Research article Open Access
Particle Distribution Algorithm for Solving Optimal Reactive Power Dispatch Problem
Lenin K1*, Ravindhranath Reddy B2, Surya Kalavathi M3
1Researcher, JNTU, Hyderabad, India.
2Deputy Executive Engineer, JNTU, Hyderabad, India.
3Professor, Department of Electrical and Electronics Engineering, JNTU, Hyderabad, India.
*Corresponding author: Lenin K, Researcher, JNTU, Hyderabad, 500 085, India, Tel: 91-9677350862; E-mail: @
Received: January 06, 2017; Accepted: January 25, 2017; Published:January 31, 2017
Citation: Pandey A, Burse K (2017) Cascade Neuro-Fuzzy Architecture Based Mobile-Robot Navigation and Obstacle Avoidance in Static and Dynamic Environments. Int J Adv Robot Automn 2(1): 1-9.
Abstract
In this paper, a novel perception of the particle swarm optimization (PSO) system called Particle Distribution Algorithm (PDA) is proposed to solve Reactive power problem. In that swarm individual particles are moved away, but discreetly characterize each particle by a field or by distribution & its behaviour absolutely dissimilar from the family of old PSO scheme. The projected PDA has been tested in standard IEEE 30 bus test system and simulation results show clearly the better performance of the proposed algorithm in reducing the real power loss.
Keywords: Particle Swarm Optimization; Particle Distribution; Optimal Reactive Power; Transmission Loss;
Introduction
Various numerical methods like the gradient method [1, 2] Newton method [3] and linear programming [4-7] have been utilized to solve the optimal reactive power dispatch problem. The problem of voltage stability and collapse play a key role in power system planning and operation [8]. Enhancing the voltage stability, voltage magnitudes within the limits alone will not be a reliable indicator to indicate that, how far an operating point is from the collapse point. The reactive power support and voltage problems are internally related to each other.This paper formulates by combining both the real power loss minimization and maximization of Static Voltage Stability Margin (SVSM) as the objectives. Numerous Evolutionary algorithms have been already utilized to solve the reactive power flow problem [9-20]. This paper proposes Particle Distribution Algorithm (PDA) to solve reactive power dispatch problem. Proposed algorithm has better exploration and exploitation capabilities in searching the global near optimal solution & each particle can be abstracted by a field or distribution. The parameters of the distribution are modernized by using a scheme alike to Thompson’s sampling, leading to a completely new and unique perspective on particle swarm systems [21-40]. Proposed PDA has been evaluated in standard IEEE 30 bus test system. Simulation results show that our proposed approach outperforms all the entitled reported algorithms in minimization of real power loss and voltage profile index is enhanced.
Voltage Stability Evaluation
Modal analysis for voltage stability evaluation
Modal analysis is one among best methods for voltage stability enhancement in power systems. The steady state system power flow equations are given by.
[ ΔP ΔQ ]=[ J pθ       J pv   J       J QV       ][ Δθ ΔV ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaWdaeaafaqabeGabaaabaWdbiabfs5aejaadcfaa8aabaWd biabfs5aejaadgfaaaaacaGLBbGaayzxaaGaeyypa0ZaamWaa8aaba qbaeqabiqaaaqaa8qacaqGkbWdamaaBaaaleaapeGaaeiCaiaabI7a caqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckaa8aabeaak8qaca qGkbWdamaaBaaaleaapeGaaeiCaiaabAhaa8aabeaak8qacaGGGcaa paqaa8qacaqGkbWdamaaBaaaleaapeGaaeyCaiaabI7aa8aabeaak8 qacaGGGcGaaiiOaiaacckacaGGGcGaaeiOaiaabQeapaWaaSbaaSqa a8qacaqGrbGaaeOvaaWdaeqaaOWdbiaacckacaGGGcGaaiiOaiaacc kacaGGGcaaaaGaay5waiaaw2faamaadmaapaqaauaabeqaceaaaeaa peGaeuiLdqKaeqiUdehapaqaa8qacqqHuoarcaWGwbaaaaGaay5wai aaw2faaaaa@6804@
Where
ΔP = Incremental change in bus real power.
ΔQ = Incremental change in bus reactive Power injection
Δθ = incremental change in bus voltage angle.
ΔV = Incremental change in bus voltage Magnitude
Jpθ , JPV , JQθ , JQV jacobian matrix are the sub-matrixes of the System voltage stability is affected by both P and Q.
To reduce (1), let ΔP = 0 , then.
ΔQ=[ J QV J J 1 J PV ]ΔV= J R ΔV MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGrbGaeyypa0ZaamWaa8aabaWdbiaabQeapaWaaSba aSqaa8qacaqGrbGaaeOvaaWdaeqaaOWdbiabgkHiTiaabQeapaWaaS baaSqaa8qacaqGrbGaaeiUdaWdaeqaaOWdbiaabQeapaWaaSbaaSqa a8qacaqGqbGaaeiUd8aadaahaaadbeqaa8qacqGHsislcaaIXaaaaa WcpaqabaGcpeGaaeOsa8aadaWgaaWcbaWdbiaabcfacaqGwbaapaqa baaak8qacaGLBbGaayzxaaGaeuiLdqKaaeOvaiabg2da9iaabQeapa WaaSbaaSqaa8qacaqGsbaapaqabaGccqqHuoarpeGaaeOvaaaa@526E@
ΔV= J 1 ΔQ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGwbGaeyypa0JaaeOsa8aadaahaaWcbeqaa8qacqGH sislcaaIXaaaaOGaeyOeI0IaeuiLdqKaaeyuaaaa@3F4D@
Where
J R =( J QV J J 1 JPV ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGkbWdamaaBaaaleaapeGaaeOuaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacaqGkbWdamaaBaaaleaapeGaaeyuaiaabAfaa8aabe aak8qacqGHsislcaqGkbWdamaaBaaaleaapeGaaeyuaiaabI7aa8aa beaak8qacaqGkbWdamaaBaaaleaapeGaaeiuaiaabI7apaWaaWbaaW qabeaapeGaeyOeI0IaaGymaaaaaSWdaeqaaOWdbiaabQeacaqGqbGa aeOvaaGaayjkaiaawMcaaaaa@49D3@
