Research article Open Access
Waning Of Real Power Loss By Modified Particle Swarm Optimization
Lenin K1*, Ravindhranath Reddy B2, Surya Kalavathi M3
1Researcher, Jawaharlal Nehru Technological University, Kukatpally.
2Deputy Executive Engineer, Jawaharlal Nehru Technological University, Kukatpally.
3Professor of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University, Kukatpally.
*Corresponding author: Lenin K, Researcher, Jawaharlal Nehru Technological University, Kukatpally, Hyderabad, 500 085, India, Tel: 91-9677350862; E-mail: @
Received: December 03, 2016; Accepted: December 20, 2016; Published: January 11, 2017
Citation: Lenin K, Reddy BR, Kalavathi MS (2017) Waning Of Real Power Loss By Modified Particle Swarm Optimization. Int J Adv Robot Automn 2(1): 1-5.
Abstract
This paper presents Modified Particle Swarm Optimization (MPSO) algorithm for solving reactive power problem. A nonlinear decreasing weight factor used to change the fundamental ways of Particle Swarm Optimization (PSO). To allow full play to the approximation capability of the function of Back Propagation neural network & overcome the main shortcomings of its liability to fall into local extreme value this study proposed a concept of modified PSO algorithm. Back Propagation network jointly to optimize the original weight, threshold value of network and incorporating the PSO algorithm into Back Propagation network to establish a Modified PSOBP network system. Proposed method MPSO progresses convergence speed and the capability to search optimal value. In order to evaluate the proposed MPSO algorithm, it has been tested on IEEE 30 bus system and compared to other reported standard algorithms and simulation results show that (MPSO) is more efficient than other algorithms in reducing the real power loss & voltage profiles are within the limits.

