- Email: [email protected]
Multi-area Economic Dispatch Using Improved Particle Swarm Optimization
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 75 (2015) 1087 – 1092
The 7th International Conference on Applied Energy – ICAE2015
Multi-area Economic Dispatch using Improved Particle Swarm Optimization Vinay K. Jadouna, Nikhil Guptaa, K. R. Niazia, Anil Swarnkara, R.C. Bansalb* a
Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur,302017, India Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa
b
Abstract This paper presents improved PSO (IPSO) to solve Multi Area Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal value of power generation and interchange of power through tie-lines interconnecting areas in such a way that total fuel cost of thermal generating units of all areas is minimized while satisfying operational constraints. The control equation of the proposed PSO is modified by suggesting improved cognitive component of the particle’s velocity by suggesting preceding experience. The operating parameters of the control equation are also modified to maintain a better balance between cognitive and social behavior of the swarm. The effectiveness of the proposed method has been tested on four areas, 40 generators test system. The application results show that IPSO is very promising to solve large-dimensional MAED problem. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
© 2015 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection peer-review of under responsibility of ICAE Peer-reviewand/or under responsibility Applied Energy Innovation Institute
Keywords: multi-area economic dispatch; fuel cost minimization; particle swarm optimization; tie-line capacity.
1. Introduction The conventional single area economic dispatch (SAED) is trailing its relevance in the present competitive business scenario of deregulated environment in power sector. Areas of individual power systems are interconnected to operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated area [1]. Each area has its own generation cost and load pattern. The power generation utilities and power pools can stagger their generations to optimize the cost * Corresponding author. Tel.: +27 12 4205446; fax: +27 12 3625000. E-mail address: [email protected].
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Applied Energy Innovation Institute doi:10.1016/j.egypro.2015.07.493
1088
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
of unit energy generation from fossil fuel plants. This can be accomplished through multi-area economic dispatch (MAED). MAED is an extension of SAED that aims to determine the optimal power generation schedule and interexchange of power in such a way that minimizes the overall fuel cost of all thermal generating units while satisfying several operational and network constraints. However, the system security imposes restriction on the inter-area power transactions through tie-lines. In fact, the complexity of MAED problem arises due to the stringent area power balance constraints, tie-line constraints and other operational constraints [2]. Some early efforts to attempt MAED problem can be briefly stated as: Jayabarathi et al. [3] solved multi-area economic dispatch problems with tie-line constraints using evolutionary programming. Chen and Chen [4] presented direct search method for solving economic dispatch problem considering transmission capacity constraints. Manoharan et al. [5] proposed covariance matrix adapted evolutionary strategy for MAED problems where a Karush Kuhun Tucker (KKT) optimality criterion is applied to guarantee the optimal convergence. Wang and Singh [6] used particle swarm optimization (PSO) for this problem where tie-line transfer capacities and area spinning reserve sharing are incorporated to ensure security and improve reliability, respectively. Zhu [7] presented a new nonlinear optimization neural network approach to study the problem of security-constrained interconnected MAED. Manisha et al. [2] formulated MAED problem with various constraints and also compared the solution quality of differential evolution (DE) variants with an improved PSO strategy. Basu [8] applied artificial bee colony optimization (ABCO) to solve MAED problem with a variety of system constraints. The author claimed that ABCO is capable to generate better quality solution