J R   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGkbWdamaaBaaaleaapeGaaeOuaaWdaeqaaOWdbiaabckaaaa@394F@ is called the reduced Jacobian matrix of the system.
Modes of Voltage instability
Voltage Stability characteristics of the system have been identified by computing the Eigen values and Eigen vectors.
Let J R =ξΛη MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGkbWdamaaBaaaleaapeGaaeOuaaWdaeqaaOWdbiabg2da9iaa b67acqqHBoatcaqG3oaaaa@3D28@ Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
^ = diagonal eigenvalue matrix of JR and J R 1 =ξ Λ 1 η MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGkbWdamaaBaaaleaapeGaaeOua8aadaahaaadbeqaa8qacqGH sislcaaIXaaaaaWcpaqabaGcpeGaeyypa0JaaeOVdiabfU5am9aada ahaaWcbeqaa8qacqGHsislcaaIXaaaaOGaae4Tdaaa@4126@ From (5) and (8), we have ΔV=ξ Λ 1 ηΔQ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGwbGaeyypa0JaaeOVdiabfU5am9aadaahaaWcbeqa a8qacqGHsislcaaIXaaaaOGaae4Tdiabfs5aejaabgfaaaa@4189@ or ΔV= I ξ i η i λ i ΔQ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGwbGaeyypa0Zaaybuaeqal8aabaWdbiaabMeaaeqa n8aabaWdbiabggHiLdaakmaalaaapaqaa8qacaqG+oWdamaaBaaale aapeGaaeyAaaWdaeqaaOWdbiaabE7apaWaaSbaaSqaa8qacaqGPbaa paqabaaakeaapeGaae4Ud8aadaWgaaWcbaWdbiaabMgaa8aabeaaaa GcpeGaeuiLdqKaaeyuaaaa@46E5@ Where ξi is the ith column right eigenvector and η the ith row left eigenvector of JR.
λi is the ith Eigen value of JR.
The ith modal reactive power variation is, Δ Q mi = K i ξ i MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGrbWdamaaBaaaleaapeGaaeyBaiaabMgaa8aabeaa k8qacqGH9aqpcaqGlbWdamaaBaaaleaapeGaaeyAaaWdaeqaaOWdbi aab67apaWaaSbaaSqaa8qacaqGPbaapaqabaaaaa@405E@ where, K i = j ξ ij 2 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGlbWdamaaBaaaleaapeGaaeyAaaWdaeqaaOWdbiabg2da9maa wafabeWcpaqaa8qacaqGQbaabeqdpaqaa8qacqGHris5aaGccaqG+o WdamaaBaaaleaapeGaaeyAaiaabQgapaWaaWbaaWqabeaapeGaaGOm aaaaaSWdaeqaaOWdbiabgkHiTiaaigdaaaa@42F7@ Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is Δ V mi =[ 1/ λ i ]Δ Q mi MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGwbWdamaaBaaaleaapeGaaeyBaiaabMgaa8aabeaa k8qacqGH9aqpdaWadaWdaeaapeGaaGymaiaac+cacaqG7oWdamaaBa aaleaapeGaaeyAaaWdaeqaaaGcpeGaay5waiaaw2faaiabfs5aejaa bgfapaWaaSbaaSqaa8qacaqGTbGaaeyAaaWdaeqaaaaa@463B@ If | λi | =0 then the ith modal voltage will collapse
In (10), let ΔQ = ek where ek has all its elements zero except the kth one being 1. Then, ΔV=  i η 1 k  ξ 1     λ 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaqGwbGaeyypa0JaaeiOamaawafabeWcpaqaa8qacaqG Pbaabeqdpaqaa8qacqGHris5aaGcdaWcaaWdaeaacqaH3oaAdaWgaa WcbaWdbiaaigdacaqGRbGaaeiOaiaabckacaqG+oWdamaaBaaameaa peGaaGymaaWdaeqaaSWdbiaabckacaqGGcGaaeiOaaWdaeqaaaGcba WdbiaabU7apaWaaSbaaSqaa8qacaaIXaaapaqabaaaaaaa@4C30@ η 1k   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaabaaaaaaaaapeGaaGymaiaabUgacaqGGcGaaeiOaaWdaeqa aaaa@3BEC@ k th element of η 1      MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaabaaaaaaaaapeGaaGymaiaabckacaqGGcGaaeiOaiaabcka caqGGcaapaqabaaaaa@3E67@
V –Q sensitivity at bus k V K Q K = i η 1 k  ξ 1     λ 1  = i P ki λ 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaaeOva8aadaWgaaWcbaWdbiaabUea a8aabeaaaOqaa8qacqGHciITcaqGrbWdamaaBaaaleaapeGaae4saa Wdaeqaaaaak8qacqGH9aqpdaGfqbqabSWdaeaapeGaaeyAaaqab0Wd aeaapeGaeyyeIuoaaOWaaSaaa8aabaGaeq4TdG2aaSbaaSqaa8qaca aIXaGaae4AaiaabckacaqGGcGaaeOVd8aadaWgaaadbaWdbiaaigda a8aabeaal8qacaqGGcGaaeiOaiaabckaa8aabeaaaOqaa8qacaqG7o WdamaaBaaaleaapeGaaGymaaWdaeqaaaaak8qacaqGGcGaeyypa0Za aybuaeqal8aabaWdbiaabMgaaeqan8aabaWdbiabggHiLdaakmaala aapaqaa8qacaqGqbWdamaaBaaaleaapeGaae4AaiaabMgaa8aabeaa aOqaa8qacaqG7oWdamaaBaaaleaapeGaaGymaaWdaeqaaaaaaaa@5B42@
Problem Formulation
The objectives of the reactive power dispatch problem is to minimize the system real power loss and maximize the Static Voltage Stability Margins (SVSM).
Minimization of Real Power Loss
Minimization of the real power loss (Ploss) in transmission lines is mathematically stated as follows. P loss= k=1 k=(i,j) n g k( V i 2 + V j 2 2 V i  V cos θ ij ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbWdamaaBaaaleaapeGaaeiBaiaab+gacaqGZbGaae4Caiab g2da9aWdaeqaaOWdbmaawahabeWcpaqaauaabeqaceaaaeaapeGaam 4Aaiabg2da9iaaigdaa8aabaWdbiaadUgacqGH9aqpcaGGOaGaamyA aiaacYcacaWGQbGaaiykaaaaa8aabaWdbiaab6gaa0WdaeaapeGaey yeIuoaaOGaae4za8aadaWgaaWcbaWdbiaabUgacaGGOaGaaeOva8aa daqhaaadbaWdbiaabMgaa8aabaWdbiaaikdaaaWccqGHRaWkcaqGwb WdamaaDaaameaapeGaaeOAaaWdaeaapeGaaGOmaaaaliabgkHiTiaa ikdacaqGwbWdamaaBaaameaapeGaaeyAaaWdaeqaaSWdbiaabckaca qGwbWdamaaBaaameaapeGaaeOAaiaabckaciGGJbGaai4Baiaacoha caqG4oWdamaaBaaabaWdbiaabMgacaqGQbaapaqabaaabeaal8qaca GGPaaapaqabaaaaa@6179@
Where n is the number of transmission lines, gk is the i and bus j, and θij is the voltage angle difference between bus i and bus j.
Minimization of Voltage Deviation
Minimization of the voltage deviation magnitudes (VD) at load buses is mathematically stated as follows.

