Multi-area economic dispatch with reserve sharing using dynamically controlled particle swarm optimization

Electrical Power and Energy Systems 73 (2015) 743–756

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Multi-area economic dispatch with reserve sharing using dynamically controlled particle swarm optimization Vinay Kumar Jadoun, Nikhil Gupta ⇑, K.R. Niazi, Anil Swarnkar Malaviya National Institute of Technology, Jaipur, India

a r t i c l e

i n f o

Article history: Received 29 May 2014 Received in revised form 25 May 2015 Accepted 3 June 2015

Keywords: Multi-area economic dispatch Fuel cost minimization Particle swarm optimization Pooling spinning reserve Contingency spinning reserve Tie-line capacity

a b s t r a c t A dynamically controlled PSO (DCPSO) is proposed to solve Multi-area Economic Dispatch (MAED) problem with reserve sharing. The objective of MAED problem is to determine the optimal value of power generation and interchange of power through tie-lines interconnecting the areas in such a way that the total fuel cost of thermal generating units of all areas is minimized while satisfying operational and spinning reserve constraints. The control equation of the proposed PSO is augmented by introducing improved cognitive and social components of the particle’s velocity. The parameters of the governing equation are dynamically controlled using exponential functions. The overall methodology effectively regulates the velocities of particles during their whole course of flight in such a way that results in substantial improvement of the performance of PSO. The effectiveness of the proposed method has been investigated on multi-area 4 generators, 40 generators and 140 generators test systems with multiple constraints such as reserve sharing and tie-line power. The application results show that the proposed DCPSO method is very promising for large-dimensional MAED problems. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In the competitive and deregulated business environment, the objective of power pool is to achieve least cost operating strategy of electricity while meeting the demand and other constraints. This is possible by transmitting power from an area which has cheaper sources of generation to another area having costlier generation. Generally power pool has several generation areas. Areas of individual utility are interconnected through tie-lines to operate with maximum reliability, reserve sharing, improved security and less production cost than operated as isolated area [1]. Each area has its own generation cost, load pattern and spinning reserves. Therefore, the aim of multi area economic dispatch (MAED) is to determine the optimal power generation schedule and interexchange of power between the areas in such a way that minimizes the overall fuel cost of all generating units in different areas while satisfying operational and network constraints. The system security concern imposes additional restriction on the inter-area power transactions through tie-lines. In a power pool, the economy and reliability of power generation can be further enhanced if the ⇑ Corresponding author. Tel.: +91 1412529063 (O), +91 9414055654 (M); fax: +91 1412529063. E-mail addresses: [email protected] (V.K. Jadoun), nikhil2007_mnit@ yahoo.com (N. Gupta), [email protected] (K.R. Niazi), [email protected] (A. Swarnkar). http://dx.doi.org/10.1016/j.ijepes.2015.06.008 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

spinning reserves are also shared mutually among the interconnected areas. This however increases the complexity of the problem. In fact, the complexity of MAED problem arises due to the stringent area power balance constraints, tie-line constraints and area reserve constraints, in addition to the other operational constraints [2]. Some early efforts to attempt MAED problem can be briefly stated as: Shoults et al. [3] considered import and export constraints between areas, Romano et al. [4] presented Dantzig–Wolfe decomposition principle, Desell et al. [5] applied linear programming, Wang and Shahidehpour [6] proposed a decomposition approach using expert systems, etc. In recent years the modern artificial intelligence based techniques have shown potential to solve such complex combinatorial constrained optimization problems due to their ability to obtain global or near global optima. Jayabarathi et al. [7] solved multi-area economic dispatch problems with tie-line constraints using evolutionary programming. Chen and Chen [8] presented direct search method for solving economic dispatch problem considering transmission capacity constraints. Manoharan et al. [9] proposed covariance matrix adapted evolutionary strategy for MAED problems where a Karush–Kuhn–Tuck er (KKT) optimality criterion is applied to guarantee the optimal convergence. Wang and Singh [10] used particle swarm optimization (PSO) for this problem where tie-line transfer capacities and area spinning reserve sharing are incorporated to ensure security

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Nomenclature aij, bij, cij the cost coefficients of the jth generator in area i (US$, US$ MW1, US$ MW2) C1, C2 acceleration coefficients for best and social experience of PSO C1b, C1p acceleration coefficients for best and preceding experience eij and fij the valve point effect coefficients of the jth generator in area i (US$, MW1) gbestt best particle during tth iteration grmst root mean square experience of the swarm during tth iteration itr current iteration count itrmax maximum iteration count itrmin minimum iteration count k ratio of dynamic cognitive and social acceleration coefficients kw ratio of maximum and minimum bound of the inertia weight M number of areas NGi number of generating units in the system in area i NGi number of generating units in the ith area pbestn best position of nth particle achieved based on its own experience PD total real power demand of the system (MW) PDi total real power demand of area i (MW) PGi total real power generation in area i (MW) PGij real power output of the jth generator in area i (MW) max Pmin minimum/maximum generation limits of jth generaGij =P Gij tor in area i (MW) max Pmin minimum/maximum tie-line power limit from area i Tim =P Tim to area m (MW) pprecedingn preceding position of nth particle achieved based on its just previous experience PTim tie-line real power flow from area i to area m (MW) rand1 () and rand2 () random numbers in [0, 1]

and improve reliability, respectively. Zhu [11] presents a new nonlinear optimization neural network approach to study the problem of security-constrained interconnected MAED. Sharma et al. [2] formulated MAED problem with various constraints and compares the solution quality of differential evolution (DE) variants with an improved PSO strategy. Basu [12] applied artificial bee colony optimization (ABCO) to solve MAED problem with a variety of system constraints. Out of these, PSO is a swarm evolutionary based meta-heuristic technique which has proven potential to optimize problems having non-smooth, non-convex and discrete functions. Researchers attracted towards the PSO due to its advantages in terms of its simplicity, convergence speed, and robustness. However, PSO has inherent tendency of local trapping. Several modified versions of PSO have been reported in the recent past to enhance its performance by modulating inertia weight [13–15], improvising cognitive and social behavior [14,16,17], using constriction factor approach [18,19], modifying the control equation of PSO [13,20–24], or squeezing the search space [23,24], etc. However, some of these suggested versions of PSO require several experimentations for parameter setting or needs 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. In this paper, a dynamically controlled PSO (DCPSO) method is proposed to efficiently solve MAED problem which minimizes the overall fuel cost and spinning reserve requirement of each area while meeting the load demand and satisfying security and other