than other established meta-heuristic techniques. PSO is a swarm evolutionary based meta-heuristic technique in which the movement of the particles is governed by two stochastic acceleration coefficients, i. e., cognitive and social components and the inertia. Several modified versions of PSO have been reported in the recent past to enhance its performance by modulating inertia weight [9-10], improvising cognitive and social behavior [10-11], using constriction factor approach [12], modifying the control equation of the PSO [9], [13-15], or squeezing the search space [15], etc. However, these suggested versions of PSO require several experimentations for parameter setting or requires some additional mechanism to avoid local trapping or to regulate particle’s velocity in order to maintain a better balance between cognitive and social behavior of the swarm. This paper attempts to overcome the drawbacks of some existing PSO methods and presents an improved PSO (IPSO) method to efficiently solve large dimensional multi-constraints optimization problems like MAED. Several measures have been incorporated in the control equation of the conventional PSO for dynamic control of its parameters so that a proper balance is maintained between cognitive and social behavior of the swarm. The proposed method effectively regulates the velocity of particles during their flights. The effectiveness of the proposed method has been tested for MAED on 40generators test system considering various operational constraints. The application results are very promising when compared with other established methods. 2. Problem formulation The objective function for the MAED problem can be stated as to M NGi
Minimize F ( PGij )
¦ ¦ (a
ij
i 1 j 1
bij PGij cij PGij ) | eij sin( fij ( PGij min - PGij )) | 2
(1)
1089
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
where aij, bij, cij, are the cost coefficients, and eij and fij are the valve point effect coefficients of the jth generator in area i, PGij is the real power output of the jth generator in area i, M is the number of areas and NGi is the number of generating units in the system in area i. Subject to the following constraints: 2.1 Power balance constraints NGi
¦P
Gij
j 1
PDi
¦P
k ,k zi
Tik
; iM
(2)
where PDi is the power demand of area i; PTik is the tie line real power transfer from area i to area k. PTik is positive when power flows from area i to area k and is negative when power flows from area k to area i. 2.2 Generator Constraints min max PGij d PGij d PGij
(3)
2.3 Tie-line Capacity Constraints PTijmax d PTij d PTijmax
(4)
3. Proposed IPSO The conventional PSO is initialized with a population of random solutions and searches for optima by updating particle positions. The velocity of the particle is influenced by three components namely, initial, cognitive and social components. An appropriate balance is required between cognitive and social behavior of the swarm to regulate particle’s velocity throughout the computational process. Further, during early iterations, the cognitive component must dominate over the social component to ensure global exploration of the search space. However, during later iterations, the social component must dominate over the cognitive one so as to divert all particles towards the global best to improve local exploitation. This is essential for a good balance between exploration and exploitation as suggested by [16]. Therefore, a modified control equation is suggested as below: vik 1 W u vik C1b u rand1 () u
s kj 1
s kj vkj 1 u 't
pbesti k - sik s k - pprecedingi k gbest k - sik C1 p u rand 2 () u i kS u C2 u rand3 () u 't 't 't
(5) (6)
where, W is the inertia weight, vik is the velocity of ith particle at kth iteration, rand1( ) and rand2( ) are random numbers between 0 and 1, sik is the position of ith particle at kth iteration, C1, C2 are the acceleration coefficients, pbestik is the best position of ith particle achieved based on its own experience, gbestk is the best particle position based on overall swarm experience, Δt is the time step, usually set to 1 second. In (5), the cognitive component is divided into best and preceding experience of the particle whereas the strength of the social component has been weakened using constriction factor ks. In addition, the inertia weight is updated exponentially as described below.
