Ke Tang and Xin Yao, Differential evolution for high-dimensional function optimization, in: IEEE Congress on Evolutionary Computation (CEC2007), 2007: 3523–3530.

[49]

Washington University. http://www.ee.washington.edu/research/pstca/.

[50]

D. Zaharie, ―Influence of crossover on the behavior of Differential Evolution Algorithms,‖ Applied Soft Computing 9 (2009) 1126–1138.

[51]

J. Tvrdík, Adaptation in differential evolution: A numerical comparison. Applied Soft Computing, 9 (3), 1149-1155, 2009.

[52]

Mallipeddi, R., Suganthan, P. N., Pan, Q. K., & Tasgetiren, M. F. (2011). Differential evolution algorithm with ensemble of parameters and mutation strategies. Applied Soft Computing, 11(2), 1679-1696 .

[53]

Z. Daniela, A comparative analysis of crossover variants in differential evolution, in: Proceedings of the International Multiconference on Computer Science and Information Technology, 2007, pp. 171–181.

[54]

Zaharie, D. (2007). A comparative analysis of crossover variants in differential evolution. Proceedings of IMCSIT 2007, 171-181.

[55]

Gjorgiev, B., Kančev, D., Čepin, M., & Volkanovski, A. (2015). Multi-objective unit commitment with introduction of a methodology for probabilistic assessment of generating capacities availability. Engineering Applications of Artificial Intelligence, 37, 236-249. 18

[56]

Li, Y. F., Pedroni, N., & Zio, E. (2013). A Memetic Evolutionary Multi-Objective Optimization Method for Environmental Power Unit Commitment. Power Systems, IEEE Transactions on, 28(3), 2660-2669.

[57]

A. M. Shaheen, R. El Sehiemy, and S. Farrag, ―Integrated Strategies of Backtracking Search Optimizer for Solving Reactive Power Dispatch Problem,‖ IEEE systems Journal, 2016, doi: 10.1109/JSYST.2016.2573799

[58]

Bouchekara, H. R. E. H., Chaib, A. E., Abido, M. A., & El-Sehiemy, R. A. (2016). Optimal power flow using an Improved Colliding Bodies Optimization algorithm. Applied Soft Computing, 42, 119-131.

Figures

19

Start Read input data Perform Newton Raphason load flow calculations for base case to get the voltage profile and power losses Create the initial population, then compute the objective function of initial population and determine the best fit vector

Perform mutation process Set parameter values Yes at max or min limits

Check if Max/Min limits of parameters are violated

No Perform crossover process Perform selection process Determine the best, average and worst objective function value of the generation

No

Stop criteria is satisfied Yes

Run the final power flow using the parameters of the best vector to obtain the final voltage profile and power losses

End Figure 1 Flowchart of MDEA for solving ORPM problem

20

13

12

14 11

G

10

1 9

6

8

G

C

7 5

4

2 3

G Figure 2 Single line diagram of the IEEE 14-bus power system

28

27

29 30

15

26

25

23

24

18

19

17

20 16

14 13

12

10

3

22

9

11

1

21

4

6 7

2

5

Figure 3 Single line diagram of the IEEE 30-bus power system 21

8

Figure 4 Single line diagram of the IEEE 57-bus system

Figure 5 Single line diagram of the 52-bus actual system (WDN)

22

0.38

0.93 0.92 0.91 0.9 0.89 0

50

100

150

200

0.84 Fitness function

0.94

Fitness function

Fitness function

0.95

0.37 0.36 0.35 0.34 0

50

Iteration number

100

0.82 0.8 0.78 0.76 0

150

50

100

150

Iteration number

Iteration number

Figure 6 Convergence curves of objective function for IEEE 14-bus test system case studies 1 through 3 0.5

0.94 0.92 0.9 0.88 0.86 0

100

200

300

0.73

Fittness function

0.96

Fitness function

Fitness function

0.98

0.4 0.3 0.2 0.1 0

400

50

100

150

0.72 0.71 0.7 0.69 0.68 0.67 0

200

50

Iteration number

Iteration number

100

150

200

Iteration number

1

1.2

0.9

1.1 1 0.9 0.8 0

50

100

150

200

Iteration number

250

300

1.6

Fitness function

1.3 Fitness function

Fitness function

Figure 7 Convergence curves of fitness function for ORPM for Cases 1 through 3 without switchable devices