RCim Sci stn Sij Spi

pool reserve contributed from area i to area m (MW) contingency spinning reserve in the ith area (MW) position of nth particle at tth iteration available reserve on the jth unit of ith area (MW) pooling spinning reserve in the ith area (MW) v tn velocity of nth particle at tth iteration W inertia weight Wmin/Wmax minimum/maximum value of inertia weight Dt time step (s) f1 and f2 exponential constriction functions g ratio of current and maximum iteration count gt the value of g at which cognitive and social behavior equalizes l constant l1, l2 coefficients of exponent terms Abbreviations ABCO Artificial Bee Colony Optimization DCPSO Dynamically Controlled Particle Swarm Optimization DE Differential Evolution DEC Differential Evolution with Chaotic Sequences ED Economic Dispatch EP Evolutionary Programming KKT Karush–Kuhn–Tucker MAED Multi-Area Economic Dispatch PSO Particle Swarm Optimization PSO-TVAC PSO-Time Varying Acceleration Coefficients TVCR Time Varying Crossover Rate RCGA Real-Coded Genetic Algorithm RMS Root Mean Square SD Standard Deviation TVDE Time-Varying Differential Evolution TVM-DE Time-Varying Mutation Differential Evolution

operational constraints. Several measures have been incorporated in the control equation of the conventional PSO and the PSO operators are dynamically controlled by introducing exponential functions. Moreover, the cognitive behavior of the swarm is improved by employing a new concept of preceding experience that takes into account the immediate previous experience of particle to improve cognitive behavior and the communication of the particle with the swarm is improved by introducing root mean square (RMS) component of velocity in the social behavior of the particle. The proposed method effectively regulates the velocity of particles during their flights so as to ensure global exploration and also facilitates local exploitation. The effectiveness of the proposed method has been investigated on three different test generating systems considering various operational constraints like valve-point loading effect, power balance, tie-line capacity and area spinning reserve constraints. The performance of the proposed method has also been considered with other established methods. Problem formulation The generator cost function is generally considered as quadratic when valve-point effects are neglected. However, large turbine generators usually have a number of fuel admission valves which are operated in sequence to meet out the increased generation. The opening of a valve increases the throttling losses rapidly thus the incremental heat rate rises suddenly. This valve-point effect introduces ripples in the heat-rate curves and can be modeled as

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sinusoidal function in the cost function. Therefore, the objective function for the MAED problem may be stated as:

updates its previous velocity and position vectors according to the following model of [25].

Minimize FðPGij Þ

v tþ1 ¼ W v tn þ C 1  rand1 ðÞ  n

¼

N Gi M X X

ðaij þ bij PGij þ cij P2Gij Þ þ jeij sinðf ij ðPmin Gij  P Gij ÞÞj

ð1Þ

i¼1 j¼1

In area i, the total power generation of all generators must be equal to its area power demand PDi with the consideration of imported and exported power [1] and can be stated as:

i 2 f1; 2; . . . ; Mg

ð2Þ

k;k–i

Generator constraints For stable operation, power output of each generator is restricted within its minimum and maximum limits. The generator power limits are expressed as: max Pmin Gij 6 P Gij 6 P Gij

gbest  stn Dt

ð6Þ ð7Þ

where Dt is the time step, usually set to 1 s. W is the inertia weight which is allowed to decrease linearly as follows:

Area power balance constraints

j¼1

t



stþ1 ¼ stn þ v ntþ1  Dt n

Subject to the following constraints.

N Gi X X PGij ¼ PDi þ PTim ;

pbest n  stn þ C 2  rand2 ðÞ Dt

ð3Þ

Tie-line security constraints

W ¼ W min þ

ðW max  W min Þ  ðitrmax  itrÞ itrmax

ð8Þ

An appropriate regulation of particle’s velocity requires a proper balance between cognitive and social behaviors of the swarm. Initially, the cognitive component must dominate over the social component to ensure global exploration of the search space. However, during later part of the journey, 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 [26]. Therefore, a modified control equation is proposed for dynamically regulating particle’s velocity by suggesting suitable exponential constriction functions f1 and f2. Moreover, the cognitive and social components are modified by considering preceding and RMS experience, respectively. The suggested control equation for the proposed DCPSO may be expressed as:

pbest n  stn þ ð1  f1 Þ Dt t s  ppreceding n  C 1p  rand2 ðÞ  n þ f2  C 2  rand3 ðÞ Dt t t gbest  sn grmst  stn þ f2  C 2  rand4 ðÞ   Dt Dt

v tþ1 ¼ W  v tn þ f1  C 1b  rand1 ðÞ  n The transfer of real tie-line power PTim from area i to area m should not exceed the security limits of the tie-lines, i.e. max Pmax Tim 6 P Tim 6 P Tim

ð4Þ

Area spinning reserve constraints In a power pool, generally a fixed reserve is kept in each area to meet the contingency requirement of that area. This reserve may be called as contingency spinning reserve of the area. A Pool reserve is kept to meet the emergency requirement of the power pool such as loss of generation in any area of the pool. This pool reserve can either be kept in each area as supplemental reserve or may be contributed by multiple areas of the pool. When the pool reserve is kept in each area, the total specified spinning reserve in each area is the sum of contingency reserve and supplemental reserve of that area. In Ref. [10], the contingency reserve of an area and its contribution to pool (supplemental) reserve are combined and termed as specified/required spinning reserve of that area. However, if only contingency reserve of an area is kept as specified reserve of that area and the pool reserve is shared among all areas of the pool, it may results in less spinning reserve requirement in each area and thereby reduces the overall reserve requirement of the pool. It is therefore proposed that the spinning reserve requirement of an area i should satisfies the following equation: N Gi X X Sij P Sci þ Spi þ RC im j¼1