1090
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
3.1 Inertia Weight Update The trend of linear modulation of inertia weight of [17] is followed to solve optimization problems using PSO by many researchers till date [13-14], etc. For large scale optimization problems, there exist large numbers of local optima in close vicinity of the global optima. Therefore, the exploitation potential of the search algorithm should be sufficient to obtain better solutions. Therefore, the inertia weight has been intuitively varied exponentially with respect to iterations. Modulations suggested to update the inertia weight is governed by the following relation: W = exp(–ηloge(Wmax/Wmin)); η = itr/itrmax
(7)
3.2 Updating Preceding Experience The cognitive behavior was split in [14] by considering also the worst experience in addition to the best experience of the particle to provide some additional diversity, but it results in poor local exploitation unless supported by a local random search. Therefore, the concept of preceding experience is suggested where the current fitness of each particle is compared with its fitness value in the preceding iteration, and if it is found less, it will be treated as the preceding experience. The preceding experience of the particle produces much less diversity than the worst experience and thus can provide better exploration and exploitation of the search space without employing any additional local random search or else. 3.3 Modified Social Experience It can be realized that the inertia imparted to particles became quite weak during later iterations while employing exponential modulations in inertia weight. This restricting particles’ velocities which are being grooming probably around the global optima. The preceding experience just fine tunes the best experience of the particles to pull them towards the global optima. Therefore, the social component must be weakened otherwise the particle may fly with moderate high velocities and eventually miss the global or near global optima. Thus the social component is made weak by proposing a constriction factor ks in the control equation. The optimal value of this factor however can be evaluated through experimentations. These alterations in the control equation of the conventional PSO regulates particles’ velocities without any additional formulation as reported in many improved versions of PSO [11], [16], yet preserving diversity due to the stochastic nature of cognitive and social behaviors of the swarm. 4. Simulation results and discussion The proposed algorithm is tested on four areas, 40 generators system [8] consists of 10 thermal units in each area with non-convexity in generator cost function due to valve-point loading effects. All areas are interconnected through six tie-lines. The load demand for area 1 is 15%, area 2 is 40%, area 3 is 30% and area 4 is15 % of the total load demand of 10500 MW. The acceleration coefficients C1b, C1p and C2 for this test system are taken as 1.6, 0.4 and 2.0 respectively, as in [14]. The value of maximum and minimum bounds of the inertia weight is taken as 0.9 and 0.1, respectively. The population size for this system is set as 50 and the maximum iterations are considered as 1500. The proposed algorithm has been developed using MATLAB and the simulations have been carried on a personal computer of Intel i5, 3.2 GHz, and 4 GB RAM.
1091
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
The optimal value of ks is determined by experimentations on this system. For the purpose, the proposed PSO runs with ks varied from 0.05 to 0.25 with a step size of 0.05. It has been observed that the best value of the average fuel cost and standard deviation (STD) is obtained when ks is 0.10. Therefore, ks is selected as 0.10 for simulations. The proposed IPSO is now applied on this system and the application results obtained after 100 trails are presented in Table 1. The table also shows that IPSO is computationally much less demanding than all other methods. In order to highlight the effect of each modifications suggested in the control equation of the PSO, the variants of PSO so obtained are classified as ‘b’, ‘c’ and ‘d’; ‘a’ refers to the conventional PSO, ‘b’ refers to ‘a’ with exponential modulations in inertia weight, ‘c’ refers to ‘b’ with preceding experience added in the cognitive component and ‘d’ refers to the proposed IPSO. A comparison of the set of convergence characteristics for best and average fuel cost is shown in Figures 1 and 2, respectively. Table 1. Comparison results Method RCGA [8] EP [8] DE [8] ABCO [8] IPSO
Best fuel cost ($/hr) 129911.8 124574.5 124544.1 124009.4 122459.5
Average fuel cost ($/hr) 123228.3
Worst fuel Cost ($/hr) 125565.3
CPU time (s) 160.5 144.5 134.8 126.9 47.8
It can be observed from Fig. 1 that while subsequently modifying the inertia weight, cognitive and the social component in the control equation of the conventional PSO, the convergence characteristics are progressively improved by avoiding more and more local trappings. It can also be observed from the figure that except IPSO, the initial shape of convergence is more or less same. Therefore, the introduction of the constriction factor in the social component of particle’s velocity is found to be one of the key factors that is responsible for better convergence in IPSO. A similar conclusion can be drawn from Fig. 2. It can be observed from the figure that in IPSO alone, the particles are not identifying the promising region during early iterations. Therefore, it offers advantage to exploit this region meticulously and thus produces better quality solutions. 5. Conclusions This paper attempts to overcome the drawbacks of the existing PSO methods and presents an improved PSO (IPSO) method. Efforts have been made to modify PSO parameters in such a fashion that ensures a proper balance between exploration and exploitation ability of PSO and result in faster global convergence, higher solution quality, and stronger robustness. The applicability of the proposed method has been investigated to solve complex MAED problem of test generating system of large dimension with a variety of operational constraints. The application results show that the proposed method is efficient and is not trapped in local minima. The comparison of application results show that the proposed method is capable of producing better quality solution and is computationally more efficient than other the established techniques. It is noteworthy that the proposed IPSO does not require additional mechanism to avoid local trapping, and empirical formula to bound particle’s velocity, squeezing the search space. Moreover, it is independent of the initial state of particles.
