An intelligent framework to solve the non-convex economic dispatch problem with practical limitations

Abstract

In this paper, a realistic formulation for the non-convex economic dispatch problem is proposed. It considers different practical constraints including ramp rate limits, valve loading effect, prohibited operating zones, spinning reserve and multi-fuel options. In this regard, a new optimization method based on the bat algorithm is proposed to solve the problem. Meanwhile, because the proposed problem is complex, nonlinear, and constrained, a new self-adaptive modification method, called modified BA or shortly MBA, is proposed. The satisfying performance of the proposed optimization method is examined using IEEE 15-unit, 40-unit and 100-unit test systems. Comparative studies demonstrate the consistent superiority of the proposed method over other alternative optimization techniques widely used in the literature for solving the economic dispatch problem.

Keywords

Economic dispatch modified bat algorithm self-adaptive modification mechanism nonlinear constrained optimization

1 Introduction

In recent years, the optimization algorithms have become one of the main aspects of most of the engineering problems especially in the field of power systems [1]. In fact, finding most optimal operating condition in the new competitive electricity markets has become a tedious and hard task. This situation necessitates more efforts devoted to developing powerful and practical techniques to overcome these problems. The key motivation here is to find the most cost-effective solutions for system operation. In this regard, economic dispatch (ED) is one of the most popular and timely issues in the area of power engineering. ED generally aims to minimize the total network costs while satisfying different operating and security constraints [2, 3]. ED in its simple form is an optimization problem with only constraints of output power of generators and the power and demand balance (i.e. Equation 2). Nevertheless, the practical form of ED can be much more nonlinear and complex than its simple type. This is due to inclusion of different nonlinear characteristics as well as practical constraints such as the valve loading effects, ramp rate limits, prohibited operating zones (POZs), spinning reserve and multi-fuel options. The valve loading effects would change the quadratic cost function of the ED to a more complete nonlinear non-smooth objective function incorporating a sinusoidal term [4]. In this regard, several optimization methods such as dynamic programming (DP) [5], evolutionary programming (EP) [6], improved fast EP (IFEP) [7], tabu search (TS) [8] and hybrid EP combined with sequential quadratic programming (SQP) [9] are proposed which have considered valve loading effects. On the other hand, the POZ constraint will change the continuous shape of the problem search space to a discontinuous one [10]. The POZ is generated as the result of mechanical limitations of the power plant components. Here some methods such as decomposition technique (DT) [11], DP method [12], GA [13] and PSO [14] have solved the ED with POZ effect. The reserve constrained ED is employed to generate an extra power more than the demand for extraordinary situations in the network (like failures). Some of the works which have considered this constraint in the ED are Bender’s decomposition (BD) [15], HNN [16] and PSO with the sequential quadratic programming (SQP) [17]. In addition, in some networks, some units may use multiple fuels (gas or oil), which leads to the multi-fuel constraints. It is evident that the cost function coefficient for a specific generator varies from one fuel type to the other. In the area of heuristic techniques, evolutionary programming (EP) [18], hierarchical method (HM) [19] and HNN [20] can be named. Nevertheless, while each of these works has provided useful and timely results in the ED problem, but neglecting some of the practical limitations and constraints has reduced their research. Meanwhile, in the recent years, there are some works which have considered a quantity or all of these constraints in the ED problem which amongst them can be named new PSO (NPSO) with local random search (NPSO-LRS) [21], GA with multiplier updating method (IGAMUM) [22] and real-coded genetic algorithm (RCGA) [23]. Here, the complexity and high nonlinearity of the proposed ED problem has resulted that most of the above mentioned methods could not solve the problem optimally. In other words, the lack of a sufficient powerful optimization tool to investigate the practical ED problem is evident from the results of these works.