Keywords: Modified Particle Swarm Optimization; Back Propagation; optimal reactive power; Transmission loss;
Introduction
Reactive power problem is one of the difficult optimization problems in power systems. Various mathematical techniques have been adopted to solve this optimal reactive power dispatch problem. These include the gradient method [1, 2], Newton method [3] and linear programming [4-7].The gradient and Newton methods suffer from the difficulty in handling inequality constraints. To apply linear programming, the input- output function is to be expressed as a set of linear functions which may lead to loss of accuracy. Recently global optimization techniques such as genetic algorithms have been proposed to solve the reactive power flow problem [8, 9]. In recent years, the problem of voltage stability and voltage collapse has become a major concern in power system planning and operation. To enhance the voltage stability, voltage magnitudes alone will not be a reliable indicator of how far an operating point is from the collapse point [10]. The reactive power support and voltage problems are intrinsically related. Hence, this paper formulates the reactive power dispatch as a multi-objective optimization problem with loss minimization and maximization of Static Voltage Stability Margin (SVSM) as the objectives. The Particle Swarm Optimization (PSO) [11-15] was based on the optimal algorithm of swarm intelligence and it guides optimal search through swarm intelligence producing by the corporation and competition among particles. The artificial Neural Network possesses the ability of processing database in parallel, organizing and learning by itself. It has made great achievements in many fields. Especially neural network technique has been used in reliability and few researchers apply the method [16-20] to predict failures. Theoretically, Network [21- 32] may approach any continual nonlinear function. However, affected greatly by the sample, it may fall into local minimum value, thus it can’t ensure that it converge the minimum value in overall situations. In our paper PSO algorithm is introduced into BP network optimization by using iterate algorithm of particle swarm instead of gradient correction algorithm of BP network. This method can shorten the time of training network and improve the convergence speed of BP algorithm. However, according to different system object, if basic particle swarm algorithm is modified, the performance of BP network is improved and its practicality is better. For balancing between the global search and the local search and improving the precision of the result, this study proposed the nonlinear strategies for decreasing inertia weight based on the idea of the existing linear decreasing inertia weight. The result of simulation shows that the optimizing methods speed the rapidity of convergence obviously and the precision of simulation increase greatly. The performance of Modified Particle Swarm Optimization (MPSO) has been evaluated in standard IEEE 30 bus test system and the simulation results show the best performance of the proposed algorithm.
Objective Function
Active power loss
The objective of the reactive power dispatch problem is to minimize the active power loss and can be written in equations as follows:
F= P L = kNbr   g k ( V i 2 + V j 2 2 V i V j cosθ ij ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgbGaeyypa0Jaaeiua8aadaWgaaWcbaWdbiaabYeaa8aabeaa k8qacqGH9aqpdaGfqbqabSWdaeaapeGaae4AaiabgIGiolaab6eaca qGIbGaaeOCaaqab0WdaeaapeGaeyyeIuoaaOGaaeiOaiaabckacaqG NbWdamaaBaaaleaapeGaae4AaaWdaeqaaOWdbmaabmaapaqaa8qaca qGwbWdamaaDaaaleaapeGaaeyAaaWdaeaapeGaaGOmaaaakiabgUca RiaabAfapaWaa0baaSqaa8qacaqGQbaapaqaa8qacaaIYaaaaOGaey OeI0IaaGOmaiaabAfapaWaaSbaaSqaa8qacaqGPbaapaqabaGcpeGa aeOva8aadaWgaaWcbaWdbiaabQgaa8aabeaak8qacaqGJbGaae4Bai aabohacaqG4oWdamaaBaaaleaapeGaaeyAaiaabQgaa8aabeaaaOWd biaawIcacaGLPaaaaaa@5C0F@
Where F- objective function, PL – power loss, gk - conductance of branch,Vi and Vj are voltages at buses i,j, Nbr- total number of transmission lines in power systems.
Voltage profile improvement
To minimize the voltage deviation in PQ buses, the objective function (F) can be written as:
F= P L + ω v ×VD MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaeyypa0Jaamiua8aadaWgaaWcbaWdbiaadYeaa8aabeaa k8qacqGHRaWkcqaHjpWDpaWaaSbaaSqaa8qacaWG2baapaqabaGcpe Gaey41aqRaamOvaiaadseaaaa@41DA@
Where VD - voltage deviation, ω v MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWG2baapaqabaaaaa@3938@ - is a weighting factor of voltage deviation.
And the Voltage deviation given by:
VD= i=1 Npq | V i 1 | MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGwbGaamiraiabg2da9maawahabeWcpaqaa8qacaWGPbGaeyyp a0JaaGymaaWdaeaapeGaamOtaiaadchacaWGXbaan8aabaWdbiabgg HiLdaakmaaemaapaqaa8qacaWGwbWdamaaBaaaleaapeGaamyAaaWd aeqaaOWdbiabgkHiTiaaigdaaiaawEa7caGLiWoaaaa@4804@
Where Npq- number of load buses
Equality Constraint