1092
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
Fig. 1. Convergence characteristics for best fuel cost
Fig. 2. Convergence characteristics for average fuel cost
References [1]. Sudhakar AVV, Chandram K, Jayalaxmi A. Multi area economic dispatch using secant method. Journal of Electrical Engineering & Technology 2013; 8 (4):708-715,. [2]. Sharma M, Pandit M, Srivastava L. Reserve constrained multi-area economic dispatch employing differential evolution with time-varying mutation. Int J. Electrical Power and Energy Systems 2011; 33:753–766. [3]. Jayabarathi T, Sadasivam G, Ramachandran V. Evolutionary programming based multi-area economic dispatch with tie line constraints. Electr Mach Power Syst. 2000; 28:1165–76. [4]. Chen CL, Chen N. Direct search method for solving economic dispatch problem considering transmission capacity constraints. IEEE Trans Power Syst. 2001; 16(4): 764–769. [5]. Manoharan PS, Kannan PS, Baskar S, Iruthayarajan MW. Evolutionary algorithm solution and KKT based optimality verification to multi-area economic dispatch. Int J Electrical Power Energy Syst. 2009; 31: 365–373. [6]. Wang L, Singh C. Reserve-constrained multi-area environmental/economic dispatch based on particle swarm optimization with local search. Engineering Applications of Artificial Intelligence 2009; 22: 298–307. [7]. Zhu JZ. Multiarea Power Systems Economic Power Dispatch Using a Nonlinear Optimization Neural Network Approach. Electric Power Components and Systems 2003; 31 (6): 553-563. [8]. Basu M. Artificial bee colony optimization for multi-area economic dispatch. Int J. Electrical Power and Energy Systems 2013; 49: 181–187. [9]. Vlachogiannis JG, Lee KY. Economic load dispatch–a comparative study on heuristic optimization techniques with an improved coordinated aggregation based PSO. IEEE Trans. Power Systems 2009; 24(2):991–1001. [10]. Wang Y, Zhou J, Zhou C, Wang Y, Qin H, Lu Y. An improved self-adaptive PSO technique for short-term hydrothermal scheduling. Expert Systems with Applications 2012; 39: 2288–2295. [11]. Ivatloo BM. Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients. Electric Power Systems Research 2013; 95: 9-18. [12]. Yu B, Yuan X, Wang J, Short-term hydro-thermal scheduling using particle swarm optimization method. Energy Conversion and Management 2007; 48: 1902–1908. [13]. Safari A, Shayeghi H. Iteration particle swarm optimization procedure for economic load dispatch with generator constraints. Expert Systems with Applications 2011; 38: 6043–6048. [14]. Selvakumar A I, Thanushkodi K. A new particle swarm optimization solution to non-convex economic dispatch problems, IEEE Trans. Power Syst. 2007; 22 (1): 42-51. [15]. Park J B, Lee K S, Shin J R, Lee KY. A Particle Swarm Optimization for Economic Dispatch with Non-smooth Cost Functions. IEEE Trans. Power Systems 2005; 20 (1): 34-42. [16]. Jiejin C, Xiaoqian M, Lixiang L, Haipeng P. Chaotic particle swarm optimization for economic dispatch considering the generator constraints. Energy Conversion & Management 2007; 48:645–653. [17]. Shi Y, Eberhart R C. Empirical study of particle swarm optimization. in Proc. 1999 IEEE Congress on Evolutionary Computation, Piscataway, NJ, USA, IEEE Press, p. 1945–1950.