According to the above discussion, the main purpose of this paper is to first introduce a complete formulation for the ED problem. After that, since the problem at hand is a complex nonlinear optimization problem, a powerful optimization tool is proposed to solve the problem suitably while escaping from the local optima as well as the premature convergence. In this regard, we introduce a new modified optimization algorithm based on the bat algorithm (BA) [24]. The original BA makes use of the behavior of the bat animals to find their food or prey. While BA has many interesting characteristics such as a simple concept, a small number of adjusting parameters, and implementation ease, it may be trapped in some local optima. Also, a new self-adaptive modification method including three sub-modification techniques is proposed which can increase both the diversity of the bat population and the convergence of the algorithm during the optimization effectively. It is called modified BA or shortly MBA. The feasibility and satisfying performance of the proposed method is examined using five IEEE standard test systems.

2 Practical ED formulation

In this section, the practical formulation for the ED problem including the objective function and the constraints are described. It is worth noting that some papers consider the polar form of AC load flow with the choice of voltage magnitude and voltage phase angle as state variables. It is worth noting that there are some papers that use other forms of AC power flow such as the rectangular form. This paper has focused on ED and thus the load flow equations are ignored.

2.1 Classical ED

The classical ED objective function is as below [2]:

$\begin{matrix} Min fun (X) = Cost (X) = \sum_{i = 1}^{n} F_{i} (P_{i}) \\ = \sum_{i = 1}^{n} (a_{i} + b_{i} P_{i} + c_{i} P_{i}^{2}) \\ X = [P_{1}, P_{2}, P_{3}, \dots P_{n}] \end{matrix}$ (1)

The above objective function is subject to the power and demand balance as well as the maxim power production capacity as follows [2]:

$\begin{matrix} \sum_{i = 1}^{n} P_{i} = D + P_{Loss} \\ P_{i}^{min} \leq P_{i} \leq P_{i}^{max} \end{matrix}$ (2)

In the ED problem, the network power losses may be neglected or approximated using B loss matrix as follows [21]: $P_{Loss} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} P_{i} B_{ij} P_{i} + \sum_{i = 1}^{n} B_{0 i} P_{i} + B_{00}$ (3)

This equation shows the resistive power losses are approximated as a second order function of generation. In the economic dispatch approaches, the penalty factors or the incremental losses are always the key points in solution algorithm and the B-coefficient method is possibly the most popular technique.

2.2 Practical ED problem

As mentioned before, the proposed ED formulation includes practical constraints. Considering the valve loading effects will change the classical ED objective function as follows [5 –9]:

$\begin{matrix} Cost (X) & = & \sum_{t =}^{N_{t}} \sum_{i = 1}^{n} F_{i} (P_{i}^{t}) \\ = & \sum_{t =}^{N_{t}} \sum_{i = 1}^{n} (a_{i} + b_{i} P_{i}^{t} + c_{i} P_{i}^{t 2} \\ + | e_{i} sin (f_{i} (P_{i}^{min} - P_{i}^{t}))) \end{matrix}$ (4)

In addition to the valve loading effect, there is another constraint which can affect the nature of the ED classical objective function. If there are multiple power units, it may happen that they use different fuels. This can be formulated as below [18 –20]: $\begin{matrix} F_{i} (P_{i}^{t}) \\ = {\begin{matrix} a_{i 1} + b_{i 1} P_{i}^{t} + c_{i 1} P_{i}^{t 2}; Fuel 1 : P_{i}^{min} \leq P_{i}^{t} \leq P_{i 1}^{max} \\ a_{i 2} + b_{i 2} P_{i}^{t} + c_{i 2} P_{i}^{t 2}; Fuel 2 : P_{i 2}^{min} \leq P_{i}^{t} \leq P_{i 2}^{max} \\ \dots \\ a_{ij} + b_{ij} P_{i}^{t} + c_{ij} P_{i}^{t 2}; Fuel j : P_{ij}^{min} \leq P_{i}^{t} \leq P_{ij}^{max} \end{matrix} \end{matrix}$ (5)

Considering the valve loading effect, the objective function for each fuel type will become as follows:

$\begin{matrix} F_{i} (P_{i}^{t}) = a_{ij} + b_{ij} P_{i}^{t} + c_{ij} P_{i}^{t 2} \\ + | e_{ij} sin (f_{ij} (P_{ij}^{min} - P_{i}^{t})) | \\ Fuel j : P_{ij}^{min} \leq P_{i}^{t} \leq P_{i}^{max}; j = 1, 2, \dots, n_{f} \end{matrix}$ (6)