The equality constraint of the problem is indicated by the power balance equation as follows:
P G = P D + P L MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbWdamaaBaaaleaapeGaam4raaWdaeqaaOWdbiabg2da9iaa dcfapaWaaSbaaSqaa8qacaWGebaapaqabaGcpeGaey4kaSIaamiua8 aadaWgaaWcbaWdbiaadYeaa8aabeaaaaa@3E25@
Where PG- total power generation, PD - total power demand.
Inequality Constraints
The inequality constraint implies the limits on components in the power system in addition to the limits created to make sure system security. Upper and lower bounds on the active power of slack bus (Pg), and reactive power of generators (Qg) are written as follows:
P gslack min P gslack P gslack max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbWdamaaDaaaleaapeGaam4zaiaadohacaWGSbGaamyyaiaa dogacaWGRbaapaqaa8qacaWGTbGaamyAaiaad6gaaaGccqGHKjYOca WGqbWdamaaBaaaleaapeGaam4zaiaadohacaWGSbGaamyyaiaadoga caWGRbaapaqabaGcpeGaeyizImQaamiua8aadaqhaaWcbaWdbiaadE gacaWGZbGaamiBaiaadggacaWGJbGaam4AaaWdaeaapeGaamyBaiaa dggacaWG4baaaaaa@53B4@ Q gi min Q gi Q gi max  , i N g MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGrbWdamaaDaaaleaapeGaam4zaiaadMgaa8aabaWdbiaad2ga caWGPbGaamOBaaaakiabgsMiJkaadgfapaWaaSbaaSqaa8qacaWGNb GaamyAaaWdaeqaaOWdbiabgsMiJkaadgfapaWaa0baaSqaa8qacaWG NbGaamyAaaWdaeaapeGaamyBaiaadggacaWG4baaaOGaaeiOaiaacY cacaqGGcGaamyAaiabgIGiolaad6eapaWaaSbaaSqaa8qacaWGNbaa paqabaaaaa@5017@
Upper and lower bounds on the bus voltage magnitudes (Vi) is given by:
V i min V i V i max  , iN MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGwbWdamaaDaaaleaapeGaamyAaaWdaeaapeGaamyBaiaadMga caWGUbaaaOGaeyizImQaamOva8aadaWgaaWcbaWdbiaadMgaa8aabe aak8qacqGHKjYOcaWGwbWdamaaDaaaleaapeGaamyAaaWdaeaapeGa amyBaiaadggacaWG4baaaOGaaeiOaiaacYcacaqGGcGaamyAaiabgI Giolaad6eaaaa@4C1C@
Uper and lower bounds on the transformers tap ratios (Ti) is given by:
T i min T i T i max  ,i N T MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaDaaaleaapeGaamyAaaWdaeaapeGaamyBaiaadMga caWGUbaaaOGaeyizImQaamiva8aadaWgaaWcbaWdbiaadMgaa8aabe aak8qacqGHKjYOcaWGubWdamaaDaaaleaapeGaamyAaaWdaeaapeGa amyBaiaadggacaWG4baaaOGaaeiOaiaacYcacaWGPbGaeyicI4Saam Ota8aadaWgaaWcbaWdbiaadsfaa8aabeaaaaa@4C26@
Upper and lower bounds on the compensators (Qc) is given by:
Q c min Q c Q C max  ,i N C MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGrbWdamaaDaaaleaapeGaam4yaaWdaeaapeGaamyBaiaadMga caWGUbaaaOGaeyizImQaamyua8aadaWgaaWcbaWdbiaadogaa8aabe aak8qacqGHKjYOcaWGrbWdamaaDaaaleaapeGaam4qaaWdaeaapeGa amyBaiaadggacaWG4baaaOGaaeiOaiaacYcacaWGPbGaeyicI4Saam Ota8aadaWgaaWcbaWdbiaadoeaa8aabeaaaaa@4BDA@
Where N is the total number of buses, Ng is the total number of generators, NT is the total number of Transformers, Nc is the total number of shunt reactive compensators.
Particle Swarm Optimization
Particle Swarm Optimization Algorithm (PSO) is a population based optimization tool where the system is initialized with a population of random particles and the algorithm searches for optima by updating generations. Suppose in the D-dimensional objects searching space, there is a community composed of N particle. The “I” particle represent a D-dimensional vector,
X i =( X i1 , X i2 ,.. X id ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGybWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacaWGybWdamaaBaaaleaapeGaamyAaiaaigdaa8aabe aak8qacaGGSaGaamiwa8aadaWgaaWcbaWdbiaadMgacaaIYaaapaqa baGcpeGaaiilaiaac6cacaGGUaGaamiwa8aadaWgaaWcbaWdbiaadM gacaWGKbaapaqabaaak8qacaGLOaGaayzkaaaaaa@46E4@ It means that the “i” particle represents its position in this space. Every position of particle “X” is a potential solution. If we put “x” into objective function, we can know the adaptive value. We can know whether the “x” is the optimal answer based on the adaptive value. The speed of particle is also a D-dimensional, it also recorded as v i =( v i1 , v i2 ,.. v id ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacaWG2bWdamaaBaaaleaapeGaamyAaiaaigdaa8aabe aak8qacaGGSaGaamODa8aadaWgaaWcbaWdbiaadMgacaaIYaaapaqa baGcpeGaaiilaiaac6cacaGGUaGaamODa8aadaWgaaWcbaWdbiaadM gacaWGKbaapaqabaaak8qacaGLOaGaayzkaaaaaa@475C@ We record the particle I to the h times, the optimal position was P i =( P i1 , P i2 ,.. P id ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacaWGqbWdamaaBaaaleaapeGaamyAaiaaigdaa8aabe aak8qacaGGSaGaamiua8aadaWgaaWcbaWdbiaadMgacaaIYaaapaqa baGcpeGaaiilaiaac6cacaGGUaGaamiua8aadaWgaaWcbaWdbiaadM gacaWGKbaapaqabaaak8qacaGLOaGaayzkaaaaaa@46C4@
All the particles to the h times, the optimal position was P gd =( P i1 , P i2 ,.. P id ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbWdamaaBaaaleaapeGaam4zaiaadsgaa8aabeaak8qacqGH 9aqpdaqadaWdaeaapeGaamiua8aadaWgaaWcbaWdbiaadMgacaaIXa aapaqabaGcpeGaaiilaiaadcfapaWaaSbaaSqaa8qacaWGPbGaaGOm aaWdaeqaaOWdbiaacYcacaGGUaGaaiOlaiaadcfapaWaaSbaaSqaa8 qacaWGPbGaamizaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@47AB@ The basic formulas are as follows:
