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Improved Particle Swarm Optimization for Multi-area Economic Dispatch with Reserve Sharing Scheme
9th 9th IFAC IFAC symposium symposium on on Control Control of of Power Power and and Energy Energy Systems Systems Indian Institute of Technology 9th IFAC symposium on Control of Power and Indian Institute of Technology Available online Systems at www.sciencedirect.com 9th IFAC symposium on Control of Power and Energy Energy Systems December 9-11, 2015. Delhi, India Indian Institute Institute of Technology Technology Indian of December 9-11, 2015. Delhi, India December December 9-11, 9-11, 2015. 2015. Delhi, Delhi, India India
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Improved Improved Particle Particle Swarm Swarm Optimization Optimization for for Multi-area Multi-area Economic Economic Dispatch Dispatch with with Improved for Multi-area Economic Dispatch with Improved Particle Particle Swarm Swarm Optimization Optimization for Multi-area Economic Dispatch with Reserve Sharing Scheme Reserve Sharing Scheme Reserve Reserve Sharing Sharing Scheme Scheme Vinay K. K. Jadoun*. Jadoun*. Nikhil Nikhil Gupta**, Gupta**, K. K. R. R. Niazi**, Niazi**, Anil Anil Swarnkar**, Swarnkar**, R.C. R.C. Bansal*** Bansal*** Vinay Niazi**, Anil Swarnkar**, R.C. Bansal*** Vinay Niazi**, Anil Swarnkar**, R.C. Bansal*** Vinay K. K. Jadoun*. Jadoun*. Nikhil Nikhil Gupta**, Gupta**, K. K. R. R. *Department of *Department of of Electrical Electrical Engineering Engineering National National Institute Institute of Technology, Technology, Hamirpur, Hamirpur, Himachal Himachal Pradesh Pradesh 177005 177005 *Department of Electrical Engineering National Institute of Technology, Hamirpur, Himachal India (Tel: +917891377638; e-mail: vjadounmnit@ gmail.com). *Department of Electrical Engineering National Institute of Technology, Hamirpur, Himachal Pradesh Pradesh 177005 177005 India (Tel: +917891377638; e-mail: vjadounmnit@ gmail.com). India (Tel: e-mail: gmail.com). **Department Malaviya Institute Jaipur, IndiaEngineering, (Tel: +917891377638; +917891377638; e-mail: vjadounmnit@ vjadounmnit@ gmail.com). **Department of of Electrical Electrical Engineering, Malaviya National National Institute of of Technology, Technology, Jaipur, Rajasthan Rajasthan 302017 302017 **Department of Engineering, Institute Jaipur, India [email protected], [email protected], [email protected]) **Department of Electrical Electrical Engineering, Malaviya Malaviya National National Institute of of Technology, Technology, Jaipur, Rajasthan Rajasthan 302017 302017 India (e-mails: (e-mails: [email protected], [email protected], [email protected]) India (e-mails: [email protected], [email protected], [email protected]) ***Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa, India (e-mails: [email protected], [email protected], [email protected]) ***Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa, (e-mail: [email protected]) ***Department and Engineering, ***Department of of Electrical, Electrical, Electronics Electronics and Computer Computer Engineering, University University of of Pretoria, Pretoria, South South Africa, Africa, (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) Abstract: This This paper paper presents presents an an improved improved particle particle swarm swarm optimization optimization (IPSO) (IPSO) to to solve solve Multi-area Multi-area Abstract: Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal Abstract: This paper presents an improved particle swarm optimization (IPSO) to solve Multi-area Abstract: This paper presentsproblem. an improved particle swarm optimization (IPSO) to solvethe Multi-area Economic Dispatch (MAED) The objective of MAED problem is to determine optimal generating schedule of thermal units and inter-area power transactions in such a way that total fuel cost is is Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal Economic Dispatch (MAED) objective oftransactions MAED problem determine generating schedule of thermal problem. units and The inter-area power in suchisatoway that totalthe fueloptimal cost generating schedule of thermal units and inter-area power transactions in such a way that total fuel cost is optimized while satisfying tie-line, spinning reserve and other operational constraints. The spinning generating schedule of thermal units and inter-area power in such aconstraints. way that total cost is optimized while satisfying tie-line, spinning reserve andtransactions other operational Thefuel spinning optimized while tie-line, spinning reserve and other constraints. The reserve requirements requirements for reserve reserve sharing provisions are investigated by considering considering contingency and optimized while satisfying satisfying tie-line, spinning reserve are andinvestigated other operational operational constraints. The spinning spinning reserve for sharing provisions by contingency and reserve requirements for reserve sharing provisions are investigated by considering contingency and pooling spinning reserves. The control equation of IPSO is modified by suggesting improved cognitive reserve requirements for reserve sharing provisions are is investigated bysuggesting considering contingency and pooling spinning reserves. The control equation of IPSO modified by improved cognitive pooling spinning control of IPSO by suggesting improved component of the the reserves. particle’sThe velocity by equation suggesting preceding experience. The operators of IPSO IPSOcognitive are also also pooling spinning reserves. The control equation ofpreceding IPSO is is modified modified by The suggesting improved cognitive component of particle’s velocity by suggesting experience. operators of are component the velocity by operators of are modified to toof maintain proper balance between preceding cognitive experience. and social social The behavior of the the swarm. The component ofmaintain the particle’s particle’s velocity by suggesting suggesting preceding experience. The operators of IPSO IPSO are also also modified aa proper balance between cognitive and behavior of swarm. The effectiveness of the the proposed proposed method has been been testedcognitive on four four areas, areas, 16 generators generators and 40 generators test modified to aa proper balance between and behavior of the swarm. modified to maintain maintain proper balance between cognitive and social social behaviorand of 40 thegenerators swarm. The The effectiveness of method has tested on 16 test systems. The application results show that IPSO is very promising to solve MAED problem. effectiveness of the proposed method has been tested on four areas, 16 generators and 40 generators test effectiveness of the proposed method has been four areas, generators and 40 generators test systems. The application results show that IPSOtested is veryonpromising to 16 solve MAED problem. systems. The application results show that IPSO is very promising to solve MAED problem. systems. The application results show that IPSO is very promising to solve MAED problem. © 2015, IFAC (Internationalspinning Federation of Automatic Control) Hosting dispatch, by Elsevierpooling Ltd. All spinning rights reserved. Keywords: Contingency reserve, multi-area economic reserve, Keywords: Contingency spinning reserve, multi-area economic dispatch, pooling spinning reserve, Keywords: Contingency spinning reserve, multi-area economic dispatch, pooling spinning reserve, particle swarm optimization, reserve sharing. Keywords: Contingency spinning particle swarm optimization, reservereserve, sharing. multi-area economic dispatch, pooling spinning reserve, particle particle swarm swarm optimization, optimization, reserve reserve sharing. sharing.
1. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Areas Areas of of individual individual power power systems systems are are interconnected interconnected to to Areas of individual power systems are interconnected to operate with maximum reliability, reserve sharing, improved Areas ofwith individual systems are sharing, interconnected to operate maximumpower reliability, reserve improved operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated stability and production cost than operated isolated area et 2013). its generation stability and less less production costarea thanhas operated as isolated area (Sudhakar (Sudhakar et al., al., 2013). Each Each area has its own own as generation cost and pattern. The power and area al., area has generation area (Sudhakar (Sudhakar et al., 2013). 2013). Each areageneration has its its own ownutilities generation cost and load load et pattern. The Each power generation utilities and power pools can stagger their generations to optimize the cost cost and load pattern. The power generation utilities and cost and load generation utilities and power pools canpattern. stagger The their power generations to optimize the cost power pools can stagger their generations to optimize the cost of unit energy generation from fossil fuel plants. This can be power can stagger their optimize of unit pools energy generation fromgenerations fossil fuel to plants. Thisthe cancost be of unit energy generation from fossil fuel plants. This can accomplished through multi-area economic dispatch (MAED) of unit energy through generation from fossil fuel plants. This can be be accomplished multi-area economic dispatch (MAED) accomplished through dispatch that aims the power schedule accomplished through multi-area multi-area economic dispatch (MAED) (MAED) that aims to to determine determine the optimal optimaleconomic power generation generation schedule that to optimal power generation schedule and inter-exchange of power in aa way that that aims aims to determine determine the optimal power generation schedule and inter-exchange of the power in such such way that minimizes minimizes and inter-exchange of power in such a way that minimizes the overall fuel cost of all thermal generating units and inter-exchange of power in such a way that minimizes the overall fuel cost of all thermal generating units while while the overall fuel of while satisfying and network constraints. the overall several fuel cost costoperational of all all thermal thermal generating units while satisfying several operational and generating network units constraints. However, system security restrictions on satisfying several operational and constraints. satisfying the several operational and network network constraints. However, the system security imposes imposes restrictions on the the inter-area transactions Therefore, However, the security imposes restrictions on However, power the system system security through imposes tie-lines. restrictions on the the inter-area power transactions through tie-lines. Therefore, inter-area through Therefore, complexity arises due area balance inter-area power power transactions through tie-lines. tie-lines. Therefore, complexity arises transactions due to to the the stringent stringent area power power balance complexity due to power balance constraints, tie-line and other operational complexity arises arises due constraints to the the stringent stringent area power balance constraints, tie-line constraints and area other operational constraints, tie-line constraints and other operational constraints (Sharma et al., 2011). On the other hand, constraints, (Sharma tie-line et constraints and theother constraints al., 2011). On other operational hand, area area constraints (Sharma et al., On the area spinning reserves have optimized the guidelines constraints (Sharma et to al.,be 2011). On within the other other hand, area spinning reserves have to be2011). optimized within the hand, guidelines spinning reserves have to be optimized within the guidelines of regulatory bodies to maintain adequate reliability of power spinning reserves have to be optimized within the guidelines of regulatory bodies to maintain adequate reliability of power of regulatory generation. of regulatory bodies bodies to to maintain maintain adequate adequate reliability reliability of of power power generation. generation. generation. Some Some early early efforts efforts to to attempt attempt this this problem problem can can be be briefly briefly stated as: Shoults et al. (1980) considered import and export Some early efforts to attempt this problem can be Some as: early effortset to this problem can and be briefly briefly stated Shoults al. attempt (1980) considered import export stated as: Shoults et al. (1980) considered import and export constraints between areas, Romano et al. (1981) presented stated as: Shoults et al. (1980) considered export constraints between areas, Romano et al. import (1981)and presented constraints between areas, Romano et al. (1981) presented Dantzig–Wolfe decomposition principle, Desell et al. (1984). constraints between areas, Romano et al. (1981) presented Dantzig–Wolfe decomposition principle, Desell et al. (1984). Dantzig–Wolfe decomposition principle, Desell applied Wang and (1992) Dantzig–Wolfe decomposition principle, Desell et et al. al. (1984). (1984). applied linear linear programming, programming, Wang and Shahidehpour Shahidehpour (1992) applied linear programming, Wang and Shahidehpour (1992) proposed a decomposition approach using expert systems, applied linear programming,approach Wang andusing Shahidehpour (1992) proposed a decomposition expert systems, proposed a decomposition approach using expert systems, etc. Modern artificial intelligence-based techniques proposed a decomposition approach using expert systems, etc. Modern artificial intelligence-based techniques have have etc. artificial techniques have shown their solve combinatorial etc. Modern Modern artificialto intelligence-based techniques have shown their potential potential tointelligence-based solve such such complex complex combinatorial shown shown their their potential potential to to solve solve such such complex complex combinatorial combinatorial
constrained constrained optimization optimization problems. problems. Jayabarathi Jayabarathi et et al. al. (2000) (2000) solved MAED problem with constraints using constrained optimization Jayabarathi et constrained optimization problems. Jayabarathi et al. al. (2000) (2000) solved MAED problemproblems. with tie-line tie-line constraints using evolutionary programming. Chen and Chen (2001) presented solved MAED problem with tie-line constraints using solved MAED problem with tie-line using evolutionary programming. Chen and Chenconstraints (2001) presented evolutionary programming. Chen and Chen (2001) presented direct search method by considering transmission capacity evolutionary Chen and Chen (2001) presented direct search programming. method by considering transmission capacity direct method capacity constraints. al. proposed direct search searchManoharan method by byet considering transmission capacity constraints. Manoharan etconsidering al. (2009) (2009) transmission proposed covariance covariance constraints. Manoharan et al. (2009) proposed covariance matrix adapted evolutionary strategy where a Karush Kuhun constraints. Manoharan et al. (2009)where proposed covariance matrix adapted evolutionary strategy a Karush Kuhun matrix adapted evolutionary strategy where a Karush Kuhun Tucker optimality criterion is applied to guarantee the matrix adapted evolutionary wheretoa Karush Kuhun Tucker optimality criterion strategy is applied guarantee the Tucker optimality criterion is applied to guarantee the optimal convergence. Zhu (2003) presented a new nonlinear Tucker convergence. optimality criterion is applied the optimal Zhu (2003) presentedtoa guarantee new nonlinear optimal convergence. Zhu (2003) presented a new nonlinear optimization using neural networks to solve the problem with optimal convergence. Zhu (2003) presented a new nonlinear optimization using neural networks to solve the problem with security-constraints. Recently, et al. optimization networks to with optimization using using neural neural networksManisha to solve solve the the problem with security-constraints. Recently, Manisha et problem al. (2011) (2011) formulated MAED problem with reserve constraints using security-constraints. Recently, Manisha et security-constraints. Recently, et al. al. (2011) (2011) formulated MAED problem withManisha reserve constraints using several of differential using time formulated MAED with reserve constraints formulated MAED problem withevolution reserve (DE) constraints using several variants variants of problem differential evolution (DE) using using time several variants of differential evolution (DE) using time varying mutations. Basu (2013) applied artificial bee colony several variants ofBasu differential evolution (DE) using time varying mutations. (2013) applied artificial bee colony varying mutations. Basu (2013) applied artificial bee colony optimization (ABCO) with a variety of system constraints, varying mutations. Basu (2013) applied bee colony optimization (ABCO) with a variety of artificial system constraints, optimization with aa variety of but reserve sharing employed. The multi-area optimization (ABCO) with not variety of system system constraints, but reserve (ABCO) sharing was was not employed. The constraints, multi-area but reserve sharing was not employed. The multi-area dispatch problem with economic and emission objectives has but reserve sharing was not employed. The multi-area dispatch problem with economic and emission objectives has dispatch problem with economic and emission objectives has been attempted by Wang and Singh (2009) using PSO. They dispatch problem with economic and emission objectives has been attempted by Wang and Singh (2009) using PSO. They formulated the by considering tie-line transfer been Wang (2009) PSO. They been attempted attempted byproblem Wang and and Singh (2009) using using PSO. They formulated the by problem by Singh considering tie-line transfer capacities spinning sharing. However, the formulated the problem by considering tie-line transfer formulated thearea problem by reserve considering tie-line transfer capacities and and area spinning reserve sharing. However, the norms of regulatory bodies enforce utilities to keep a definite capacities and area spinning reserve sharing. However, the capacities and area bodies spinning reserve sharing. However, the norms of regulatory enforce utilities to keep a definite norms of regulatory bodies enforce utilities to keep a definite amount of contingency spinning reserve in each area to meet norms ofofregulatory bodies enforce utilities to keep amount contingency spinning reserve in each areaa definite to meet amount of spinning area out its contingencies. The concept maintaining the amount of contingency contingency spinning reserve inof each area to to meet meet out its own own contingencies. The reserve conceptin ofeach maintaining the out its own contingencies. The concept of maintaining the contingency spinning reserve along with the pooling reserve out its own spinning contingencies. concept of maintaining the contingency reserveThe along with the pooling reserve contingency spinning reserve along with the pooling reserve is yet not reported in the literature. contingency spinning reserve along with the pooling reserve is yet not reported in the literature. is is yet yet not not reported reported in in the the literature. literature. PSO PSO is is aa swarm swarm intelligence-based intelligence-based optimization optimization technique technique in in which movement of is by PSO aa swarm intelligence-based optimization technique in PSO is is the swarm intelligence-based optimization technique in which the movement of the the particles particles is governed governed by two two stochastic coefficients, i. and which movement the governed two which the the acceleration movement of of the particles particles iscognitive governed bysocial two stochastic acceleration coefficients, i. e. e. is cognitive andby social components and weight. are attracted stochastic coefficients, i. social stochastic acceleration acceleration coefficients, i. e. e. cognitive cognitive and social components and the the inertia inertia weight. Researchers Researchers areand attracted components and the inertia weight. Researchers are attracted towards PSO due to its simplicity, convergence speed, and components weight. Researchers attracted towards PSOand duethetoinertia its simplicity, convergenceare speed, and towards PSO due to its simplicity, convergence speed, and robustness. But, PSO has inherent tendency of local trapping towards PSO duePSO to its convergence speed, and robustness. But, hassimplicity, inherent tendency of local trapping robustness. But, PSO has inherent tendency of local trapping due to poor exploitation and this leads to premature robustness. But,exploitation PSO has inherent localpremature trapping due to poor and tendency this leadsof to due due to to poor poor exploitation exploitation and and this this leads leads to to premature premature
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convergence. Several modified versions of PSO have been reported in the recent past to enhance its performance by modulating inertia weight (Park et al., 2010), improvising cognitive and social behavior (Ratnaweera et al., 2004), using constriction factor approach (Wang and Singh, 2008), modifying the control equation of the PSO (Selvakumar and Thanushkodi, 2007); Roy and Ghoshal, 2008), or squeezing the search space (Barisal, 2013), etc. However, most of these variants require several experimentations for parameter setting or some additional mechanism to avoid local trapping or to regulate particle’s velocity in order to maintain a proper balance between cognitive and social behavior of the swarm.
Nm
P
Gmj
j 1
PT m k ;
mM
(2)
2.2 Generator Constraints m in
m ax
(3)
PG m j PG m j PG m j
2.3 Tie-line Capacity Constraints m ax
m ax
(4)
PT m j PT m j PT m j
2.4 Area Spinning Reserve Constraints Generally a pooling spinning reserve is maintained in a power pool to meet the emergency requirement of the power pool such as surplus power demand or loss of generation in any area of the pool. However, in accordance with the norms of regulatory bodies, some emergency spinning reserve is to be kept in each area to encounter its own contingency. This reserve may be called as contingency spinning reserve of the area. The pooling spinning reserve can either be kept in one area as the supplementary reserve or it may be contributed by multiple areas of the pool. When this reserve is kept in each area, the total specified spinning reserve of each area is the sum of contingency reserve and the supplementary reserve of that area. In (Wang and Singh, 2009), the pooling spinning reserve is kept in each area, but the contingency spinning reserve was not maintained. However, if only contingency reserve of an area is kept as specified reserve of that area and the pooling reserve is shared among all areas, it will result in less spinning reserve requirement in each area and thereby reduces the overall reserve requirement of the power pool. It is therefore proposed that the spinning reserve requirement of an area m should satisfy the following relation:
2. PROBLEM FORMULATION The generator cost function is generally considered as quadratic when valve-point loading effects are neglected. However, this introduces ripples in the heat-rate curves and can be modeled as sinusoidal function in the cost function. Therefore, the objective function for the MAED problem may be stated as to Min M
k ,k m
where, PDm is the power demand of area m; PTmk is the tie line real power transfer from area m to area k. PTmk is positive when power flows from area m to area k and is negative when power flows from area k to area m.