ScienceDirect Energy Procedia 75 (2015) 1087 – 1092
The 7th International Conference on Applied Energy – ICAE2015
Multi-area Economic Dispatch using Improved Particle Swarm Optimization Vinay K. Jadouna, Nikhil Guptaa, K. R. Niazia, Anil Swarnkara, R.C. Bansalb* a
Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur,302017, India Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa
b
Abstract This paper presents improved PSO (IPSO) to solve Multi Area Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal value of power generation and interchange of power through tie-lines interconnecting areas in such a way that total fuel cost of thermal generating units of all areas is minimized while satisfying operational constraints. The control equation of the proposed PSO is modified by suggesting improved cognitive component of the particle’s velocity by suggesting preceding experience. The operating parameters of the control equation are also modified to maintain a better balance between cognitive and social behavior of the swarm. The effectiveness of the proposed method has been tested on four areas, 40 generators test system. The application results show that IPSO is very promising to solve large-dimensional MAED problem. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
© 2015 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection peer-review of under responsibility of ICAE Peer-reviewand/or under responsibility Applied Energy Innovation Institute
Keywords: multi-area economic dispatch; fuel cost minimization; particle swarm optimization; tie-line capacity.
1. Introduction The conventional single area economic dispatch (SAED) is trailing its relevance in the present competitive business scenario of deregulated environment in power sector. Areas of individual power systems are interconnected to operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated area [1]. Each area has its own generation cost and load pattern. The power generation utilities and power pools can stagger their generations to optimize the cost * Corresponding author. Tel.: +27 12 4205446; fax: +27 12 3625000. E-mail address: [email protected].
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Applied Energy Innovation Institute doi:10.1016/j.egypro.2015.07.493
1088
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
of unit energy generation from fossil fuel plants. This can be accomplished through multi-area economic dispatch (MAED). MAED is an extension of SAED that aims to determine the optimal power generation schedule and interexchange of power in such a way that minimizes the overall fuel cost of all thermal generating units while satisfying several operational and network constraints. However, the system security imposes restriction on the inter-area power transactions through tie-lines. In fact, the complexity of MAED problem arises due to the stringent area power balance constraints, tie-line constraints and other operational constraints [2]. Some early efforts to attempt MAED problem can be briefly stated as: Jayabarathi et al. [3] solved multi-area economic dispatch problems with tie-line constraints using evolutionary programming. Chen and Chen [4] presented direct search method for solving economic dispatch problem considering transmission capacity constraints. Manoharan et al. [5] proposed covariance matrix adapted evolutionary strategy for MAED problems where a Karush Kuhun Tucker (KKT) optimality criterion is applied to guarantee the optimal convergence. Wang and Singh [6] used particle swarm optimization (PSO) for this problem where tie-line transfer capacities and area spinning reserve sharing are incorporated to ensure security and improve reliability, respectively. Zhu [7] presented a new nonlinear optimization neural network approach to study the problem of security-constrained interconnected MAED. Manisha et al. [2] formulated MAED problem with various constraints and also compared the solution quality of differential evolution (DE) variants with an improved PSO strategy. Basu [8] applied artificial bee colony optimization (ABCO) to solve MAED problem with a variety of system constraints. The author claimed that ABCO is capable to generate better quality solution than other established meta-heuristic techniques. PSO is a