An efficient fitness-based differential evolution algorithm and a constraint handling technique for dynamic economic emission dispatch

Energy 186 (2019) 115801

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An efficient fitness-based differential evolution algorithm and a constraint handling technique for dynamic economic emission dispatch Xin Shen a, Dexuan Zou a, *, Na Duan a, Qiang Zhang a, b a b

School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, PR China School of Information and Electrical Engineering, Xuzhou Open University, Xuzhou, Jiangsu, 221116, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 October 2018 Received in revised form 23 May 2019 Accepted 21 July 2019 Available online 24 July 2019

In this paper, an efficient fitness-based differential evolution (EFDE) algorithm and a constraint handling technique for dynamic economic emission dispatch (DEED) are proposed. In EFDE, there are three improvements compared to the standard differential evolution (DE) algorithm. First, an archive containing the current and previous population is established to provide more candidate solutions. Second, two mutation strategies are used to generate mutant individuals, where the population similarity is introduced to choose a suitable one between DE/rand/1 and DE/best/1. The fitness-based mutation operation is efficient to balance the exploration and exploitation ability of EFDE. Third, EFDE adopts a randombased mutation factor, and the crossover rate with the learning ability is developed to produce more excellent solutions. In addition, the infeasible solutions can be effectively avoided by the proposed repair technique. Four cases are selected to judge the performance of the proposed EFDE and constraint handling technique. For the fuel cost and emission minimizations of four DEED cases, a normalized approach (NA) is used to help EFDE to find the best compromise solutions in the evolution process. According to the simulation results, EFDE exhibits a huge advantage in comparison with the other approaches for the single-objective and multi-objective optimization problems. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Dynamic economic emission dispatch Efficient fitness-based differential evolution Constraint handling Archive Mutation Improved control parameters

1. Introduction Dynamic economic dispatch (DED) [1] is a meaningful but complicated real-time optimization problem that is required to allocate the appropriate output power to the generating units over a dispatch period in power systems. Moreover, the main goal of DED is to minimize the fuel cost while meeting the load demand and practical system constraints, which can bring considerable economic and social benefits. In many studies, the mathematical model of DED is a quadratic function, and the input-output characteristics of generating units is smooth and convex in this case [2]. However, in reality, the generating units usually perform non-smooth and non-convex, which is caused by the frequent opening and closing operation of multiple valves in steam turbines. The steam valve state changing is

* Corresponding author. E-mail addresses: [email protected] (X. Shen), [email protected] (D. Zou), [email protected] (N. Duan), [email protected] (Q. Zhang). https://doi.org/10.1016/j.energy.2019.07.131 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

to maintain the power balance, but it also results in the existence of ripples in the fuel cost function, which makes the objection function highly non-linear. Therefore, to obtain more accurate DED model, such valve-point effects (VPE) can also be considered [3]. A DED problem with VPE has multiple local optimal points, which is very challenging to find the global optimal point. In power system, DED contains inequality and equality constraints, such as power generation limits, ramp rate limits, prohibited operating zones (POZs), and power balance constraints [4,5]. Each available unit should conform to power generation limits that create the feasible solution space. DED is more challenging than the static economic dispatch (SED) [6,7] in the time interval changing and ramp rate limits. The ramp rate limits are utilized to reduce the stresses of the equipment when the generating units operate from one interval to another interval. For the ramp rate limits, there are three situations when the generating units are running [8]. The unit is running in a steady state from the time interval t
1 to t. The output power of the unit increases from the time interval t
1 to t. The output power of the unit reduces from the time interval t
1 to t. The interval changing makes the dispatch problem more time-

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X. Shen et al. / Energy 186 (2019) 115801

consuming and needs to take more actual factors into account compared to SED. Owing to the physical limitation of the equipment, such as corporeal operational limitations of steam valve and shaft bearing vibrations, the output power of some generating units may keep away from certain zones, which is called POZs. POZs will make the cost curve discontinuous, which lead to the changing of the input-output characteristics of generating units [8]. Power balance constraint that is an equality constraint is equal to the sum of the total load demand and transmission losses, and it is the most difficult one when handling constraints. There are two types of power balance constraints, and they are power balance constraint without and with transmission losses. Transmission losses are related to the power generation and the topology of the network, which are incorporated into power system for the unit output with long distances [8]. With the development of society, people pay more attention to environmental issues, which are related to the sustainable development of humanity. In 1990, the clean air act amendments have forced to reduce the pollutant emissions by emission dispatch [9]. Moreover, the policies of governments around the world require power plants to be rectified to achieve energy conservation and emission reduction. Environmental issues have received considerable attention by the public. Thus, many researchers take the pollutant emission into account in the DED problems, which becomes the multi-objection optimization problems, namely dynamic economic emission dispatch (DEED) [10]. The emission that is another crucial objective in the power dispatch mainly include sulfur oxides SOx and nitrogen oxides NOx emissions [11]. DEED needs to optimize the fuel cost and emission simultaneously while the two objectives are contradictory. Specifically, if the fuel cost reduces, the emission will increase, and vice versa. The operational constraints of DEED are the same as those of DED. SOx emissions are proportional to the unit's fuel consumption. NOx emissions are generated from two different sources, nitrogen in the air and in the fuel, and the yielding process is determined by several factors including boiler temperature and air content [12]. In constraint optimization problems, the feasibility of the solutions is more important than the objective function values of the solutions. To obtain the feasible solutions, it is very necessary to deal with those DEED constraints. The penalty function method is common and popular for handling some constraints [13]. However, the penalty function method need choose a suitable penalty factor by carrying out some tedious trials and the difficulty to completely eliminate the constraint violation. Therefore, some scholars proposed several types of repair methods for meeting problem constraints, which are effective to obtain the feasible solutions [14e16] compared to the penalty function method. Many conventional optimization approaches have been applied to optimize the DED and DEED problems, such as quadratic programming method [17,18], linear programming [19], nonlinear programming [20], dynamic programming [21,22], Maclaurin series based lagrangian method [23], gradient algorithm [24], and Lagrange relaxation [25]. However, they can hardly handle the nonconvex and non-differential DED and DEED problems, because the DED and DEED problems consider more factors such as the valvepoint effects and various strong-constraints. Therefore, some researchers turn to employ the stochastic search techniques to solve the DED and DEED problems in recent years, such as particle swarm optimization (PSO) [26], genetic algorithm (GA) [27], differential evolution (DE) [28], differential harmony search algorithm [5], hybrid biogeography-based optimization with brain storm optimization [29], grey wolf optimization [30]. These methods don't rely on the initial solutions and are able to obtain the global optimum or the satisfactory feasible solutions near the global optimum with a high probability.

