This paper proposes a Modified Particle Swarm Optimization with Time Varying Acceleration Coefficients (MPSOTVAC) for solving economic load dispatch (ELD) problem. Due to prohibited operating zones (POZ) and ramp rate limits of the practical generators, the ELD problems become nonlinear and nonconvex optimization problem. Furthermore, the ELD problem may be more complicated if transmission losses are considered. Particle swarm optimization (PSO) is one of the famous heuristic methods for solving nonconvex problems. However, this method may suffer to trap at local minima especially for multimodal problem. To improve the solution quality and robustness of PSO algorithm, a new best neighbour particle called ‘rbest’ is proposed. The rbest provides extra information for each particle that is randomly selected from other best particles in order to diversify the movement of particle and avoid premature convergence. The effectiveness of MPSOTVAC algorithm is tested on different power systems with POZ, ramprate limits and transmission loss constraints. To validate the performances of the proposed algorithm, comparative studies have been carried out in terms of convergence characteristic, solution quality, computation time and robustness. Simulation results found that the proposed MPSOTVAC algorithm has good solution quality and more robust than other methods reported in previous work.
1. Introduction
Economic load dispatch (ELD) is one of the important tasks in power system operation and planning. The main purpose of ELD is to determine the real power output of scheduled generators to meet power demand at minimum cost whilst satisfying the equality and inequality constraints. Optimal combination of generator power output can reduce the cost of power plant operation significantly.
In general, the cost characteristic of generator is assumed to be convex and is represented by a single quadratic function for ELD problems. It was successfully solved by mathematical programming methods based on derivative information of cost function
[1]
. However, the cost function of a practical generator becomes highly nonlinear and discontinuous due to prohibited operating zones (POZ) and ramprate limits of the generator
[2
,
3]
. Therefore, ELD problems with equality and inequality constraints are nonconvex and very difficult to solve using a mathematical approach. Conventional methods such as gradient method, lambda iteration, base point participation and Newton methods are unable to solve nonconvex optimization problem
[1]
. On the other hand, dynamic programming can solve nonconvex ELD problem due to no restriction on the cost function, but suffers from ‘curse of dimensionality’ when involves with high number of variables
[4]
.
Recently, modern heuristic methods such as genetic algorithm
[5
,
6]
, evolutionary programming
[7]
, differential evolution
[8]
, ant colony optimization
[9]
, tabu search
[10]
, simulated annealing
[11]
, neural network
[12]
, and particle swarm optimization (PSO)
[13

18]
have been successfully applied to nonconvex ELD problems. However, these approaches are not always promising a global optimum solution and sometimes are trapped at local point.
Among these techniques, PSO is widely used for solving nonconvex ELD problem due to its simple implementation, less complexity and most of the time able to find global solution. In classical PSO, premature convergence is always occurring due to the lack of diversity of PSO algorithm. This can lead the particles to converge at a local optimal solution especially for complex and nonconvex problems with multiple minima. To overcome this problem, many types of PSO variants were proposed in ELD application
[16
,
18