v id t+1 =w v id t + c 1 r 1 t ( P id t x id t )+ c 2 r 2 t ( P gd t x id t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWdamaaDaaaleaapeGaamyAaiaadsgaa8aabaWdbiaadsha cqGHRaWkcaaIXaaaaOGaeyypa0Jaam4DaiaadAhapaWaa0baaSqaa8 qacaWGPbGaamizaaWdaeaapeGaamiDaaaakiabgUcaRiaadogapaWa aSbaaSqaa8qacaaIXaaapaqabaGcpeGaamOCa8aadaqhaaWcbaWdbi aaigdaa8aabaWdbiaadshaaaGcdaqadaWdaeaapeGaamiua8aadaqh aaWcbaWdbiaadMgacaWGKbaapaqaa8qacaWG0baaaOGaeyOeI0Iaam iEa8aadaqhaaWcbaWdbiaadMgacaWGKbaapaqaa8qacaWG0baaaaGc caGLOaGaayzkaaGaey4kaSIaam4ya8aadaWgaaWcbaWdbiaaikdaa8 aabeaak8qacaWGYbWdamaaDaaaleaapeGaaGOmaaWdaeaapeGaamiD aaaakmaabmaapaqaa8qacaWGqbWdamaaDaaaleaapeGaam4zaiaads gaa8aabaWdbiaadshaaaGccqGHsislcaWG4bWdamaaDaaaleaapeGa amyAaiaadsgaa8aabaWdbiaadshaaaaakiaawIcacaGLPaaaaaa@6449@
x id t+1 = x id t + v id t+1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWdamaaDaaaleaapeGaamyAaiaadsgaa8aabaWdbiaadsha cqGHRaWkcaaIXaaaaOGaeyypa0JaamiEa8aadaqhaaWcbaWdbiaadM gacaWGKbaapaqaa8qacaWG0baaaOGaey4kaSIaamODa8aadaqhaaWc baWdbiaadMgacaWGKbaapaqaa8qacaWG0bGaey4kaSIaaGymaaaaaa a@47F2@
Where, c 1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFJbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@381B@ &   c 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGGcacbmGaa83ya8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@393F@ : Speeding coefficient, adjusting the maxim step length that flying the best particle in whole situation and the individual best particle respectively. Appropriate c1 and c2 speed up the convergence and avoid falling into partial optimality r1&r2: Random number between 0 and 1, for controlling the weight of speed W: Inertia factor. It was oriented toward overall searching .We usually take the original value as 0.9 and make it to 0.1 with the addition and reduction of the times of iteration. It mainly used to total searching, making the searching space converge to a certain space. Then we can get the solution in high degree of accuracy by partial refined researching. With the increasing number of dimension of problems, basic PSO algorithm is easily falling into partial extreme value, thus influence the optimal function of algorithm. Someone brought up with improved algorithm. Many scholars’ research shows [11- 15] that “w” has a great influence on the algorithm of particle swarm. When the “w” is bigger, the algorithm has a strong ability in total searching and when the “W” is smaller, it is good for partial searching. Therefore, in recent years, some scholars brought up many schemes. LDW (Linearly Decreasing Inertia Weight) is given by,
w= w max t×( w max w min ) t max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bGaeyypa0Jaam4Da8aadaWgaaWcbaWdbiaad2gacaWGHbGa amiEaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8qacaWG0bGaey41aq 7aaeWaa8aabaWdbiaadEhapaWaaSbaaSqaa8qacaWGTbGaamyyaiaa dIhaa8aabeaak8qacqGHsislcaWG3bWdamaaBaaaleaapeGaamyBai aadMgacaWGUbaapaqabaaak8qacaGLOaGaayzkaaaapaqaa8qacaWG 0bWdamaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaaaaaa@4FED@
Where w max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bWdamaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaa aa@3A41@ & w min   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bWdamaaBaaaleaapeGaamyBaiaadMgacaWGUbaapaqabaGc peGaaeiOaaaa@3B7C@ The maximum and minimum value of W, t : The step of iteration, t max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bWdamaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaa aa@3A3E@ The maximum iteration step.
However, there are still problems in equation (12), so in the primary period of operation, if it detects the optimal point, it wants to converge to the optimal point promptly. However, the linear reduction slows down the speed of convergence of algorithm. In the later period of function, with the reduction of “w”, it may make the ability of total searching decline and the variety awaken. Finally it may easily fall into partial optimum. In this text, we use the PSO method of nonlinear variation weight with momentum to improve this method is given by,
w= w max t×( w max w min ) 2 θ t max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bGaeyypa0Jaam4Da8aadaWgaaWcbaWdbiaad2gacaWGHbGa amiEaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8qacaWG0bGaey41aq 7aaeWaa8aabaWdbiaadEhapaWaaSbaaSqaa8qacaWGTbGaamyyaiaa dIhaa8aabeaak8qacqGHsislcaWG3bWdamaaBaaaleaapeGaamyBai aadMgacaWGUbaapaqabaaak8qacaGLOaGaayzkaaaapaqaa8qacaaI YaWdamaaCaaaleqabaWdbiabeI7aXbaakiaadshapaWaaSbaaSqaa8 qacaWGTbGaamyyaiaadIhaa8aabeaaaaaaaa@52B5@
2θ is momentum, when in θ = t / tmax ,t is smaller, 2θ is near to 1 and w is near to w max MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bWdamaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaa aa@3A41@ ., it ensure the ability of total searching. With the increasing of t, w reduces in non linearity, ensuring the searching ability in partial areas. In the later t = t max avoiding the problems caused by the decrease of w. That is, the reduction ability of total searching and the decline of variety.