Minimize VD = k=1 nl | V k 1.0 | MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaapeGaae4Aaiabg2da9iaaigdaa8aabaWdbiaa b6gacaqGSbaan8aabaWdbiabggHiLdaakmaaemaapaqaa8qacaqGwb WdamaaBaaaleaapeGaae4AaaWdaeqaaOWdbiabgkHiTiaaigdacaGG UaGaaGimaaGaay5bSlaawIa7aaaa@45E6@
Where nl is the number of load busses and Vk is the voltage magnitude at bus k.
System Constraints
Objective functions are subjected to these constraints shown below.
Load flow equality constraints: P Gi   P Di V i j=1 nb V j [ G ij cos θ ij + B ij sin θ ij ]=0, i=1,2., nb MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbWdamaaBaaaleaapeGaae4raiaabMgacaqGGcaapaqabaGc peGaai4eGiaabckacaqGqbWdamaaBaaaleaapeGaaeiraiaabMgaa8 aabeaak8qacqGHsislcaqGwbWdamaaBaaaleaapeGaaeyAamaawaha beadpaqaa8qacaqGQbGaeyypa0JaaGymaaWdaeaapeGaaeOBaiaabk gaa4WdaeaapeGaeyyeIuoaaSGaaeOva8aadaWgaaadbaWdbiaabQga a8aabeaaaSqabaGcpeWaamWaa8aabaqbaeqabiGaaaqaa8qacaqGhb WdamaaBaaaleaapeGaaeyAaiaabQgaa8aabeaaaOqaa8qaciGGJbGa ai4BaiaacohacaqG4oWdamaaBaaaleaapeGaaeyAaiaabQgaa8aabe aaaOqaa8qacqGHRaWkcaqGcbWdamaaBaaaleaapeGaaeyAaiaabQga a8aabeaaaOqaa8qaciGGZbGaaiyAaiaac6gacaqG4oWdamaaBaaale aapeGaaeyAaiaabQgaa8aabeaaaaaak8qacaGLBbGaayzxaaGaeyyp a0JaaGimaiaacYcacaqGGcGaaeyAaiabg2da9iaaigdacaGGSaGaaG OmaiabgAci8kaac6cacaGGSaGaaeiOaiaab6gacaqGIbaaaa@6F07@
Q Gi  Q Di  V i j=1 nb V j [ G ij sin θ ij + B ij cos θ ij ]=0, i=1,2., nb MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGrbWdamaaBaaaleaapeGaae4raiaabMgacaqGGcaapaqabaGc peGaeyOeI0Iaaeyua8aadaWgaaWcbaWdbiaabseacaqGPbaapaqaba GcpeGaeyOeI0IaaeiOaiaabAfapaWaaSbaaSqaa8qacaqGPbWaaybC aeqam8aabaWdbiaabQgacqGH9aqpcaaIXaaapaqaa8qacaqGUbGaae OyaaGdpaqaa8qacqGHris5aaWccaqGwbWdamaaBaaameaapeGaaeOA aaWdaeqaaaWcbeaak8qadaWadaWdaeaafaqabeGacaaabaWdbiaabE eapaWaaSbaaSqaa8qacaqGPbGaaeOAaaWdaeqaaaGcbaWdbiGacoha caGGPbGaaiOBaiaabI7apaWaaSbaaSqaa8qacaqGPbGaaeOAaaWdae qaaaGcbaWdbiabgUcaRiaabkeapaWaaSbaaSqaa8qacaqGPbGaaeOA aaWdaeqaaaGcbaWdbiGacogacaGGVbGaai4CaiaabI7apaWaaSbaaS qaa8qacaqGPbGaaeOAaaWdaeqaaaaaaOWdbiaawUfacaGLDbaacqGH 9aqpcaaIWaGaaiilaiaabckacaqGPbGaeyypa0JaaGymaiaacYcaca aIYaGaeyOjGWRaaiOlaiaacYcacaqGGcGaaeOBaiaabkgaaaa@6F3F@
where, nb is the number of buses, PG and QG are the real and reactive power of the generator, PD and QD are the real and reactive load of the generator, and Gij and Bij are the mutual conductance and susceptance between bus i and bus j.
Generator bus voltage (VGi) inequality constraint:
V Gi  min  V Gi V Gi max ,ing   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGwbWdamaaDaaaleaapeGaae4raiaabMgacaqGGcaapaqaa8qa caqGTbGaaeyAaiaab6gaaaGccqGHKjYOcaqGGcGaaeOva8aadaWgaa WcbaWdbiaabEeacaqGPbaapaqabaGcpeGaeyizImQaaeOva8aadaqh aaWcbaWdbiaabEeacaqGPbaapaqaa8qacaqGTbGaaeyyaiaabIhaaa GccaGGSaGaaeyAaiabgIGiolaab6gacaqGNbGaaeiOaiaabckaaaa@51AE@ Load bus voltage (VLi) inequality constraint:
V Li  min  V Li V Li max ,inl MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGwbWdamaaDaaaleaapeGaaeitaiaabMgacaqGGcaapaqaa8qa caqGTbGaaeyAaiaab6gaaaGccqGHKjYOcaqGGcGaaeOva8aadaWgaa WcbaWdbiaabYeacaqGPbaapaqabaGcpeGaeyizImQaaeOva8aadaqh aaWcbaWdbiaabYeacaqGPbaapaqaa8qacaqGTbGaaeyyaiaabIhaaa GccaGGSaGaaeyAaiabgIGiolaab6gacaqGSbaaaa@4F7B@ Switchable reactive power compensations (QCi) inequality constraint:
Q Ci  min  Q Ci Q Ci max ,inc MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGrbWdamaaDaaaleaapeGaae4qaiaabMgacaqGGcaapaqaa8qa caqGTbGaaeyAaiaab6gaaaGccqGHKjYOcaqGGcGaaeyua8aadaWgaa WcbaWdbiaaboeacaqGPbaapaqabaGcpeGaeyizImQaaeyua8aadaqh aaWcbaWdbiaaboeacaqGPbaapaqaa8qacaqGTbGaaeyyaiaabIhaaa GccaGGSaGaaeyAaiabgIGiolaab6gacaqGJbaaaa@4F49@ Reactive power generation (QGi) inequality constraint:
Q Gi  min  Q Gi Q Gi max ,ing MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGrbWdamaaDaaaleaapeGaae4raiaabMgacaqGGcaapaqaa8qa caqGTbGaaeyAaiaab6gaaaGccqGHKjYOcaqGGcGaaeyua8aadaWgaa WcbaWdbiaabEeacaqGPbaapaqabaGcpeGaeyizImQaaeyua8aadaqh aaWcbaWdbiaabEeacaqGPbaapaqaa8qacaqGTbGaaeyyaiaabIhaaa GccaGGSaGaaeyAaiabgIGiolaab6gacaqGNbaaaa@4F59@ Transformers tap setting (Ti) inequality constraint:
T min  T i T i max ,int  MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGubWdamaaDaaaleaapeGaaeyAaiaabckaa8aabaWdbiaab2ga caqGPbGaaeOBaaaakiabgsMiJkaabckacaqGubWdamaaBaaaleaape GaaeyAaaWdaeqaaOWdbiabgsMiJkaabsfapaWaa0baaSqaa8qacaqG Pbaapaqaa8qacaqGTbGaaeyyaiaabIhaaaGccaGGSaGaaeyAaiabgI Giolaab6gacaqG0bGaaeiOaaaa@4E34@ Transmission line flow (SLi) inequality constraint:
S Li  min S Li max ,inl   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGtbWdamaaDaaaleaapeGaaeitaiaabMgacaqGGcaapaqaa8qa caqGTbGaaeyAaiaab6gaaaGccqGHKjYOcaqGtbWdamaaDaaaleaape GaaeitaiaabMgaa8aabaWdbiaab2gacaqGHbGaaeiEaaaakiaacYca caqGPbGaeyicI4SaaeOBaiaabYgacaqGGcGaaeiOaaaa@4BDC@ Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.