0.8 0.7 0.6 0.5 0.4 0

50

100

150

Iteration number

200

250

1.4 1.2 1 0.8 0

50

100

150

Iteration number

Figure 8 Convergence curves for cases 1 through 3 using MDEA applied to IEEE 30 bus test system

Figure 9 Convergence curves for cases 1 through 3 using MDEA applied to IEEE57-bus system

23

200

Figure 10 Convergence curves for cases 1 through 3 using MDEA applied to Western Delta network 19 test 1 test 2 test 3 adaptive settings

objective function (Ploss in MW)

18.5 18 17.5 17 16.5 16 15.5 15

5

10

15

20

25 run number

30

35

40

45

50

Figure 11 Influence of adaptive parameters of the proposed MDEA on the best power loss for 50 run 110 100 90 Test 1 Test 2

Penalty Factor

80 70 60 50 40 30 20 0

5

10

15

20 25 Population size

30

35

40

Figure 12 Penalty factor variation against population size for varied β and γ a. Test 1 when β=10 and γ= 100

b.Test 2 when β=1000; γ= 2000

24

45

17.4

12.5

17.35 17.3

12.45

Power losses (MW)

Power losses (MW)

17.25 12.4

12.35

17.2 X: 0.0116 Y: 17.11

17.15 17.1 17.05 17

12.3

16.95 12.25 0.0164

0.0166

0.0168

0.017

a.

0.0172 Voltage Dev.

0.0174

0.0176

0.0178

16.9 0.0105

0.018

0.011

0.0115

0.012

0.0125

Voltage Dev.

14-bus test system

b. 30-bus test system 38

26.96 26.955

37.9

26.95

37.8 26.945

37.7

26.94 26.935

X: 0.0159 Y: 37.62

37.6

26.93

37.5

26.925

X: 0.0257 Y: 26.92

37.4

26.92

37.3

26.915 26.91 0.0255

0.0256

0.0257

c.

0.0258

0.0259

0.026

37.2 0.0155

0.0261

57-bus test system

0.016

0.0165

0.017

0.0175

d. Western Delta real system

Figure 13 Pareto solution for the tested power systems

25

0.018

Tables Table 1 Settings of MDEA Parameters for different tested systems Parameter

IEEE 14-bus

IEEE 30-Bus

IEEE 57-bus

Western delta

Maximum number of generation

150-200

200-400

2000

500

Number of control variables

4-5

5

6

8

Number of populations

20

20

30

40

Crossover factor

0.95

0.95

0.45-0.65

0.95

Acceleration factor

1.8

1.8

1.8

1.8

Deceleration factor

1

1

1

1

Decreasing factor

1

1

1

1

Table 2 ORPM solution results using the MDEA for different case studies Variables in pu

Maximum limits

Minimum limits

Base case

Case 1

Case 2

Case 3

v g1

1.05

0.95

1.06

0.9968

1.01

1.00

vg2

1.05

0.95

1.045

0.9742

0.988

0.9862

Q sw9

0.2

0.05

0.00

0.05

0.05

0.05

Qsw14

0.2

0.05

0.00

0.05

0.155

0.05

vL6

1.06

0.94

1.07

1.00

1.01

1.01

vL7

1.06

0.94

1.062

0.99

1

1

v L8

1.06

0.94

1.09

1.02

1.03

1.03

v g6

1.06

0.94

1.07

1.022

1.015

1.02

vL7

1.06

0.94

1.062

1

1

1

v g8

1.06

0.94

1.09

1.046

0.992

1.03

Q g1

0.10

0

-.165

0

0

0

Q g2

0.50

-0.40

0.435

-0.0494

0.459

0.0157

Q g3

0.40

0

0.258

0.40

0.40

0.40

Q g6

0.24

-0.06

0.127

0.23

0.0101

0.2152

Q g8

0.24

-0.06

0.176

0.24

-0.06

0.1934

Q g1

0.10

0

-.165

0

0

0

Q g2

0.50

-.40

0.435

0.2745

0.2771

0.2745

power losses

-

-

0.134

0.12

0.1247

0.1228

Voltage deviation

-

-

0.0479

0.0249

0.0165

0.0173

% Saving in power losses

-

-

0.00

10.45%

6.94%

8.4%

-

-

0.00

48%

65.55%

63.9%

% Reduction in voltage deviation

26

Table 3 Solution of ORPM problem without considering the existence of switchable devices for different case studies Control variable setting in pu