21]
. However, most of these strategies do not produce consistent results in many different trials. In
[22]
, a new index (iteration best) are introduced to improve the solution quality for unit commitment problem based on modification of velocity equation. Later, the method in
[22]
was applied to solve ELD problem, but the robustness of this algorithm is not discussed in
[23]
. In
[16]
, the authors proposed a new PSO strategy based on the information of bad experience among particle. The updated particle will use this information to move away from the bad position that has been achieved from the population. The time varying acceleration coefficients (TVAC) approach is proposed by varying the value of the acceleration coefficient for the cognitive component (
c_{1}
) and social component (
c_{2}
) during the iterative process
[24
,
25]
. By proper tuning of these coefficients (
c_{1}
and
c_{2}
), the particles are guided towards optimum solution.
In this paper, a new PSO variant named MPSOTVAC method is proposed for solving ELD problems with the objectives to improve the solution quality, robustness and to avoid premature convergence. A new best neighbour parameter (
rbest
) is introduced into velocity equation, which is randomly selected from the best position (
pbest
) that has been obtained by another particle. This can enhance the searching behaviour and exploration capability of particles through the entire solution space. Moreover, TVAC (for
c_{1}
,
c_{2}
and
c_{3}
) can provide a balance exploration and exploitation for the particle to get a better optimum solution. To validate the proposed MPSOTVAC method, it is tested on ELD problem with generator limits, POZ, ramprate limits and transmission loss constraints. A new constraint handling has been introduced to handle POZ and ramprate limits constraints instead of used penalty factor as in
[16
,
17]
and discussed in Section 4 (step 5). The results obtained by MPSOTVAC are compared with some PSO variants in term of convergence characteristic, solution quality and robustness. For the ELD problem, the results show that MPSOTVAC approach provides lower cost and more robust than other PSO strategies and results of an existing method.
In this paper, Section 2 details mathematical formulations of ELD problem considering POZ, ramp rate limits and transmission loss. Section 3 describes the proposed MPSOTVAC approach. Section 4 presents the detail procedure of implementing the MPSOTVAC strategy for solving the nonconvex ELD problems. The simulation results and comparison study are presented and discussed in section 5. Finally, the conclusions are drawn in Section 6.
2. Formulation of ELD Problem
The primary objective of ELD problem is to minimize the total fuel cost (
F_{C}
) of thermal generator while satisfying the operational constraints of a power system. Therefore, ELD problem can be formulated based on single quadratic function as below:
where
F_{i}(P_{i})
is the fuel cost of the ith generator ($/h) which is defined by following equation:
where
Ng
is the number of generator,
P_{i}
is the active power of generator
i
(MW) and
a_{i}
,
b_{i}
and
c_{i}
are the fuel cost coefficients of
i
th generator.
In this paper, the POZ and ramp rate limits are considered as practical constraints of generator and transmission losses as network constraint. This result in the ELD becomes more complicated and nonconvex optimization problem that has multiple local minima which is difficult to find a global optimum solution. The ELD constraints are discussed as follows:
 2.1 Power balance constraint
The total generated power must meet the total load demand and transmission losses as given in (3) and (4).
where
P_{D}
is the total power demand and
P_{L}
is the transmission losses in the power network. Transmission losses in (3) can be calculated either using penalty factor or
B
loss coefficient
[1
,
26]
. In ELD problem,
B
loss coefficient is commonly used in the previous study and is adopted in this paper as following equation:
where
B_{ij}
is the
i,j
element of the loss coefficient matrix,
B_{i0}
is the
i
th element of the loss coefficient vector and
B_{00}
is the loss coefficient constant.
 2.2 Generator limit constraints
The active power output of each generator should satisfy the minimum and maximum limits as given:
where
and
are the minimum and maximum limits for
i
th generating unit.
 2.3 Ramp rate limits
In practical, adjustments of power output are not instantaneous. Increasing or decreasing of power output is restricted by ramp rate limits of the generating unit by the following conditions:
If power generation increases:
If power generation decreases:
Therefore, the effective generator limits with the presence ramp rate limits are modified as follows:
where
is the previous active power output of generator
i
(MW),
DR_{i}
and
UR_{i}
are the upper and lower ramp rate limits of generator
i
(MW/time period) respectively.
 2.4 Prohibited Operating Zones (POZ)
The generating unit may have certain zones where the operation is not allowed due to vibration in shaft bearing or problem of machine components
[2]
. Thus, discontinuous and non smooth fuel cost characteristic is produced corresponding to the POZ as illustrated in
Fig. 1
.
Fuel cost characteristic with POZ
In practical, adjustments of the output power of generator
i
must be avoided to operate within these zones. The allowable operating zones incorporated POZ constraints are formulated as follows:
where
and
are the lower and upper bounds of
z
th POZ of
i
th generator in (MW) and
Nz
is the number of POZ of
i
th generator.
3. Proposed MPSOTVAC Algorithm
 3.1 Review PSO algorithm
The particle swarm optimization is a population based optimization technique that was introduced by Kennedy and Eberhart in 1995
[27]
. This modern heuristic technique is inspired by social behaviour of the swarm of fishes and flocks of birds searching for the food. The main advantages of the PSO algorithm compared to other optimization methods are simple, easy to implement, less storage requirement and able to find a global optimum solution
[28]
.