Particle Distribution Algorithm (PDA)
In the basic particle swarm optimization (PSO) model, an individual particle consists of: A current position, a current velocity and a personal best found position. Each of these components is required in order to decide the succeeding position of the particle for the subsequent iteration. The particle’s position in the succeeding iteration depends upon its present position and the particle’s velocity. The modernized velocity, in turn, depends on the particle’s current velocity, the particle’s personal best found position and the population’s global best found position. The representation of a swarm of particles “flying” through space no longer aptly describes the high-level concept of the algorithm. Rather, the algorithm now consists of a population of “particle fields” which move throughout the space in a different way. Because the “positions” of these “particle fields” are defined as arbitrary distributions, “evaluating” a “particle field’s” existing “position” is nondeterministic, and so these “particle fields” do not certainly have to “move” to discover new points. These “particle fields” remain “stationary” in the space until either the individual’s personal best point changes, or the population’s global best point changes. This population of “particle fields” can, itself, be alleged as an arbitrary field of particles defined as a mixture distribution made up of each individual distribution. This population level distribution can be thought of as an abstract depiction of a particle swarm, representing a probability distribution of all possible particle locations for the consequent iteration. With this new outlook, it is possible to discover new directions in refining or altering the behaviour of the algorithm. With these changes taken into account, we have now moved away from the traditional PSO paradigm and arrived at a new, distinct algorithm, which will be hereafter referred to as Particle Distribution Algorithm (PDA). This algorithm consists of a population of “particle field” individuals and a “point pool” of candidate solution points. The population of particle field individuals uses PSO principles to guide the exploration of the solution space, which is carried out by creating and weighing, the pool of candidate solution points. Similar to traditional PSO algorithms, the PDA algorithm consists of an initialization phase and a simulation phase which loops until some end criteria is met, at which point the best solution found by the algorithm is returned as output. A particle field individual is designated at an arbitrary position from the population, according to some weighting scheme. Then, the point is produced by sampling the arbitrary distribution defined by the selected individual. This arbitrary distribution is constructed using the individual’s personal best, and the global best points.
Given a particle field with personal best point Pi and for which the global best point is Pg the position of the candidate solution point, c is determined according to: p m = p i + p g 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala WaaSbaaSqaaiaad2gaaeqaaOGaeyypa0ZaaSaaaeaaceWGWbGbaSaa daWgaaWcbaGaamyAaaqabaGccqGHRaWkceWGWbGbaSaadaWgaaWcba Gaam4zaaqabaaakeaacaaIYaaaaaaa@3F2C@ σ 2 =| p i p g | MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaqa aaaaaaaaWdbiaabo8apaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaiaa wEnia8qacqGH9aqpdaabdaWdaeaadaWhcaqaa8qacaqGWbWdamaaBa aaleaapeGaaeyAaaWdaeqaaaGccaGLxdcapeGaeyOeI0YdamaaFiaa baWdbiaabchapaWaaSbaaSqaa8qacaqGNbaapaqabaaakiaawEniaa WdbiaawEa7caGLiWoaaaa@47A2@ c = N ( p m σ 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4yayaala Gaeyypa0JabmOtayaalaWaaeWaaeaaceWGWbGbaSaadaWgaaWcbaGa amyBaaqabaGccqGHsislcuaHdpWCgaWcamaaCaaaleqabaGaaGOmaa aaaOGaayjkaiaawMcaaaaa@4047@
Once the candidate solution has been produced, the objective function is weighed, using this produced point as its input, in order to assign it a value. After each candidate solution in the point pool has been produced, the second phase begins. In this phase, the population of particle field individuals is modernized. Each individual updates its own best found point using the set of candidate solutions produced from its own distribution. Each individual selects the best point from the set of associated candidate solutions. If the best associated candidate solution is better than the individual’s personal best found point, the individual sets its personal best found point to be equal to that candidate solution point. The pool of candidate solutions is then “emptied”, and the simulation endures to the succeeding iteration. Once the termination criteria have been met, the global best found point is returned as output of the algorithm.