For DEED, in addition to research constraint handling and optimization approaches, scholars also devote to study how to handle the multi-objective, and there are mainly two types of approaches to treat it. One is that the multi-objective optimization problem is converted to a single-objective optimization problem by a linear combination of both objectives [31]. As a result, the best compromise solution can be obtained. Another is Pareto based approach that produces a set of nondominated solutions for decision-making, and the multi-objective algorithms have niched Pareto genetic algorithm (NPGA) [32], nondominated sorting genetic algorithm II (NSGA-II) [10], strength Pareto evolutionary algorithm (SPEA) [33]. Our multi-objective optimization approach belongs to the former. In the last two decades, most of researchers are committed to improving the stochastic search techniques, constraint handling, and multi-objective handling for the DEED problems. In the mean time, the importance of efficiency in the DEED problems also can't be neglected, because high efficiency to solve the DEED problems is beneficial to timely response to the changing of the load demand, reduce transmission losses, save cost and promote environmental protection. The efficiency also has a very importance influence on DEED considering wind power [34] and DEED considering plug-in electric vehicles charging [35]. Thus, the adopted methods should obtain relatively low cost and emission in a short period of time. According to the above discussions, we find that DEED is a multi-objective and strong-constraint optimization problem with multiple local optimal points. Although many scholars have proposed various optimization approaches, constraint handling and multi-objective handling approaches to solve the DEED problems. However, the solutions obtained by these approaches have relatively low precision and large constraint violation. To address these problems and consider computation time, we propose an improved DE version, a constraint handling technique and a multi-objective optimization approach to further improve the results of the DEED problems. The main contributions of this paper can be summarized as follows. First, an efficient fitness-based differential evolution algorithm (EFDE) is developed as the optimization approach for the DEED problem. Second, a constraint handling technique is presented to handle the DEED constraints. Third, a normalized approach (NA) is proposed to find the compromise solutions. Finally, the selected four DEED cases are used to test the performance of EFDE, the proposed constraint handling and NA. The results reveal that the proposed methods are very efficient. The remainder of this paper is planned as follows: Section 2 reviews some popular approaches for the DEED problems and some energy related problems. In Section 3, the mathematical model of the DEED problems is described in detail, and a constraint handling mechanism and a multi-objective optimization approach are proposed to obtain the feasible solutions or satisfactory solutions. Section 4 introduces the standard DE algorithm and proposed DE variant to find the global optimum for the DEED problems. Experimental results and analysis are presented in Section 5. Section 6 concludes the paper. 2. Literature review Due to many merits of stochastic search techniques, scholars tend to use and improve them based on several strategies to solve the DEED problems. Additionally, scholars devote to research constraint and multi-objective handling approaches. This section reviews methodologies that have been developed in the literatures for the DEED problems. In addition to the presented DEED problems, some energy related problems are also concerned and studied by scholars. The energy related problems that have been presented in the literatures are also reviewed in this section.

X. Shen et al. / Energy 186 (2019) 115801

Basu [36] proposed particle swarm optimization (PSO) based goal attainment method for the DEED problem. Goal attainment method (GAM) is used to convert the multi-objective optimization into a single-objective optimization. PSO is employed as the optimization technique, where the inertia weight is set to linearly adjust as the generation number increases. For the constraint handling, the last unit is calculated by solving a quadratic equation. If the last unit meets the other operational constraints, the total constraint violations are equal to 0. The performance of the proposed method is tested by optimizing the 5-unit test system with non-smooth fuel cost and emission level functions. Elaiw et al. [13] utilized two hybrid approaches including hybrid differential evolution (DE) and sequential quadratic programming (SQP) (DE-SQP) and hybrid particle swarm optimization (PSO) and SQP (PSO-SQP) to solve the dynamic economic emission dispatch problem with valve-point effects. DE or PSO act as the global search. SQP is considered as the local search, which is fine tuning to obtain the best solution. Therefore, the two hybrid approaches clearly divide the work of DE or PSO and SQP. For the constraint handing, the penalty function method is used to cope with the power balance constraint, and the components of penalty function and fitness function construct a new objective function to comprehensively judge the quality of solutions. In handling the multiobjective optimization, the weighting method is adopted to transfer a multi-objective optimization problem into a single-objective one, which doesn't consider the difference of the two objectives dimension. The 5-unit and 10-unit systems with transmission losses are selected to test the effectiveness of the proposed methods. Alsumait et al. [37] proposed an improved algorithm based on Pattern Search method (PS) to solve the DED and DEED problems. The transaction period to the next time horizon (next day) is considered to guarantee the continuity of the power system running. By optimizing the 5-unit system with transmission losses, the obtained results reveal that PS is better than EP [38] and SA [38]. Zhang et al. [39] solved the DEED problems by a multi-objective hybrid differential evolution with simulated annealing technique (MOHDE-SAT). In MOHDE-SAT, the orthogonal initialization method is merged into DE, which is beneficial for the population diversity in the initialization stage. To improve convergence rate, mutation operator is modified, and archive elitist retention strategies are adopted. Additionally, the crossover operator with simulated annealing method is also modified to avoid the premature convergence. Simulated annealing technique and entropy diversity method are commonly used to adaptively adjust the population diversity in the evolution process. For the sake of constraint handling, a constraint-handling technique was presented, where the equality constraint is dealt with coarse adjustment and fine tuning technique. After executing fine tuning, the feasible selection mechanism is operated. The fuzzy-based approach [40] is used to pick out the best compromise solution in the Pareto-optimal solutions. Niu et al. [41] introduced an efficient harmony search with new pitch adjustment (NPAHS) for the DED and DEED problems. To improve the solution quality and convergence speed, the new pitch adjustment is added to traditionally harmony search (HS), which is related to perturbation and the mean value of the harmony memory. Furthermore, a new constraint handling technique is proposed to handle the inequality and equality constraints. To be more specific, if the output power of a generating unit is beyond its bounds for the power generation limits, the output power of the unit is randomly generated within its boundary limits. The method to solve the quadratic equation is used to handle the power balance constraint, and it is different from that to solve the quadratic equation [10,14]. The new output power of the randomly chosen

3

unit is calculated by the two roots of the quadratic equation whose variable is the output power of the randomly chosen unit. When the violation of the power balance constraint is within the tolerance error, the output of the unit is desirable. Finally, by testing ten benchmark functions and three DEED cases, the performance of the proposed optimization algorithm and constraint handling method is verified. Basu et al. [10] presented a nondominated sorting genetic algorithm-II (NSGA-II) to solve the DEED problems. The approach has the advantage over some general approaches in producing the quality of Pareto front, which are fast nondominated sorting procedure, fast crowded distance estimation procedure and simple crowded-comparison operator, respectively. The output power of the last unit is obtained by solving the quadratic equation whose variable is the output power of the last unit. Additionally, scalar optimization method was also used to solve the multi-objective optimization problems, and it is compared with NSGA-II. Pandit et al. [31] developed an improved bacterial foraging algorithm (IBFA) to solve the combined static/dynamic environmental economic dispatch. IBFA adopts a parameter automation strategy and crossover operation to reduce computation cost. At the same time, the parameter automation strategy, that is a dynamic chemotactic step size, is also beneficial to enhance the exploration and exploitation of the algorithm. IBFA has the four main steps, including chemotaxis, crossover, reproduction and elimination and dispersal. In addition, the price penalty factor approach was proposed to handle the multi-objective optimization problems. To optimize the DEED problems, Jiang et al. [42] used a multiobjective differential evolution algorithm with expanded double selection and adaptive random restart (MAMODE). Expanded double selection (EDS) enhances the global search ability of MAMODE, where it expands the mutation-crossover-selection steps of DE to mutation-selection (I)-crossover-selection (II). After mutation operation, selection (I) is executed, and selection (II) is taken place after crossover operation. In the multi-objective optimization problems, the selection operation of individuals is based on the Pareto dominance approach. Adaptive random restart (ARR) uses the elitist information, where the half worse solutions of the population are replaced by a random rule. To find the feasible solutions, a dynamic heuristic constraint handling (DHCH) approach was proposed to cope with various constraints. After operating DHCH, constraint Pareto dominance is calculated to further eliminate the violation. In MAMODE, the external archive is established in which the elitist strategy is used to updated it. Finally, the four cases are selected to testify the performance of MAMODE. Mason et al. [15] applied particle swarm optimization (PSO) variants to the DEED problems, and these PSO variants contain the standard PSO (SPSO), the PSO with avoidance of worst locations (PSO AWL) and a selection of different topologies including the PSO with a gradually increasing directed neighborhood (PSO GIDN). In addition, the neighborhood topologies are introduced to enhance the performance of PSO variants. Roy and Bhui [43] presented the chemical reaction optimization (CRO) algorithm and hybrid differential evolution-based CRO (HCRO) methodology to optimize the DEED problems. HCRO is got by combining the CRO and DE methods, which improves the solution quality and accelerate the convergence of the algorithm. The two test systems are chosen to exhibit the superiority of HCRO. The results reveal that HCRO outperforms CRO in the solution quality and convergence speed. Alham and Elshahed [84] studied the DEED considering wind power uncertainty incorporating energy storage system (ESS) and demand side management (DSM). A high penetration of wind energy is helpful for reducing emissions. However, wind energy usage leads to some issues related to its variability and stochastic