Back Propagation (BP) Neural Network
The standard Back Propagation (BP) Neural Network consists of input layer, one or several hidden layers and an input layer.

The node action function of BP neural network is generally “S” function. Common activation function f (x) is derivable sigmoid function:
f( x )= 1 1+ e x MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGMbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH9aqp daWcaaWdaeaapeGaaGymaaWdaeaapeGaaGymaiabgUcaRiaadwgapa WaaWbaaSqabeaapeGaeyOeI0IaamiEaaaaaaaaaa@4072@
Error function R is:
R= ( Y mj Y j ) 2 2  ( j=1,2,..,n ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbGaeyypa0ZaaSaaa8aabaWaaubiaeqaleqabaGaaGzaVdqd baWdbiabggHiLdaakmaabmaapaqaa8qacaWGzbWdamaaBaaaleaape GaamyBaiaadQgaa8aabeaak8qacqGHsislcaWGzbWdamaaBaaaleaa peGaamOAaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8 qacaaIYaaaaaGcpaqaa8qacaaIYaaaaiaabckadaqadaWdaeaapeGa amOAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aacYcacaWGUbaacaGLOaGaayzkaaaaaa@5044@
In this formula, Y j MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383D@ is expected out Y mj MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamyBaiaadQgaa8aabeaaaaa@392F@ is actual output n issample length.
The uniform expression of weight modified formula of BP algorithm is: w ij (t+1) = w ij ( t )+η δ pj ο pj MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG3bWdamaaDaaaleaapeGaamyAaiaadQgaa8aabaWdbiaacIca caWG0bGaey4kaSIaaGymaiaacMcaaaGccqGH9aqpcaWG3bWdamaaBa aaleaapeGaamyAaiaadQgaa8aabeaak8qadaqadaWdaeaapeGaamiD aaGaayjkaiaawMcaaiabgUcaRiabeE7aOjabes7aK9aadaWgaaWcba WdbiaadchacaWGQbaapaqabaGcpeGaeq4Vd82damaaBaaaleaapeGa amiCaiaadQgaa8aabeaaaaa@4ECB@
The specific process of BP algorithm can be generalized as follows:
a. Select n samples as a training set.
b. Initialize weight and biases value in neural network. The initialized values are always random numbers between (-1, 1). Every sample in the training set needs the following processing:
c. According to the size of every connection weight, the data of input layer are weighted and input into the activation function of hidden layer and then new values are obtained. According to the size of every connection weight, the new values are weighted and input into the activation function of output layer and the output results of output layer are calculated.
d. If error exist between output result and desired result, the calculation training is wrong.
e. Adjust weight and biases value.
f. According to new weight and biases values, the output layer is calculated. The calculation doesn’t stop until the training set meets the stopping condition.
Modified Particle Swarm Optimization (MPSO)
We apply Particle Swarm Optimization (PSO) to train BP Network by optimizing the original weight and the threshold value. When the algorithm ends, we can find the point near the overall situation optimal point. In the particle swarm, every particle’s position represents weights set among the BP network during the resent iteration. The dimension of every particle is decided by the number of the weight and the threshold value serving as connecting bridge.