Input:
Function f ( ) to be optimized
Initialization range lbound, ubound
Particle field population size nPop
Candidate solution point pool size npool
Weighting function ù ( )
Output:
Point  p i MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaqa aaaaaaaaWdbiaabckacaqGWbaapaGaay51GaWaaSbaaSqaa8qacaqG Pbaapaqabaaaaa@3B26@ representing best found solution
Method:
Generate particle field population P with size n pop MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGUbWdamaaBaaaleaapeGaaeiCaiaab+gacaqGWbaapaqabaaa aa@3A39@
Generate candidate solution point pool C with size n pool MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGUbWdamaaBaaaleaapeGaaeiCaiaab+gacaqGVbGaaeiBaaWd aeqaaaaa@3B27@
For each particle field ip MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGPbGaeyicI4SaaeiCaaaa@3979@ do p i = U [ 1bound, ubound ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JabmyvayaalaWaamWaaeaa caaIXaGaamOyaiaad+gacaWG1bGaamOBaiaadsgacaGGSaGaaeiiai aabwhacaqGIbGaae4BaiaabwhacaqGUbGaaeizaaGaay5waiaaw2fa aaaa@4862@ if f( p i )<f( p j ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI caceWGWbGbaSaadaWgaaWcbaGaamyAaaqabaGccaGGPaGaeyipaWJa amOzaiaacIcaceWGWbGbaSaadaWgaaWcbaGaamOAaaqabaGccaGGPa aaaa@3FD8@ then p i p j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala WaaSbaaSqaaiaadMgaaeqaaOGaeyi0HWTabmiCayaalaWaaSbaaSqa aiaadQgaaeqaaaaa@3C9B@ End if
End for
While termination criteria not met do
For each candidate solution point  c C MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaqa aaaaaaaaWdbiaabckacaqGJbaapaGaay51GaWdbiabgIGiolaaboea aaa@3C3C@ do Select particle field ip MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGPbGaeyicI4SaamiCaaaa@397D@ with probability ω( i ) p P ω( p ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaaeyYdmaabmaapaqaa8qacaqGPbaacaGLOaGa ayzkaaaapaqaamaavacabeWcbeqaaiaaygW7a0qaa8qacqGHris5aa GccaqGWbGaeyicI4Saaeiua8aadaahaaWcbeqaa8qacaqGjpWaaeWa a8aabaWdbiaabchaaiaawIcacaGLPaaaaaaaaaaa@4555@
c N ( p i + p g 2 ,[ p i p g ] ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4yayaala Gaeyi0HWTabmOtayaalaWaaeWaaeaadaWcaaqaaiqadchagaWcamaa BaaaleaacaWGPbaabeaakiabgUcaRiqadchagaWcamaaBaaaleaaca WGNbaabeaaaOqaaiaaikdaaaGaaiilamaadmaabaGabmiCayaalaWa aSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmiCayaalaWaaSbaaSqaai aadEgaaeqaaaGccaGLBbGaayzxaaaacaGLOaGaayzkaaaaaa@499B@
s i s i { c } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Cayaala WaaSbaaSqaaiaadMgaaeqaaOGaeyi0HWTaam4CamaaBaaaleaacaWG PbaabeaakiabgQIiipaacmaabaGabm4yayaalaaacaGL7bGaayzFaa aaaa@4163@ End for
For each particle field ip MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGPbGaeyicI4SaamiCaaaa@397D@ do
Choose point c min MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4yayaala WaaSbaaSqaaiGac2gacaGGPbGaaiOBaaqabaaaaa@39EE@ from S i MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaaaaa@37E8@ which minimizes f( c min ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI caceWGJbGbaSaadaWgaaWcbaGaciyBaiaacMgacaGGUbaabeaakiaa cMcaaaa@3C3C@
if f( c min <f(p)) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI caceWGJbGbaSaadaWgaaWcbaGaciyBaiaacMgacaGGUbaabeaakiab gYda8iaadAgacaGGOaGaamiCaiaacMcacaGGPaaaaa@4079@ then
p i c min MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala WaaSbaaSqaaiaadMgaaeqaaOGaeyi0HWTabm4yayaalaWaaSbaaSqa aiGac2gacaGGPbGaaiOBaaqabaaaaa@3E72@
If f( p i )<( p j ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI caceWGWbGbaSaadaWgaaWcbaGaamyAaaqabaGccaGGPaGaeyipaWJa aiikaiqadchagaWcamaaBaaaleaacaWGQbaabeaakiaacMcaaaa@3EEE@
Then p i p j MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala WaaSbaaSqaaiaadMgaaeqaaOGaeyi0HWTabmiCayaalaWaaSbaaSqa aiaadQgaaeqaaaaa@3C9C@
End if
End if
S i ϕ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaakiabgcDiClabew9aMbaa@3C12@
End for
End
Simulation Results
The efficiency of the proposed Particle Distribution Algorithm (PDA) is demonstrated by testing it on standard IEEE-30 bus system. The IEEE-30 bus system has 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10) , (4-12) and (28-27) - are with the tap setting transformers. The lower voltage magnitude limits at all buses are 0.95 p.u. and the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference bus. The simulation results have been presented in Tables 1, 2, 3 &4. And in the Table 5 shows the proposed algorithm powerfully reduces the real power losses when compared to other given algorithms. The optimal values of the control variables along with the minimum loss obtained are given in Table 1. Corresponding to this control variable setting, it was found that there are no limit violations in any of the state variables.
Table 1: Results of PDA – ORPD optimal control variables