The ED constraints that are considered as limitations are described at below:

–Ramp Rate limits [22, 23]: ${\begin{matrix} P_{i}^{t} - P_{0 i}^{t} \leq {UR}_{i}^{t}; If generation increases \\ P_{0 i}^{t} - P_{i}^{t} \leq {DR}_{i}^{t}; If generation decreases \end{matrix}$ (7)

Similarly, the generating capacity for each power unit is amended as follows:

$\begin{array}{l} \max (P_{i}^{\min}, P_{0 i}^{t} - D R_{i}^{t}) \leq P_{i}^{t} \leq \min (P_{i}^{\max}, P_{0 i}^{t} + U R_{i}^{t}) \\ i = 1, 2, .., n; t = 1, 2, ..., N_{t} \end{array}$ (8)

Prohibited Operating Zone (POZs)

The POZ for the unit i is described as below [15 –17]:

$\begin{matrix} {\begin{matrix} P_{i}^{min} \leq P_{i}^{t} \leq P_{i, 1}^{LB} \\ P_{i, j - 1}^{UB} \leq P_{i}^{t} \leq P_{i, j}^{LB}; j = 2, 3, \dots, {NP}_{i} \\ P_{i, j}^{UB} \leq P_{i}^{t} \leq P_{i}^{max} \end{matrix}; \\ i = 1, 2, \dots, N_{GP} \end{matrix}$ (9)

–Spinning Reserve considering POZs

The spinning reserve constraint considering the POZ is stated as below [11 –14]: $S_{r, i}^{t} = {\begin{matrix} min {(P_{i}^{max} - P_{i}^{t}), S_{i}^{max}} & \forall i \in (Ω - Θ) \\ 0 & \forall i \in Θ \end{matrix}$ (10) $S_{r}^{t} = \sum_{i = 1}^{n} S_{r, i}^{t}$ (11) $S_{r}^{t} \geq S_{R}^{t}$ (12)

3 Optimization tool based on Modified Bat Algorithm (-MBA)

In this section, at first, the original bat algorithm (BA) is described.

3.1 Original BA

BA is a metaheuristic evolutionary optimization algorithm inspired from the searching behavior of the bat animals for food or prey [24]. Each bat produces and sends a signal to the surrounding environment and then waits to hear its echo. This process is called echolocation process. The original BA is constructed based on four main ideas: 1) The echolocation process will help the bats to distinguish between the food and prey; 2) Each bat at position X_i will fly with the velocity of V_i and at the same time produces a signal with the frequency and loudness of F_i and A_i respectively; 3) The frequency of F_i and its rate r_i are adjusted automatically and 4) The value of A_i is reduced from a large amount to a low amount.

Similar to the other population based optimization algorithms, BA also starts its search by generating a random population. Then, the position of each bat in the population is improved using the following equations:

$\begin{matrix} V_{i}^{new} & = & V_{i}^{old} + F_{i} (X_{G} - X_{i}); i = 1, \dots, N_{bat} \\ X_{i}^{new} & = & X_{i}^{old} + V_{i}^{new}; i = 1, \dots, N_{bat} \\ F_{i} & = & F_{i}^{min} + φ_{1} (F_{i}^{max} - F_{i}^{min}); i = 1, \dots, N_{bat} \end{matrix}$ (13)

In BA, the movement of the bats is simulated in another way too. Therefore, at first, a random value β is generated. If the random value β is larger than r_i, then a new solution is produced around the bat X_i as follows: $X_{i}^{new} = X_{i}^{old} + ɛ A_{mean}^{old}; i = 1, \dots, N_{bat}$ (14)

If β is less than r_i then a new solution $X_{i}^{new}$ is produced randomly. The new solution $X_{i}^{new}$ is accepted if the below criterion is satisfied: $[β < A_{i}] & [fun (X_{i}) < fun (X_{G})]$ (15)

Finally, the loudness and rate parameters are updated as follows:

$\begin{matrix} A_{i}^{new} & = & α A_{i}^{old} \\ r_{i}^{Iter + 1} & = & r_{i}^{0} [1 - \exp (- γ \times t)] \end{matrix}$ (16)

3.2 New MBA

In this paper we also suggest a new self-adaptive modification to improve the both local and global search ability of the algorithm. The proposed modification method consists of three sub-modification methods to let each bat choose the modification method best suiting its current condition. These modification methods are described here.