swarm evolutionary based meta-heuristic technique in which the movement of the particles is governed by two stochastic acceleration coefficients, i. e., cognitive and social components and the inertia. Several modified versions of PSO have been reported in the recent past to enhance its performance by modulating inertia weight [9-10], improvising cognitive and social behavior [10-11], using constriction factor approach [12], modifying the control equation of the PSO [9], [13-15], or squeezing the search space [15], etc. However, these suggested versions of PSO require several experimentations for parameter setting or requires some additional mechanism to avoid local trapping or to regulate particle’s velocity in order to maintain a better balance between cognitive and social behavior of the swarm. This paper attempts to overcome the drawbacks of some existing PSO methods and presents an improved PSO (IPSO) method to efficiently solve large dimensional multi-constraints optimization problems like MAED. Several measures have been incorporated in the control equation of the conventional PSO for dynamic control of its parameters so that a proper balance is maintained between cognitive and social behavior of the swarm. The proposed method effectively regulates the velocity of particles during their flights. The effectiveness of the proposed method has been tested for MAED on 40generators test system considering various operational constraints. The application results are very promising when compared with other established methods. 2. Problem formulation The objective function for the MAED problem can be stated as to M NGi
Minimize F ( PGij )
¦ ¦ (a
ij
i 1 j 1
bij PGij cij PGij ) | eij sin( fij ( PGij min - PGij )) | 2
(1)
1089
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
where aij, bij, cij, are the cost coefficients, and eij and fij are the valve point effect coefficients of the jth generator in area i, PGij is the real power output of the jth generator in area i, M is the number of areas and NGi is the number of generating units in the system in area i. Subject to the following constraints: 2.1 Power balance constraints NGi
¦P
Gij
j 1
PDi
¦P
k ,k zi
Tik
; iM
(2)
where PDi is the power demand of area i; PTik is the tie line real power transfer from area i to area k. PTik is positive when power flows from area i to area k and is negative when power flows from area k to area i. 2.2 Generator Constraints min max PGij d PGij d PGij
(3)
2.3 Tie-line Capacity Constraints PTijmax d PTij d PTijmax
(4)
3. Proposed IPSO The conventional PSO is initialized with a population of random solutions and searches for optima by updating particle positions. The velocity of the particle is influenced by three components namely, initial, cognitive and social components. An appropriate balance is required between cognitive and social behavior of the swarm to regulate particle’s velocity throughout the computational process. Further, during early iterations, the cognitive component must dominate over the social component to ensure global exploration of the search space. However, during later iterations, the social component must dominate over the cognitive one so as to divert all particles towards the global best to improve local exploitation. This is essential for a good balance between exploration and exploitation as suggested by [16]. Therefore, a modified control equation is suggested as below: vik 1 W u vik C1b u rand1 () u
s kj 1
s kj vkj 1 u 't
pbesti k - sik s k - pprecedingi k gbest k - sik C1 p u rand 2 () u i kS u C2 u rand3 () u 't 't 't
(5) (6)
where, W is the inertia weight, vik is the velocity of ith particle at kth iteration, rand1( ) and rand2( ) are random numbers between 0 and 1, sik is the position of ith particle at kth iteration, C1, C2 are the acceleration coefficients, pbestik is the best position of ith particle achieved based on its own experience, gbestk is the best particle position based on overall swarm experience, Δt is the time step, usually set to 1 second. In (5), the cognitive component is divided into best and preceding experience of the particle whereas the strength of the social component has been weakened using constriction factor ks. In addition, the inertia weight is updated exponentially as described below.