In PSO, each particle represents the possible solutions to the problem. Initially, a random population of particles (or solution) is generated in
d
dimensional (or variable) search space. A particle
i
at iteration
j
is represented as position vector
and velocity vector
. Based on the evaluation function value, each particle in current iteration has its own best position represented as
. The best particle in a population is defined as global best
. The velocity and position of each particle are updated using equations below:
where
r_{1}
and
r_{2}
are random numbers between 0 and 1,
c_{1}
is the cognitive acceleration coefficient which pushes the particles towards
pbest
,
c_{2}
is the social acceleration coefficient which push the particles towards
gbest
and
w
is the inertia weight factor.
The inertia weight controls the impact of the previous velocity on updating velocity of a particle. A proper selection of
w
can provide a good exploration and exploitation to find the optimum solution. A large initial value of
w
can provide a better global exploration while smaller values of
w
facilitates better exploitation in local search
[29]
. The linearly decreasing of
w
is computed as follows
[15]
:
where
w_{min}
and
w_{max}
are the initial and final inertia weights respectively and
j_{max}
is the maximum iteration number. For effective balance between global and local searches, the inertia weight is decreased linearly from 0.9 to 0.4 during the optimization process
[14
,
30
,
31]
.
 3.2 Review IPSO algorithm
Iteration particle swarm optimization (IPSO) was introduced by TsungYing Lee and ChungLung Chen in 2007
[22]
in order to improve the solution quality of PSO. A new index named ‘Iteration best’ was introduced in (10) to enrich searching behaviour of PSO. A modified velocity equation of the particles is
where
is the best value of fitness function that has been achieved by any particle in current iteration
j
and
c_{3}
is the stochastic acceleration coefficient that pulls the particles towards
I_{best}
.
 3.3 Proposed MPSOTVAC algorithm
In order to improve the solution quality and robustness of PSO, a novel modified PSO with time varying acceleration coefficients (MPSOTVAC) is proposed. This method introduces a new parameter named the best neighbour particles (
rbest
) in (10) based on randomizing the best position of neighbour particles. The idea is to provide the extra information to each particle, thus increasing the exploration capability and avoiding being trapped in a local optimum. In this strategy, each particle has its own
, which is randomly selected from the best position (
pbest
) of other particles.
Fig. 2
is an illustration on how to determine
rbest
value for particle 2, where the other
pbest
values (except its own
pbest
) are randomly preferred. A similar approach is applied to other particles in the swarm. The new updated velocity for proposed method is given in (14):
Determination of rbest value of the ith particle.
where,
c_{3}
is the acceleration coefficient that pulls each particle towards
rbest
.
The performance of PSO is dependent to the proper tuned parameters that results in the optimum solutions. Generally, the acceleration coefficients for cognitive (
c_{1}
) and social components (
c_{2}
) are set to constant values. The impact of acceleration coef ficients setting is reported in
[13
,
32]
. A relatively high value of the social acceleration coefficient
c_{2}
than cognitive acceleration coefficient
c_{1}
is selected, the algorithm will converge to a local optimum solution (premature convergence). However, a relatively high value of cognitive acceleration coefficient
c_{1}
compared to social acceleration coefficient
c_{2}
results in wandering of particles around the search space
[27]
.
To enhance exploration and exploitation of particle towards optimum solution, both coefficients should be varies according to the iteration number
[24]
. A large value of cognitive component and small social component in initial iteration pushes the particles to move to the entire the solution space. As iteration increases, the value of cognitive will decrease and the value of the social components will increase, which pull the articles to the global solution. The acceleration coefficients are varied according the following formulas:
where
c_{1i}
and
c_{1f}
are the initial and final values of cognitive coefficient respectively and
c_{2i}
and
c_{2f}
are the initial and final values of social coefficient respectively.
Presenting a new parameter (
rbest
) in the velocity equation in (14), will encourage the particle movement to converge at optimum solution due to extra information provided by the
rbest
value in current iteration. The time varying acceleration coefficient for
rbest
component (
c_{3}
) is using the following Eq.
[33]
:
The behaviour acceleration coefficients (
c_{1}
,
c_{2}
and
c_{3}
) of the MPSOTVAC algorithm are shown in
Fig. 3
. It assumed that, the
c_{1}
value varies from 1 to 0.2 and
c_{2}
varies from 0.2 to 1 during 100 iterations. At the initial iteration, the
c_{3}
value is increased immediately which helps the particles to explore the entire possible solutions based on the best neighbour particle (
rbest
). This can avoid the particle to rapidly converge at the local
gbest
. As iteration proceeds, the
c_{2}
value is linearly increased to encourage particles towards global gbest value. Therefore, the exploration and exploitation capability of MPSOTVAC is improved, thus providing good solution quality and consistent results near to the global optimum.
Behaviour of acceleration coefficients (c_{1},c_{2} and c_{3}) during iteration.
4. Procedure of MPSOTVAC Algorithm for ELD Problem
In this section, the proposed MPSOTVAC method for solving nonsmooth ELD problems with POZ and ramp rate limits constraints are explained. This paper also proposes a new strategy to handle POZ and ramp rate limit constraints during the optimization process without using penalty factor. The results obtained by this approach satisfy all the constraints at the minimum cost. The flowchart of the MPSOTVAC algorithm is shown in
Fig. 4
. The detailed implementation of the proposed algorithm are described as follows:
Flowchart of proposed MPSOTVAC algorithm.