-Sub-modification method 1: $θ_{i}^{new} = θ_{i}^{old} + φ_{1} \oplus τ$ (17) $τ = {Le}^{'} vy (ω) \sim {Iter}^{- ω}; (1 < ω \leq 3)$ (18)

-Sub-modification Method 2:

For each bat X_i, three bats X_{b
₁}, X_{b
₂} and X_{b
₃} are chosen randomly such that b₁ ≠ b₂ ≠ b₃ ≠ i. Now the below test bats are produced:

$\begin{matrix} X_{Test, 1} & = & X_{b_{1}} + φ_{1} \times (X_{b_{2}} - X_{b_{3}}) \\ X_{Test, 2} & = & φ_{2} \times X_{G} + φ_{3} \times (X_{G} - X_{i}) \end{matrix}$ (19)

-Sub-modification Method 3:

The third sub-modification method is an adaptive formulation to update the value of α in (16): $α^{new} = (1 / 2 Iter)^{1 / Iter} α^{old}$ (20)

The above formulation will update the value of α during the optimization process such that faster convergence would be achieved. This formula is found by several running of the algorithm experimentally.

4 Simulation results

In this section, five IEEE test systems are employed to examine the performance and accomplishment of the proposed MBA method.

Test system 1: Number of thermal units, 6; time horizon 1 h; with valve-point effects and POZs. The required data is mainly derived from [36].

Test system 2: Number of thermal units, 10; time horizon 1 h; with valve-point effects and multi-fuel option. The required data is mainly derived from [29].

Test system 3: Number of thermal units, 15; time horizon 1 h; with valve-point effects, POZs, Ramp Rate and Transmission losses. The required data is mainly derived from [17].

Test system 4: Number of thermal units, 40; time horizon 1 h; with valve-point effects and POZs. Here two scenarios are defined. Scenario 1: Here POZs constraint is neglected; Scenario 2: Here POZs constraint is considered. The required data is mainly derived from [7].

Test system 5: Number of thermal units, 100; time horizon 24 h; with valve-point effects. The required data is mainly derived from [37].

Tables 1 and 2 show the simulation results on the first test system. The simulation results for the second test system (10-unit system) are shown in Table 3. Here, for comparison purpose, the results of nine other well-known methods are shown in the table reported from [9 , 33]. As it can be seen from this table, the best, average and worst solutions of the proposed method are better than the best, average and worst results of the other methods, respectively. In fact, the worst solution of the proposed method is better than the best solution of the other alternative methods. From the computational effort, while the proposed method has not the least CPU time, but the obtained CPU time is appropriate. It is worth to note that the proposed MBA has reached to the cost function value of 624.0673 at the time much lower that 0.09266 which can overcome most of the other works in Table 3. The low value of standard deviation shows the high stability of the proposed algorithm. Table 4 shows the frequency of attaining the optimal cost function value during 100 iterations. As it can be seen from these results, the proposed algorithm has reached the optimal solution for all trails. These results show the high robustness of the proposed method. Table 5 shows the relevant power production of the 10 generators.

In Tables 6 and 7, the results of the third test are shown. This test system considers a high number of constraints including the valve-loading affect, POZs, ramp rate and transmission losses simultaneously. Here comparison is made with the results reported in [17]. The superiority of the proposed method is evident here too.

As mentioned before, for test system 4, two different scenarios are defined: 1) neglecting POZs and 2) considering POZs. The simulation results for both scenarios are shown in Tables 8–12. The solution space of this test case has many local minima, and the global minimum is hard to obtain [33]. The comparison is made with the reposted results from [7 , 33]. The superiority of the proposed method in both scenarios can be found easily. By comparison of the results in Tables 8 and 11, it is deduced that considering POZs in the ED problem will increase the total system costs, effectively. The high stability and convergence ability of the proposed algorithm can be inferred from results in Table 9.