1090
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
3.1 Inertia Weight Update The trend of linear modulation of inertia weight of [17] is followed to solve optimization problems using PSO by many researchers till date [13-14], etc. For large scale optimization problems, there exist large numbers of local optima in close vicinity of the global optima. Therefore, the exploitation potential of the search algorithm should be sufficient to obtain better solutions. Therefore, the inertia weight has been intuitively varied exponentially with respect to iterations. Modulations suggested to update the inertia weight is governed by the following relation: W = exp(–ηloge(Wmax/Wmin)); η = itr/itrmax
(7)
3.2 Updating Preceding Experience The cognitive behavior was split in [14] by considering also the worst experience in addition to the best experience of the particle to provide some additional diversity, but it results in poor local exploitation unless supported by a local random search. Therefore, the concept of preceding experience is suggested where the current fitness of each particle is compared with its fitness value in the preceding iteration, and if it is found less, it will be treated as the preceding experience. The preceding experience of the particle produces much less diversity than the worst experience and thus can provide better exploration and exploitation of the search space without employing any additional local random search or else. 3.3 Modified Social Experience It can be realized that the inertia imparted to particles became quite weak during later iterations while employing exponential modulations in inertia weight. This restricting particles’ velocities which are being grooming probably around the global optima. The preceding experience just fine tunes the best experience of the particles to pull them towards the global optima. Therefore, the social component must be weakened otherwise the particle may fly with moderate high velocities and eventually miss the global or near global optima. Thus the social component is made weak by proposing a constriction factor ks in the control equation. The optimal value of this factor however can be evaluated through experimentations. These alterations in the control equation of the conventional PSO regulates particles’ velocities without any additional formulation as reported in many improved versions of PSO [11], [16], yet preserving diversity due to the stochastic nature of cognitive and social behaviors of the swarm. 4. Simulation results and discussion The proposed algorithm is tested on four areas, 40 generators system [8] consists of 10 thermal units in each area with non-convexity in generator cost function due to valve-point loading effects. All areas are interconnected through six tie-lines. The load demand for area 1 is 15%, area 2 is 40%, area 3 is 30% and area 4 is15 % of the total load demand of 10500 MW. The acceleration coefficients C1b, C1p and C2 for this test system are taken as 1.6, 0.4 and 2.0 respectively, as in [14]. The value of maximum and minimum bounds of the inertia weight is taken as 0.9 and 0.1, respectively. The population size for this system is set as 50 and the maximum iterations are considered as 1500. The proposed algorithm has been developed using MATLAB and the simulations have been carried on a personal computer of Intel i5, 3.2 GHz, and 4 GB RAM.
1091
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
The optimal value of ks is determined by experimentations on this system. For the purpose, the proposed PSO runs with ks varied from 0.05 to 0.25 with a step size of 0.05. It has been observed that the best value of the average fuel cost and standard deviation (STD) is obtained when ks is 0.10. Therefore, ks is selected as 0.10 for simulations. The proposed IPSO is now applied on this system and the application results obtained after 100 trails are presented in Table 1. The table also shows that IPSO is computationally much less demanding than all other methods. In order to highlight the effect of each modifications suggested in the control equation of the PSO, the variants of PSO so obtained are classified as ‘b’, ‘c’ and ‘d’; ‘a’ refers to the conventional PSO, ‘b’ refers to ‘a’ with exponential modulations in inertia weight, ‘c’ refers to ‘b’ with preceding experience added in the cognitive component and ‘d’ refers to the proposed IPSO. A comparison of the set of convergence characteristics for best and average fuel cost is shown in Figures 1 and 2, respectively. Table 1. Comparison results Method RCGA [8] EP [8] DE [8] ABCO [8] IPSO
Best fuel cost ($/hr) 129911.8 124574.5 124544.1 124009.4 122459.5
Average fuel cost ($/hr) 123228.3
Worst fuel Cost ($/hr) 125565.3
CPU time (s) 160.5 144.5 134.8 126.9 47.8
It can be observed from Fig. 1 that while subsequently modifying the inertia weight, cognitive and the social component in the control equation of the conventional PSO, the convergence characteristics are progressively improved by avoiding more and more local trappings. It can also be observed from the figure that except IPSO, the initial shape of convergence is more or less same. Therefore, the introduction of the constriction factor in the social component of particle’s velocity is found to be one of the key factors that is responsible for better convergence in IPSO. A similar conclusion can be drawn from Fig. 2. It can be observed from the figure that in IPSO alone, the particles are not identifying the promising region during early iterations. Therefore, it offers advantage to exploit this region meticulously and thus produces better quality solutions. 5. Conclusions This paper attempts to overcome the drawbacks of the existing PSO methods and presents an improved PSO (IPSO) method. Efforts have been made to modify PSO parameters in such a fashion that ensures a proper balance between exploration and exploitation ability of PSO and result in faster global convergence, higher solution quality, and stronger robustness. The applicability of the proposed method has been investigated to solve complex MAED problem of test generating system of large dimension with a variety of operational constraints. The application results show that the proposed method is efficient and is not trapped in local minima. The comparison of application results show that the proposed method is capable of producing better quality solution and is computationally more efficient than other the established techniques. It is noteworthy that the proposed IPSO does not require additional mechanism to avoid local trapping, and empirical formula to bound particle’s velocity, squeezing the search space. Moreover, it is independent of the initial state of particles.