Maximum limits

Minimum limits

Initial case

Case 1

Case 2

Case 3

v g1

1.05

0.95

1.06

0.988

1.05

0.995

vg 2

1.05

0.95

1.045

0.962

1.03

0.97

v g3

1.05

0.95

1.01

0.95

1.001

0.955

v g6

1.05

0.95

1.07

1.022

1.015

1.02

v g8

1.05

0.95

1.09

1.046

0.99

1.035

power losses

-

-

0.134

0.1160

0.1347

0.1176

Voltage deviation

-

-

0.045

0.0154

0.0053

0.0139

%Saving in power losses

-

-

0.00

13.43%

-0.5%

12.23%

-

-

0.00

65.8%

88.2%

69.11%

% Reduction in voltage deviation

Table 4 Evaluation of the proposed MDEA against other optimization techniques

LP [1]

GA [1]

Power losses in pu

0.1273

0.1124

0.1111

0.1094

Voltage deviation

0.0414

0.0258

0.022

-

-

-

-

Average time of convergence in (s) Number of population

DE [34] Hybrid ABC-

PSO [1] ACO [1]

1

Proposed

DE [34]

MDEA

MDEA2

0.1237

0.1237

0.1228

0.116

0.0173

-

-

0.0173

0.0152

-

-

30.37

11.56

1.17

1.9

-

-

60

20

20

25

1

refers to the cases of considering both generators and switchable devices as control variables

2

refers to the cases of considering only generators as control variables

*

Proposed

*

ABC-DE refers to artificial bee colony assisted DE

Table 5 Statistics evaluation of IEEE 14-bus case studies for 50 run Case 1 factors PL

VD

Min

0.11596

0.0149

Case 2 Fitness

PL

VD

0.1159

0.13442

0.00527

function

Case 3 Fitness

Fitness

PL

VD

0.00527

0.11651

0.01321

0.08085

function

function

Max

0.1161

0.0155

0.11610

0.13476

0.00542

0.00542

0.11873

0.01461

0.08180

Aver

0.116

0.0153

0.11598

0.13470

0.00529

0.00529

0.11764

0.01387

0.08132

Sd

0.0000298

0.0003347

0.0000298

0.0002273

0.0000268

0.0000268

0.000923

0.000542

0.000388

27

Table 6 ORPM solution for the IEEE 30-bus test system using MDEA Variables

Maximum limits

Minimum limits

Base case

Case1

Case2

Case3

v g1

1.05

0.95

1.06

1

1.05

1.028

vg 2

1.05

0.95

1.045

0.973

1.027

1

v g5

1.05

0.95

1.01

0.95

0.995

0.9761

v g8

1.05

0.95

1.01

0.955

0.991

0.9773

v g11

1.05

0.95

1.082

1.05

1.05

1.05

v g13

1.05

0.95

1.07

1.0487

1.012

1.05

Q sw10

0.2

0.05

-

0.05

0.05

0.05

Q sw17

0.15

0.05

-

0.0501

0.05

0.0501

Qsw24

0.1

0.05

-

0.0628

0.1

0.1

v g11

1.06

0.94

1.082

1.05

1.05

1.05

v g13

1.06

0.94

1.07

1.0487

1.012

1.05

Q g1

0.10

0

-.204

0

0

0

Q g2

0.50

-0.40

0.567

0.1023

0.4297

0.2311

Q g5

0.40

-0.40

0.357

0.40

0.40

0.40

Q g8

0.40

-0.10

0.361

0.40

0.40

0.40

Q g11

0.24

-0.06

0.161

0.2286

0.144

0.1646

Q g13

0.24

-0.06

0.105

0.2219

-0.0309

0.1526

power losses

-

-

0.175

0.1564

0.1738

0.1654

Voltage deviation

-

-

0.026

0.0243

0.0105

0.0154

Saving in power losses

-

-

0.00

10.83%

0.96%

5.75%

Reduction in voltage deviation

-

-

0.00

59.77%

41%

6.89%

Table 7 Evaluation of the proposed MDEA against other optimization techniques for the IEEE 30-bus system LP [1]

GA [1]

PSO [1]

ACO [1]

DE [34]

Hybrid ABC- DE [34]

Proposed MDEA

Power loss in pu

0.2014

0.1922

0.1634

0.1631

0.162184

0.162184

0.1564

Voltage deviation in pu

0.030

0.0269

0.0257

0.0135

-

-

0.0243

Number of population

-

-

-

-

72

24

40

Average time of convergence in (s)