The last test system is a network including 100 generators. The main idea behind the use of this test system is to see the search ability of the proposed algorithm in the face of very large systems. As it can be seen from Table 13, the MBA could find a solution more optimal than the solution found by other methods reported in [37]. In addition the stability and computational burden of the proposed method is more efficient than the other methods.

It is also important to check the convergence behavior of the proposed MBA method. Figures 1 to 6 show the convergence behavior of the algorithm during 500 iterations. According to these figures, the algorithm could reach to stable condition in the first iterations. Here, as the size of the networks becomes larger, the convergence speed of the proposed method is preserved fast (in the first 100 iterations).

5 Conclusion

In this work, a sufficient formulation is proposed to reach a more practical framework for the ED problem. The proposed formulation considers power generation limits, spinning reserve, multi-fuel option, valve loading effects, ramp rate limits and POZs. A new optimization algorithm called MBA is proposed to solve this nonlinear and non-convex optimization problem. The proposed algorithm employs an adaptive mechanism using three different modification methods to effectively improve its search capability to escape from local optima. The satisfying performance of the proposed method is examined on the 15-unit, 40-unit and 100-unit IEEE test systems. It is demonstrated that the performance of the proposed method in finding the optimal solution is superior to those widely used by others in the relevant literature. Last but not the least, analyzing the MBA convergence behavior also indicates that its computational requirements are trivially nothing.

- Appendix

This section provides the POZ for all case studies. Tables 14 to 16 show the POZs for 6-unit, 15-unit and 40-unit test systems, respectively.

References

Tadayon Fard

, Esapour

, Dehghan

and Zare

, A new evolutionary-based intelligent method to solve the practical economic dispatch problem, Journal of Intelligent & Fuzzy Systems28 (2015), 271–277.

Niknam

, Azizipanah-Abarghooee

, Sedaghati

and Kavousi-Fard

, An Enhanced hybrid particle swarm optimization and simulated annealing for practical economic dispatch, Energy Education Science and Technology Part A: Energy Science and Research30(1) (2012), 397–408.

Kavousi-Fard

, Abbasi

and Baziar

, A novel adaptive modified harmony search algorithm to solve multi-objective environmental/economic dispatch, Journal of Intelligent & Fuzzy Systems26(6) (2014), 2817–2823.

Akbari-Zadeh

M.R.

, Kavousi-Fard

, Hoseinzadeh

, Baziar

and Saleh

, A new intelligent method for optimal coordination of vehicle-to-grid plug-in electric vehicles in power systems, Journal of Intelligent & Fuzzy Systems28 (2015), 1291–1298.

Kavousi-Fard

, Akbari-Zadeh

M.R.

, Kavousi-Fard

and Rostami

M.A.

, Effect of wind turbine on the economic load dispatch problem considering the wind speed Uncertainty, Journal of Intelligent and Fuzzy Systems28 (2014), 693–705.

Yang

H.T.

, Yang

P.C.

and Huang

C.L.

, Evolutionary programming based economic dispatch for units with nonsmooth fuel cost functions, IEEE Trans Power Syst11 (1996), 112–118.

Sinha

, Chakrabarti

and Chattopadhyay

P.K.

, Evolutionary programming techniques for economic load dispatch, IEEE Evol Comput7(1) (2003), 83–94.

Lin

W.M.

, Cheng

F.S.

and Tsay

M.T.

, An improved Tabu search for economic dispatch with multiple minima, IEEE Trans Power Syst17(1) (2002), 108–112.

Kavousi-Fard

, Akbari-Zadeh

M.R.

, Dehghan

and Kavousi-Fard

, A novel sufficient bio-inspired optimization method based on modified krill herd algorithm to solve the economic load dispatch, International Journal of Bio-Inspired Computation6(2) (2014), 416–423.

10.

Hooshmand

R.A.

, Parastegari

and Morshed

M.J.

, Emission, reserve and economic load dispatch problem with non-smooth and non-convex cost functions using the hybrid bacterial foraging-Nelder–Mead algorithm, Applied Energy89(1) (2012), 443–453.

11.

Lee

F.N.

and Breipohl

A.M.