Control Variables

Variable Setting

V1

V2

V5

V8

V11

V13

T11

T12

T15

T36

Qc10

Qc12

Qc15

Qc17

Qc20

Qc23

Qc24

Qc29

Real power loss

SVSM

1.047

1.046

1.041

1.030

1.004

1.031

1.00

1.00

1.01

1.01

3

3

2

0

2

3

3

2

4.2869

0.2479

Table 2: Results of PDA -Voltage Stability Control Reactive Power Dispatch Optimal Control Variables

Control Variables

Variable Setting

V1

V2

V5

V8

V11

V13

T11

T12

T15

T36

Qc10

Qc12

Qc15

Qc17

Qc20

Qc23

Qc24

Qc29

Real power loss

SVSM

1.049

1.047

1.044

1.032

1.005

1.034

0.090

0.090

0.090

0.090

3

3

2

3

0

2

2

3

4.9896

0.2484

Table 3: Voltage Stability under Contingency State

Sl.No

Contingency

ORPD Setting

VSCRPD Setting

1

28-27

0.1419

0.1434

2

4-12

0.1642

0.1650

3

1-3

0.1761

0.1772

4

2-4

0.2022

0.2043

Optimal Reactive Power Dispatch problem together with voltage stability constraint problem was handled in this case as a multi-objective optimization problem where both power loss and maximum voltage stability margin of the system were optimized simultaneously. Table 2 indicates the optimal values of these control variables. Also it is found that there are no limit violations of the state variables. It indicates the voltage stability index has increased from 0.2479 to 0.2484, an advance in the system voltage stability. To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained in case 1 and case 2. The Eigen values equivalents to the four critical contingencies are given in Table 3. From this result it is observed that the Eigen value has been improved considerably for all contingencies in the second case.
Conclusion
In this paper, Particle Distribution Algorithm (PDA) has been successfully applied to solve Optimal Reactive Power Dispatch problem. The proposed algorithm has been tested on the standard IEEE 30 bus system. Simulation results show the heftiness of projected Particle Distribution Algorithm (PDA) method in decreasing the real power loss & voltage profile index has been enhanced. The control variables obtained after the optimization by Particle Distribution Algorithm (PDA) is within the limits.
Table 4: Limit Violation Checking Of State Variables