-

-

-

-

41

24

4.6

28

Table 8 Statistics evaluation of the IEEE 30-bus case studies for 30 run

factors

Case 1

Case 2

Case 3

PL

VD

Fitness function

PL

VD

Fitness function

PL

VD

Fitness function

Min

0.15644

0.02418

0.15644

0.17365

0.01051

0.01051

0.16496

0.01438

0.13959

Max

0.15658

0.02491

0.15658

0.17505

0.01066

0.01066

0.16774

0.01572

0.14171

Aver

0.15648

0.02434

0.15648

0.17392

0.01056

0.01056

0.16597

0.01516

0.14034

Sd

0.022172

0.000347

0.0000655

0.000421 0.000077

0.000077

0.001479

0.00112

0.001091

Table 9 ORPM solution for the IEEE 57-bus system using MDEA Variables

Maximum limits

Minimum limits

Base case

Case 1

Case 2

Case 3

v g1

1.06

0.94

1.04

1.04

1.04

1.04

vg 2

1.06

0.94

1.0100

1.0276

1.0265

1.0268

v g3

1.06

0.94

0.9850

1.0148

1.0040

1.0182

v g6

1.06

0.94

0.9800

1.0245

1.0036

1.0300

v g8

1.06

0.94

0.9800

1.0136

1.0206

1.0283

v g9

1.06

0.94

1.0150

0.9861

0.9858

0.9935

v g12

1.06

0.94

1.0100

1.0276

1.0265

1.0268

Q g1

2.0000

-1.4

1.2885

0.7038

0.7899

0.6248

Q g2

0.5000

-0.17

-0.0075

0.4812

0.5565

0.4020

Q g3

0.6000

-0.1

-0.0090

0.4095

0.2474

0.4130

Q g6

0.2500

-0.08

0.0087

0.4348

0.1173

0.4087

Q g8

2.0000

-1.4

0.6210

0.1999

0.2321

0.1087

Q g9

0.0900

-.03

0.0229

1.0499

1.4014

1.3200

Q g12

1.5500

-1.5

1.2863

0.0789

0.0332

0.0645

power losses

-

-

0.2787

0.2687

0.2718

0.2690

Voltage deviation

-

-

0.0246

0.0257

0.0255

0.0262

Reactive power loss

-

-

1.47

1.4162

1.4272

1.42

Saving in power losses

-

-

0.00

3.58%

2.47%

3.4%

Reduction in voltage deviation

-

-

0.00

-4.47%

-3.65%

-6.5%

Reduction in reactive power losses

-

-

0.00

4.08%

2.91%

3.4%

29

Table 10 ORPM solution for Western Delta network using MDEA Variables

Maximum limits

Minimum limits

Base case

Case 1

Case 2

Case 3

v g1

1.05

0.95

1.05

1.05

1.05

1.05

vg 2

1.05

0.95

1

1.0387

1.0298

1.0304

v g3

1.05

0.95

1

1.0500

1.0500

1.0500

vg 4

1.05

0.95

1

1.0499

1.0086

1.0086

v g5

1.05

0.95

1

1.0491

1.0231

1.0324

v g6

1.05

0.95

1

1.0192

1.0049

1.0049

v g7

1.05

0.95

1

1.0164

1.0145

1.0145

v g8

1.05

0.95

1

1.0096

1.0263

1.0262

Q g1

2.0000

0

0.5671

0.2984

0.3743

0.3620

Q g2

2.0000

0

0.2251

0.2981

0.3742

0.3619

Q g3

2.0000

0

1.0450

1.0536

1.1162

1.1133

Qg 4

2.0000

0

1.1489

1.1826

1.0445

1.0422

Q g5

2.0000

0

0.6794

1.0504

0.8288

0.9560

Q g6

2.0000

0

0.8147

0.6469

0.2445

0.1481

Q g7

2.5000

0

0.8362

0.8184

0.9321

0.9323

Q g8

2.5000

0

0.6014

0.4028

0.8943

0.8807

power losses

-

-

0.4046

0.3706

0.3853

0.3818

Voltage deviation

-

-

0.0265

0.0214

0.0150

0.0152

Saving in power losses

-

-

0.00

8.4%

4.77%

5.56%

-

-

0.00

19.24%

43.39%

42.64%

Reduction in voltage deviation

Table 11 Best compromise solution for variant test systems on the basis of variant weights Test system