In this paper, the MAED problem is attempted with the consideration of pooling and contingency spinning reserves by proposing an improved PSO (IPSO). Several modifications are suggested in IPSO to maintain a proper balance between cognitive and social behavior of the swarm. The cognitive behaviour is improved by introducing preceding experience that encompasses just previous experience of the particle, whereas the social component is diluted by proposing constriction factor. In addition, the inertia weight is dynamically controlled by introducing exponential function. The effectiveness of the proposed method is investigated on four-areas 16 generators and fourareas 40 generators test systems considering various operational constraints such as power balance, valve-point loading, tie-line capacity and spinning reserves, etc. The application results are presented and compared with other established valve-point loading methods.
F ( PG m j )
PD m
N Gm
(a
2
mj
b m j PG m j c m j PG m j ) | e m j sin( f m j ( PG m j m in - PG m j )) |
(1)
N Gm
m 1 j 1
j 1
Where, amj, bmj, cmj, are the cost coefficients, and emj and fmj are the coefficient of valve point loading effect of the jth generator in area m, PGmj is the real power output of the jth generator in area m, M is the number of areas and NGm is the number of generating units in the system in area m. The problem constraints are briefly described in the following sub-sections.
S m j S cm S pm
R C mk
(5)
k ,k m
In (5), NGm is the number of generating units in the mth area; Smj is the available reserve on the jth unit of mth area, Scm is the contingency spinning reserve in the mth area, Spm is the pooling spinning reserve share of the mth area and RCmk is the pool reserve contributed from area m to area k. 3. PROPOSED IPSO
2.1 Power Balance Constraints
The conventional PSO is initialized with a population of random solutions and searches for optima by updated particle positions. The velocity of the particle is influenced by three components, namely inertia weight, cognitive and social components. The inertia weight imparting momentum and the cognitive and social components provide directive control to particles. A proper balance between cognitive and social
In area m, the total power generation of all generators must be equal to the area power demand PDm with the consideration of imported and exported power (Sudhakar et al., 2013) and can be stated as:
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behavior of the swarm is desired throughout the computational process to regulate particle’s velocities. Therefore, following model of the control equation is suggested for the proposed IPSO as given in equation (6).
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 restricts particles’ velocities which are probably being grooming 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 diluted, otherwise the particle may fly with moderate high velocities and eventually misses the global or near global optima. Thus the social component is made weak by proposing constriction factor ks in the control equation of IPSO. The optimal value of this factor however can be evaluated through experimentations. These aforementioned modifications in the control equation of the proposed IPSO provide better control of particles’ velocities, yet preserving diversity due to the stochastic nature of cognitive and social behaviors of the swarm.
k
k 1
vi
k
W v i C 1b rand 1 ()
pbest i - s i t
k
C 1 p rand 2 () k 1
sj
k
k 1
sj vj
s i - ppreceding i t
k
k S C 2 rand 3 ()
t
(6)
gbest i - s i t
(7)
Where, 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, pbesti is the best position of ith particle achieved based on its own experience, gbesti is the best particle position based on overall swarm experience, Δt is the time step, usually set to 1 second and W is the inertia weight in (6), the cognitive component is bifurcated into best and preceding experience of the particle, whereas the social component has been diluted using the constriction factor ks. In addition, the inertia weight is updated exponentially, instead of linearly, as suggested in the standard PSO. These modifications are briefly described in the following sections.
3.4 Particle Encoding and Initialization The solution of MAED problem constitutes a set of optimal generating schedule and the power flow of tie-lines connected to that area for the desired objective satisfying defined operational constraints. Therefore, the particles are encoded as a set of current generations and the connected tieline power flows of that area in MW, as shown in Fig. 1.
3.1 Inertia Weight Update
PG11 PG21 PGm1 PT12
The trend of linear modulation of inertia weight of (Shi and Eberhart, 1999) is followed to solve optimization problems using PSO by many researchers till date (Selvakumar and Thanushkodi, 2007). For large-scale optimization problems, there exist numerous local optima in the close vicinity of the global optima. The exploitation potential of the search algorithm must be sufficient to obtain better solutions. Therefore, the modulations of inertia weight are intuitively varied exponentially in IPSO as given below. W = exp(–ηloge(Wmax/Wmin)); η = itr/itrmax
163
PG12 PG22 PGm2 PT13
…. …. …. ….
PG1j PG2j PGmj PT23
…. …. …. ….
…. …. …. ….
PG1N PG2N PGiN PTmk
Fig. 1. Particle encoding for the proposed IPSO The initial population is randomly created with predefined p number of particles to maintain diversity. Each of these particles satisfies problem constraints defined by (2)-(5). The fitness of each particle is evaluated using (1) and then pbest, ppreceding, and gbest are initialized. The initial velocity of particles is assumed to be zero.
(8)
Where, Wmin and Wmax are the respective minimum and maximum bounds of the inertia weigh, itrmax is the maximum number of iterations and itr is the current number of iteration.
4. SIMULATION RESULTS AND DISCUSSION The proposed algorithm is investigated on four areas, 16 generators and four areas, 40 generators systems. The value of acceleration coefficients C1b, C1p and C2 are taken as 1.6, 0.4 and 2.0 respectively, as in (Selvakumar and Thanushkodi, 2007). The value of maximum and minimum bounds of the inertia weight is taken as 0.9 and 0.1, respectively. The swarm size and maximum iterations have been obtained after usual tradeoff. A swarm size of 20 and 50 is taken for these test system, respectively and the maximum iteration count is universally taken as 1500. The proposed algorithm is developed using MATLAB and the simulations have been carried on a personal computer of Intel i5, 3.2 GHz, and 4 GB RAM.
3.2 Updating Preceding Experience The cognitive behavior of PSO was proposed to split into worst and best experience of the particle to provide some additional diversity (Selvakumar and Thanushkodi, 2007). But it results in poor local exploitation unless supported by suggesting a local random search by the authors. Therefore, the concept of preceding experience is suggested in IPSO, 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 provides better exploration and exploitation of the search space without employing additional local random search.
The optimal value of the constriction factor ks is determined by experimentations. For the purpose, the proposed IPSO runs with ks varied from 0.05 to 0.25 with a step size of 0.05. 163
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It has been observed that the best value of the average fuel cost is obtained when ks is 0.10, and is used for simulations. The proposed IPSO is now applied with three different cases. In case 1, all areas are assumed not be interconnected by tielines and every area has to individually satisfy its own reserve requirement. In case 2, the areas are interconnected but individual area reserves are not mutually shared, whereas in case 3, the areas are interconnected and proposed reserve sharing is allowed.
close proximity. The table also shows the average CPU time is reasonable for IPSO. Table 1. Comparison results of best fuel cost ($/hr) Method MOPSO (Wang and Singh, 2009) DEC2 (Sharma et al., 2011) IPSO
Case 1 2191.140
Case 2 2166.820
Case 3 -
2181.261 2154.793
2152.965 2145.407
2127.812 2126.927
Table 2. Quality solution using IPSO 4.1 Test system 1: Four area-16 generators system This test system consists of four areas, each with 4 thermal units and transmission losses are neglected for simplicity. All the four areas are interconnected through six tie-lines as shown in Fig. 2. The detailed data of this system may be referred from (Wang and Singh, 2009). The system base MVA is taken as 100 MVA. The area power demand is 0.3, 0.5, 0.4, and 0.6 p.u., respectively. The lower and upper capacity limit of tie-lines is considered as in (Wang and Singh, 2009).
Case
Best fuel cost ($/hr)
Average fuel cost ($/hr)
Worst fuel Cost ($/hr)
1 2 3
2154.793 2145.407 2126.927
2154.829 2145.407 2126.927
2154.837 2145.407 2126.927
CPU time (s) 2.62 2.88 2.78
Table 3. Different reserves for case 3 Reserve
Area 1 Area 2 Area 3 Area 4
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Fig. 2. Four areas 16 generators system For case 1 and case 2, the value of specified spinning reserve for each area is taken as 30% of area power demand (Wang and Singh, 2009). However, for case 3, the contingency spinning reserve for each area is taken as 7% of its power demand and the pooling spinning reserve is taken as 30% of the load demand of area 4, having the highest load demand. The comparison results obtained using proposed IPSO are presented in Table 1. It can be observed from the table that for case 1, IPSO provides fuel cost which is 1.66 % less than MOPSO (Wang and Singh, 2009) and is also 1.21 % less than DEC2 (Sharma et al., 2011). For case 2, the best fuel cost obtained by IPSO is 2145.407 $/hr which is 0.99 % less than MOPSO of (Wang and Singh, 2009) and is 0.35% less than DEC2 of (Sharma et al., 2011). It has been found that the fuel cost is reduced from 2154.793 $/hr to 2145.407 $/hr when compared with case 1. Therefore, inter-area transaction reduces fuel cost. The table also shows that the best result obtained for case 3 using IPSO. It can be observed that IPSO provides less fuel cost than DEC2 of (Sharma et al., 2011). It can also be observed that this fuel cost is less as obtained in case 2. Thus, more economy may be achieved using proposed reserve sharing. The quality of solutions obtained after 100 trials of IPSO is presented in Table 2. The table shows that the proposed method is capable to generate better quality solution as the best, average and the worst fuel costs are in
Contingency spinning reserve (p.u.)