State variables

limits

ORPD

VSCRPD

Lower

upper

Q1

-20

152

1.3422

-1.3269

Q2

-20

61

8.9900

9.8232

Q5

-15

49.92

25.920

26.001

Q8

-10

63.52

38.8200

40.802

Q11

-15

42

2.9300

5.002

Q13

-15

48

8.1025

6.033

V3

0.95

1.05

1.0372

1.0392

V4

0.95

1.05

1.0307

1.0328

V6

0.95

1.05

1.0282

1.0298

V7

0.95

1.05

1.0101

1.0152

V9

0.95

1.05

1.0462

1.0412

V10

0.95

1.05

1.0482

1.0498

V12

0.95

1.05

1.0400

1.0466

V14

0.95

1.05

1.0474

1.0443

V15

0.95

1.05

1.0457

1.0413

V16

0.95

1.05

1.0426

1.0405

V17

0.95

1.05

1.0382

1.0396

V18

0.95

1.05

1.0392

1.0400

V19

0.95

1.05

1.0381

1.0394

V20

0.95

1.05

1.0112

1.0194

V21

0.95

1.05

1.0435

1.0243

V22

0.95

1.05

1.0448

1.0396

V23

0.95

1.05

1.0472

1.0372

V24

0.95

1.05

1.0484

1.0372

V25

0.95

1.05

1.0142

1.0192

V26

0.95

1.05

1.0494

1.0422

V27

0.95

1.05

1.0472

1.0452

V28

0.95

1.05

1.0243

1.0283

V29

0.95

1.05

1.0439

1.0419

V30

0.95

1.05

1.0418

1.0397

Table 5: Comparison of Real Power Loss

Method

Minimum loss

Evolutionary programming [41]

5.0159

Genetic algorithm [42]

4.665

Real coded GA with Lindex as SVSM  [43]

4.568

Real coded genetic algorithm [44]