The concrete process of MPSO is narrated as follows:
a. Initialization: n i MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGUbWdamaaBaaaleaapeGaamyAaaWdaeqaaaaa@3851@ is the number of neurons in the hidden layer no representing the number of neurons in input layer. So, the dimension of particle swarm D is:
D= n i × n h + n h × n ° + n h ++ n ° MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGebGaeyypa0JaamOBa8aadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacqGHxdaTcaWGUbWdamaaBaaaleaapeGaamiAaaWdaeqaaOWdbi abgUcaRiaad6gapaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaey41 aqRaamOBa8aadaWgaaWcbaWdbiabgclaWcWdaeqaaOWdbiabgUcaRi aad6gapaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaey4kaSIaey4k aSIaamOBa8aadaWgaaWcbaWdbiabgclaWcWdaeqaaaaa@4F78@
b. Setting fitness function of particle swarm’ in this text, we choose mean square error in BP Neural Network as fitness function of particle swarm:
E= 1 M K m j1 n o ( y kj y kj ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGfbGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad2ea aaWaaybCaeqal8aabaWdbiaadUeaa8aabaWdbiaad2gaa0Wdaeaape GaeyyeIuoaaOWaaybCaeqal8aabaWdbiaadQgacqGHsislcaaIXaaa paqaa8qacaWGUbWdamaaBaaameaapeGaam4BaaWdaeqaaaqdbaWdbi abggHiLdaakmaabmaapaqaa8qacaWG5bWdamaaBaaaleaapeGaam4A aiaadQgaa8aabeaak8qacqGHsislpaWaaCbiaeaapeGaamyEa8aada WgaaWcbaWdbiaadUgacaWGQbaapaqabaaabeqaaiabgkHiTaaaaOWd biaawIcacaGLPaaaaaa@5030@ y kj MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaBaaaleaapeGaam4AaiaadQgaa8aabeaaaaa@394D@ The output in theory based on sample K
y kj MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadMhapaWaaSbaaSqaa8qacaWGRbGaamOAaaWdaeqa aaqabeaacqGHsislaaaaaa@3A78@ The virtual output based on sample K
M: The number of Neural Network

