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 . However, the cost function of a practical generator becomes highly nonlinear and discontinuous due to prohibited operating zones (POZ) and ramp-rate 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 . 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 .
Recently, modern heuristic methods such as genetic algorithm [5, 6], evolutionary programming , differential evolution , ant colony optimization , tabu search , simulated annealing , neural network , 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 , 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  was applied to solve ELD problem, but the robustness of this algorithm is not discussed in . In , 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 (c1) and social component (c2) during the iterative process [24, 25]. By proper tuning of these coefficients (c1 and c2), the particles are guided towards optimum solution.
In this paper, a new PSO variant named MPSO-TVAC 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 c1, c2 and c3) can provide a balance exploration and exploitation for the particle to get a better optimum solution. To validate the proposed MPSO-TVAC method, it is tested on ELD problem with generator limits, POZ, ramp-rate limits and transmission loss constraints. A new constraint handling has been introduced to handle POZ and ramp-rate limits constraints instead of used penalty factor as in [16, 17] and discussed in Section 4 (step 5). The results obtained by MPSO-TVAC are compared with some PSO variants in term of convergence characteristic, solution quality and robustness. For the ELD problem, the results show that MPSO-TVAC 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 MPSO-TVAC approach. Section 4 presents the detail procedure of implementing the MPSO-TVAC 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 (FC) 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 Fi(Pi) is the fuel cost of the ith generator ($/h) which is defined by following equation:
where Ng is the number of generator, Pi is the active power of generator i (MW) and ai, bi and ci are the fuel cost coefficients of ith 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 PD is the total power demand and PL 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 Bij is the i,j element of the loss coefficient matrix, Bi0 is the ith element of the loss coefficient vector and B00 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 ith 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), DRi and URi 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 . Thus, discontinuous and non smooth fuel cost characteristic is produced corresponding to the POZ as illustrated in Fig. 1.
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 zth POZ of ith generator in (MW) and Nz is the number of POZ of ith generator.
3. Proposed MPSO-TVAC 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 . 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 .
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 r1 and r2 are random numbers between 0 and 1, c1 is the cognitive acceleration coefficient which pushes the particles towards pbest, c2 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 . The linearly decreasing of w is computed as follows :
where wmin and wmax are the initial and final inertia weights respectively and jmax 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 Tsung-Ying Lee and Chung-Lung Chen in 2007  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 c3 is the stochastic acceleration coefficient that pulls the particles towards Ibest.
3.3 Proposed MPSO-TVAC algorithm
In order to improve the solution quality and robustness of PSO, a novel modified PSO with time varying acceleration coefficients (MPSO-TVAC) 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):
Fig. 2.Determination of rbest value of the i-th particle.
where, c3 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 (c1) and social components (c2) 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 c2 than cognitive acceleration coefficient c1 is selected, the algorithm will converge to a local optimum solution (premature convergence). However, a relatively high value of cognitive acceleration coefficient c1 compared to social acceleration coefficient c2 results in wandering of particles around the search space .
To enhance exploration and exploitation of particle towards optimum solution, both coefficients should be varies according to the iteration number . 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 c1i and c1f are the initial and final values of cognitive coefficient respectively and c2i and c2f 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 (c3) is using the following Eq. :
The behaviour acceleration coefficients (c1, c2 and c3) of the MPSO-TVAC algorithm are shown in Fig. 3. It assumed that, the c1 value varies from 1 to 0.2 and c2 varies from 0.2 to 1 during 100 iterations. At the initial iteration, the c3 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 c2 value is linearly increased to encourage particles towards global gbest value. Therefore, the exploration and exploitation capability of MPSO-TVAC is improved, thus providing good solution quality and consistent results near to the global optimum.
Fig. 3.Behaviour of acceleration coefficients (c1,c2 and c3) during iteration.