ð5Þ

m;m–i

Proposed DCPSO 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. Each particle

ð9Þ

In the proposed DCPSO, the inertia weight W is modified by proposing a n exponentially decaying function to regulate the tradeoff between the global exploration and the local exploitation of the swarm. The preceding experience of particles pprceedingn is considered to improve the cognitive component by memorizing just previous experience whereas an aggregate experience of the swarm is embedded in grmst to enhance the social component of particles. Further, dynamic acceleration coefficients have been introduced using constriction functions f1 and f2 in order to dynamically regulate the cognitive and social behaviors of the swarm. These modifications are discussed in the following sections. Updating inertia weight The trend of linear modulation of inertia weight of [27] is followed to solve ED problems using PSO by many researchers till date [20,21,28–30], etc. In the proposed method, the inertia weight has been allowed to vary in accordance to an exponential decaying function and the modulations suggested to update the inertia weight is governed by the following relation:

W ¼ expðg loge kw Þ

ð10Þ

where g = itr/itrmax; itrmin 6 itr 6 itrmax, and kw be selected in accordance of the desired minimum and maximum bounds of the inertia weight. In this paper, the value of kw is the ratio of maximum and minimum bound of the inertia weight. Updating preceding experience The cognitive behavior was split in [21] by considering also the worst experience in addition to the best experience of the particle

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to provide some additional diversity, but it results in poor local exploitation unless supported by a local random search. Therefore in the proposed method, 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 less diversity than the worst experience and thus provides better exploration and exploitation of the search space without employing any additional local random search or else. Updating RMS experience In PSO, only local and global best positions are transparent to other particles [19]. This poor communication among particles may leads to lack of diversity and thus results in poor performance, especially while dealing with large dimensional problems. One way to improve communication of the particles is to add RMS

PG11 PG21 …. PGi1

PG12 PG22 …. PGi2

…. …. …. ….

component of all particles’ velocities in the control equation, as suggested in (9). This results in global sharing of information and particles profit from the discoveries and previous experience of all other companions during their search. Dynamic control of acceleration coefficients In conventional PSO, cognitive and social behaviors are governed by static acceleration coefficients. However, many researchers [14,16,17] suggested their dynamic control to regulate particle’s velocity. In the present work, the acceleration coefficients are dynamically controlled by introducing two exponential constriction functions f1 and f2. These constriction functions regulate the cognitive and social behaviors of the swarm, thus limiting particles’ velocities during their whole course of flight and are given as:

f1 ¼ expðl1 gÞ

PG1j PG2j …. PGij

Fig. 1. Particle encoding for the proposed MPSO.

Fig. 2. Constraint handling algorithm.

ð11Þ

… … …. …

PG1N PG2N …. PGiN

V.K. Jadoun et al. / Electrical Power and Energy Systems 73 (2015) 743–756

f2 ¼ k  expðl2 gÞ;

k ¼ f1 C 1b =f2 C 2

ð12Þ

where k is the ratio of proposed dynamic cognitive and social acceleration coefficients. For identical values of these coefficients at g = gt

  k ¼ ðC 1b =C 2 Þ exp gt ðl1þ l2 Þ

ð13Þ

Next, for social behavior to be ke at the end of search

k ¼ ðke =C 2 Þ expðl2 Þ

satisfies problem constraints defined by (2)–(5). Infeasible particle, whenever appeared, are corrected by employing a constrained handling algorithm as described later in the section. The fitness of each particle is evaluated using (1) and then pbest, ppreceding, gbest and grms are initialized. The initial velocity of particles is assumed to be zero. Constrained handling

ð14Þ

Thus, from (13) and (14)

l2 ¼ ð1  gt Þ=gt  ðgt l1 þ loge ðke =C 1b ÞÞ

747

ð15Þ

For the given values of C1b, C2, l1 and gt, the value of l2 can be obtained corresponding to the optimal value of ke. These alterations in the control equation of the conventional PSO regulates particles’ velocity within predefined bounds without any additional formulation as reported in many improved versions of PSO [17,24,26,31], yet preserving diversity due to the stochastic nature of cognitive and social behaviors of the swarm. Particle encoding and initialization The solution of an MAED problem is the set of most optimal generations for the desired objective (s) bounded by certain operational constraints. In the proposed PSO, the particles are encoded in real numbers as the set of current generations in MW, as shown in Fig. 1. The initial population is randomly created with predefined number of particles to maintain diversity. Each of these particles

The velocity and position update may create infeasible solutions. In proposed method, infeasible individuals are not rejected but are corrected to feasible ones using a constrained handling algorithm. For this purpose, the power generated by each generator is adjusted by their respective bounded generation limits, tie-line and area spinning reserve constraints as given in Eqs. (3)–(5). If the generated power is less or more than its minimum or maximum generation level then the corresponding generation is set at minimum or maximum bound limits as in (3). Similarly, if the transfer of real tie-line power from area i to area k exceeds its limit then the corresponding tie-line power is set at tie-line bound limits as mentioned in (4) for security consideration. For area spinning reserve constraint, every area has to fulfill their respective reserve requirement as per (5), and if not satisfied, then the difference in amount of power is distributed equally among all generating units of that area till Eq. (5) is satisfied. The power balance error is calculated using (2). The error is equally divided among all generators and the procedure is repeated till it reduced to a predefined mismatch value e. In this work, e is taken as 0.001. The flow chart of the constraint handling algorithm is shown in Fig. 2.

Fig. 3. Flow chart of proposed DCPSO.