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X. Shen et al. / Energy 186 (2019) 115801

characteristic. Demand side activities are first considered in all energy policy decisions. DSM programs are classified as load management, strategic conservation, new uses, electrification, and customer generation. There are some advantages about DSM programs, such as reducing emissions and strengthening the security of the power system. Due to the introducing of the wind energy, ESS is used to charge at off peak period then discharge during peak period. The model of the proposed DEED is very complicated including two objective functions considering wind energy, various inequality and equality constraints incorporating wind energy. Ran and Miao [85] solved uncertainty quantification and economic dispatch model with windesolar energy. Wind and solar power are considered into the power system. The forecasting deviation of wind speed, solar radiation and load is easy to lead to uncertainties that are power fluctuation. A framework that includes joint probabilistic models of several uncertainties and the risk models of load shed under two different scenarios is established. 3. Dynamic economic emission dispatch (DEED)

Fig. 1. The cost function without and with VPE.

The goal of DEED is to minimize the total fuel cost and pollutant emissions simultaneously during a specified dispatch period while satisfying various constraints to obtain feasible solutions in power systems. However, there are two challenges including multiobjective and constraint handling, which are the key components studied in this paper. The two types of objective functions, corresponding constraints, a constraint handling mechanism and a multi-objective optimization approach of DEED are described in detail.

3.1.1. Cost function The practical power system generally exists multi-valve steam turbines. It changes the input-output characteristic of the units when steam valves open or close, which lead to generate the valvepoint effects (VPE). There are two versions for the cost function. One is a quadratic function without VPE. Another is a quadratic function added a sinusoidal function with VPE. The above two kinds of cost functions are given by Refs. [1,31]: T X D X

T X D     X aj P 2jt þ bj Pjt þ cj Cjt Pjt ¼

t¼1 j¼1

minfc ¼

T X D X t¼1 j¼1

  x ESO Pjt ¼ Sj1 þ Sj2 Pjt þ Sj3 P 2jt jt

(3)

where Sj1 , Sj2 and Sj3 denote the SOx emission coefficients of unit j. x ESO ðPjt Þ denotes the SOx emission of the jth unit based on real jt

3.1. Objective functions

minfc ¼

The mathematical model of SOx emission is a quadratic function, which is proportional to the thermal unit's fuel consumption [11,44]. For the tth time interval, the SOx emission function of the jth unit is expressed by:

(1)

t¼1 j¼1 T P D    P aj Pjt2 þ bj Pjt þ cj Cjt Pjt ¼ t¼1 j¼1

       þej sin fj Pjmin
Pjt 

power output at interval t. Because NOx emission function is highly nonlinear in power output P and the rate of change in its derivative is not always increasing while it is U-shaped [11], NOx emissions are more difficult to evaluate than SOx emissions. Mathematically, NOx emissions of a unit are represented by an equation that is the combination of one straight line and two exponentials terms. Owing to the limited number of data points, Gent and Lamont [45] presented a simplified NOx emissions calculation form, and NOx emissions of a unit are equal to the sum of one straight line and one exponential components, and it is given by:

    x ENO Pjt ¼ Nj1 þ Nj2 Pjt þ Nj3 exp Nj4 Pjt jt

(4)

where Nj1 , Nj2 , Nj3 and Nj4 denote the NOx emission coefficients of

(2)

where Cjt denotes the cost of unit j for the tth time interval. Pjt denotes the output power of unit j for the tth time interval. fc denotes the total fuel cost of all the generating units over T time intervals. Pjmin denotes the minimum power generation of unit j. T denotes the number of dividing a specified period, and D denotes the number of the generating units. aj , bj , cj , ej and fj stand for the cost coefficients of unit j. Furthermore, the cost function without and with VPE are shown in Fig. 1. In this paper, Eq. (2) is taken as the cost function.

x unit j. ENO ðPjt Þ denotes the NOx emission of the jth unit based on jt

real power output at interval t. The total pollutant emissions of a unit are expressed by adding NOx and SOx emissions, and it can be stated by:

      Ejt Pjt ¼ Sj1 þ Nj1 þ Sj2 þ Nj2 Pjt þ Sj3 P 2jt þ Nj3 exp Nj4 Pjt (5) Therefore, in all intervals, the emission function consists of a quadratic and an exponential function for all generating units, and it is shown as follows [46]:

minfe ¼

T X D X t¼1 j¼1

3.1.2. Emission function The another objective function of DEED is the emission function. The pollutant emissions mainly contain sulfur oxides SOx and nitrogen oxides NOx [11].

T X D     X  Ejt Pjt ¼ aj P 2jt þ bj Pjt þ gj þ hj exp dj Pjt t¼1 j¼1

(6) where Ejt denotes the emission of unit j for the tth time interval. fe denotes the total emissions of all the generating units over T time

X. Shen et al. / Energy 186 (2019) 115801

intervals. aj , bj , gj , hj and dj stand for the emission coefficients of unit j.

3.2. Constraints The constraints of DEED are divided into the inequality and equality constraints. To be more specific, the inequality constraints include power generation limits, generating unit ramp rate limits and prohibited operating zones (POZs). The equality constraints refer to power balance constraints. These constraints are introduced below [41].

The power balance constraint with transmission losses for the tth ðt ¼ 1; 2; /; TÞ time interval is given by:

Pjt
PLt
PDt ¼ 0

(4) Prohibited operating zones Due to the physical operation limitations, some generating units can't run in certain zones, namely prohibited operating zones (POZs). Therefore, for a unit with POZs, its advisable operating zones may be divided into several discontinuous subregions (as Fig. 2), and these zones are given by:

8 > l > > < Pjmin  Pjt  Pj1 u l Pjt 2 Pjk
1  Pjt  Pjk > > u > : Pjnpzj  Pjt  Pj max

k ¼ 2; 3; /; npzj

(13)

where P ujk
1 and P ljk represent the upper and lower bounds of the

(1) Power balance constraints

D X

5

(7)

ðk
1Þ th and kth POZ of unit j, respectively. npzj is the number of POZs of unit j.

3.3. A constraint handling mechanism

j¼1

where PDt is the load demand of power system for the tth time interval. PLt is the transmission loss for the tth time interval, which is usually expressed by Kron's loss formula [47] as follows.