Step 1: Initialization of the Swarm.

The active power output of the generator is defined as a variable (or dimension) for ELD problem. For a population size ofNpop, the particles are randomly generated between the generator limits in (5) and satisfy all constraints in (6) and (7). Theith particle forNggenerator number is represented by

Step 2: Evaluation Function.

The fitness of each particle is evaluated based on the defined evaluation function. The evaluation function should minimize the total cost function and satisfy the constraints. Commonly, the penalty factor method is widely implemented in solving ELD problem, which is adopted here. In this method, the penalty function is integrated with the objective function in order to satisfy the power balance constraint in (3). The penalty parameter must be chosen carefully to distinguish between feasible and infeasible solutions. The evaluation functionf(Pi) is defined as

where,kis the penalty factor for the total active power which does not satisfy the power balance constraints.

Step 3: Initialization ofpbest,gbestandrbest

The fitness value of each particle is calculated using (20). Initial particles in Step 1 are set as initialpbestvalues. The best fitness function among thepbestvalue is defined asgbest. Then,rbestof each particle is randomly selected from other best particles.

Step 4: Update Velocities and Particles Position.

The velocities are updated using (14) within the range of. The maximum velocity ofdth dimension is computed by

where,Ris the chosen number of interval indth dimension. The maximum velocityis set to 20% of the dynamic range of each variable (PmaxPmin). Then, every particle in the swarm is moved to a new position using (11).

Step 5: Constraints Handling.

The updated position in Step 4 may have violated from inequality constraints in (8) and (9) due to the over or under velocity. If the updated position ofith particle indth dimension (or generator) is larger/lower than the effective maximum/minimum, the updated position is set to the effective maximum/minimum. This approach ensures that the particles in a swarm are moved around feasible solution only. The adjustments of the updated position to satisfy both constraints are

where,Pdmin_newandPdmax_neware the effective minimum and maximum ofdgenerator.

If the generator outputiis violated the POZ constraints in (9), it will be pushed to the nearest boundary ofzth POZ as follows :

where,Pi,zmeanis the average value of the zth POZ which calculated as follows:

Step 6: Update the Swarm.

The updated particles are evaluated using (20). If the current value is better than the previouspbest, the current value is stored aspbest. Otherwise, it is remained as the previouspbest. Thegbestvalue is updated as similar to thepbest. Then, therbestvalue for each particle is defined.

Step 7: Termination Condition.