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Elitism and termination criterion In stochastic based algorithms like PSO, the solution with the best fitness in the current iteration may be lost in the next iteration. Therefore, the particle with the best fitness is kept preserved for the next iteration. The algorithm is terminated when either all particles converge to a single position or the predefined maximum iteration count is exhausted. The flow chart of proposed DCPSO is shown in Fig. 3.

reserve and tie-line constraints and the obtained results are compared without and with reserve sharing schemes between different areas. Case study 1

Parameter

Case study 1

Case study 2

Case study 3

Total power demand (MW) Tie-line limit (MW) Area load demand (%) Specified area spinning reserve (%) Area contingency spinning reserve (%) Pooling spinning reserve Wmin/Wmax C1p/C1b/C2 itrmin/itrmax Popsize

1120

10,500

49,342

This case study consists of two-area system with four generating units as shown in Fig. 4, and the detailed data may be referred from [9]. The load demand on two areas is taken as 70% and 30% of the total power demand. The specified spinning reserves are taken as 40% and 30% of the respective area load demand. Table 2 shows the comparison of results with tie-line capacity of 300 MW and when reserve sharing is not employed. The authors of [2] used several different methods for solving this problem. Among these methods only TVDE, DE, TVM-DE and DEC2 have better quality results in terms of zero standard deviation (SD). It can be observed from the table that proposed PSO generates better value of minimum fuel cost than all methods of [2]. It may also be seen from the table that the proposed method is capable of generating better quality solution than other versions of PSO and DE of as reflected from its standard deviation, best fuel cost and worst fuel cost. The optimal generating schedule in MW obtained using the proposed method is 368.92 MW, 115.08 MW for area 1 and 295.99 MW, 340.01 MW for area 2 with 300 MW flowing through the tie-line from area 2 to area 1. In order to investigate the effect of proposed reserve sharing, the contingency spinning reserve for each area is taken as 7% of its load demand which is reasonable to meet out local contingency requirements. In extreme emergency, when this reserve is not sufficient then the requirement is met by pooling spinning reserve. In this work, the pooling spinning reserve is assumed equal to the specified spinning reserve of the heavily loaded area. The contingency and pooling spinning reserves for this case study are given in Table 3. The optimal generating schedule in MW obtained using proposed PSO is 372.34 MW, 113.38 MW for area 1 and 294.73 MW, 339.28 MW for area 2 with tie-line power of 298.28 MW which is being flowing from area 2 to area 1. The available reserves for this solution are also shown in the table. The system fuel cost obtained with reserve sharing option is 10564.911658 $/h which is 1.98 $/h less than when reserve sharing is not employed. The average computational time taken by the proposed DCPSO is 5.23 s.

300 70/30 40/30

200/100 15/40/30/15 20/20/20/20

500 7/20/18/25/30 40/18/18/30/7

Table 2 Comparison results with specified spinning reserve constraint for case study 1.

7/7

7/7/7/7

7/7/7/7/7

Simulation results The proposed algorithm is tested on three generating systems, namely, two area 4 generators system [9], four area 40 generators system [12] and five area 140 generators system. The five area 140 generators system has been designed by considering data from [15]. The design parameters of the system and the control parameters of DCPSO considered for these case studies are presented in Table 1. The value of acceleration coefficients C1b, C1p and C2 used for all these test systems are taken from [21]. The swarm size and maximum iteration count have been obtained after usual tradeoff. The proposed algorithm has been developed using MATLAB and the simulations have been carried out on a personal computer of Intel i5, 3.2 GHz, and 4 GB RAM. The coefficient of the exponent l1 is selected to 5, as beyond that, the term exp (–l1g) is not perceptible at the end of search. Further, the most appropriate value of gt is obtained as 2/3 after experimentations. For this value of gt the optimized value of ke is found to be 0.2, with a corresponding value of 3.9617 for l2. These optimal values are obtained on the basis of average fuel cost obtained after 100 independent trials of DCPSO. Tie-line limits, area-wise specified and contingency spinning reserves considered for these systems are also shown in the table which are being expressed as the percentage of respective area power demand. The proposed method is applied to solve MAED problem with area

Table 1 System design parameters and control parameters of DCPSO for case studies.

l1 gt ke

– 0.1/0.9 0.4/1.6/2.0 1/1000 20 5.0 2/3 0.2

25% of PD of area 2 0.1/0.9 0.4/1.6/2.0 1/2500 50 5.0 2/3 0.2

25% of PD of area 4 0.1/0.9 0.4/1.6/2.0 1/5000 50 5.0 2/3 0.2

Variant

Best fuel cost ($/h)

Worst fuel cost ($/h)

SD

PSO [2] PSO_TVAC [2] TVDE [2] TVCR1 [2] TVCR2 [2] TVCR3 [2] DE [2] TVM-DE [2] DEC1 [2] DEC2 [2] DEC3 [2] Proposed DCPSO

10568.2696 10567.0059 10566.9942 10566.9942 10566.9942 10566.9942 10566.9942 10566.9942 10566.9942 10566.9942 10566.9942 10566.8883

10641.5278 10613.7863 10566.9942 10597.9979 10566.9942 10664.0845 10566.9942 10566.9942 10567.1873 10566.9942 10567.1913 10566.8883

91.472 11.051 0.0 6.855 0.035 36.311 0.0 0.0 0.008 0.0 0.038 0.0

Table 3 Contingency and pooling spinning reserves for case study 1.

Fig. 4. Two-area, four-generators system.

Reserve

Area 1

Area 2

Contingency spinning reserve (MW) Pooling spinning reserve (MW) Available reserve (MW)

54.88 313.6 314.28

23.52 105.99

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Case study 2

Fig. 5. Four-area, 40 generators system.