PLt ¼

D X D X

Pjt Bji Pit þ

j¼1 i¼1

D X

B0j Pjt þ B00

where Bji , B0j and B00 represent the loss coefficients of ation units. In addition to the above power balance considering transmission losses, the power balance neglecting transmission losses is usually used as a version, and it can be expressed as follows. D X

(8)

j¼1

Pjt
PDt ¼ 0

the generconstraint constraint simplified

(9)

Due to the strong-constraints of DEED, it is very necessary to deal with those constraints to make the generated solutions feasible as the generation number increases. So far, the penalty function method is popular to handle various complex constraints, which helps to penalize the infeasible solutions to choose the feasible ones. However, the penalty function method has some disadvantages, such as the tedious trials to choose a suitable penalty factor and the difficulty to completely eliminate the constraint violation. Therefore, an efficient repair method for the constraint handling and penalty function method combined are used to guarantee all solutions feasible. For the proposed constraint handling mechanism, it also can be applied to other heuristic algorithms including GA, PSO and HS for the DEED problems. In other words, this paper provides an efficient constraint handling framework. The constraint handling mechanism is stated in detail as follows.

j¼1

(2) Power generation limits The power generation of each generating unit has an upper and lower limit, which can be expressed as follows:

Pjmin  Pjt  Pj max

3.3.1. The repair method for the power generation limits and ramp rate limits handling First, in the Gth ðG ¼ 1; 2; /; Gmax Þ generation, it should be pointed out that all of Pjt ðj ¼ 1; 2; /; D; t ¼ 1; 2; /; TÞ values are updated within their ranges that consider the power generation limits and ramp rate limits. Furthermore, the new bounds of every variable are given by:

(10)

where Pjmin and Pj max represent the minimum and maximum power generation of unit j, respectively. (3) Generating unit ramp rate limits For the tth ðt ¼ 2; 3; /; TÞ time interval, the ramp rate of each unit can't exceed the maximum ramp rate, which is to ensure the safe operation of the equipment. In other words, each unit is required to operate in a feasible zone when the time interval of unit running changes, and it can be explained by:

Pjt
Pjt
1  URj

(11)

Pjt
1
Pjt  DRj

(12)

where URj and DRj are ramp up and ramp down limits of unit j, respectively.

Fig. 2. The cost function with POZs.

6

X. Shen et al. / Energy 186 (2019) 115801

Pjtmin ¼

Pjtmax ¼

8
; jmin  : max Pjmin ; Pjt
1
DRj ;

If t ¼ 1 Otherwise

0

(14)

2 Bzz Pzt þ @2

X

1

Bzj Pjt þ B0z
1APzt þ @PDt þ

j2Z ‾ z

8
; jmax 

 : min Pjmax ; Pjt
1 þ URj ;

If t ¼ 1 Otherwise

þ

(15)

0

X

j2Z ‾ z

B0j Pjt


X

1

X X

Pjt Bji Pit

j2Z ‾ z i2Z ‾ z

Pjt þ B00 A ¼ 0

j2Z ‾ z

(19) where Pjtmin and Pjtmax stand for the new lower and upper bounds of unit j for the tth time interval ðj ¼ 1; 2; /; D; t ¼ 1; 2; /; TÞ. Then, any variable Pjt beyond their new bound will be limited to the bound which it exceeds for an algorithm. Namely, all of Pjt values are repaired by:

8 > > < Pjmin ; Pjt ¼ Pjmax ; > > : Pjt ;

P

If Pjt  Pjmin If Pjt  Pjmax Otherwise

(16)

where Eq. (16) makes the power generation limits and ramp rate limits satisfied, and set t ¼ 1. The repair method can ensure that the two inequality constraints are completely met, and it is more simple and efficient than the traditional penalty function method.

3.3.2. The repair method for the power balance constraint handling Compared to the above inequality constraints, the power balance constraint is more difficult to handle in reducing and eliminating its violation. The repair method for the power balance constraint handling is proposed, and its steps are described as follows. Step 1. Set an archive Z ¼ fzj1; 2; /; Dg to store D units in the interval t. Set t ¼ 1. Step 2. Randomly select a component z from the set Z. Then, the Pzt value is calculated by the following formulas, which is based on the D
1 variables in the interval t. Due to the difference of the DEED cases with and without transmission losses, the Pzt value is separately calculated for the two situations. Situation 1: The Pzt value is calculated for the DEED problems with transmission losses, and Eq. (7) is converted into a quadratic equation (as Eq. (19)), which only has an unknown variable Pzt . The new quadratic equation is derived from Refs. [16,48], and the Pzt and PLt values are firstly calculated by:

Pzt ¼ PDt þ PLt


X

where Z z denotes the set that excludes z from the set Z. The quadratic equation can be represented as WP 2zt þ UPzt þ V ¼ 0. The coefficients of the quadratic equation are, respectively, W ¼ Bzz , P P P U¼ 2 Bzj Pjt þ B0z
1 and V ¼ PDt þ Pjt Bji Pit þ

Pjt

(17)

j2Z z

j2Z z

P

B0j Pjt


j2Z z

j2Z z i2Z z

Pjt þ B00 . It is universally known that the two roots

j2Z z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2
4WV and P
U
U 2
4WV . Next, are calculated by Pzt1 ¼
Uþ 2W zt2 ¼ 2W there are three explanations on choosing one suitable root from the two roots.

(i) If both roots of the quadratic equation violate the bounds that are Pztmin and Pztmax , the one with smaller constraint violation of ½Pztmin ; Pztmax  will be updated by Eq. (16) and be selected as the Pzt value. At this time, the power balance constraint hasn't been met in the interval t. Set 4t ¼ 1, and go to Step 3. (ii) Only if one out of the two roots lies in the range ½Pztmin ; Pzt max , the root will be selected as the Pzt value in the interval t, which makes the power balance constraint completely met. Set 4t ¼ 0, and go to Step 5. (iii) If both roots of the quadratic equation are within the range ½Pztmin ; Pzt max , there are two cases to be discussed.

Case 1. For the DEED problems without POZs, one out of the two roots is randomly selected as the Pzt value in the interval t. Set 4t ¼ 0 and go to Step 5. Case 2. For the DEED problems with POZs, if both roots of the quadratic equation don't lie in the POZs, one out of the two roots is also randomly selected as the Pzt value in the interval t. In this situation, all the inequality constraints violations are equal to 0. Only if one out of the two roots doesn't lie in the POZs, it will replace the Pzt value, which meets all the inequality constraints. If both roots of the quadratic equation fall into the POZs, the one with smaller violation related to the POZs will be selected as the Pzt value. Set 4t ¼ 0 and go to Step 5. Situation 2: The Pzt value is calculated for the DEED problems without transmission losses ðPLt ¼ 0Þ, and it is expressed by the following formula.

PLt ¼

X X j2Z z i2Z z

Pjt Bji Pit þ 2Pzt

X j2Z z

Bzj Pjt þ Bzz P 2zt þ

X

B0j Pjt þ B0z Pzt þ B00

(18)

j2Z z

Pzt ¼ PDt


X j2Z z

Pjt

(20)

X. Shen et al. / Energy 186 (2019) 115801

In addition, for the situation 1 and 2, Pzt is reserved if Pzt meets its new bounds, and set 4t ¼ 0. Go to Step 5. If Pzt isn't within the range ½Pztmin ; Pztmax , it is repaired by Eq. (16). Set 4t ¼ 1 and go to Step 3. Step 3. Exclude z from Z, and go to Step 2 if Z is nonempty; Otherwise, go to Step 4. Step 4. By the above first repair, the power balance constraint is very likely to be met. However, if 4t is equal to 1, the equality constraints haven't been satisfied. Therefore, it is necessary to further handle the remaining violations. The second repair is operated, and its process is shown in Algorithm 1. If 4t is equal to 0, go to Step 5.