A maximum iteration is applied as the stopping criteria for the algorithm. If the maximum iteration is reached, then MPSOTVAC algorithm is stopped and the best solution is selected. Otherwise, the algorithm returns to 4.
5. Simulation Results and Performance Analysis
The proposed MPSOTVAC method is tested on 6, 15 and 38generator ELD problems with different sizes and complexity. To validate the effectiveness of the proposed algorithm, the test results are compared with PSO and IPSO after 50 different runs.
The obtained results are compared with the results reported in previous work. In this study, the parameter setting used for every case study is listed in
Table 1
. The simulation was performed using MATLAB 7.6 on Core 2 Quad processor, 2.66 GHz and 4 GB RAM.
Parameter setting for the selected algorithm
Parameter setting for the selected algorithm
 5.1 Test system
The first test system consists of six generators with POZ, ramp rate limits and load demand of 1263MW. The cost data and
B
loss coefficients are given in
[14]
. All the generators have ramp rate limits and POZ. The best result reported to date is $15450
[16]
.
The second test system consists of 15 generators with ramp rate limit and POZ. A load demand of 2630 MW is considered in this case. The input data are taken from
[14]
. This system has many local minima and the optimum cost reported to date is $32704.50
[34]
. All the generators have ramp rate limits and four generators with POZ. The transmission losses are considered both test systems and are calculated using (4).
The third test system consists 38 generators and 6000 MW of load demand. The input data are given in
[35]
.
 5.2 Parameter tuning for MPSOTVAC
The performance of PSO algorithm is influenced by the setting of cognitive and social coefficients (
c_{1}
and
c_{2}
). The best combination of
c_{1}
and
c_{2}
depends on the problem. To determine the best combination of
c_{1}
and
c_{2}
, the different range of
c_{1}
and
c_{2}
is tested. The value of
c_{3}
varies according to
c_{1}
and
c_{2}
as in (17).
Table 2
shows the results of the best, worst and mean costs after 50 independent runs for test system 1 and 2. Most of the combinations of
c_{1}
and
c_{2}
produce results near to the global optimum solution. However, the combination of
c_{1i}
=
c_{2f}
=1.0 and
c_{1f}
=
c_{2i}
=0.2 is found to be the optimum results than others. The same combination of
c_{1}
and
c_{2}
also provides the best results for the 38 generators system.
Influence of acceleration coefficient on MPSOTVAC performance
Influence of acceleration coefficient on MPSOTVAC performance
Tables 3
and
4
show the performances of MPSOTVAC for different population size according to the number of dimension and complexicty of the problem.It clearly shows that the proposed MPSOTVAC can obtain the global or near to a global solution for all population size while other PSO methods reach near to the global solution with a large number of population. The population sizes of 30 and 150 were found the best results for both systems respectively. Meanwhile, population size of 200 was found to be the best results for 38 generators system.
Statistical Results of various PSO algorithms (6generator system)
Statistical Results of various PSO algorithms (6generator system)
Statistical Results of various PSO algorithms (15generator system)
Statistical Results of various PSO algorithms (15generator system)
 5.3 Convergence characteristic
The convergence characteristics of three different PSO strategies are shown in
Figs. 5
and
6
. It shows that the PSO and IPSO have loss diversify and converge at local minimal after certain iterations. However, the MPSO_TVAC method can reach near to the global value due the extra information provided by other best neighbour particle (
rbest
) and proper tuning of TVAC values. In the early iteration,
rbest
value in (14) helps every particle to explore the entire search space and high value of
c_{2}
exploits the best solution in the latter iteration. It will lead the algorithm to find near to the global optimum effectively.
Convergence characteristic of three PSO strategies for 6generator system.
Convergence characteristic of three PSO strategies for 15generator system.
 5.4 Solution quality
Tables 3
and
4
show the best, worst, mean and standard deviation (SD) cost obtained from 50 runs for three PSO approaches. The best, mean and SD cost obtained by the MPSOTVAC is lower than other methods, which demonstrates the high solution quality of the proposed method.
Tables 5