Table 4 Comparison results for case study 2. Method

Best fuel cost ($/h)

Average fuel cost ($/h)

Worst fuel Cost ($/h)

CPU time (s)

RCGA [12] EP [12] DE [12] ABCO [12] Proposed DCPSO

129911.8 124574.5 124544.1 124009.4 121948.8

– – – – 123450.8

– – – – 125772.4

160.5 144.5 134.8 126.9 64.8

Table 5 Optimal generating schedule for case study 2 (without reserve constraints). Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

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

110.8011 110.8008 97.35774 179.7211 87.81154 105.3997 299.9927 284.5989 284.6013 130.0002 94.95654 100.2370 304.5653 394.2854 394.2873 394.2863

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

493.0715 490.8852 517.7637 515.6676 523.2810 523.2794 523.2789 523.2784 523.2788 523.279 10.0000 10.00008 10.00000 87.79690 159.7347 159.7383

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 – –

189.9993 164.8000 191.9640 164.7999 110.0000 110.0000 89.11779 511.2790 199.9932 11.11496 95.0222 200.000 99.9994 81.4122 – –

This is a four-area 40 units generating system with non-convexity in cost function due to valve-point loading effects 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. 5. The figure also shows area power demands as a percentage of the total power demand. The detailed data may be referred from [12]. 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. For the sake of comparison, the proposed method is applied on the system with the consideration of tie-line constraints alone, however the reserve constraints will be dealt later on. The results obtained using DCPSO are presented and compared with established population based techniques in Table 4. It can be observed from the table that DCPSO is providing minimum fuel cost. In fact, the fuel cost obtained using DCPSO is less than 6.13%, 2.11%, 2.08% and 1.66% than those obtained using RCGA, EP, DE and ABCO methods of [12], respectively. The table also shows that DCPSO is computationally more efficient than other methods. The optimal dispatch of thermal generators obtained by the proposed PSO can be referred from Table 5. The negative sign of tie-line power indicates that it is actually flowing in opposite direction to that shown in Fig. 5. The problem is now extended by considering spinning reserve of each area as 20% of its power demand. However, proposed reserve sharing has not been included. The solution obtained using proposed PSO provides an overall fuel cost of 122919.282461 $/h. This is found to be 970.48 $/h more than when the reserve constraints were not considered. Moreover, the proposed reserve sharing is employed with contingency spinning reserve of 7% of power demand for each area and a pooling spinning reserve of 25% of the power demand of area 2. The optimal generating schedule and the corresponding tie-line flows obtained using proposed PSO is presented in Table 6. The optimum fuel cost obtained for this solution is 121992.37 $/h which is 926.91 $/h less than when reserves were not shared. Thus reserve sharing causes a significant reduction in the overall fuel cost of the system. The average computational time taken by the proposed DCPSO is 74.23 s. Table 7 provides a quick reference to check the validity of the reserve sharing constraints

Table 7 Contingency, pooling and available reserves for case study 2. Reserve

Area 1

Area 2

Area 3

Area 4

Contingency spinning reserve (MW) Pooling spinning reserve (MW) Available reserve (MW)

110.25 1050 289.66

294

220.5

110.25

1131.5

589.53

211.28

Table 6 Optimal generating schedule for case study 2 (with reserve constraints). Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, 9 1, 10

110.7992 110.8029 97.39970 179.7327 87.80040 140.0000 259.5998 284.6000 284.5996 130.0025

2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 2, 9 2, 10

315.5228 94.00390 304.5194 394.2734 394.2793 214.7587 489.2794 489.2809 511.2791 511.2792

3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, 9 3, 10

523.2771 523.2794 523.2757 523.2789 523.2787 523.2789 10.00000 10.00000 10.00000 87.79980

4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 4, 7 4, 8 4, 9 4, 10

159.7349 189.9999 159.7332 164.8054 164.8000 199.9457 109.9998 89.11540 89.30150 511.2794

T1, 2 T1, 3 T1, 4 T2, 3 T2, 4 T3, 4 – – – –

196.0418 0.01500 85.7196 187.240 98.2412 79.7555 – – – –

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imposed on the solution obtained. The table shows that the available reserve for each area is more than its respective contingency spinning reserve requirements. It can also depict from the table that the sum of available reserve is maintained higher than the sum of pooling and contingency spinning reserves.

Fig. 6. Five-area 140 generators system.

Case study 3 In order to investigate the effectiveness of the proposed DCPSO, a large scale test system is proposed for this case study. This system consists of 140 generating units with valve-points effect and the data has been taken from [15]. The total power demand for this system is 49,342 MW. These 140 generating units are assumed to be divided into five areas which are interconnected through eight tie-lines as shown in Fig. 6. The tie-line capacity is considered as 500 MW for each tie-line. The other details for this system may be referred from the Appendix A. When no reserve sharing is employed, the specified spinning reserves are assumed as 40% for area 1, 18% for both areas 2 and 3, 30% for area 4 and 7% for area 5 of their respective area power demand. The overall fuel cost obtained using proposed method is 1721134.304603 $/h. Furthermore, the proposed reserve sharing is incorporated with contingency spinning reserve of 7% of each area’s power demand. The pooling spinning reserve is assumed equal to 25% of the power demand of area 4. The fuel cost obtained using proposed PSO is 1720328.204903 $/h. It is found to be 806.0997 $/h less than when reserve sharing was not employed. The average computational time incurred using proposed DCPSO is 410.94 s. The optimal generating schedule and corresponding tie-line flows obtained for this solution are presented in Table 8. It can be observed from the table that the area-wise total optimal generation in MW is 1953.96, 10368.44, 9376.56, 10840.44 and 16802.59, respectively. For this optimal generation, the contingency & pooling spinning reserves for this system and the available reserve for each area for the optimal generating schedule so obtained are presented in Table 9. The table shows that the reserve constraints for inter-area transactions are well satisfied. Therefore,

Table 8 Optimal generating schedule for case study 3. Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

Area, unit

Power (MW)

1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, 9 1, 10 1, 11 1, 12 1, 13 1, 14 1, 15 1, 16 1, 17 1, 18 1, 19 1, 20 1, 21 1, 22 1, 23 1, 24 1, 25 1, 26 1, 27 1, 28 1, 29 – – – – – –

94.000531 94.009933 94.005231 274.487104 244.542600 249.168148 95.628656 95.002377 116.00262 175.000831 2.001867 4.002436 15.094327 9.014251 12.000344 10.000224 112.205833 4.000176 5.247233 5.001473 50.009763 5.004553 42.0094 43.523464 41.00592 17.002764 8.372683 7.000536 29.612985 – – – – – –