Pooling spinning reserve (p.u.)
0.0210 0.1800 0.0350 0.0280 0.0420 Table 4. Optimal solution
Case 1 Power (p.u.) 0.120000 0.060000 0.010000 0.110000 0.250000 0.120000 0.051900 0.078100 0.061000 0.019500 0.019500 0.300000 0.096100 0.111500 0.289100 0.103300 -
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Case 2 Power (p.u.) 0.120000 0.060000 0.051300 0.068700 0.250000 0.120000 0.109300 0.020700 0.052800 0.027700 0.019500 0.300000 0.110000 0.000500 0.161600 0.300000 0.039900 -0.026800 -0.033000 0.028200 0.036600 0.024200
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Available reserve (p.u.)
0.0600 0.2490 0.8011 0.4830 Case 3 Power (p.u.) 0.120000 0.100000 0.090000 0.120000 0.250000 0.120000 0.120600 0.010400 0.013100 0.013100 0.072700 0.300000 0.110000 0.004000 0.300000 0.013000 0.001000 0.001000 0.128000 0.001000 0.001000 0.001000
Table 3 provides a quick reference to check the validity of the reserve sharing constraints imposed for the best solution obtained using IPSO. The table shows that the available reserve for each area is more than its respective contingency spinning reserve requirement and the sum of available reserves is maintained higher than the pooling spinning reserve requirement. It may be depicted from the table that the total spinning reserve to be maintained in case 3 is 0.306 pu, whereas it was 0.54 pu for case 2 Thus, inter-area reserve 164
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sharing provides significant reduction in the spinning reserve of the power pool. The optimal solution obtained using proposed IPSO can be referred from Table 4.
165
from the table that the sum of available reserve of all areas is maintained higher than the sum of pooling and contingency spinning reserves of all areas. The optimal generating schedule and the corresponding tie-line flows obtained for MAED of case 3 is presented in Table 7. It can be observed from the table that the optimal solution satisfies all the problem constraints.
4.2 Test system 2: Four area-40 generators system This is a four-area 40 units system (Basu, 2013) with nonconvexity in the cost function due to valve-point loading and with negligible transmission losses. Each area consists of 10 generating units and all four areas are interconnected through six tie-lines as shown in Fig. 3. The figure also shows area power demands as a percentage of the total power demand (PD) of 10500 MW. The area wise fuel cost coefficient data may be referred from (Basu, 2013). The tie-line limit from area 1 to area 2, from area 1 to area 3 and from area 2 to area 3 or vice versa is taken as 200 MW and that for the remaining each tie-line is taken as 100 MW.
Table 6. Different reserves for case 3 Reserve
Area 1 Area 2 Area 3 Area 4
Contingency spinning reserve (MW) 110.25 294 220.5 110.25
Pooling spinning reserve (MW) 1050
Available reserve (MW) 180.2478 1134.753900 718.1963 188.8054
Table 7. Optimal generation schedule for test system 2 Area, Unit 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 1,10 2,1 2,2 2,3 2,4 2,5 2,6
Fig. 3. Four area 40 generators system For case 1 and 2, the specified spinning reserves for each area is assumed as 20% of its power demand. For case 3, the contingency reserve for each area is considered as 7% and the pooling spinning reserve is assumed as 25% of power demand of area 2. The best results obtained by proposed IPSO to solve MAED problem for all the three cases are presented in Table 5.
Power (MW) 114.0000 114.0000 120.0000 179.7520 97.00000 140.0000 300.0000 300.0000 300.0000 130.0002 94.0036 94.0243 125.0087 484.0400 394.2794 484.0393
Area, Unit 2,7 2,8 2,9 2,10 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 3,10 4,1 4,2
Power (MW) 489.2894 489.2793 511.2876 549.9945 523.2809 523.2837 523.2816 523.285 433.5287 523.2794 10.00000 10.00010 10.00000 48.86430 190.0000 190.0000
Area, Unit 4,3 4,4 4,5 4,6 4,7 4,8 4,9 4,10 T1,2 T1,3 T1,4 T2,3 T2,4 T3,4 -
Power (MW) 190.0000 168.8838 200.0000 170.7907 110.0000 110.0000 110.0000 421.5201 187.5148 132.2127 -99.9747 -199.992 -97.2462 -88.9747 -
Table 5. Solution quality of IPSO Case 1 2 3
Best fuel cost ($/hr) 124555.862 123938.042 123039.428
Average fuel cost ($/hr) 125584.018 124295.194 124081.214
Worst fuel cost ($/hr) 128178.932 127743.357 126654.260
CPU time (s) 38.92 45.52 46.82
It can be observed from the table that in case 2 the best fuel cost 123938.042 $/hr whereas it is 123039.428 $/hr for case 3. Therefore, there is a significant saving of 898.6 $/hr when proposed reserve sharing is employed. Table also shows that the solution quality obtained using IPSO method is good for all the three cases as the best, average and the worst fuel cost are in the closed proximity. The best fuel cost lies in the band of less than 1% from the average fuel cost. The table also shows the CPU time for all the three cases which seems to be reasonable for this dimension of the problem. No other comparisons are available in the literature for this test system. The Table 6 provides the contingency, pooling spinning and available reserves for the solution obtained using IPSO. The table shows that the available reserve for each area obtained by the optimal solution is over and above than its respective contingency spinning reserve requirement. It can also be seen
Fig. 4.Convergence characteristics for best fuel cost In order to highlight the effect of each modification suggested in the control equation of the conventional 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 PSO and its variants for test system 2 is shown in Fig. 4. It can be observed from the figure that while subsequently modifying the inertia weight, cognitive and the social component, the convergence characteristics are progressively improved by avoiding more and more local 165
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trappings. It can also be observed from the figure that except IPSO, the initial shape of the convergence is more or less same. Therefore the use of the constriction factor in the social component of particle’s velocity is one of the key factors for its better convergence.
Park, J. B., Jeong, Y. W., Shin, J. R. and Lee, K.Y. (2010). An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Trans. Power Syst., 25 (1), 156-166, Feb. Ratnaweera, A., Halgamuge, S. K., and Watson, H. C. (2004).Self-Organizing Hierarchical Particle Swarm Optimizer with Time-Varying Acceleration Coefficients. IEEE Trans. Evolutionary Computation, 8 (3), 240- 255, June. Romano, R., Quintana, V. H., Lopez, R. and Valadez, V. (1981). Constrained economic dispatch of multi-area systems using the Dantzig–Wolfe decomposition principle. IEEE Trans. Power Apparatus Syst., 100 (4), 2127–37. Roy, R. and Ghoshal, S.P. (2008). A novel crazy swarm optimized economic load dispatch for various types of cost functions. Int. J. Electrical Power and Energy Systems, 30, 242–253. Selvakumar, A. I., and Thanushkodi, K. (2007). A new particle swarm optimization solution to non-convex economic dispatch problems. IEEE Trans. Power Syst., 22 (1), 42-51, Feb. Sharma, M., Pandit, M. and Srivastava, L. (2011). Reserve constrained multi-area economic dispatch employing differential evolution with time-varying mutation. Int. J. Electrical Power and Energy Systems, 33, 753–766. Shi, Y. and Eberhart, R. C. (1999). Empirical study of particle swarm optimization. in Proc. 1999 IEEE Congress on Evolutionary Computation, Piscataway, NJ, USA, IEEE Press, 1945–1950. Shoults, R. R., Chang, S. K., Helmick, S. and Grady, W. M. (1980). A practical approach to unit commitment, economic dispatch and savings allocation for multiplearea pool operation with import/export constraints. IEEE Trans Power Apparatus Syst., 99 (2), 625–35. Sudhakar, AVV., Chandram, K. and Jayalaxmi, A. (2013). Multi area economic dispatch using secant method. Journal of Electrical Engineering & Technology, 8 (4), 708-715. Wang, C., Shahidehpour, S. M. (1992). A decomposition approach to non-linear multi area generation scheduling with tie-line constraints using expert systems. IEEE Trans. Power Syst., 7 (4), 1409–1418. Wang, L. and Singh, C. (2009). Reserve-constrained multiarea environmental/economic dispatch based on particle swarm optimization with local search, Engineering Applications of Artificial Intelligence, 22, 298–307. Wang, L., and Singh, C. (2008). Stochastic economic emission load dispatch through a modified particle swarm optimization algorithm. Electric Power Systems Research, 78, 1466–1476. Zhu, J. Z. (2003). Multiarea Power Systems Economic Power Dispatch Using a Nonlinear Optimization Neural Network Approach. Electric Power Components and Systems, 31 (6), 553-563.