6
present the best generator output obtained by three PSO algorithms. Due to limited space, only the comparison of the best generation cost for 38 generator system shown in
Table 7
. It shows that the generation cost obtained by MPSOTVAC is better than other PSO while satisfying all the operational constraints.
Best simulation result for 6generator system
Best simulation result for 6generator system
Best simulation result for 15generator system
Best simulation result for 15generator system
Best simulation result for 38generator system
Best simulation result for 38generator system
 5.5 Robustness test
The performance of heuristic method such as PSO algorithm cannot be evaluated by a single run due to the inherent randomness involved in the optimization process. Therefore, the robustness of each PSO algorithm are evaluated based on 50 different runs. The algorithm is robust when it capable to produce consistence results. The best results obtained by three PSO variants after 50 runs are plotted in
Figs. 7
and
8
. It can be seen that the MPSOTVAC method achieves consistent result at the lowest cost in every run compared to other PSO methods. Moreover, the smallest SD obtained by the MPSOTVAC in
Tables 3
and
4
also shows that the MPSOTVAC is more robust than other PSO methods.
Best result of variant PSO algorithms for 50 runs (6generator system).
Best result of variant PSO algorithms for 50 runs (15generator system).
 5.6 Comparison of best solution
The best result archived by the MPSOTVAC for 6generators system is compared with the previous publiccations of GA
[14]
, PSO
[14]
, PSO_LRS
[16]
, NPSO
[16]
, NPSO_LRS
[16]
and PSOTVAC
[36]
in
Table 8
. The results show that the MPSOTVAC provides the minimum cost with less computational time compared to other methods . For 15generators system, the results obtained by the MPSOTVAC are compared with GA
[14]
, PSO
[14]
, BF
[37]
, SOH_PSO
[17]
,GAAPI
[6]
, PSOMSAF
[38]
, PSOTVAC
[36]
and FA
[34]
in
Table 9
. It shows that MPSOTVAC can produce a better cost and less computational time compared to other methods. Similarly, the results obtained by MPSOTVAC for 38generators system are compared with PSO, IPSO and PSO_TVAC
[25]
in
Table 7
. From these results, it clearly shows that the proposed MPSOTVAC has been found to be successful in solving ELD with generators constraints.
Comparison among different methods after 50 trials (6generator system)
^{a} The solution provided in [36] is violated the equality constraints (ΣP_{i}≠P_{D}+P_{L}). ‘’: Not reported in the refereed literature.
Comparison among different methods after 50 trials (15generator system)
^{a} The solution provided in [36] is violated the equality constraints (ΣP_{i}≠P_{D}+P_{L}). ‘’: not reported in the refereed literature
6. Conclusion
This paper has proposed a MPSOTVAC algorithm for solving nonconvex ELD problem considering generator limit, POZ, ramp rate limits and transmission loss. The proposed algorithm introduced a new best neighbour particle (
rbest
) in velocity equation which helps the particle to explore the entire solution space thus avoiding a premature convergence. Moreover, the used of time varying acceleration coefficients (TVAC) for
c_{1}
,
c_{2}
and
c_{3}
enhanced exploration and exploitation of the proposed algorithm. The MPSOTVAC performances have been compared with some PSO variants for three benchmark power systems after 50 different runs. The simulation results have shown that the MPSOTVAC has the ability to obtain lower generation cost and is more robust compared to PSO and other method reported in literature. These studies validate the effectiveness and applicability of the proposed algorithm for solving ELD problems.
Acknowledgements
This work is supported by the Ministry of Higher Education Malaysia under the Exploratory Research Grant Scheme (ERGS), Grant Code: ER0252011A.
BIO
Mohd Noor Abdullah received his B.Eng. (Hons) in Electrical Engineering and M.Eng. in Electrical (Power) from Universiti Teknologi Malaysia(UTM) in 2008 and 2010 respectively. He is currently pursuing his Ph.D in University of Malaya, Kuala Lumpur Malaysia and serving as a tutor in Department of Electrical Power, Faculty of Electrical and Electronics Engineering, Universiti Tun Hussein Onn Malaysia. His research interests are economic load dispatch, renewable energy and optimization techniques.
Ab Halim Abu Bakar received his B.Sc. in Electrical Engineering in 1976 from Southampton University UK and M. Eng and PhD from University Technology Malaysia in 1996 and 2003. He has 30 years of utility experience in Malaysia before joining academia. Currently he is a Lecturer in the Department of Electrical Engineering, University Malaya, Malaysia. Dr. Halim is a Member of IEEE, CIGRE, IET and a Chartered Engineer. His research interests include power system protection and power system transient.