Algorithm 1: The second repair process for the power balance constraint 1 Initialize a set ZZ (ZZ ¼ 1; 2; /; D). Set V tpl ¼ D P

2

D P j¼1

Pjt
PLt
PDt and V tp ¼

Pjt
PDt ;

j¼1 If V tpl

(or V tp ) < 0

3 Randomly select a component zz from the set ZZ; 4 While Pzzt ¼ Pzztmin 5 Remove zz from the set ZZ, and let the new set be ZZ 0 . Then randomly select a new component zz0 from the set ZZ 0 , and zz) zz0 , ZZ) ZZ 0 ; 6 End while 7 If ZZ 0 is empty 8 Break; 9 End if 10 Calculate the adjustment limit of the component zz, namely DAL ¼ Pzzt
Pzztmin ; 11 Else if V tpl (or V tp ) < 0 12 Randomly select a component zz from the set ZZ; 13 While Pzzt ¼ Pzztmax 14 Remove zz from the set ZZ, and let the new set be ZZ 0 . Then randomly select a new component zz0 from the set ZZ 0 , and zz) zz0 , ZZ) ZZ 0 ; 15 End while 16 If ZZ 0 is empty 17 Break; 18 End if 19 Calculate the adjustment limit of the component zz, namely DAL ¼ Pzztmax
Pzzt ; 20 End if         21 If V tpl  (orV tp ) > DAL 22

Pzzt ¼ Pzzt
signðV tpl Þ  DAL (or Pzzt ¼ Pzzt
signðV tp Þ  DAL);

234t ¼ 1; 24 Else 25

Pzzt ¼ Pzzt
V tpl (or Pzzt ¼ Pzzt
V tp );

264t ¼ 0; 27 End if

Step 5. t¼tþ1 if t  T, go to Step 2. Otherwise, terminate the above repair process. Finally, in order to make the above repair process understandable, the flowchart of the repair method for the power balance constraint handling is shown in Fig. 3. In this section, the first repair is executed to handle the power balance constraint without and with transmission losses. Its advantage is that the power balance constraint violation can completely be repaired in some cases. Specially, for situation 1, the quadratic equation is solved to obtain the Pzt value, which makes the power balance constraint violation equal to 0 in (ii) and (iii). In

7

(i), the repaired Pzt value reduces the power balance constraint violation. For situation 2, the power balance constraint violation is reduced and even equal to 0. After the first repair, if the equality constraint hasn't been satisfied, the second repair is operated. Its purpose is to further repair the equality constraint violations. The second repair is an important supplement to the first repair for the equality constraint handling.

3.3.3. The penalty function method for the remaining constraint violations By the repair process for the power balance constraint, the equality constraint has a great chance to be met. However, its violation can still exist in a few cases, which set 4t ¼ 1. If the equality constraint violation is strictly equal to 0 after implementing the above repair process, the flag variable 4t is set to 0. When 4t is equal to 1, the penalty function method is used to handle the remaining equality constraint violation, which has a potential to obtain the desirable feasible solutions. If the remaining equality constraint violation isn't handled, the obtained solutions are still infeasible. If the penalty function method is directly adopted to handle the remaining equality constraint violation that is likely to exist, the experimental environment can effect the objective function value when the equality constraint has been met. Therefore, the rule to select penalty function method for the equality constraint is that the equality constraint hasn't been satisfied. In addition, the penalty function method is also used to handle the POZs, which gradually eliminates the violation related to the POZs as the evolution progresses. The constraint violation related to the power balance constraint is calculated by Eq. (21) or (22).

Vpl ¼

T X t¼1

Vp ¼

T X t¼1

D   X   4t ,  Pjt
PLt
PDt 

(21)

j¼1

 D  X   4t ,  Pjt
PDt 

(22)

j¼1

where Vpl and Vp denote the constraint violations related to the power balance constraint with and without transmission losses, respectively. 4t is set to 0 or 1 for the tth time interval. For the DEED problems considering the POZs, the constraint violation related to the POZs is calculated according to the literature [48]. Furthermore, a commonly penalty function method is utilized to form the new objective function with the penalty function term, and it is calculated by Eq. (23) or (24).

  fp ¼ f þ x, Vpl þ Vpoz

(23)

  fp ¼ f þ x, Vp þ Vpoz

(24)

where f denotes the fitness function including the fuel cost function fc , emission function fe or the following multi-objective function fa (as Eq. (25)). fp denotes the comprehensive objective function with the fitness function and penalty function, which determines the quality of the generated solutions. x is a penalty factor. Vpoz represents the total constraint violation considering the POZs for the entire dispatch period. In addition, Vpoz is equal to 0 for the DEED problems without POZs.

8

X. Shen et al. / Energy 186 (2019) 115801

Fig. 3. The flowchart of the repair method.

3.4. Multi-objective optimization: a normalized approach (NA) DED is a single-objective optimization problem, and it only minimizes the cost function. Different from the DED problem, DEED aims to optimize the two objectives simultaneously, the cost and emission, which is a difficult work for evaluating the solutions. Now, it is very popular that a multi-objective optimization problem converts into a single-objective one by the weighting approach [13]. It is unreasonable to construct a single-objective function by simply adding the two functions, because their dimension has the difference. To be more specific, the dimensions of cost and emission are $/h and ton/h, respectively. In order to treat the two objectives at a same level, a normalized approach (NA) for the multi-objective optimization is proposed, and it is given by:

" f a ¼ u1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ,v ,v " u A u A uX uX 2 C ðxa Þ þ u2 Eðxa Þ t E2 ðxa Þ Cðxa Þ t a¼1

a¼1

(25) where fa is the new objection function considering the fuel cost and emission. xa ¼ ðP1 ; P2 ; /; PD Þ is the ath ða ¼ 1; 2; /; AÞ candidate solution, and Cðxa Þ and Eðxa Þ are its fuel cost and emission values, respectively. A is the number of candidate solutions. u1 and u2 represent the weighting factors. Currently, it is not yet certain whether the fuel cost is more important than the emission. Therefore, u1 and u2 are set to 0.5, which are to treat the two objectives equally in the new objective function. There are some additionally explanations on the NA. After carrying out the NA, each of the new cost and emission function values lies in the range ½0; 1, and the quality of each solution in a set of candidate solutions f1; 2; /; Ag doesn't change. Thus, it is reasonable to construct a single-objective function by adding the two components. In other words, if the a1 th solution is superior to the

X. Shen et al. / Energy 186 (2019) 115801

a2 th solution before carrying out the NA, the normalized a1 th solution is also superior to the normalized a2 th solution.

Different evolution (DE) algorithm [49] is one of the most popular and competitive evolution algorithms at present. Due to its high-performance and versatility, DE has been widely used in many real-word problems, such as electricity energy consumption forecasting [50,51], electricity price prediction [52], ethylene plant optimization [53], power load forecasting [54], generalized assignment problem [55], reliability problem [56], optimal reactive power dispatch [57], and vehicle routing problem [58], etc. The standard DE and its one variant are introduced in this section. Similar to most evolution algorithms, the population of DE consists of NP individuals containing D variables xij ði ¼ 1; 2;/;NP; j ¼ 1; 2; /; DÞ. First, all individuals are randomly generated in the solution space. Then, the three important operators of DE are orderly executed, and they are mutation, crossover, and selection. These operators are introduced in detail as follows.

Mutation: After obtaining the initial population, the mutant individual vG i is produced by the mutation operation for each parent individual xG i ði ¼ 1; 2; /; NP; G ¼ 1; 2; /; Gmax Þ in the current population. G represents the generation number, and its maximum generation number is represented as Gmax . Currently, DE/rand/1 is the most common adopted mutation strategy in DE, and it can be expressed by:

¼

  G þ F, xG i1
xi2

(26)

where the control parameter F denotes mutation factor, and it plays a significant role in maintaining the population diversity. The individual indexes i1 , i2 and i3 are three randomly selected integers in ½1; NP, and i1 si2 si3 si. Furthermore, another five types of popular mutation strategies are expressed as follows [59]: (1) DE/rand/2

    G G G G G vG i ¼ xi3 þ F, xi1
xi2 þ F, xi4
xi5

(27)

(2) DE/best/1

  G G G vG i ¼ xibest þ F, xi1
xi2

(28)

(3) DE/best/2

    G G G G G vG i ¼ xibest þ F, xi1
xi2 þ F, xi4
xi5

(29)

(4) DE/rand-to-best/1

    G G G G G vG i ¼ xi3 þ F, xibest
xi3 þ F, xi1
xi2

(5) DE/current-to-best/1

(31)

selected integers in ½1; NP are different from i and ibest for Eqs. (2731).