5. CONCLUSIONS This paper presents multi-area economic dispatch (MAED) problem of power systems with proposed spinning reserve sharing scheme by suggesting an improved particle swarm optimization (IPSO) method. In addition to pooling spinning reserve, a contingency spinning reserve for each area is proposed as per the norms of the regulatory bodies. The proposed IPSO method efficiently handles with stringent tieline security and spinning reserve constraints for inter-area power transactions. The application results shows that more economy in power generation and spinning reserves requirements can be achieved by sharing pooling spinning reserve among concerned areas while keeping the contingency spinning reserve intact in each area to enhance reliability of power generation. The parameters of PSO are redefined in such a fashion that ensures a proper balance between exploration and exploitation ability of PSO and thus result in faster global convergence, higher solution quality, and stronger robustness. It has been observed that each suggested modification in PSO is contributed to enhance its performance. The proposed IPSO method is capable to provide better quality solution and is computationally efficient. It is noteworthy that the proposed IPSO does not require additional mechanism to avoid local trapping, empirical formula to bound particle’s velocity or squeezing the search space, and it is quite independent of the initial state of particles. REFERENCES Barisal, A. K. (2013). Dynamic search space squeezing strategy based intelligent algorithm solutions to economic dispatch with multiple fuels. Int. J. Electrical Power and Energy Systems, 45, 50–59. Basu, M. (2013). Artificial bee colony optimization for multiarea economic dispatch. Int. J. Electrical Power and Energy Systems, 49, 181–187. Chen, C. L. and Chen, N. (2001). Direct search method for solving economic dispatch problem considering transmission capacity constraints. IEEE Trans. Power Syst., 16 (4), 764–769. Desell, A. L., McClelland, E. C., Tammar, K. and Horne, P. R. V. (1984). Transmission constrained production cost analysis in power system planning. IEEE Trans. Power Apparatus Syst., 103 (8), 2192–2198. Jayabarathi, T., Sadasivam, G., and Ramachandran, V. (2000). Evolutionary programming based multi-area economic dispatch with tie line constraints. Electr Mach Power Syst., 28, 1165–1176. Manoharan, P. S., Kannan, P. S., Baskar, S. and Iruthayarajan, M. W. (2009). Evolutionary algorithm solution and KKT based optimality verification to multiarea economic dispatch. Int. J. Electr Power Energy Syst., 31, 365–373. 166
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Improved Improved Particle Particle Swarm Swarm Optimization Optimization for for Multi-area Multi-area Economic Economic Dispatch Dispatch with with Improved for Multi-area Economic Dispatch with Improved Particle Particle Swarm Swarm Optimization Optimization for Multi-area Economic Dispatch with Reserve Sharing Scheme Reserve Sharing Scheme Reserve Reserve Sharing Sharing Scheme Scheme Vinay K. K. Jadoun*. Jadoun*. Nikhil Nikhil Gupta**, Gupta**, K. K. R. R. Niazi**, Niazi**, Anil Anil Swarnkar**, Swarnkar**, R.C. R.C. Bansal*** Bansal*** Vinay Niazi**, Anil Swarnkar**, R.C. Bansal*** Vinay Niazi**, Anil Swarnkar**, R.C. Bansal*** Vinay K. K. Jadoun*. Jadoun*. Nikhil Nikhil Gupta**, Gupta**, K. K. R. R. *Department of *Department of of Electrical Electrical Engineering Engineering National National Institute Institute of Technology, Technology, Hamirpur, Hamirpur, Himachal Himachal Pradesh Pradesh 177005 177005 *Department of Electrical Engineering National Institute of Technology, Hamirpur, Himachal India (Tel: +917891377638; e-mail: vjadounmnit@ gmail.com). *Department of Electrical Engineering National Institute of Technology, Hamirpur, Himachal Pradesh Pradesh 177005 177005 India (Tel: +917891377638; e-mail: vjadounmnit@ gmail.com). India (Tel: e-mail: gmail.com). **Department Malaviya Institute Jaipur, IndiaEngineering, (Tel: +917891377638; +917891377638; e-mail: vjadounmnit@ vjadounmnit@ gmail.com). **Department of of Electrical Electrical Engineering, Malaviya National National Institute of of Technology, Technology, Jaipur, Rajasthan Rajasthan 302017 302017 **Department of Engineering, Institute Jaipur, India [email protected], [email protected], [email protected]) **Department of Electrical Electrical Engineering, Malaviya Malaviya National National Institute of of Technology, Technology, Jaipur, Rajasthan Rajasthan 302017 302017 India (e-mails: (e-mails: [email protected], [email protected], [email protected]) India (e-mails: [email protected], [email protected], [email protected]) ***Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa, India (e-mails: [email protected], [email protected], [email protected]) ***Department of Electrical, Electronics and Computer Engineering, University of Pretoria, South Africa, (e-mail: [email protected]) ***Department and Engineering, ***Department of of Electrical, Electrical, Electronics Electronics and Computer Computer Engineering, University University of of Pretoria, Pretoria, South South Africa, Africa, (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) Abstract: This This paper paper presents presents an an improved improved particle particle swarm swarm optimization optimization (IPSO) (IPSO) to to solve solve Multi-area Multi-area Abstract: Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal Abstract: This paper presents an improved particle swarm optimization (IPSO) to solve Multi-area Abstract: This paper presentsproblem. an improved particle swarm optimization (IPSO) to solvethe Multi-area Economic Dispatch (MAED) The objective of MAED problem is to determine optimal generating schedule of thermal units and inter-area power transactions in such a way that total fuel cost is is Economic Dispatch (MAED) problem. The objective of MAED problem is to determine the optimal Economic Dispatch (MAED) objective oftransactions MAED problem determine generating schedule of thermal problem. units and The inter-area power in suchisatoway that totalthe fueloptimal cost generating schedule of thermal units and inter-area power transactions in such a way that total fuel cost is optimized while satisfying tie-line, spinning reserve and other operational constraints. The spinning generating schedule of thermal units and inter-area power in such aconstraints. way that total cost is optimized while satisfying tie-line, spinning reserve andtransactions other operational Thefuel spinning optimized while tie-line, spinning reserve and other constraints. The reserve requirements requirements for reserve reserve sharing provisions are investigated by considering considering contingency and optimized while satisfying satisfying tie-line, spinning reserve are andinvestigated other operational operational constraints. The spinning spinning reserve for sharing provisions by contingency and reserve requirements for reserve sharing provisions are investigated by considering contingency and pooling spinning reserves. The control equation of IPSO is modified by suggesting improved cognitive reserve requirements for reserve sharing provisions are is investigated bysuggesting considering contingency and pooling spinning reserves. The control equation of IPSO modified by improved cognitive pooling spinning control of IPSO by suggesting improved component of the the reserves. particle’sThe velocity by equation suggesting preceding experience. The operators of IPSO IPSOcognitive are also also pooling spinning reserves. The control equation ofpreceding IPSO is is modified modified by The suggesting improved cognitive component of particle’s velocity by suggesting experience. operators of are component the velocity by operators of are modified to toof maintain proper balance between preceding cognitive experience. and social social The behavior of the the swarm. The component ofmaintain the particle’s particle’s velocity by suggesting suggesting preceding experience. The operators of IPSO IPSO are also also modified aa proper balance between cognitive and behavior of swarm. The effectiveness of the the proposed proposed method has been been testedcognitive on four four areas, areas, 16 generators generators and 40 generators test modified to aa proper balance between and behavior of the swarm. modified to maintain maintain proper balance between cognitive and social social behaviorand of 40 thegenerators swarm. The The effectiveness of method has tested on 16 test systems. The application results show that IPSO is very promising to solve MAED problem. effectiveness of the proposed method has been tested on four areas, 16 generators and 40 generators test effectiveness of the proposed method has been four areas, generators and 40 generators test systems. The application results show that IPSOtested is veryonpromising to 16 solve MAED problem. systems. The application results show that IPSO is very promising to solve MAED problem. systems. The application results show that IPSO is very promising to solve MAED problem. © 2015, IFAC (Internationalspinning Federation of Automatic Control) Hosting dispatch, by Elsevierpooling Ltd. All spinning rights reserved. Keywords: Contingency reserve, multi-area economic reserve, Keywords: Contingency spinning reserve, multi-area economic dispatch, pooling spinning reserve, Keywords: Contingency spinning reserve, multi-area economic dispatch, pooling spinning reserve, particle swarm optimization, reserve sharing. Keywords: Contingency spinning particle swarm optimization, reservereserve, sharing. multi-area economic dispatch, pooling spinning reserve, particle particle swarm swarm optimization, optimization, reserve reserve sharing. sharing.
1. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Areas Areas of of individual individual power power systems systems are are interconnected interconnected to to Areas of individual power systems are interconnected to operate with maximum reliability, reserve sharing, improved Areas ofwith individual systems are sharing, interconnected to operate maximumpower reliability, reserve improved operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated operate with maximum reliability, reserve sharing, improved stability and less production cost than operated as isolated stability and production cost than operated isolated area et 2013). its generation stability and less less production costarea thanhas operated as isolated area (Sudhakar (Sudhakar et al., al., 2013). Each Each area has its own own as generation cost and pattern. The power and area al., area has generation area (Sudhakar (Sudhakar et al., 2013). 2013). Each areageneration has its its own ownutilities generation cost and load load et pattern. The Each power generation utilities and power pools can stagger their generations to optimize the cost cost and load pattern. The power generation utilities and cost and load generation utilities and power pools canpattern. stagger The their power generations to optimize the cost power pools can stagger their generations to optimize the cost of unit energy generation from fossil fuel plants. This can be power can stagger their optimize of unit pools energy generation fromgenerations fossil fuel to plants. Thisthe cancost be of unit energy generation from fossil fuel plants. This can accomplished through multi-area economic dispatch (MAED) of unit energy through generation from fossil fuel plants. This can be be accomplished multi-area economic dispatch (MAED) accomplished through dispatch that aims the power schedule accomplished through multi-area multi-area economic dispatch (MAED) (MAED) that aims to to determine determine the optimal optimaleconomic power generation generation schedule that to optimal power generation schedule and inter-exchange of power in aa way that that aims aims to determine determine the optimal power generation schedule and inter-exchange of the power in such such way that minimizes minimizes and inter-exchange of power in such a way that minimizes the overall fuel cost of all thermal generating units and inter-exchange of power in such a way that minimizes the overall fuel cost of all thermal generating units while while the overall fuel of while satisfying and network constraints. the overall several fuel cost costoperational of all all thermal thermal generating units while satisfying several operational and generating network units constraints. However, system security restrictions on satisfying several operational and constraints. satisfying the several operational and network network constraints. However, the system security imposes imposes restrictions on the the inter-area transactions Therefore, However, the security imposes restrictions on However, power the system system security through imposes tie-lines. restrictions on the the inter-area power transactions through tie-lines. Therefore, inter-area through Therefore, complexity arises due area balance inter-area power power transactions through tie-lines. tie-lines. Therefore, complexity arises transactions due to to the the stringent stringent area power power balance complexity due to power balance constraints, tie-line and other operational complexity arises arises due constraints to the the stringent stringent area power balance constraints, tie-line constraints and area other operational constraints, tie-line constraints and other operational constraints (Sharma et al., 2011). On the other hand, constraints, (Sharma tie-line et constraints and theother constraints al., 2011). On other operational hand, area area constraints (Sharma et al., On the area spinning reserves have optimized the guidelines constraints (Sharma et to al.,be 2011). On within the other other hand, area spinning reserves have to be2011). optimized within the hand, guidelines spinning reserves have to be optimized within the guidelines of regulatory bodies to maintain adequate reliability of power spinning reserves have to be optimized within the guidelines of regulatory bodies to maintain adequate reliability of power of regulatory generation. of regulatory bodies bodies to to maintain maintain adequate adequate reliability reliability of of power power generation. generation. generation. Some Some early early efforts efforts to to attempt attempt this this problem problem can can be be briefly briefly stated as: Shoults et al. (1980) considered import and export Some early efforts to attempt this problem can be Some as: early effortset to this problem can and be briefly briefly stated Shoults al. attempt (1980) considered import export stated as: Shoults et al. (1980) considered import and export constraints between areas, Romano et al. (1981) presented stated as: Shoults et al. (1980) considered export constraints between areas, Romano et al. import (1981)and presented constraints between areas, Romano et al. (1981) presented Dantzig–Wolfe decomposition principle, Desell et al. (1984). constraints between areas, Romano et al. (1981) presented Dantzig–Wolfe decomposition principle, Desell et al. (1984). Dantzig–Wolfe decomposition principle, Desell applied Wang and (1992) Dantzig–Wolfe decomposition principle, Desell et et al. al. (1984). (1984). applied linear linear programming, programming, Wang and Shahidehpour Shahidehpour (1992) applied linear programming, Wang and Shahidehpour (1992) proposed a decomposition approach using expert systems, applied linear programming,approach Wang andusing Shahidehpour (1992) proposed a decomposition expert systems, proposed a decomposition approach using expert systems, etc. Modern artificial intelligence-based techniques proposed a decomposition approach using expert systems, etc. Modern artificial intelligence-based techniques have have etc. artificial techniques have shown their solve combinatorial etc. Modern Modern artificialto intelligence-based techniques have shown their potential potential tointelligence-based solve such such complex complex combinatorial shown shown their their potential potential to to solve solve such such complex complex combinatorial combinatorial
constrained constrained optimization optimization problems. problems. Jayabarathi Jayabarathi et et al. al. (2000) (2000) solved MAED problem with constraints using constrained optimization Jayabarathi et constrained optimization problems. Jayabarathi et al. al. (2000) (2000) solved MAED problemproblems. with tie-line tie-line constraints using evolutionary programming. Chen and Chen (2001) presented solved MAED problem with tie-line constraints using solved MAED problem with tie-line using evolutionary programming. Chen and Chenconstraints (2001) presented evolutionary programming. Chen and Chen (2001) presented direct search method by considering transmission capacity evolutionary Chen and Chen (2001) presented direct search programming. method by considering transmission capacity direct method capacity constraints. al. proposed direct search searchManoharan method by byet considering transmission capacity constraints. Manoharan etconsidering al. (2009) (2009) transmission proposed covariance covariance constraints. Manoharan et al. (2009) proposed covariance matrix adapted evolutionary strategy where a Karush Kuhun constraints. Manoharan et al. (2009)where proposed covariance matrix adapted evolutionary strategy a Karush Kuhun matrix adapted evolutionary strategy where a Karush Kuhun Tucker optimality criterion is applied to guarantee the matrix adapted evolutionary wheretoa Karush Kuhun Tucker optimality criterion strategy is applied guarantee the Tucker optimality criterion is applied to guarantee the optimal convergence. Zhu (2003) presented a new nonlinear Tucker convergence. optimality criterion is applied the optimal Zhu (2003) presentedtoa guarantee new nonlinear optimal convergence. Zhu (2003) presented a new nonlinear optimization using neural networks to solve the problem with optimal convergence. Zhu (2003) presented a new nonlinear optimization using neural networks to solve the problem with security-constraints. Recently, et al. optimization networks to with optimization using using neural neural networksManisha to solve solve the the problem with security-constraints. Recently, Manisha et problem al. (2011) (2011) formulated MAED problem with reserve constraints using security-constraints. Recently, Manisha et security-constraints. Recently, et al. al. (2011) (2011) formulated MAED problem withManisha reserve constraints using several of differential using time formulated MAED with reserve constraints formulated MAED problem withevolution reserve (DE) constraints using several variants variants of problem differential evolution (DE) using using time several variants of differential evolution (DE) using time varying mutations. Basu (2013) applied artificial bee colony several variants ofBasu differential evolution (DE) using time varying mutations. (2013) applied artificial bee colony varying mutations. Basu (2013) applied artificial bee colony optimization (ABCO) with a variety of system constraints, varying mutations. Basu (2013) applied bee colony optimization (ABCO) with a variety of artificial system constraints, optimization with aa variety of but reserve sharing employed. The multi-area optimization (ABCO) with not variety of system system constraints, but reserve (ABCO) sharing was was not employed. The constraints, multi-area but reserve sharing was not employed. The multi-area dispatch problem with economic and emission objectives has but reserve sharing was not employed. The multi-area dispatch problem with economic and emission objectives has dispatch problem with economic and emission objectives has been attempted by Wang and Singh (2009) using PSO. They dispatch problem with economic and emission objectives has been attempted by Wang and Singh (2009) using PSO. They formulated the by considering tie-line transfer been Wang (2009) PSO. They been attempted attempted byproblem Wang and and Singh (2009) using using PSO. They formulated the by problem by Singh considering tie-line transfer capacities spinning sharing. However, the formulated the problem by considering tie-line transfer formulated thearea problem by reserve considering tie-line transfer capacities and and area spinning reserve sharing. However, the norms of regulatory bodies enforce utilities to keep a definite capacities and area spinning reserve sharing. However, the capacities and area bodies spinning reserve sharing. However, the norms of regulatory enforce utilities to keep a definite norms of regulatory bodies enforce utilities to keep a definite amount of contingency spinning reserve in each area to meet norms ofofregulatory bodies enforce utilities to keep amount contingency spinning reserve in each areaa definite to meet amount of spinning area out its contingencies. The concept maintaining the amount of contingency contingency spinning reserve inof each area to to meet meet out its own own contingencies. The reserve conceptin ofeach maintaining the out its own contingencies. The concept of maintaining the contingency spinning reserve along with the pooling reserve out its own spinning contingencies. concept of maintaining the contingency reserveThe along with the pooling reserve contingency spinning reserve along with the pooling reserve is yet not reported in the literature. contingency spinning reserve along with the pooling reserve is yet not reported in the literature. is is yet yet not not reported reported in in the the literature. literature. PSO PSO is is aa swarm swarm intelligence-based intelligence-based optimization optimization technique technique in in which movement of is by PSO aa swarm intelligence-based optimization technique in PSO is is the swarm intelligence-based optimization technique in which the movement of the the particles particles is governed governed by two two stochastic coefficients, i. and which movement the governed two which the the acceleration movement of of the particles particles iscognitive governed bysocial two stochastic acceleration coefficients, i. e. e. is cognitive andby social components and weight. are attracted stochastic coefficients, i. social stochastic acceleration acceleration coefficients, i. e. e. cognitive cognitive and social components and the the inertia inertia weight. Researchers Researchers areand attracted components and the inertia weight. Researchers are attracted towards PSO due to its simplicity, convergence speed, and components weight. Researchers attracted towards PSOand duethetoinertia its simplicity, convergenceare speed, and towards PSO due to its simplicity, convergence speed, and robustness. But, PSO has inherent tendency of local trapping towards PSO duePSO to its convergence speed, and robustness. But, hassimplicity, inherent tendency of local trapping robustness. But, PSO has inherent tendency of local trapping due to poor exploitation and this leads to premature robustness. But,exploitation PSO has inherent localpremature trapping due to poor and tendency this leadsof to due due to to poor poor exploitation exploitation and and this this leads leads to to premature premature
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convergence. Several modified versions of PSO have been reported in the recent past to enhance its performance by modulating inertia weight (Park et al., 2010), improvising cognitive and social behavior (Ratnaweera et al., 2004), using constriction factor approach (Wang and Singh, 2008), modifying the control equation of the PSO (Selvakumar and Thanushkodi, 2007); Roy and Ghoshal, 2008), or squeezing the search space (Barisal, 2013), etc. However, most of these variants require several experimentations for parameter setting or some additional mechanism to avoid local trapping or to regulate particle’s velocity in order to maintain a proper balance between cognitive and social behavior of the swarm.