Nasrudin Abd Rahim was born in Johor, Malaysia, in 1960. He received his B.Sc. (Hons.) and M.Sc. degrees from the University of Strathclyde, Glasgow, U.K. and Ph.D. degree from HeriotWatt University, Edinburgh, U. K., in 1995. He is currently a Professor with the Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia, and is also the Director and Founder of the UM Power Energy Dedicated Advanced Centre (UMPEDAC), University of Malaya. His research interests include power electronics, realtime control systems, electrical drives, and renewable energy systems. Dr. Nasrudin is a Fellow of the Academy of Sciences Malaysia, Fellow of the Institution of Engineering and Technology and is a Chartered Engineer.
Hazlie Mokhlis received his B. Eng in Electrical Engineering in 1999 and M. Eng. Sc in 2002 from University of Malaya, Malaysia. He obtained PhD degree from the University of Manchester, UK in 2009. Currently he is a Senior Lecturer in the Department of Electrical Engineering, University of Malaya. He is also an associate member of UM Power Energy Dedicated Advanced Research Centre (UMPEDAC), University of Malaya. His main research interest is in distribution automation area and power system protection. He is a member of IEEE.
Hazlee A. Illias was born in Kuala Lumpur, Malaysia in 1983. He received the B. Eng in electrical and electronic engineering from the University of Malaya, Malaysia in 2006 and the PhD degree from the University of Southampton, UK in 2011.Currently he is a Lecturer in the Department of Electrical Engineering, University Malaya, Malaysia. His main research interests include partial discharge modeling and measurements in cavity voids within solid dielectric materials, partial discharge simulation using Finite Element Analysis method and load flow analysis.
Jasrul Jamani Jamian obtained his B. Eng. (Hons) in Electrical Engineering in 2008 and M. Eng. Electrical (Power) in 2010 from Universiti Teknologi Malaysia (UTM), Johor Bahru, Malaysia where he is currently pursues his Ph.D. degree. His current research interests include smart grid system, power system stability, renewable energy application and their control method. He is a student member of IEEE.
A.J. Wood
,
B.F. Wollenberg
1996
Power Generation, Operation and Control
2nd ed.
John Wiley and Sons
New York
Orero S.O.
,
Irving M.R.
1996
Economic dispatch of generators with prohibited operating zones: a genetic algorithm approach
IEE Proceedings Generation, Transmission and Distribution
143
529 
34
DOI : 10.1049/ipgtd:19960626
Wang C.
,
Shahidehpour S.M.
1993
Effects of ramprate limits on unit commitment and economic dispatch
IEEE Transactions on Power Systems
8
1341 
50
DOI : 10.1109/59.260859
Shoults R.R.
,
Venkatesh S.V.
,
Helmick S.D.
,
Ward G.L.
,
Lollar M.J.
1986
A Dynamic Programming Based Method for Developing Dispatch Curves When Incremental Heat Rate Curves Are NonMonotonically Increasing
IEEE Transactions on Power Systems
1
10 
6
Amjady N.
,
NasiriRad H.
2009
Economic dispatch using an efficient realcoded genetic algorithm
IET Generation, Transmission & Distribution
3
266 
78
DOI : 10.1049/ietgtd:20080469
Ciornei I.
,
Kyriakides E.
2012
A GAAPI Solution for the Economic Dispatch of Generation in Power System Operation
IEEE Transactions on Power Systems
27
233 
42
DOI : 10.1109/TPWRS.2011.2168833
Sinha N.
,
Chakrabarti R.
,
Chattopadhyay P.K.
2003
Evolutionary programming techniques for economic load dispatch
IEEE Transactions on Evolutionary Computation
7
83 
94
DOI : 10.1109/TEVC.2002.806788
Sharma M.
,
Pandit M.
,
Srivastava L.
2011
Reserve constrained multiarea economic dispatch employing differential evolution with timevarying mutation
International Journal of Electrical Power & Energy Systems
33
753 
66
DOI : 10.1016/j.ijepes.2010.12.033
Pothiya S.
,
Ngamroo I.
,
Kongprawechnon W.
2010
Ant colony optimisation for economic dispatch problem with nonsmooth cost functions
International Journal of Electrical Power & Energy Systems
32
478 
87
DOI : 10.1016/j.ijepes.2009.09.016
Sangiamvibool W.
,
Pothiya S.
,
Ngamroo I.
2011
Multiple tabu search algorithm for economic dispatch problem considering valvepoint effects
International Journal of Electrical Power & Energy Systems
33
846 
54
DOI : 10.1016/j.ijepes.2010.11.011
Wong K.P.
,
Fung C.C.
1993
Simulated annealing based economic dispatch algorithm
IEE Proceedings C Generation, Transmission and Distribution
140
509 
15
DOI : 10.1049/ipc.1993.0074
Kasangaki V.B.A.
,
Sendaula H.M.
,
Biswas S.K.
1997
Stochastic Hopfield artificial neural network for unit commitment and economic power dispatch
Electric Power Systems Research
42
215 
23
DOI : 10.1016/S03787796(96)012102
Wang Y.
,
Zhou J.
,
Lu Y.
,
Qin H.
,
Wang Y.
2011
Chaotic selfadaptive particle swarm optimization algorithm for dynamic economic dispatch problem with valvepoint effects
Expert Systems with Applications