2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 2, 9 2, 10 2, 11 2, 12 2, 13 2, 14 2, 15 2, 16 2, 17 2, 18 2, 19 2, 20 2, 21 2, 22 2, 23 2, 24 2, 25 2, 26 2, 27 2, 28 – – – – – – –

71.00009 120 125 125 90.000125 90 449.911453 486.184742 465.520722 488.471491 470.2013 494.523801 460.823514 470.929326 260 466.864181 450.426049 394.084662 482.715137 292.160208 376.405123 448.486108 394.792279 474.064997 495.984448 352.971509 535.451504 536.470668 – – – – – – –

3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, 9 3, 10 3, 11 3, 12 3, 13 3, 14 3, 15 3, 16 3, 17 3, 18 3, 19 3, 20 3, 21 3, 22 3, 23 3, 24 3, 25 3, 26 3, 27 3, 28 – – – – – – –

500.999928 500.999502 505.999764 505.999923 505.996931 505.999846 499.9997 499.999819 240.999469 240.999871 773.998586 768.999725 5.73619 3.000008 249.999889 249.999899 247.577256 244.82037 249.999601 249.999968 249.999869 249.999868 312.225275 311.145326 173.123877 167.311869 180.348959 180.281507 – – – – – – –

4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 4, 7 4, 8 4, 9 4, 10 4, 11 4, 12 4, 13 4, 14 4, 15 4, 16 4, 17 4, 18 4, 19 4, 20 4, 21 4, 22 4, 23 4, 24 4, 25 4, 26 4, 27 4, 28 4, 29 4, 30 4, 31 4, 32 4, 33 4, 34 4, 35

175.694565 198.008805 311.952977 306.591218 163.002704 95.057854 508.540352 510.994822 489.957727 394.774629 489.99292 489.993376 130.001489 234.715084 146.779338 388.329624 534.951614 527.967607 509.438313 177.927444 364.843169 566.778275 530.99471 530.98281 541.994304 56.010506 115.015947 115.000607 115.006465 207.004971 207.0113 176.711673 176.320226 175.023286 177.070384

5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 5, 7 5, 8 5, 9 5, 10 5, 11 5, 12 5, 13 5, 14 5, 15 5, 16 5, 17 5, 18 5, 19 5, 20 T1, 2 T1, 3 T1, 5 T2, 4 T2, 5 T3, 4 T3, 5 T4, 5 – – – – – – –

579.996576 644.998087 819.350453 977.999727 681.998623 719.999324 717.998454 719.996697 853.137077 958 772.766107 751.497132 1012.995169 1019.997763 953.998722 951.999058 1005.998747 818.26342 1020.999495 820.602997 499.9966 499.9860 499.9988 499.9994 499.9531 495.0164 499.9992 500.0000 – – – – – – –

V.K. Jadoun et al. / Electrical Power and Energy Systems 73 (2015) 743–756 Table 9 Contingency and pooling spinning reserves for case study 3. Reserve

Area 1

Area 2

Area 3

Area 4

Area 5

Contingency spinning reserve (MW) Pooling spinning reserve (MW) Available reserve (MW)

241.78

690.79

621.71

863.49

1036.18

1903.56

1769.44

4192.56

1153.41

3083.88 1911.04

the proposed DPSO method seems to be promising to solve large scale MAED problems. Discussion The outcomes of MAED problem solutions obtained without and with inter area reserve sharing options are compared in Table 10. The table shows that when inter-area reserve sharing is allowed, the spinning reserve capacity is significantly reduced. This encourages power generation from the cheaper units, thus

Table 10 Comparison of without and with reserve sharing capacity and percentage reserve capacity reduction. Case study

1 2 3

Spinning reserve capacity (MW)

% Reduction

Without reserve sharing

With reserve sharing

414.4 2100 9493

313.6 1050 3083.88

24.32 50.00 67.51

751

reflects reduction in overall fuel cost. In addition, the spinning charges incurred due to reduction in spinning reserve capacity further reduces the cost of unit generation which is not included in this study. In order to show the impact of each modification suggested in the control equation of PSO, a set of convergence characteristics for the case study 3 are presented in Fig. 7. In the figure, the variants of PSO so obtained are classified as ‘b’, ‘c’, ‘d’ and ‘e’; ‘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, ‘d’ refers to ‘c’ with RMS experience added in the social component and ‘e’ refers to the proposed DCPSO. From the figure it can be observed that the convergence of PSO is substantially improved in ‘b’, when inertia weight is modulated using exponential function and then it further subsequently improved by the addition of each of the other modifications proposed in DCPSO. Finally; when constriction functions are incorporated in DCPSO, it enhances both exploration and exploitation potentials of the search. Fig. 8 shows an enlarged view of that portion of Fig. 7 which is not legible. It is clearly shown from the figure that every modification suggested for the proposed DCPSO is contributing towards improving its performance. In DCPSO, particles explore the promising region around first five hundred iterations and then exploiting this region meticulously to avoid multiple local trappings. This can be explicitly depicted from the enlarged views presented in Figs. 9 and 10. Similar conclusions may be drawn from Fig. 11 which depicts the overall movement of the swarm in the search space. Therefore, the modifications suggested in the control equation of the conventional PSO maintains a proper balance between cognitive and social behavior of the swarm and thus DCPSO generates better quality solutions.

Fig. 7. Convergence characteristic for best fuel cost.

Fig. 8. An enlarged view of Fig. 7.