4.1. Different evolution algorithm

xG i3

    G G G G þ F, xG vG i ¼ xi þ F, xibest
xi i1
xi2

where xG ibest stands for the global best individual at generation G. All

4. Different evolution algorithm and its one variant

vG i

9

(30)

Crossover: G G G Crossover operation is used to build trial vector uG i ðui ¼ ðui1 ;ui2 ; /; uG iD Þ; i ¼ 1; 2; /; NPÞ, whose components are picked from the G G mutant vector vG i and parent vector xi , and each component of ui ði ¼ 1; 2; /; NPÞ is obtained according to the following equation.

uG ij ¼

8 > < vG ;

If randðÞ  CR or j ¼ jD

G > : xij ;

Otherwise

ij

(32)

where CR denotes the crossover rate, which is another important control parameter. jD is a randomly generated integer between 1 and D. randðÞ is a randomly generated number in the interval ½0; 1. Selection: In order to choose the better one between the trial individual uG i and its parent individual xG i ði ¼ 1; 2; /; NPÞ for the offspring, the selection operation is carried out, which is similar to a greedy strategy. For the minimization problem, the individuals with smaller fitness value will survive into the next generation for both G the trial individual uG i and its parent individual xi ði ¼ 1; 2; /; NPÞ, and this operation is described by:

xGþ1 i

¼

8 <

uG i ; : xG i ;

   G If f uG i < f xi Otherwise

(33)

where f ð,Þ denotes the fitness function, which is used to evaluate the performance of individuals. 4.2. An efficient fitness-based differential evolution algorithm Typically, the effectiveness of the standard DE algorithm is determined by the most appropriate mutation strategy and control parameters F and CR for a given problem. Clearly, it may expend a huge amount of computation costs. Also, different mutation strategies and control parameters values can be more effective to help DE approach the optimal solution than a single mutation strategy with the fixed parameter settings during different stages of evolution. Therefore, we propose the fitness-based mutation operation in which one out of two mutation strategies is selected to generate the mutant vector for each target individual according to the current population similarity, a random-based F and a self-adaptive CR. Additionally, to improve DE's global search capacity, it is necessary to maintain the population diversity in the evolution process. In the mean time, its execution efficiency is also an important consideration. Motivated by these observations, an archive combing the current and previous population is constructed, and it is updated after each optimization interval. Based on the above discussions, an efficient fitness-based differential evolution algorithm (EFDE) is developed, and its core idea is elucidated as follows. (1) An archive combining the current and previous population

10

X. Shen et al. / Energy 186 (2019) 115801

In the mutation operation, the traditional way of selecting the parent individuals from the current population is easy to make the population get trapped into a local optimum. To address this problem, EFDE establishes an archive that provides the information of the current and previous population for the offspring. More candidate individuals are provided to maintain the population diversity and enhance the global optimization capability of EFDE. More specifically, the archive AR that stores the previous population is firstly set up, and the current population is copied to it after each evolutionary interval I0 . For the initialization of the archive AR, its components are generated by replicating the initial population to AR. Second, individuals of the current population and archive AR are merged to construct a new archive AAR. Based on these observations, the archive AAR doesn't update frequently in the evolution process. Thus, it doesn't lead to excessive computation cost. In addition, individuals of the archive AAR are only used in DE/rand/1. (2) The fitness-based mutation operation In DE, the mutation strategy plays an important role in its convergence performance, including the convergence rate and global search capacity. DE/rand/1 is said to be the most successful mutation strategy in the literature [60]. Its base vector is a random individual in the current population, which guides the target inG dividual to approach it. The difference vector xG i1
xi2 is used for perturbation, which controls the search direction and step. Based on the scheme, DE/rand/1 is beneficial to maintain the population diversity, thereby enhancing DE's exploration ability. However, due to the selecting way of the base vector, DE/rand/1 usually performs slow convergence speed. Compared to DE/rand/1, the base vector of greedy strategy DE/best/1 is the best individual in the current population, which guides the population search toward the current best solution. Therefore, DE/best/1 converges fast. Nevertheless, the fast convergence may lose the diversity and global exploration ability of the population, and it is easy to get trapped to some local optima. In addition to DE/rand/1 and DE/best/1, the characteristics of other mutation strategies that have been presented in Section 4.1 can be obtained from the literature [61]. At present, to trade off the population diversity and convergence speed, many scholars are committed to researching multistrategies mutation, and they select several mutation strategies of different characteristics. Some set up a pool that stores selected mutation strategies to pick out one when the mutation operation is implemented [61,62]. Some set a probability rule to select a mutation strategy among all possible mutation strategies, which contain the probability rules based on the number of generations [63] and fitness of individuals [64]. Motivated by these researches, a probability rule adopting fitness-based population similarity is developed, and an appropriate mutation strategy is selected based on the population similarity between DE/rand/1 and DE/best/1 for each individual. The higher population similarity is, the poorer population diversity is. The population similarity is calculated based on the fitness values of the current all individuals, and it is given by:

    G f xG i
f xibest     i¼1 G f xG iworst
f xibest

individual, which means that the population similarity is high. At this time, DE/best/1 is adopted to speed up the convergence of EFDE in high probability. Otherwise, if the current individuals are far from the current best individual, the SG value is large, suggesting that the population similarity is low. DE/rand/1 with the archive AAR is used to enhance the exploration ability of EFDE in high probability. SG can be described as Fig. 4. It can be seen that most of the SG values are within the range of ½0:32; 0:63 in the evolution process, and the distribution of these values is uniform. Therefore, in each generation, DE/rand/1 and DE/ best/1 can be selected one of them in a steady state. Furthermore, the population diversity and convergence speed of the algorithm can be effectively balanced based on the population similarity in the evolution process. In addition, to accelerate the convergence and find the global optimum, DE/best/1 is only used in the late stage of evolution. Here, a threshold Gf is set to 90%  Gmax , which explains that DE/best/1 is forced to use as the mutation strategy when the number of generation reaches Gf . The mutant individual is generated as follows.

8   > < sG þ F, sG
sG ; i i i 3 1 2   vG i ¼> G G : xibest þ F, xG i1
xi2 ;

If randðÞ < SG and G < Gf Otherwise

G G where sG i1 , si2 and si3 are three individuals randomly generated from

the archive AAR in the Gth generation, and i1 si2 si3 si. (3) The improved control parameters In the standard DE, the choice of numerical values for control parameters F and CR highly depends on the assumptions of the problems and scenarios. F controls the scale of the difference vector, which is closely related to the convergence speed [61]. CR is more sensitive to problems with different characteristics. Unlike the fixed parameters, many scholars are committed to making the two control parameters adaptively change to improve the performance of the algorithm [64,65]. Inspired by these scholars, we adopt a different learning mechanism to generate CR, and utilize a randombased mutation factor F to find numerous potential candidate solutions. To be more specific, the F value is generated by a random number that is within the range ½0; 1 in each generation for each target individual, which accelerates the convergence and enhances

XNP SG ¼

(34)

NP

G where f ðxG ibest Þ and f ðxiworst Þ represent the fitness value of the best and

worst individuals at generation G, respectively. SG is the population similarity at generation G. Eq. (34) shows that SG keeps a low value when the current individuals are close to the current best

(35)

Fig. 4. Population similarity.SG

X. Shen et al. / Energy 186 (2019) 115801

the exploration ability of the algorithm. The CR value of each individual is generated based on the quality of the corresponding individual, and the quality of individual i ði ¼ 1; 2; /; NPÞ is determined by the following formula.