Nm
P
Gmj
j 1
PT m k ;
mM
(2)
2.2 Generator Constraints m in
m ax
(3)
PG m j PG m j PG m j
2.3 Tie-line Capacity Constraints m ax
m ax
(4)
PT m j PT m j PT m j
2.4 Area Spinning Reserve Constraints Generally a pooling spinning reserve is maintained in a power pool to meet the emergency requirement of the power pool such as surplus power demand or loss of generation in any area of the pool. However, in accordance with the norms of regulatory bodies, some emergency spinning reserve is to be kept in each area to encounter its own contingency. This reserve may be called as contingency spinning reserve of the area. The pooling spinning reserve can either be kept in one area as the supplementary reserve or it may be contributed by multiple areas of the pool. When this reserve is kept in each area, the total specified spinning reserve of each area is the sum of contingency reserve and the supplementary reserve of that area. In (Wang and Singh, 2009), the pooling spinning reserve is kept in each area, but the contingency spinning reserve was not maintained. However, if only contingency reserve of an area is kept as specified reserve of that area and the pooling reserve is shared among all areas, it will result in less spinning reserve requirement in each area and thereby reduces the overall reserve requirement of the power pool. It is therefore proposed that the spinning reserve requirement of an area m should satisfy the following relation:
2. PROBLEM FORMULATION The generator cost function is generally considered as quadratic when valve-point loading effects are neglected. However, this introduces ripples in the heat-rate curves and can be modeled as sinusoidal function in the cost function. Therefore, the objective function for the MAED problem may be stated as to Min M
k ,k m
where, PDm is the power demand of area m; PTmk is the tie line real power transfer from area m to area k. PTmk is positive when power flows from area m to area k and is negative when power flows from area k to area m.
In this paper, the MAED problem is attempted with the consideration of pooling and contingency spinning reserves by proposing an improved PSO (IPSO). Several modifications are suggested in IPSO to maintain a proper balance between cognitive and social behavior of the swarm. The cognitive behaviour is improved by introducing preceding experience that encompasses just previous experience of the particle, whereas the social component is diluted by proposing constriction factor. In addition, the inertia weight is dynamically controlled by introducing exponential function. The effectiveness of the proposed method is investigated on four-areas 16 generators and fourareas 40 generators test systems considering various operational constraints such as power balance, valve-point loading, tie-line capacity and spinning reserves, etc. The application results are presented and compared with other established valve-point loading methods.
F ( PG m j )
PD m
N Gm
(a
2
mj
b m j PG m j c m j PG m j ) | e m j sin( f m j ( PG m j m in - PG m j )) |
(1)
N Gm
m 1 j 1
j 1
Where, amj, bmj, cmj, are the cost coefficients, and emj and fmj are the coefficient of valve point loading effect of the jth generator in area m, PGmj is the real power output of the jth generator in area m, M is the number of areas and NGm is the number of generating units in the system in area m. The problem constraints are briefly described in the following sub-sections.
S m j S cm S pm
R C mk
(5)
k ,k m
In (5), NGm is the number of generating units in the mth area; Smj is the available reserve on the jth unit of mth area, Scm is the contingency spinning reserve in the mth area, Spm is the pooling spinning reserve share of the mth area and RCmk is the pool reserve contributed from area m to area k. 3. PROPOSED IPSO
2.1 Power Balance Constraints
The conventional PSO is initialized with a population of random solutions and searches for optima by updated particle positions. The velocity of the particle is influenced by three components, namely inertia weight, cognitive and social components. The inertia weight imparting momentum and the cognitive and social components provide directive control to particles. A proper balance between cognitive and social
In area m, the total power generation of all generators must be equal to the area power demand PDm with the consideration of imported and exported power (Sudhakar et al., 2013) and can be stated as:
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behavior of the swarm is desired throughout the computational process to regulate particle’s velocities. Therefore, following model of the control equation is suggested for the proposed IPSO as given in equation (6).
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 restricts particles’ velocities which are probably being grooming 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 diluted, otherwise the particle may fly with moderate high velocities and eventually misses the global or near global optima. Thus the social component is made weak by proposing constriction factor ks in the control equation of IPSO. The optimal value of this factor however can be evaluated through experimentations. These aforementioned modifications in the control equation of the proposed IPSO provide better control of particles’ velocities, yet preserving diversity due to the stochastic nature of cognitive and social behaviors of the swarm.
k
k 1
vi
k
W v i C 1b rand 1 ()
pbest i - s i t
k
C 1 p rand 2 () k 1
sj
k
k 1
sj vj
s i - ppreceding i t
k
k S C 2 rand 3 ()
t
(6)
gbest i - s i t
(7)
Where, 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, pbesti is the best position of ith particle achieved based on its own experience, gbesti is the best particle position based on overall swarm experience, Δt is the time step, usually set to 1 second and W is the inertia weight in (6), the cognitive component is bifurcated into best and preceding experience of the particle, whereas the social component has been diluted using the constriction factor ks. In addition, the inertia weight is updated exponentially, instead of linearly, as suggested in the standard PSO. These modifications are briefly described in the following sections.
3.4 Particle Encoding and Initialization The solution of MAED problem constitutes a set of optimal generating schedule and the power flow of tie-lines connected to that area for the desired objective satisfying defined operational constraints. Therefore, the particles are encoded as a set of current generations and the connected tieline power flows of that area in MW, as shown in Fig. 1.
3.1 Inertia Weight Update
PG11 PG21 PGm1 PT12
The trend of linear modulation of inertia weight of (Shi and Eberhart, 1999) is followed to solve optimization problems using PSO by many researchers till date (Selvakumar and Thanushkodi, 2007). For large-scale optimization problems, there exist numerous local optima in the close vicinity of the global optima. The exploitation potential of the search algorithm must be sufficient to obtain better solutions. Therefore, the modulations of inertia weight are intuitively varied exponentially in IPSO as given below. W = exp(–ηloge(Wmax/Wmin)); η = itr/itrmax
163
PG12 PG22 PGm2 PT13
…. …. …. ….
PG1j PG2j PGmj PT23
…. …. …. ….
…. …. …. ….
PG1N PG2N PGiN PTmk
Fig. 1. Particle encoding for the proposed IPSO The initial population is randomly created with predefined p number of particles to maintain diversity. Each of these particles satisfies problem constraints defined by (2)-(5). The fitness of each particle is evaluated using (1) and then pbest, ppreceding, and gbest are initialized. The initial velocity of particles is assumed to be zero.
(8)
Where, Wmin and Wmax are the respective minimum and maximum bounds of the inertia weigh, itrmax is the maximum number of iterations and itr is the current number of iteration.
4. SIMULATION RESULTS AND DISCUSSION The proposed algorithm is investigated on four areas, 16 generators and four areas, 40 generators systems. The value of acceleration coefficients C1b, C1p and C2 are taken as 1.6, 0.4 and 2.0 respectively, as in (Selvakumar and Thanushkodi, 2007). The value of maximum and minimum bounds of the inertia weight is taken as 0.9 and 0.1, respectively. The swarm size and maximum iterations have been obtained after usual tradeoff. A swarm size of 20 and 50 is taken for these test system, respectively and the maximum iteration count is universally taken as 1500. The proposed algorithm is developed using MATLAB and the simulations have been carried on a personal computer of Intel i5, 3.2 GHz, and 4 GB RAM.
3.2 Updating Preceding Experience The cognitive behavior of PSO was proposed to split into worst and best experience of the particle to provide some additional diversity (Selvakumar and Thanushkodi, 2007). But it results in poor local exploitation unless supported by suggesting a local random search by the authors. Therefore, the concept of preceding experience is suggested in IPSO, 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 provides better exploration and exploitation of the search space without employing additional local random search.
The optimal value of the constriction factor ks is determined by experimentations. For the purpose, the proposed IPSO runs with ks varied from 0.05 to 0.25 with a step size of 0.05. 163
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It has been observed that the best value of the average fuel cost is obtained when ks is 0.10, and is used for simulations. The proposed IPSO is now applied with three different cases. In case 1, all areas are assumed not be interconnected by tielines and every area has to individually satisfy its own reserve requirement. In case 2, the areas are interconnected but individual area reserves are not mutually shared, whereas in case 3, the areas are interconnected and proposed reserve sharing is allowed.
close proximity. The table also shows the average CPU time is reasonable for IPSO. Table 1. Comparison results of best fuel cost ($/hr) Method MOPSO (Wang and Singh, 2009) DEC2 (Sharma et al., 2011) IPSO
Case 1 2191.140
Case 2 2166.820
Case 3 -
2181.261 2154.793
2152.965 2145.407
2127.812 2126.927
Table 2. Quality solution using IPSO 4.1 Test system 1: Four area-16 generators system This test system consists of four areas, each with 4 thermal units and transmission losses are neglected for simplicity. All the four areas are interconnected through six tie-lines as shown in Fig. 2. The detailed data of this system may be referred from (Wang and Singh, 2009). The system base MVA is taken as 100 MVA. The area power demand is 0.3, 0.5, 0.4, and 0.6 p.u., respectively. The lower and upper capacity limit of tie-lines is considered as in (Wang and Singh, 2009).