c. Using the improved particle swarm algorithm to optimize the weight and the threshold value of BP network.

Coming to the optimal weight and the threshold value based on equation given below, g best =[ h 1 , h 2 ,.., h n h , o 1 , o 2 ,.., o n o , ih 1 , ih 2 ,.., ih n i × n h , ho 1 , ho 2 ,.., h o h × n o ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGNbWdamaaBaaaleaapeGaamOyaiaadwgacaWGZbGaamiDaaWd aeqaaOWdbiabg2da9maadmaapaqaauaabeqaceaaaeaapeGaamiAa8 aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiAa8aadaWg aaWcbaWdbiaaikdaa8aabeaak8qacaGGSaGaaiOlaiaac6cacaGGSa GaamiAa8aadaWgaaWcbaWdbiaad6gapaWaaSbaaWqaa8qacaWGObaa paqabaaaleqaaOWdbiaacYcacaqGVbWdamaaBaaaleaapeGaaGymaa WdaeqaaOWdbiaacYcacaqGVbWdamaaBaaaleaapeGaaGOmaaWdaeqa aOWdbiaacYcacaGGUaGaaiOlaiaacYcacaqGVbWdamaaBaaaleaape GaaeOBa8aadaWgaaadbaWdbiaab+gaa8aabeaaaSqabaGcpeGaaiil aiaabMgacaqGObWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacY cacaqGPbGaaeiAa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGG SaGaaiOlaiaac6cacaGGSaGaaeyAaiaabIgapaWaaSbaaSqaa8qaca qGUbWdamaaBaaameaapeGaaeyAaaWdaeqaaSWdbiabgEna0kaab6ga paWaaSbaaWqaa8qacaqGObaapaqabaaaleqaaOWdbiaacYcaa8aaba WdbiaabIgacaqGVbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaa cYcacaqGObGaae4Ba8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qaca GGSaGaaiOlaiaac6cacaGGSaGaaeiAa8aadaWgaaWcbaWdbiaab+ga paWaaSbaaWqaa8qacaqGObaapaqabaWcpeGaey41aqRaaeOBa8aada WgaaadbaWdbiaab+gaa8aabeaaaSqabaaaaaGcpeGaay5waiaaw2fa aaaa@7B20@
h i ( =1,2,.., n k ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGObWdamaaBaaaleaapeGaaeyAaaWdaeqaaOWdbmaabmaapaqa a8qacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6caca GGSaGaaeOBa8aadaWgaaWcbaWdbiaabUgaa8aabeaaaOWdbiaawIca caGLPaaaaaa@424D@ The threshold value in the hidden layer
o i ( i=1,2,.., n o )  MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGVbWdamaaBaaaleaapeGaaeyAaaWdaeqaaOWdbmaabmaapaqa a8qacaqGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6caca GGUaGaaiilaiaab6gapaWaaSbaaSqaa8qacaqGVbaapaqabaaak8qa caGLOaGaayzkaaGaaeiOaaaa@4467@ The threshold value in the output layer
ih i ( i1,2,.., n i × ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGPbGaaeiAa8aadaWgaaWcbaWdbiaabMgaa8aabeaak8qadaqa daWdaeaapeGaaeyAaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaai OlaiaacYcacaqGUbWdamaaBaaaleaapeGaaeyAaaWdaeqaaOWdbiab gEna0cGaayjkaiaawMcaaaaa@4534@ The weight between the hidden layer and the input layer
ho i ( i=1,2,.., n h × n o ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGObGaae4Ba8aadaWgaaWcbaWdbiaabMgaa8aabeaak8qadaqa daWdaeaapeGaaeyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcaca GGUaGaaiOlaiaacYcacaqGUbWdamaaBaaaleaapeGaaeiAaaWdaeqa aOWdbiabgEna0kaab6gapaWaaSbaaSqaa8qacaqGVbaapaqabaaak8 qacaGLOaGaayzkaaaaaa@4896@ The weight between the hidden layer and the output layer
d. Let the optimal weight and the threshold value be as the original weight and then put them into Neural Network for training. Adjusting the weight and the threshold value based on BP algorithm until the function index of the network’s Mean Square Error (MSE) < e. “e” is the preset expected index.
Simulation Results
Validity of proposed Modified particle swarm optimization (MPSO) algorithm has been verified by testing in IEEE 30-bus, 41 branch system and it has 6 generator-bus voltage magnitudes, 4 transformer-tap settings, and 2 bus shunt reactive compensators. Bus 1 is taken as slack bus and 2, 5, 8, 11 and 13 are considered as PV generator buses and others are PQ load buses. Control variables limits are given in Table 1. In Table 2 the power limits of generators buses are listed.