4.5015

Proposed PDA  method

4.2869

ReferencesTop
  1. Alsac O and Scott B. Optimal load flow with steady state security. IEEE Transaction. 1973;93(3):745-751. DOI: 10.1109/TPAS.1974.293972.
  2. Lee KY, Paru YM, Oritz JL. A united approach to optimal real and reactive power dispatch. IEEE Transactions on Power Apparatus and Systems. 1985;104(5):1147-1153. DOI: 10.1109/TPAS.1985.323466.
  3. Monticelli A, Pereira MVF. Granville S. Security constrained optimal power flow with post contingency corrective rescheduling. IEEE Transactions on Power Systems. 1987;2(1):175-182. DOI: 10.1109/TPWRS.1987.4335095.
  4. Deeb N, Shahidehpur SM. Linear reactive power optimization in a large power network using the decomposition approach. IEEE Transactions on Power Systems. 1990;5(2):428-435.
  5. Hobson E. Network consrained reactive power control using linear programming. IEEE Transactions on Power Systems. 1980;99(3):868-877. DOI: 10.1109/TPAS.1980.319715.
  6. Lee KY, Park YM, Oritz JL. Fuel –cost optimization for both real and reactive power dispatches. IEE Proc. 1984;131(3):85-93. DOI: 10.1049/ip-c:19840012.
  7. Mangoli MK,  Lee KY.  Optimal real and reactive power control using linear programming. Electr Power Syst Res. 1993;26(1):1-10.
  8. Canizares CA, De Souza ACZ, Quintana VH. Comparison of performance indices for detection of proximity to voltage collapse. IEEE Transactions on Power Systems. 1996;11(3):1441-1450. DOI: 10.1109/59.535685.
  9. Berizzi A, Bovo C, Merlo M, Delfanti M. A GA approach to compare ORPF objective functions including secondary voltage regulation. Electric Power Systems Research. 2012;84(1):187-194.
  10. Devaraj D, Yeganarayana B. Genetic algorithm based optimal power flow for security enhancement. IEE Proceedings-Generation, Transmission and Distribution. 2005;152(6):899-905. DOI: 10.1049/ip-gtd:20045234.
  11. Berizzi A, Bovo C, Merlo M, Delfanti M. A GA approach to compare ORPF objective functions including secondary voltage regulation. Electric Power Systems Research. 2012;84(1):187-194.
  12. Yang CF, Lai GG, Lee CH, Su CT, Chang GW. Optimal setting of reactive compensation devices with an improved voltage stability index for voltage stability enhancement. International Journal of Electrical Power and Energy Systems. 2012;37(1):50-57.
  13. Roy P, Ghoshal S, Thakur S. Optimal var control for improvements in voltage profiles and for real power loss minimization using biogeography based optimization. International Journal of Electrical Power and Energy Systems. 2012;43(1):830-838.
  14. Venkatesh B, Sadasivam G, Khan M. A new optimal reactive power scheduling method for loss minimization and voltage stability margin maximization using successive multi-objective fuzzy lp technique. IEEE Transactions on Power Systems. 2000;15(2):844-851. DOI: 10.1109/59.867183.
  15. Yan W, Lu S, Yu D. A novel optimal reactive power dispatch method based on an improved hybrid evolutionary programming technique. IEEE Transactions on Power Systems. 2004;19(2):913-918. DOI: 10.1109/TPWRS.2004.826716.
  16. Yan W, Liu F, Chung C, Wong K. A hybrid genetic algorithminterior point method for optimal reactive power flow. IEEE Transactions on Power Systems. 2006;21(3):1163-1169. DOI: 10.1109/TPWRS.2006.879262.
  17. Yu J, Yan W, Li W, Chung C, Wong K. An unfixed piecewiseoptimal reactive power-flow model and its algorithm for ac-dc systems. IEEE Transactions on Power Systems. 2008;23(1):170-176. DOI: 10.1109/TPWRS.2007.907387.
  18. Capitanescu F. Assessing reactive power reserves with respect to operating constraints and voltage stability. IEEE Transactions on Power Systems. 2011;26(4):2224-2234. DOI: 10.1109/TPWRS.2011.2109741.
  19. Hu Z, Wang X, Taylor G. Stochastic optimal reactive power dispatch: Formulation and solution method. International Journal of Electrical Power and Energy Systems. 2010;32(6):615-621.
  20. Kargarian A, Raoofat M, Mohammadi M. Probabilistic reactive power procurement in hybrid electricity markets with uncertain loads. Electric Power Systems Research. 2012;82(1):68-80.
  21. Angeline PJ. Using selection to improve particle swarm optimization. presented at the IEEE International Conference on Computational Intelligence. 1998. DOI: 10.1109/ICEC.1998.699327.
  22. Clerc M, Kennedy J. The particle swarm - explosion, stability, and convergence in a multidimensional complex space.  IEEE Transactions on Evolutionary Computation. 2002;6(1):58-73. DOI: 10.1109/4235.985692.
  23. Bell N. Swarm optimization using agents modeled as distributions. MCS Thesis, Carelton University. 2014.
  24. Eberhart R, Shi Y. Comparing inertia weights and constriction factors in particle swarm optimization. Congress on Evolutionary Computation. 2000;1:84-88.
  25. Higashi N, Iba H. Particle swarm optimization with Gaussian mutation. presented at the IEEE Swarm Intelligence Symposium, Indianapolis. 2003. DOI: 10.1109/SIS.2003.1202250.
  26. Kennedy J. Bare bones particle swarms. presented at the IEEE Swarm Intelligence Symposium. 2003. DOI: 10.1109/SIS.2003.1202251.
  27. Kennedy J, Eberhart R. Particle swarm optimization. presented at the IEEE International Conference on Neural Networks. 1995. DOI: 10.1109/ICNN.1995.488968.
  28. Kennedy J, Mendes R. Population structure and particle swarm performance. Congress on Evolutionary Computation. 2002;2:1671-1676. DOI: 10.1109/CEC.2002.1004493.
  29. Locatelli M. A note on the griewank test function.  Journal of Global Optimization. 2003;25(2):169-174. DOI: 10.1023/A:1021956306041.
  30. Lovbjerg M, Rasmussen TK, Krink T. Hybrid particle swarm optimiser with breeding and subpopulations. presented at the Genetic and Evolutionary Computation Conference. 2001.
  31. Mendes R, Kennedy J, Neves J. The fully informed particle swarm: Simpler, maybe better. IEEE Transactions on Evolutionary Computation. 2004;8(3):204-210. DOI: 10.1109/TEVC.2004.826074.
  32. Monson CK, Seppi KD. Exposing origin-seeking bias in pso. presented at the Conference on Genetic and Evolutionary Computation, GECCO ’05. 2005:241-248. Doi:10.1145/1068009.1068045.
  33. Ozcan E, Mohan CK. Analysis of a simple particle swarm optimization system. Intelligent Engineering Systems Through Artificial Neural Networks. 1988;8:253-258.
  34. Ozcan E, Mohan CK.  Particle swarm optimization: Surfing the waves. Congress on Evolutionary Computation. 1999;3:1939-1944. DOI: 10.1109/CEC.1999.785510.
  35. Ratnaweera A, Halgamuge S,  Watson HC. Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Transactions on Evolutionary Computation. 2004;8(3):240-255.
  36. Settles M, Soule T. Breeding swarms: A ga/pso hybrid. presented at the Conference on Genetic and Evolutionary Computation, GECCO ’05. 2005:161-168. Doi:10.1145/1068009.1068035.
  37. Shi Y, Eberhart R. Empirical study of particle swarm optimization. Congress on Evolutionary Computation. 1999;3:1945-1950. DOI: 10.1109/CEC.1999.785511.
  38. Spears WM, Green DT, Spears DF. Biases in particle swarm optimization. Int J Swarm Intell Res. 2010;1(2):34-57. DOI: 10.4018/978-1-4666-1592-2.ch002.
  39. Zhan ZH, Zhang J, Chung HSH. Adaptive particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Swarm Intelligence Symposium, 2003. 2009;39(6):1362-1381. DOI: 10.1109/TSMCB.2009.2015956.
  40. Nathan B, John Oommen B. Particle field optimization: A new paradigm for swarm intelligence. in Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems. 2015:257-265.
  41. Wu QH, Ma JT. Power system optimal reactive power dispatch using evolutionary programming. IEEE Transactions on power systems. 1995;10(3):1243-1248 . DOI: 10.1109/59.466531.
  42. Durairaj S, Devaraj D, Kannan PS. Genetic algorithm applications to optimal reactive power dispatch with voltage stability enhancement. IE(I) Journal-EL. 2006;87.
  43. Devaraj D. Improved genetic algorithm for multi – objective reactive power dispatch problem. European Transactions on electrical power. 2007;17:569-581.
  44. Aruna Jeyanthy P, Devaraj D. Optimal Reactive Power Dispatch for Voltage Stability Enhancement Using Real Coded Genetic Algorithm. International Journal of Computer and Electrical Engineering, 2010;2(4):1793-8163.    
 
Listing : ICMJE   

Creative Commons License Open Access by Symbiosis is licensed under a Creative Commons Attribution 3.0 Unported License