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Conclusions The multi-area economic dispatch (MAED) provides more economy of power generation if area spinning reserves are also shared mutually among the interconnected areas, keeping their respective contingency reserves intact. However, the system security imposes restriction on the inter-area power transactions through tie-lines making MAED highly complex combinatorial constrained optimization problem with continuous decision variables where complexity arises due to the stringent area power balance constraints, tie-line constraints and area reserve constraints, in addition to the other operational constraints related to economic dispatch. In this paper, a dynamically controlled PSO (DCPSO) method has been proposed to solve complex MAED problem. The proposed method aims to overcome some drawbacks of the existing PSO methods and efforts have been made to dynamically choose PSO parameters

in such a manner that ensures a proper balance between exploration and exploitation capability of PSO and result in faster global convergence, better solution quality, and stronger robustness. The proposed method has been applied on three generating systems of small to large scale dimensions with specified spinning reserve constraints and inter-area reserve sharing provisions. The application results show that the proposed method is consistently efficient and is usually not trapped in local minima. A comparison of the proposed method with other established methods has also been carried out. The comparison results show that the proposed method is capable of producing better quality solution and is computationally more efficient than other established methods. It is noteworthy that DCPSO does not require additional mechanism to avoid local trapping, to bound particle’s velocity, and squeezing the search space. Moreover, it is independent of the initial state of particles. It has been observed that when inter-area reserve sharing

Fig. 9. An enlarged view of Fig. 8 for DCPSO.

Fig. 10. An enlarged view of Fig. 9 for DCPSO.

Fig. 11. Convergence characteristic for average fuel cost.

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is allowed, the overall fuel cost is reduced and it also causes a significant reduction in total spinning reserve requirement. The overall application results show that the proposed DCPSO method is promising tool to solve large-dimensional complex MAED

problems. The proposed method can be extended to solve multi-objective MAED problems with the inclusion of more objectives and constraints like network losses, environmental constraints, network congestion constraints, etc.

Appendix A . Area-wise data of generating units for case study 3

Area, unit

Generator

A

b

C

d

E

Pmin

1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, 9 1, 10 1, 11 1, 12 1, 13 1, 14 1, 15 1, 16 1, 17 1, 18 1, 19 1, 20 1, 21 1, 22 1, 23 1, 24 1, 25 1, 26 1, 27 1, 28 1, 29 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 2, 9 2, 10 2, 11 2, 12 2, 13 2, 14 2, 15 2, 16 2, 17 2, 18 2, 19 2, 20 2, 21 2, 22 2, 23

OIL#01 OIL#02 OIL#03 OIL#04 OIL#05 OIL#06 OIL#07 OIL#08 OIL#09 OIL#10 OIL#11 OIL#12 OIL#13 OIL#14 OIL#15 OIL#16 OIL#17 OIL#18 OIL#19 OIL#20 OIL#21 OIL#22 OIL#23 OIL#24 OIL#25 OIL#26 OIL#27 OIL#28 OIL#29 COAL#01 COAL#02 COAL#03 COAL#04 COAL#05 COAL#06 COAL#07 COAL#08 COAL#09 COAL#10 COAL#11 COAL#12 COAL#13 COAL#14 COAL#15 COAL#16 COAL#17 COAL#18 COAL#19 COAL#20 COAL#21 COAL#22 COAL#23

1269.132 1269.132 1269.132 4965.124 4965.124 4965.124 2243.185 2290.381 1681.533 6743.302 394.398 1243.165 1454.74 1011.051 909.269 689.378 1443.792 535.553 617.734 90.966 974.447 263.81 1335.594 1033.871 1391.325 4477.11 57.794 57.794 1258.437 1220.645 1315.118 874.288 874.288 1976.469 1338.087 1818.299 1133.978 1320.636 1320.636 1320.636 1106.539 1176.504 1176.504 1176.504 1176.504 1017.406 1017.406 1229.131 1229.131 1229.131 1229.131 1267.894

89.83 89.83 89.83 64.125 64.125 64.125 76.129 81.805 81.14 46.665 78.412 112.088 90.871 97.116 83.244 95.665 91.202 104.501 83.015 127.795 77.929 92.779 80.95 89.073 161.288 161.829 84.972 84.972 16.087 61.242 41.095 46.31 46.31 54.242 61.215 11.791 15.055 13.226 13.226 13.226 14.498 14.651 14.651 14.651 14.651 15.669 15.669 14.656 14.656 14.656 14.656 14.378

0.014355 0.014355 0.014355 0.030266 0.030266 0.030266 0.024027 0.00158 0.022095 0.07681 0.953443 0.000044 0.072468 0.000448 0.599112 0.244706 0.000042 0.085145 0.524718 0.176515 0.063414 2.740485 0.112438 0.041529 0.000911 0.005245 0.234787 0.234787 1.111878 0.032888 0.00828 0.003849 0.003849 0.042468 0.014992 0.007039 0.003079 0.005063 0.005063 0.005063 0.003552 0.003901 0.003901 0.003901 0.003901 0.002393 0.002393 0.003684 0.003684 0.003684 0.003684 0.004004

0 0 0 0 0 0 0 600 0 1200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 700 0 0 0 0 600 0 0 0 0 800 0 0 0 0 0 0 600 0

0 0 0 0 0 0 0 0.07 0 0.043 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.08 0 0 0 0 0.055 0 0 0 0 0.06 0 0 0 0 0 0 0.05 0

94 94 94 244 244 244 95 95 116 175 2 4 15 9 12 10 112 4 5 5 50 5 42 42 41 17 7 7 26 71 120 125 125 90 90 280 280 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260

Pmax 203 203 203 379 379 379 190 189 194 321 19 59 83 53 37 34 373 20 38 19 98 10 74 74 105 51 19 19 40 119 189 190 190 190 190 490 490 496 496 496 496 506 509 506 505 506 506 505 505 505 505 505

(continued on next page)

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Appendix A (continued)

Area, unit

Generator

A

b

C

d

E

Pmin

2, 24 2, 25 2, 26 2, 27 2, 28 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, 9 3, 10 3, 11 3, 12 3, 13 3, 14 3, 15 3, 16 3, 17 3, 18 3, 19 3, 20 3, 21 3, 22 3, 23 3, 24 3, 25 3, 26 3, 27 3, 28 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 4, 7 4, 8 4, 9 4, 10 4, 11 4, 12 4, 13 4, 14 4, 15 4, 16 4, 17 4, 18 4, 19 4, 20 4, 21 4, 22 4, 23 4, 24 4, 25 4, 26 4, 27 4, 28