1092
Vinay K. Jadoun et al. / Energy Procedia 75 (2015) 1087 – 1092
Fig. 1. Convergence characteristics for best fuel cost
Fig. 2. Convergence characteristics for average fuel cost
References [1]. Sudhakar AVV, Chandram K, Jayalaxmi A. Multi area economic dispatch using secant method. Journal of Electrical Engineering & Technology 2013; 8 (4):708-715,. [2]. Sharma M, Pandit M, Srivastava L. Reserve constrained multi-area economic dispatch employing differential evolution with time-varying mutation. Int J. Electrical Power and Energy Systems 2011; 33:753–766. [3]. Jayabarathi T, Sadasivam G, Ramachandran V. Evolutionary programming based multi-area economic dispatch with tie line constraints. Electr Mach Power Syst. 2000; 28:1165–76. [4]. Chen CL, Chen N. Direct search method for solving economic dispatch problem considering transmission capacity constraints. IEEE Trans Power Syst. 2001; 16(4): 764–769. [5]. Manoharan PS, Kannan PS, Baskar S, Iruthayarajan MW. Evolutionary algorithm solution and KKT based optimality verification to multi-area economic dispatch. Int J Electrical Power Energy Syst. 2009; 31: 365–373. [6]. Wang L, Singh C. Reserve-constrained multi-area environmental/economic dispatch based on particle swarm optimization with local search. Engineering Applications of Artificial Intelligence 2009; 22: 298–307. [7]. Zhu JZ. Multiarea Power Systems Economic Power Dispatch Using a Nonlinear Optimization Neural Network Approach. Electric Power Components and Systems 2003; 31 (6): 553-563. [8]. Basu M. Artificial bee colony optimization for multi-area economic dispatch. Int J. Electrical Power and Energy Systems 2013; 49: 181–187. [9]. Vlachogiannis JG, Lee KY. Economic load dispatch–a comparative study on heuristic optimization techniques with an improved coordinated aggregation based PSO. IEEE Trans. Power Systems 2009; 24(2):991–1001. [10]. Wang Y, Zhou J, Zhou C, Wang Y, Qin H, Lu Y. An improved self-adaptive PSO technique for short-term hydrothermal scheduling. Expert Systems with Applications 2012; 39: 2288–2295. [11]. Ivatloo BM. Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients. Electric Power Systems Research 2013; 95: 9-18. [12]. Yu B, Yuan X, Wang J, Short-term hydro-thermal scheduling using particle swarm optimization method. Energy Conversion and Management 2007; 48: 1902–1908. [13]. Safari A, Shayeghi H. Iteration particle swarm optimization procedure for economic load dispatch with generator constraints. Expert Systems with Applications 2011; 38: 6043–6048. [14]. Selvakumar A I, Thanushkodi K. A new particle swarm optimization solution to non-convex economic dispatch problems, IEEE Trans. Power Syst. 2007; 22 (1): 42-51. [15]. Park J B, Lee K S, Shin J R, Lee KY. A Particle Swarm Optimization for Economic Dispatch with Non-smooth Cost Functions. IEEE Trans. Power Systems 2005; 20 (1): 34-42. [16]. Jiejin C, Xiaoqian M, Lixiang L, Haipeng P. Chaotic particle swarm optimization for economic dispatch considering the generator constraints. Energy Conversion & Management 2007; 48:645–653. [17]. Shi Y, Eberhart R C. Empirical study of particle swarm optimization. in Proc. 1999 IEEE Congress on Evolutionary Computation, Piscataway, NJ, USA, IEEE Press, p. 1945–1950.