, Reserve constrained economic dispatch with prohibited operating zones, IEEE Trans Power Syst8(1) (1993), 246–254.

12.

Liang

Z.-X.

, Glover

J.D.

and A zoom feature for a dynamic programming solution to economic dispatch including transmission losses, IEEE Trans Power Syst7(2) (1992), 544–550.

13.

Chiang

C.-L.

, Genetic-based algorithm for power economic load dispatch, IET Gen, Transm, Distrib1(2) (2007), 261–269.

14.

Niknam

, A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem, Applied Energy87(1) (2010), 327–339.

15.

Moya

O.E.

, A spinning reserve, load shedding, and economic dispatch solution by Bender’s decomposition, IEEE Trans Power Syst20(1) (2005), 384–388.

16.

Liang

R.-H.

, A neural-based redispatch approach to dynamic generation application, IEEE Trans Power Syst14(4) (1999), 1388–1393.

17.

Victoire

T.A.A.

and Jeyakumar

A.E.

, Reserve constrained dynamic dispatch of units with valve-point effects, IEEE Trans Power Syst20(3) (2005), 1273–1282.

18.

Jayabarathi

and Sadasivam

, Evolutionary programming-based economic dispatch for units with multiple fuel options, Eur Trans Elect Power10(3) (2000), 167–170.

19.

Lin

C.E.

and Viviani

G.L.

, Hierarchical economic dispatch for piecewise quadratic cost functions, IEEE Trans Power App SystPAS-103 (1984), 1170–1175.

20.

D.N.

and Ongsakul

, Economic dispatch with multiple fuel types by enhanced augmented Lagrange Hopfield network, Applied Energy91(1) (2012), 281–289.

21.

Selvakumar

A.I.

and Thanushkodi

, A new particle swarm optimization solution to nonconvex economic dispatch problems, IEEE Trans Power Syst22(1) (2007), 42–51.

22.

C.-T.

and Chiang

C.-L.

, Nonconvex power economic dispatch by improved genetic algorithm with multiplier updating method, Elect Power Comput Syst32(3) (2004), 257–273.

23.

Amjady

and Nasiri-Rad

, Nonconvex economic dispatch with AC constraints by a new real coded genetic algorithm, IEEE Trans Power System24(3) (2009), 1–8.

24.

Komarasamy

and Wahi

, An optimized K-means clustering technique using bat algorithm, European Journal of Scientific Research84(2) (2012), 263–273.

25.

Ahmed

and Salam

, A Maximum Power Point Tracking (MPPT) for PV system using Cuckoo Search with partial shading capability, Applied Energy119 (2014), 118–130.

26.

Niknam

, Azizipanah-Abarghooee

and Roosta

, Reserve constrained dynamic economic dispatch: A new fast self-adaptive modified firefly algorithm, IEEE Syst J6(2) (2012), 635–646.

27.

Chaturvedi

K.T.

, Pandit

and Srivastava

, Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch, IEEE Trans Power Syst23(3) (2008), 1079–1087.

28.

Jiejin

, Xiaqqian

, Lixiang

and Haipeng

, Chaotic particle swarm optimization for economic dispatch considering the generator constraints, Energy Convers Manage48 (2007), 645–653.

29.

Panigrahi

B.K.

and Pandi

V.R.

, Bacterial foraging optimization: Nelder–Mead hybrid algorithm for economic load dispatch, IET Gener Transm Distrib2(4) (2008), 556–565.

30.

Yang

X.S.

, Sadat Hosseini

S.S.

and Gandomi

A.H

, Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect, Applied Soft Computing12 (2012), 1180–1186.

31.

Park

J.-B.

, Lee

K.-S.

, Shin

J.-R.

and Lee

K.Y.

, A particle swarm optimization for economic dispatch with nonsmooth cost functions, IEEE Trans Power Syst20(1) (2005), 34–42.

32.

Pereira-Neto

, Unsihuay

and Saavedra

O.R.

, Efficient evolutionary strategy optimization procedure to solve the nonconvex economic dispatch problem with generator constraints, Proc Inst Elect.

33.

Ling

S.H.

, Leung

F.H.F.