COAL#24 COAL#25 COAL#26 COAL#27 COAL#28 COAL#29 COAL#30 COAL#31 COAL#32 COAL#33 COAL#34 COAL#35 COAL#36 COAL#37 COAL#38 COAL#39 COAL#40 LNG#01 LNG#02 LNG_CC#01 LNG_CC#02 LNG_CC#03 LNG_CC#04 LNG_CC#05 LNG_CC#06 LNG_CC#07 LNG_CC#08 LNG_CC#09 LNG_CC#10 LNG_CC#11 LNG_CC#12 LNG_CC#13 LNG_CC#14 LNG_CC#15 LNG_CC#16 LNG_CC#17 LNG_CC#18 LNG_CC#19 LNG_CC#20 LNG_CC#21 LNG_CC#22 LNG_CC#23 LNG_CC#24 LNG_CC#25 LNG_CC#26 LNG_CC#27 LNG_CC#28 LNG_CC#29 LNG_CC#30 LNG_CC#31 LNG_CC#32 LNG_CC#33 LNG_CC#34 LNG_CC#35 LNG_CC#36 LNG_CC#37 LNG_CC#38 LNG_CC#39 LNG_CC#40 LNG_CC#41 LNG_CC#42

1229.131 975.926 1532.093 641.989 641.989 911.533 910.533 1074.81 1074.81 1074.81 1074.81 1278.46 861.742 408.834 408.834 1288.815 1436.251 669.988 134.544 3427.912 3751.772 3918.78 3379.58 3345.296 3138.754 3453.05 5119.3 1898.415 1898.415 1898.415 1898.415 2473.39 2781.705 5515.508 3478.3 6240.909 9960.11 3671.997 1837.383 3108.395 3108.395 7095.484 3392.732 7095.484 7095.484 4288.32 13813.001 4435.493 9750.75 1042.366 1159.895 1159.895 1303.99 1156.193 2118.968 779.519 829.888 2333.69 2028.945 4412.017 2982.219

14.656 16.261 13.362 17.203 17.203 15.274 15.212 15.033 15.033 15.033 15.033 13.992 15.679 16.542 16.542 16.518 15.815 75.464 129.544 56.613 54.451 54.736 58.034 55.981 61.52 58.635 44.647 71.584 71.584 71.584 71.584 85.12 87.682 69.532 78.339 58.172 46.636 76.947 80.761 70.136 70.136 49.84 65.404 49.84 49.84 66.465 22.941 64.314 45.017 70.644 70.959 70.959 70.302 70.662 71.101 37.854 37.768 67.983 77.838 63.671 79.458

0.003684 0.001619 0.005093 0.000993 0.000993 0.002473 0.002547 0.003542 0.003542 0.003542 0.003542 0.003132 0.001323 0.00295 0.00295 0.000991 0.001581 0.90236 0.110295 0.024493 0.029156 0.024667 0.016517 0.026584 0.00754 0.01643 0.045934 0.000044 0.000044 0.000044 0.000044 0.002528 0.000131 0.010372 0.007627 0.012464 0.039441 0.007278 0.000044 0.000044 0.000044 0.018827 0.010852 0.018827 0.018827 0.03456 0.08154 0.023534 0.035475 0.000915 0.000044 0.000044 0.001307 0.000392 0.000087 0.000521 0.000498 0.001046 0.13205 0.096968 0.054868

0 0 0 0 0 0 0 0 0 600 0 0 0 0 0 0 600 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1200 0 1000 0 0 0 0 0 0 0 0 0 0 0 1000

0 0 0 0 0 0 0 0 0 0.043 0 0 0 0 0 0 0.043 0 0 0 0 0 0 0 0 0 0 0 0.043 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.03 0 0.05 0 0 0 0 0 0 0 0 0 0 0 0.05

260 280 280 280 280 260 260 260 260 260 260 260 260 120 120 423 423 3 3 160 160 160 160 160 160 160 160 165 165 165 165 180 180 103 198 100 153 163 95 160 160 196 196 196 196 130 130 137 137 195 175 175 175 175 330 160 160 200 56 115 115

Pmax 505 537 537 549 549 501 501 506 506 506 506 500 500 241 241 774 769 19 28 250 250 250 250 250 250 250 250 504 504 504 504 471 561 341 617 312 471 500 302 511 511 490 490 490 490 432 432 455 455 541 536 540 538 540 574 531 531 542 132 245 245

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Area, unit

Generator

A

4, 29 4, 30 4, 31 4, 32 4, 33 4, 34 4, 35 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 5, 7 5, 8 5, 9 5, 10 5, 11 5, 12 5, 13 5, 14 5, 15 5, 16 5, 17 5, 18 5, 19 5, 20

LNG_CC#43 LNG_CC#44 LNG_CC#45 LNG_CC#46 LNG_CC#47 LNG_CC#48 LNG_CC#49 NUCLEAR#01 NUCLEAR#02 NUCLEAR#03 NUCLEAR#04 NUCLEAR#05 NUCLEAR#06 NUCLEAR#07 NUCLEAR#08 NUCLEAR#09 NUCLEAR#10 NUCLEAR#11 NUCLEAR#12 NUCLEAR#13 NUCLEAR#14 NUCLEAR#15 NUCLEAR#16 NUCLEAR#17 NUCLEAR#18 NUCLEAR#19 NUCLEAR#20

2982.219 3174.939 3218.359 3723.822 3551.405 4322.615 3493.739 226.799 382.932 156.987 154.484 332.834 326.599 345.306 350.372 370.377 367.067 124.875 130.785 878.746 827.959 432.007 445.606 467.223 475.94 899.462 1000.367

b

C 79.458 93.966 94.723 66.919 68.185 60.821 68.551 2.842 2.946 3.096 3.04 1.709 1.668 1.789 1.815 2.726 2.732 2.651 2.798 1.595 1.503 2.425 2.499 2.674 2.692 1.633 1.816

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115 207 207 175 175 175 175 360 415 795 795 578 615 612 612 758 755 750 750 713 718 791 786 795 795 795 795

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