38
14231 
7
G. ZweLee
2003
Particle swarm optimization to solving the economic dispatch considering the generator constraints
IEEE Transactions on Power Systems
18
1187 
95
DOI : 10.1109/TPWRS.2003.814889
Park J.B.
,
Lee K.S.
,
Shin J.R.
,
Lee K.Y.
2005
A particle swarm optimization for economic dispatch with nonsmooth cost functions
IEEE Transactions on Power Systems
20
34 
42
DOI : 10.1109/TPWRS.2004.831275
Selvakumar A.I.
,
Thanushkodi K.
2007
A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems
IEEE Transactions on Power Systems
22
42 
51
DOI : 10.1109/TPWRS.2006.889132
Chaturvedi K.T.
,
Pandit M.
,
Srivastava L.
2008
SelfOrganizing Hierarchical Particle Swarm Optimization for Nonconvex Economic Dispatch
IEEE Transactions on Power Systems
23
1079 
87
DOI : 10.1109/TPWRS.2008.926455
Cai J.
,
Ma X.
,
Li L.
,
Haipeng P.
2007
Chaotic particle swarm optimization for economic dispatch considering the generator constraints
Energy Conversion and Management
48
645 
53
DOI : 10.1016/j.enconman.2006.05.020
Vo Ngoc D.
,
Schegner P.
,
Ongsakul W.
2011
A newly improved particle swarm optimization for economic dispatch with valve point loading effects
2011 IEEE Power and Energy Society General Meeting
1 
8
Park J.B.
,
Jeong Y.W.
,
Shin J.R.
,
Lee K.Y.
2010
An Improved Particle Swarm Optimization for Nonconvex Economic Dispatch Problems
IEEE Transactions on Power Systems
25
156 
66
DOI : 10.1109/TPWRS.2009.2030293
Lu H.
,
Sriyanyong P.
,
Song Y.H.
,
Dillon T.
2010
Experimental study of a new hybrid PSO with mutation for economic dispatch with nonsmooth cost function
International Journal of Electrical Power & Energy Systems
32
921 
35
DOI : 10.1016/j.ijepes.2010.03.001
Safari A.
,
Shayeghi H.
2011
Iteration particle swarm optimization procedure for economic load dispatch with generator constraints
Expert Systems with Applications
38
6043 
8
DOI : 10.1016/j.eswa.2010.11.015
Ratnaweera A.
,
Halgamuge S.K.
,
Watson H.C.
2004
Selforganizing hierarchical particle swarm optimizer with timevarying acceleration coefficients
IEEE Transactions on Evolutionary Computation
8
240 
55
DOI : 10.1109/TEVC.2004.826071
Chaturvedi K.T.
,
Pandit M.
,
Srivastava L.
2009
Particle swarm optimization with time varying acceleration coefficients for nonconvex economic power dispatch
International Journal of Electrical Power & Energy Systems
31
249 
57
DOI : 10.1016/j.ijepes.2009.01.010
Saadat H.
2002
Power System Analysis
2 ed.
McGraw Hill
Kennedy J.
,
Eberhart R.
1995
Particle swarm optimization
IEEE International Conference on Neural Networks
4
1942 
8
Niknam T.
,
Mojarrad H.D.
,
Nayeripour M.
2010
A new fuzzy adaptive particle swarm optimization for nonsmooth economic dispatch
Energy
35
1764 
78
DOI : 10.1016/j.energy.2009.12.029
Shi Y.
,
Eberhart R.C.
1999
Empirical study of particle swarm optimization
1999 CEC 99 Proceedings of the 1999 Congress on Evolutionary Computation
3
1950 
Jeyakumar D.N.
,
Jayabarathi T.
,
Raghunathan T.
2006
Particle swarm optimization for various types of economic dispatch problems
International Journal of Electrical Power & Energy Systems
28
36 
42
DOI : 10.1016/j.ijepes.2005.09.004
Eberhart R.C.
,
Shi Y.
2000
Comparing inertia weights and constriction factors in particle swarm optimization
Proceedings of the Congress on Evolutionary Computation
1
84 
8
Yuan X.
,
Su A.
,
Yuan Y.
,
Nie H.
,
Wang. L.
2009
An improved PSO for dynamic load dispatch of generators with valvepoint effects
Energy
34
67 
74
DOI : 10.1016/j.energy.2008.09.010
Mohammadiivatloo B.
,
Rabiee A.
,
Ehsan M.
2012
Timevarying acceleration coefficients IPSO for solving dynamic economic dispatch with nonsmooth cost function
Energy Conversion and Management
56
175 
83
DOI : 10.1016/j.enconman.2011.12.004
Yang X.S.
,
Sadat Hosseini S.S.
,
Gandomi A.H.
2012
Firefly Algorithm for solving nonconvex economic dispatch problems with valve loading effect
Applied Soft Computing
12
1180 
6
DOI : 10.1016/j.asoc.2011.09.017
Sydulu M.
A very fast and effective noniterative “lambda logic based” algorithm for economic dispatch of thermal units
TENCON 99 Proceedings of the IEEE Region 10 Conference1999
2
1434 
7
Abedinia O.
,
Amjady N.
,
Kiani K.
2012
Optimal Complex Economic Load Dispatch Solution Using Particle Swarm Optimization with Time Varying Acceleration Coefficient
International Review of Electrical Engineering
7
8 
Panigrahi B.K.
,
Pandi V.R.
2008
Bacterial foraging optimisation: NelderMead hybrid algorithm for economic load dispatch
IET Generation, Transmission & Distribution
2
556 
65
DOI : 10.1049/ietgtd:20070422
Subbaraj P.
,
Rengaraj R.
,
Salivahanan S.
,
Senthilkumar T.R.
2010
Parallel particle swarm optimization with modified stochastic acceleration factors for solving large scale economic dispatch problem
International Journal of Electrical Power & Energy Systems
32
1014 
1023
DOI : 10.1016/j.ijepes.2010.02.003