    f xG
f xG i i best    IG i ¼  G f xiworst
f xG ibest

11

individuals at generation G. According to Fig. 5, CRG poor obtains more G from CRG best with a high probability. In contrast, CRgood learns less

from CRG best with a high probability. 4.3. The flowchart of the EFDE algorithm

(36)

where I G i denotes the quality of the ith individual at generation G. Obviously, if the current individual i is superior, its fitness value is

To fully illustrate the computational steps of the EFDE algorithm, a simple algorithm procedure is presented in Fig. 6. Where pop0 represents the initial population. popG represents

close to that of the current best individual, meaning that I G i has a relatively small value. Furthermore, a probability based on the quality of individuals is given to select a suitable strategy of generating CR. In the Gth generation, the CR value of each individual is generated by:

CRG i ¼

8 > >  > > 5 G G > > > < l,CRibest þ ð1
lÞ,N; If Ii  ðð1
lÞ,randðÞ þ l > > 5 > G > > > ð1
lÞ,CRibest þ l,N; > :

(37)

Otherwise

where N is the number of problem variables. CRG i denotes the crossover rate of the ith individual at generation G. CRG ibest denotes the crossover rate of the best individual at generation G. l is a local parameter, and it is equal to 0.6 in EFDE. Thus, ðð1
lÞ,randðÞ þ lÞ is a random number in the interval [0.6,1]. Eq. (37) shows how G much CRG i ði ¼ 1; 2; /; NPÞ learns from CRibest , and the remaining 5 component of CRG i obtains from that of the fixed value N . It can be concluded that CR owned by the best individual is first-rate among all crossover rates, which is utilized to guide the other individuals

evolution. For the poor individuals, they have larger I G i values, so their CR have greater chance to learn more from CR owned by the best individual to improve the performance of offspring. In the initial process, CRi ði ¼ 1; 2; /; NPÞ is randomly generated in the range ½CRmin ; CRmax . CRmin and CRmax denote the minimum and maximum values of CR, respectively. The algorithm is effectively improved by adjusting CR. Visually, the diagram of the generating crossover rate is shown in Fig. 5. G Where CRG poor and CRgood represent, respectively, the crossover

rates of the poor and good individuals in the Gth generation. I G poor and I G good represent, respectively, the quality of the poor and good

Fig. 5. The diagram of the generating crossover rates for the poor and good individuals.

Fig. 6. The flowchart of EFDE.

12

X. Shen et al. / Energy 186 (2019) 115801

the population at generation G. ModðG; I0 Þ represents a modulo function with modulus I0 . 5. Experimental results and analysis 5.1. Problem cases and parametric setup In this section, to evaluate the performance of EFDE, four DEED cases and two outstanding DE variants are selected from the literature. To be more specific, these DEED cases are listed as follows. (1) the 5-unit system considering transmission losses [38]; (2) the 5unit system considering transmission losses and POZs [38,66]; (3) the 10-unit system considering transmission losses [10]; (4) the 10unit system neglecting transmission losses and POZs [10,39]. These cases include three DEED problems considering transmission losses, one DEED problem neglecting transmission losses, and one DEED problem considering transmission losses and POZs, which contain a variety of features in the practical DEED problems. The detailed data of four cases are available from Refs. [10,38,66]. The two DE variants from the literature are the improving differential evolution with a successful-parent-selecting framework (SPS-DE) [67] and memory based differential evolution algorithm (MBDE) [68], respectively. SPS-DE modifies the mutation by utilizing a new parentselecting method. This method that introduces the successful solutions is combined with DE, which provides a successful-parentselecting framework. At each generation, the successful solutions are stored to the archive A. When stagnation is happening to DE, the parent individuals are selected from the recently updated solutions instead of the set of original population. Therefore, the framework is helpful to increase the probabilities of generating successful solutions. However, the archive A is updated at each generation, which reduces the efficiency of the algorithm. To save time costs, EFDE updates the archive AAR after each fixed interval. MBDE adopted two “swarm operators” including swarm mutation and swarm crossover, which are based on pBEST and gBEST mechanism of PSO. These swarm operators are helpful to enhance the global searching ability of the algorithm. In addition, the elitism mechanism is used to select the better half from the population generated by merging the initial population with the recently updated population. Although the quality of candidate solutions can be improved in the early stage, it may result in premature convergence in the middle and late evolutionary stages. Due to these shortcomings of SPS-DE and MBDE, the goal of EFDE is to improve them and seek the optimal solution of DEED problems. Additionally, SPSDE and MBDE also have many merits as mentioned above, which make them competitive. They are used to compare with EFDE, which show that EFDE is excellent. The parameter settings of the proposed EFDE and two comparison algorithms are presented as follows. For SPS-DE, the stagnation tolerance Q is set to 32, and the SPS framework is incorporated with the standard DE with the strategy DE/rand/1/bin, mutation factor F ¼ 0:6 and crossover rate CR ¼ 0:3. For MBDE, crossover rate CR ¼ 0:9. The parameters of SPS-DE and MBDE are from the corresponding literatures and adopt the trial-and-error method to set for better performance. For EFDE, the interval I0 ¼ Gmax =100, CRmin and CRmax are, respectively, equal to 0.1 and 1.1. The local parameter l is equal to 0.6, and Gf ¼ 90%  Gmax . The parameter settings of EFDE also adopt the trial-and-error method for better performance, and this paper shows the parameter setting process of Gf and l. For I0 and Gf , their values are related to Gmax . For l, its value should be greater than 0.5. In addition, the penalty factor x is set to 1010 , and the weight coefficients u1 and u2 are equal to 0.5 for all test cases. In the following all experiments, the

population size NP is set to 20. The number of units is equal to 5, 5, 10 and 10 for cases 1e4, respectively. T is equal to 24, and each time interval is equal to 1. The maximum generation number Gmax is set to 5000, 5000, 20000 and 20000 for cases 1e4, respectively. The experimental results are obtained in 20 independent runs. About experimental environment, Matlab8.3 simulation software is used to implement all experiments, and the experimental computer is configured as Intel (R) Core (TM) i5-2450 M CPU @ 2.50GHZ.