Table 3 shows the proposed MPSO approach successfully kept the control variables within limits.Table 4 narrates about the performance of the proposed MPSO algorithm. Figure 1 show about the voltage deviations during the iterations and Table 5
Table 1: Primary Variable Limits (Pu)

 

Variables

 

Min.

 

Max.

Category

Generator Bus

0.90

1.11

Continuous

Load Bus

0.91

1.01

Continuous

Transformer-Tap

0.92

1.01

Discrete

Shunt Reactive Compensator

-0.10

0.30

Discrete

Table 2: Generators Power Limits

Bus

Pg

Pgmin

Pgmax

Qgmin

1

96.00

49

200

-19

2

79.00

18

79

-19

5

49.00

14

49

-11

8

21.00

11

31

-14

11

21.00

11

28

-12

13

21.00

11

39

-14

Table 3: After optimization values of control variables

Control  Variables

 

MPSO

V1

1.0541

V2

1.0452

V5

1.0263

V8

1.0374

V11

1.0745

V13

1.0575

T4,12

0.00

T6,9

0.01

T6,10

0.90

T28,27

0.91

Q10

0.10

Q24

0.10

Real power loss

4.2809

Voltage deviation

0.9085

Table 4: Performance of MPSO algorithm

Iterations

32

Time taken (secs)

8.91

Real power loss

4.2809

Table 5: Comparison of results

Techniques

Real power loss (MW)

SGA(Wu et al., 1998) [33]

4.98

PSO(Zhao et al., 2005) [34]

4.9262

LP(Mahadevan et al., 2010) [35]

5.988

EP(Mahadevan et al., 2010) [35]

4.963

CGA(Mahadevan et al., 2010) [35]

4.980

AGA(Mahadevan et al., 2010) [35]

4.926

CLPSO(Mahadevan et al., 2010) [35]

4.7208

HSA (Khazali et al., 2011) [36]

4.7624

BB-BC (Sakthivel et al., 2013) [37]

4.690

MCS(Tejaswini sharma et al.,2016) [38]

4.87231

Proposed MPSO

4.2809

Figure 1:Voltage Deviation (VD) Characteristics
list out the overall comparison of the results of optimal solution obtained by various methods.
Conclusion
In this paper an inventive approach Modified Particle Swarm Optimization (MPSO) algorithm is used to solve reactive power problem, by considering various constraints. The efficiency of the projected Modified particle swarm optimization (MPSO) method is demonstrated by testing it on standard IEEE 30-bus system and the real power loss has been considerably reduced when compared to other reported standard algorithms & voltage profile are well within the specified limits .
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