, An Improved genetic algorithm with average-bound crossover and wavelet mutation operations, Soft Comput11(1) (2007), 7–31.

34.

Ling

S.H.

, Iu

H.H.C.

, Chan

K.Y.

, Lam

H.K.

, Yeung

B.C.W.

and Leung

F.H.

, Hybrid particle swarm optimization with wavelet mutation and its industrial applications, IEEE Trans Syst, Man, Cybern38(3) (2008), 743–763.

35.

Wang

S.-K.

, Chiou

J.-P.

and Liu

C.-W.

, Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm, IET Gen, Transm, Distrib1(5) (2007), 793–803.

36.

Selvakumar

A.I.

and Thanushkodi

, Anti-predatory particle swarm optimization: Solution to nonconvex economic dispatch problems, Electr Power Syst Res78 (2008), 2–10.

37.

Bhattacharya

and Chattopadhyay

P.K.

, Solving complex economic load dispatch problems using biogeography-based optimization, Expert Syst Appl37 (2010), 3605–3615.

38.

Alsumait

J.S.

, Sykulski

J.K.

and Al-Othman

A.K.

, A hybrid GA–PS–SQP method to solve power system valve-point economic dispatch problems, Appl Energy87 (2010), 1773–1781.

39.

Al-Sumait

J.S.

, Al-Othmam

A.K.

and Sykulski

J.K.

, Application of pattern search method to power system valve-point economic load dispatch, Electr Power Energy Syst29(10) (2007), 720–730.

40.

Niknam

, Doagou Mojarrad

and Zeinoddini Meymand

, Non-smooth economic dispatch computation by fuzzy and self adaptive particle swarm optimization, Appl Soft Comput11(2) (2011), 2805–2817.

41.

Lin

C.E.

and Viviani

G.L.

Hierarchical economic dispatch for piecewise quadratic cost functions, IEEE Trans Power App SystPAS- 103(6) (1984), 1170–1175.

42.

Baziar

, Kavousi-Fard

and Zare

A novel self adaptive modification approach based on bat algorithm for optimal management of renewable MG, Journal of Intelligent Learning Systems and Applications5 (2013), 11–18.

43.

Kavousi-Fard

and Akbari-Zadeh

M.R.

A hybrid method based on wavelet, ANN and ARIMA model for short-term load forecasting, Journal of Experimental & Theoretical Artificial Intelligence26(2) (2014), 167–182.

44.

Abido

M.A.

Multiobjective evolutionary algorithms for electric power dispatch problem, IEEE Trans Evol Comput10(3) (2006), 315–329.

45.

Kavousi-Fard

, Khosravi

and Nahavadi

A new fuzzy based combined prediction interval for wind power forecasting, IEEE Trans on Power System (31) (2015), 18–26.

46.

Secui

D.C.

The chaotic global best artificial bee colony algorithm for the multi-area economic/emission dispatch Original Research Article, Energy93(2) (2015), 2518–2545.

47.

Niknam

, Kavousi-Fard

and Ostadi

Impact of hydrogen production and thermal energy recovery of PEMFCPPs on optimal management of renewable micro-grids, IEEE Trans on Industrial Informatics11(5) (2015), 1190–1197.

48.

Manoharan

P.S.

, Kannan

P.S.

, Baskar

and Iruthayarajan

M.W.

Penalty parameter-less constraint handling scheme based evolutionary algorithm solutions to economic dispatch, IET Gen Transm Distrib2(4) (2008), 78–490.

49.

Kavousi-Fard

, Niknam

and Fotuhi-Firuzabad

Stochastic reconfiguration and optimal coordination of V2G plug-in electric vehicles considering correlated wind power generation, IEEE Trans on Sustainable Energy6(3) (2015), 822–830.

50.

Mellal

M.A.

and Williams

E.J.

Cuckoo optimization algorithm with penalty function for combined heat and power economic dispatch problem, Energy93(2) (2015), 1711–1718.

51.

Kavousi-Fard

, Niknam

and Fotuhi-Firuzabad

A novel stochastic framework based on cloud theory and Θ-modified bat algorithm to solve the distribution feeder reconfiguration, IEEE Trans on Smart Grid7(2) (2015), 740–750.