4. Procedure of MPSO-TVAC Algorithm for ELD Problem
In this section, the proposed MPSO-TVAC method for solving non-smooth 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 MPSO-TVAC algorithm is shown in Fig. 4. The detailed implementation of the proposed algorithm are described as follows:
Fig. 4.Flowchart of proposed MPSO-TVAC 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 of Npop, the particles are randomly generated between the generator limits in (5) and satisfy all constraints in (6) and (7). The ith particle for Ng generator 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 function f(Pi) is defined as
where, k is the penalty factor for the total active power which does not satisfy the power balance constraints.Step 3: Initialization of pbest, gbest and rbestThe fitness value of each particle is calculated using (20). Initial particles in Step 1 are set as initial pbest values. The best fitness function among the pbest value is defined as gbest. Then, rbest of 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 of dth dimension is computed by
where, R is the chosen number of interval in d-th dimension. The maximum velocity is set to 20% of the dynamic range of each variable (Pmax-Pmin). 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 of ith particle in dth 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_new and Pdmax_new are the effective minimum and maximum of d-generator.If the generator output i is violated the POZ constraints in (9), it will be pushed to the nearest boundary of z-th POZ as follows :
where, Pi,zmean is the average value of the z-th 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 previous pbest, the current value is stored as pbest. Otherwise, it is remained as the previous pbest. The gbest value is updated as similar to the pbest. Then, the rbest value 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 MPSO-TVAC algorithm is stopped and the best solution is selected. Otherwise, the algorithm returns to 4.
5. Simulation Results and Performance Analysis
The proposed MPSO-TVAC method is tested on 6, 15 and 38-generator 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.
Table 1.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 . All the generators have ramp rate limits and POZ. The best result reported to date is $15450 .
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 . This system has many local minima and the optimum cost reported to date is $32704.50 . 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 .
5.2 Parameter tuning for MPSO-TVAC
The performance of PSO algorithm is influenced by the setting of cognitive and social coefficients (c1 and c2). The best combination of c1 and c2 depends on the problem. To determine the best combination of c1 and c2, the different range of c1 and c2 is tested. The value of c3 varies according to c1 and c2 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 c1 and c2 produce results near to the global optimum solution. However, the combination of c1i=c2f=1.0 and c1f=c2i=0.2 is found to be the optimum results than others. The same combination of c1 and c2 also provides the best results for the 38 generators system.
Table 2.Influence of acceleration coefficient on MPSO-TVAC performance
Tables 3 and 4 show the performances of MPSO-TVAC for different population size according to the number of dimension and complexicty of the problem.It clearly shows that the proposed MPSO-TVAC 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.
Table 3.Statistical Results of various PSO algorithms (6-generator system)
Table 4.Statistical Results of various PSO algorithms (15-generator 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 c2 exploits the best solution in the latter iteration. It will lead the algorithm to find near to the global optimum effectively.
Fig. 5.Convergence characteristic of three PSO strategies for 6-generator system.
Fig. 6.Convergence characteristic of three PSO strategies for 15-generator 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 MPSO-TVAC 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 MPSO-TVAC is better than other PSO while satisfying all the operational constraints.
Table 5.Best simulation result for 6-generator system
Table 6.Best simulation result for 15-generator system
Table 7.Best simulation result for 38-generator 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 MPSO-TVAC method achieves consistent result at the lowest cost in every run compared to other PSO methods. Moreover, the smallest SD obtained by the MPSO-TVAC in Tables 3 and 4 also shows that the MPSO-TVAC is more robust than other PSO methods.
Fig. 7.Best result of variant PSO algorithms for 50 runs (6-generator system).
Fig. 8.Best result of variant PSO algorithms for 50 runs (15-generator system).
5.6 Comparison of best solution
The best result archived by the MPSO-TVAC for 6-generators system is compared with the previous publiccations of GA , PSO , PSO_LRS , NPSO , NPSO_LRS  and PSO-TVAC  in Table 8. The results show that the MPSO-TVAC provides the minimum cost with less computational time compared to other methods . For 15-generators system, the results obtained by the MPSO-TVAC are compared with GA , PSO , BF  , SOH_PSO ,GA-API , PSO-MSAF , PSO-TVAC  and FA  in Table 9. It shows that MPSO-TVAC can produce a better cost and less computational time compared to other methods. Similarly, the results obtained by MPSO-TVAC for 38-generators system are compared with PSO, IPSO and PSO_TVAC  in Table 7. From these results, it clearly shows that the proposed MPSO-TVAC has been found to be successful in solving ELD with generators constraints.
Table 8.a The solution provided in  is violated the equality constraints (ΣPi≠PD+PL). ‘-’: Not reported in the refereed literature.
Table 9.a The solution provided in  is violated the equality constraints (ΣPi≠PD+PL). ‘-’: not reported in the refereed literature
This paper has proposed a MPSO-TVAC 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 c1, c2 and c3 enhanced exploration and exploitation of the proposed algorithm. The MPSO-TVAC performances have been compared with some PSO variants for three benchmark power systems after 50 different runs. The simulation results have shown that the MPSO-TVAC 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.