Power losses

Voltage deviation

14-bus test system

12.28 MW

0.0173

30- bus test system

17.11 MW

0.0116

57-bus test system

26.92 MW

0.0257

Western Delta real system

37.62 MW

0.0159

30

Table 12: Influence of MDEA parameter variation for 50 run applied to the 30-bus test system Case

Test 1

Test 2

Test 3

Test 4

Penalty Factor

10

5

5

Adaptive

Scaling Factor

0.05

1

0.05

Adaptive

Minimum power loss

15.79154

15.19186

15.08497

15.64379

Average power loss

17.29886

17.16557

17.13488

15.64745

Maximum power loss

18.79606

18.88306

18.68845

15.6577

Standard variation

0.778218

0.867133

0.766701

0.003661

Table 13 Parameters variation at different runs (w1=0.5, w2=0.5) Run 8 System

IEEE 30-bus system





Run 10 Total objective function





Run 15 Total objective function





Total objective function

5

50

0.0921

5

50

0.0923

5

50

0.0923

100

100

0.0921

100

100

0.0922

100

100

0.0923

10

500

0.0922

10

500

0.0922

10

500

0.0922

1

100

0.0922

1

100

0.0922

1

100

0.0921

0.1

10000

0.0921

0.1

10000

0.0922

0.1

10000

0.0921

0.01

100

0.0918

0.01

100

0.0921

0.01

100

0.0920

10

1000000

1.0145

10

1000000

1.0097

10

1000000

1.0103

1

10000000

1.0138

1

10000000

1.09

1

10000000

1.01

IEEE 57-bus

0.001

100000

1.0145

0.001

100000

1.097

0.001

100000

1.002

system

0.0001

100

1.4117

0.0001

100

1.79

0.0001

100

1.6

0.1

1

1.32

0.1

1

1.34

0.1

1

1.35

0.01

100000

1

0.01

100000

1

0.01

100000

0.99

100

100

0.864

2

200

0.8642

0.05

5

0.8643

0.1

10000

0.864

1

10000

0.8640

0.5

50

0.8643

0.001

10

0.8639

10

100

0.8639

5

5000

0.8643

10

100

0.8638

0.001

0.01

0.8640

1

1000

0.8643

2

200

0.863

100

100

0.863

0.1

100

0.8643

0.01

10

0.7585

0.01

10

0.7586

0.01

10

0.7585

WDN

31

Table 14 Parameters variation for equal priority weighting factors (w1=0.5, w2=0.5) 8 runs

system

IEEE 30-bus system

IEEE 57- bus system

WDN

10 runs

S

b

d





0.5

0.5

0.5

0.01

100

0.6

0.6

0.6

0.01

1

0.9

0.9

0.01

1.5

0.9

0.9

1.8

1

1

0.5

0.5

0.6

0.6

1

Total objective

Total objective

S

B

d





0.0921

0.5

0.5

0.5

0.01

100

0.0927

100

0.092

0.6

0.6

0.6

0.01

100

0.0924

100

0.092

1

0.9

0.9

0.01

100

0.0924

0.01

100

0.0919

1.5

0.9

0.9

0.01

100

0.0923

0.01

100

0.0918

1.8

1

1

0.01

100

0.0921

0.5

0.01

100000

0.9973

0.5

0.5

0.5

0.01

100000

1.01

0.6

0.01

100000

0.9994

0.6

0.6

0.6

0.01

100000

1.01

0.9

0.9

0.01

100000

1.0125

1

0.9

0.9

0.01

100000

1.01

1.5

0.9

0.9

0.01

100000

1.0105

1.5

0.9

0.9

0.01

100000

1.015

1.8

1

1

0.01

100000

1

1.8

1

1

0.01

100000

1

0.5

0.5

0.5

0.01

10

0.7623

0.5

0.5

0.5

0.01

10

0.7668

0.6

0.6

0.6

0.01

10

0.76

0.6

0.6

0.6

0.01

10

0.7668

1

0.9

0.9

0.01

10

0.76

1

0.9

0.9

0.01

10

0.7668

1.5

0.9

0.9

0.01

10

0.7598

1.5

0.9

0.9

0.01

10

0.7611

1.8

1

1

0.01

10

0.7585

1.8

1

1

0.01

10

0.7586

function

32

function