Case
Best fuel cost ($/hr)
Average fuel cost ($/hr)
Worst fuel Cost ($/hr)
1 2 3
2154.793 2145.407 2126.927
2154.829 2145.407 2126.927
2154.837 2145.407 2126.927
CPU time (s) 2.62 2.88 2.78
Table 3. Different reserves for case 3 Reserve
Area 1 Area 2 Area 3 Area 4
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Fig. 2. Four areas 16 generators system For case 1 and case 2, the value of specified spinning reserve for each area is taken as 30% of area power demand (Wang and Singh, 2009). However, for case 3, the contingency spinning reserve for each area is taken as 7% of its power demand and the pooling spinning reserve is taken as 30% of the load demand of area 4, having the highest load demand. The comparison results obtained using proposed IPSO are presented in Table 1. It can be observed from the table that for case 1, IPSO provides fuel cost which is 1.66 % less than MOPSO (Wang and Singh, 2009) and is also 1.21 % less than DEC2 (Sharma et al., 2011). For case 2, the best fuel cost obtained by IPSO is 2145.407 $/hr which is 0.99 % less than MOPSO of (Wang and Singh, 2009) and is 0.35% less than DEC2 of (Sharma et al., 2011). It has been found that the fuel cost is reduced from 2154.793 $/hr to 2145.407 $/hr when compared with case 1. Therefore, inter-area transaction reduces fuel cost. The table also shows that the best result obtained for case 3 using IPSO. It can be observed that IPSO provides less fuel cost than DEC2 of (Sharma et al., 2011). It can also be observed that this fuel cost is less as obtained in case 2. Thus, more economy may be achieved using proposed reserve sharing. The quality of solutions obtained after 100 trials of IPSO is presented in Table 2. The table shows that the proposed method is capable to generate better quality solution as the best, average and the worst fuel costs are in
Contingency spinning reserve (p.u.)
Pooling spinning reserve (p.u.)
0.0210 0.1800 0.0350 0.0280 0.0420 Table 4. Optimal solution
Case 1 Power (p.u.) 0.120000 0.060000 0.010000 0.110000 0.250000 0.120000 0.051900 0.078100 0.061000 0.019500 0.019500 0.300000 0.096100 0.111500 0.289100 0.103300 -
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Case 2 Power (p.u.) 0.120000 0.060000 0.051300 0.068700 0.250000 0.120000 0.109300 0.020700 0.052800 0.027700 0.019500 0.300000 0.110000 0.000500 0.161600 0.300000 0.039900 -0.026800 -0.033000 0.028200 0.036600 0.024200
Unit 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 PT1,2 PT1,3 PT1,4 PT2,3 PT2,4 PT3,4
Available reserve (p.u.)
0.0600 0.2490 0.8011 0.4830 Case 3 Power (p.u.) 0.120000 0.100000 0.090000 0.120000 0.250000 0.120000 0.120600 0.010400 0.013100 0.013100 0.072700 0.300000 0.110000 0.004000 0.300000 0.013000 0.001000 0.001000 0.128000 0.001000 0.001000 0.001000
Table 3 provides a quick reference to check the validity of the reserve sharing constraints imposed for the best solution obtained using IPSO. The table shows that the available reserve for each area is more than its respective contingency spinning reserve requirement and the sum of available reserves is maintained higher than the pooling spinning reserve requirement. It may be depicted from the table that the total spinning reserve to be maintained in case 3 is 0.306 pu, whereas it was 0.54 pu for case 2 Thus, inter-area reserve 164
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sharing provides significant reduction in the spinning reserve of the power pool. The optimal solution obtained using proposed IPSO can be referred from Table 4.
165
from the table that the sum of available reserve of all areas is maintained higher than the sum of pooling and contingency spinning reserves of all areas. The optimal generating schedule and the corresponding tie-line flows obtained for MAED of case 3 is presented in Table 7. It can be observed from the table that the optimal solution satisfies all the problem constraints.
4.2 Test system 2: Four area-40 generators system This is a four-area 40 units system (Basu, 2013) with nonconvexity in the cost function due to valve-point loading and with negligible transmission losses. Each area consists of 10 generating units and all four areas are interconnected through six tie-lines as shown in Fig. 3. The figure also shows area power demands as a percentage of the total power demand (PD) of 10500 MW. The area wise fuel cost coefficient data may be referred from (Basu, 2013). The tie-line limit from area 1 to area 2, from area 1 to area 3 and from area 2 to area 3 or vice versa is taken as 200 MW and that for the remaining each tie-line is taken as 100 MW.
Table 6. Different reserves for case 3 Reserve
Area 1 Area 2 Area 3 Area 4
Contingency spinning reserve (MW) 110.25 294 220.5 110.25
Pooling spinning reserve (MW) 1050
Available reserve (MW) 180.2478 1134.753900 718.1963 188.8054
Table 7. Optimal generation schedule for test system 2 Area, Unit 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 1,10 2,1 2,2 2,3 2,4 2,5 2,6
Fig. 3. Four area 40 generators system For case 1 and 2, the specified spinning reserves for each area is assumed as 20% of its power demand. For case 3, the contingency reserve for each area is considered as 7% and the pooling spinning reserve is assumed as 25% of power demand of area 2. The best results obtained by proposed IPSO to solve MAED problem for all the three cases are presented in Table 5.
Power (MW) 114.0000 114.0000 120.0000 179.7520 97.00000 140.0000 300.0000 300.0000 300.0000 130.0002 94.0036 94.0243 125.0087 484.0400 394.2794 484.0393
Area, Unit 2,7 2,8 2,9 2,10 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 3,10 4,1 4,2
Power (MW) 489.2894 489.2793 511.2876 549.9945 523.2809 523.2837 523.2816 523.285 433.5287 523.2794 10.00000 10.00010 10.00000 48.86430 190.0000 190.0000
Area, Unit 4,3 4,4 4,5 4,6 4,7 4,8 4,9 4,10 T1,2 T1,3 T1,4 T2,3 T2,4 T3,4 -
Power (MW) 190.0000 168.8838 200.0000 170.7907 110.0000 110.0000 110.0000 421.5201 187.5148 132.2127 -99.9747 -199.992 -97.2462 -88.9747 -
Table 5. Solution quality of IPSO Case 1 2 3
Best fuel cost ($/hr) 124555.862 123938.042 123039.428
Average fuel cost ($/hr) 125584.018 124295.194 124081.214
Worst fuel cost ($/hr) 128178.932 127743.357 126654.260
CPU time (s) 38.92 45.52 46.82
It can be observed from the table that in case 2 the best fuel cost 123938.042 $/hr whereas it is 123039.428 $/hr for case 3. Therefore, there is a significant saving of 898.6 $/hr when proposed reserve sharing is employed. Table also shows that the solution quality obtained using IPSO method is good for all the three cases as the best, average and the worst fuel cost are in the closed proximity. The best fuel cost lies in the band of less than 1% from the average fuel cost. The table also shows the CPU time for all the three cases which seems to be reasonable for this dimension of the problem. No other comparisons are available in the literature for this test system. The Table 6 provides the contingency, pooling spinning and available reserves for the solution obtained using IPSO. The table shows that the available reserve for each area obtained by the optimal solution is over and above than its respective contingency spinning reserve requirement. It can also be seen
Fig. 4.Convergence characteristics for best fuel cost In order to highlight the effect of each modification suggested in the control equation of the conventional 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 PSO and its variants for test system 2 is shown in Fig. 4. It can be observed from the figure that while subsequently modifying the inertia weight, cognitive and the social component, the convergence characteristics are progressively improved by avoiding more and more local 165
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trappings. It can also be observed from the figure that except IPSO, the initial shape of the convergence is more or less same. Therefore the use of the constriction factor in the social component of particle’s velocity is one of the key factors for its better convergence.
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5. CONCLUSIONS This paper presents multi-area economic dispatch (MAED) problem of power systems with proposed spinning reserve sharing scheme by suggesting an improved particle swarm optimization (IPSO) method. In addition to pooling spinning reserve, a contingency spinning reserve for each area is proposed as per the norms of the regulatory bodies. The proposed IPSO method efficiently handles with stringent tieline security and spinning reserve constraints for inter-area power transactions. The application results shows that more economy in power generation and spinning reserves requirements can be achieved by sharing pooling spinning reserve among concerned areas while keeping the contingency spinning reserve intact in each area to enhance reliability of power generation. The parameters of PSO are redefined in such a fashion that ensures a proper balance between exploration and exploitation ability of PSO and thus result in faster global convergence, higher solution quality, and stronger robustness. It has been observed that each suggested modification in PSO is contributed to enhance its performance. The proposed IPSO method is capable to provide better quality solution and is computationally efficient. It is noteworthy that the proposed IPSO does not require additional mechanism to avoid local trapping, empirical formula to bound particle’s velocity or squeezing the search space, and it is quite independent of the initial state of particles. REFERENCES Barisal, A. K. (2013). Dynamic search space squeezing strategy based intelligent algorithm solutions to economic dispatch with multiple fuels. Int. J. Electrical Power and Energy Systems, 45, 50–59. Basu, M. (2013). Artificial bee colony optimization for multiarea economic dispatch. Int. J. Electrical Power and Energy Systems, 49, 181–187. Chen, C. L. and Chen, N. (2001). Direct search method for solving economic dispatch problem considering transmission capacity constraints. IEEE Trans. Power Syst., 16 (4), 764–769. Desell, A. L., McClelland, E. C., Tammar, K. and Horne, P. R. V. (1984). Transmission constrained production cost analysis in power system planning. IEEE Trans. Power Apparatus Syst., 103 (8), 2192–2198. Jayabarathi, T., Sadasivam, G., and Ramachandran, V. (2000). Evolutionary programming based multi-area economic dispatch with tie line constraints. Electr Mach Power Syst., 28, 1165–1176. Manoharan, P. S., Kannan, P. S., Baskar, S. and Iruthayarajan, M. W. (2009). Evolutionary algorithm solution and KKT based optimality verification to multiarea economic dispatch. Int. J. Electr Power Energy Syst., 31, 365–373. 166