5.2. Comparison among three DE variants In this section, the performance of EFDE is compared with that of the other two reported DEs on solving the single-objective and multi-objective dispatch problems, where the three DEs use the same constraint handling method and NA for a given problem. When the fuel cost is only optimized, the minimum Cmin , maximum Cmax , mean Cmean , median Cmedian , standard deviation Cstd , average running time (ART), and average number of iterations that the total constraint violation reaches the tolerance error (AIV) of the twenty costs are used to assess the performance of the three DEs. Additionally, the marks ‘‘ þ ", ‘‘
" and ‘‘ ¼ " obtained by combining Wilcoxon rank-sum test at the 0.05 significance level [69e71] and the winning rate are used to show the performance of the two compared algorithms in 20 independent runs. When the emission is only optimized, the minimum Emin , maximum Emax , mean Emean , median Emedian , standard deviation Estd , ART, and AIV of the twenty emissions are used as indicators to evaluate the three DEs. The marks ‘‘ þ ", ‘‘
" and ‘‘ ¼ " are also used to show the performance of the two compared algorithms in 20 independent runs. When the multi-objective DEED cases are optimized, the Cmean , Emean , Cstd , Estd , ART, and AIV in 20 independent runs are adopted to evaluate the quality of compromise solutions of the three DEs. In addition, the two comprehensive indexes Nimin and Nistd ði ¼ 1; 2; 3Þ are used to compare the comprehensive performance of the three DEs for a multi-objective optimization problem. ART is used to evaluate the efficiency of an algorithm's execution. AIV is used to assess how fast a method finds the desirable solutions. The marks ‘‘ þ ", ‘‘
" and ‘ ‘ ¼ " represent that the performance of EFDE is, respectively, better than, worse than, and the same as that of a comparison algorithm, which are listed in the last columns of Tables 1e3. It is explained that Wilcoxon rank-sum test at the 0.05 significance level can only obtain whether the proposed EFDE and each of the two reported DEs have significant difference. To be more specific, if the two algorithms have no significant difference, ‘‘ ¼ " is marked for the EFDE and one of the two reported DEs. On the contrary, if the two algorithms have significant difference, winning rates of the EFDE and one of the two reported DEs will be calculated to generate the winner. Suppose sm ðm ¼ 1; 2; /; MÞ is a solution obtained by the H algorithm for a problem, and sn ðn ¼ 1; 2; /; NÞ is a solution obtained by the K algorithm for the same problem, where M and N are the number of solutions. If sm ðm ¼ 1; 2; /; MÞ is better than sn ðn ¼ 1; 2; /; NÞ, the H algorithm scores one point. Otherwise, the K algorithm scores one point. If sm ðm ¼ 1; 2; /; MÞ is equal to sn ðn ¼ 1; 2; /;NÞ, the H and K algorithm, respectively, score 0.5 point. The H and K algorithm are compared up to M  N times. On all comparisons, the winning rate of the H (K) algorithm is defined as its total points divided by M  N. If the winning rate of the H algorithm is higher than that of the K algorithm, the H algorithm will be winner for the case, and vice versa. For the all selected cases, the tolerance error of the total constraint violation is set to 0.0001. Table 1 lists the results of three DE variants for the fuel cost minimization of four DEED cases. The Cmin , Cmax , Cmean , Cmedian , and Cstd obtained by EFDE are smaller than those obtained by the other two DEs for all cases, which indicates that EFDE is superior to its

X. Shen et al. / Energy 186 (2019) 115801

13

Table 1 The results of three DE variants for the fuel cost minimization of four DEED cases. Cases

Algorithms

ART (min)

AIV

Cmin

Cmax

Cmean

Cmedian

Cstd

Case1: the 5-unit system considering transmission losses [38] Case2: the 5-unit system considering transmission losses and POZs [38,66] Case3: the 10-unit system considering transmission losses [10] Case4: the 10-unit system without transmission losses or POZs [10,39]

SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE

3.73 3.46 3.35 4.43 3.55 3.55 24.11 16.45 17.28 7.04 8.40 5.63

1 1 1 2598 5000 145 1 1 1 1 1 1

43952.118875 48319.635111 43047.851168 50268.396417 50946.600138 43431.576531 2469980.185316 2601963.956486 2463022.017655 2343764.799204 2482843.791884 2337199.608281

45556.080296 50430.178695 43325.179120 51964.825213 53121.774536 44057.254743 2471945.225847 2711708.165011 2463243.079898 2346298.659953 2595664.800111 2337491.986469

44872.221032 49458.222194 43167.757203 51088.027357 52360.288535 43673.148804 2470977.138204 2658326.172403 2463163.052493 2344941.988070 2536958.904738 2337327.750117

44878.696491 49409.698714 43149.799900 51174.845168 52491.381015 43675.198500 2470874.266696 2657550.928930 2463170.585529 2344804.536108 2535631.148090 2337302.358019

462.870299 507.594890 64.046984 462.877703 604.787698 161.878956 526.740117 27383.526077 65.215729 616.935332 34152.083018 93.321841

Table 2 The results of three DE variants for the emission minimization of four DEED cases. Cases

Algorithms

ART (min)

AIV

Emin

Emax

Emean

Emedian

Estd

Case1: the 5-unit system considering transmission losses [38] Case2: the 5-unit system considering transmission losses and POZs [38,66] Case3: the 10-unit system considering transmission losses [10] Case4: the 10-unit system without transmission losses or POZs [10,39]

SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE

3.57 3.64 3.54 4.51 3.78 3.61 20.93 17.79 15.81 6.54 8.74 5.36

1 1 1 1915 5000 137 1 1 1 1 1 1

17855.693833 18688.822728 17852.958340 23015.751097 20082.454549 17858.900947 292417.029736 328468.581102 291816.137769 261246.350897 297235.425431 260700.953361

17858.338263 20198.971471 17852.958342 26603.311658 27902.648645 17924.713901 292731.577308 361299.910951 291818.242461 261628.882070 316941.706716 260702.129532

17857.256412 19120.637790 17852.958342 24698.120682 24392.545568 17864.440994 292574.343330 344507.142192 291816.536970 261454.027037 306003.045365 260701.221951

17857.276161 19027.203184 17852.958342 24625.979089 24724.346467 17859.151395 292583.214314 344428.497234 291816.461864 261459.869908 306601.872836 260701.075982

0.602735 407.608005 0 840.402792 1839.525653 16.866227 87.075449 8793.996786 0.467550 83.281319 5665.110588 0.355726

Table 3 The results of three DE variants for the fuel cost and emission minimizations of four DEED cases. Cases

Algorithms

ART (min)

AIV

Cmin

Emin

N imin

Cstd

Estd

N istd

Case1: the 5-unit system considering transmission losses [38] Case2: the 5-unit system considering transmission losses and POZs [38,66] Case3: the 10-unit system considering transmission losses [10] Case4: the 10-unit system without transmission losses or POZs [10,39]

SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE SPS-DE MBDE EFDE

3.87 8.84 3.54 8.32 16.73 6.78 22.01 64.91 16.16 6.53 17.14 5.23

1 1 1 2224 4636 134 1 1 1 1 1 1

45918.406478 50501.977074 45242.515604 50778.771075 51675.584137 45406.143765 2535138.720482 2622443.184124 2532717.411389 2424096.277256 2475942.800021 2422486.555346

18312.978085 18460.066994 18417.618036 22957.670589 19866.674934 18381.022610 295225.923964 315793.968856 294992.589715 263436.117144 280507.677480 263550.784888

0.567726 0.598021 0.565241 0.620376 0.582074 0.524482 0.567500 0.596980 0.567005 0.569086 0.593518 0.569018

111.025213 504.212885 33.564618 441.815926 364.787940 104.823818 1801.987461 29242.952580 100.368980 2857.424724 32884.769169 136.965507

19.185260 197.577853 10.715717 1352.721748 3452.772795 64.172452 177.190749 5650.826603 11.384074 302.813400 7060.419043 15.630291

0.155549 0.984209 0.059389 0.561631 0.778621 0.098635 0.046423 0.998804 0.002720 0.064707 0.997658 0.003181

two competitors in the global optimization ability and stability. Furthermore, it is clearly observed that the Cmax value of EFDE is even smaller than the Cmin values of another two algorithms for the four selected DEED cases. Thus, EFDE has stronger search ability in each run compared to the other two DEs. For all cases, all algorithms have significant difference, and the two comparison algorithms are worse than EFDE according to the combining Wilcoxon rank-sum test statistical analysis and winning rate methods. In terms of the ART, EFDE has lower computation cost than SPS-DE and MBDE for cases 1 and 4. For Case 2, the ART of EFDE is the same as that of MBDE, and smaller than that of SPS-DE. For the 10unit system considering transmission losses, the ART of EFDE is larger than that of MBDE, and smaller than that of SPS-DE. According to the criterion AIV, all three DEs can quickly search the satisfactory feasible solutions within the tolerance error for cases 1, 3 and 4, which indicates that the proposed constraint handling technique is effective to cope with the existing constraints. For the 5-unit system considering transmission losses and POZs, EFDE is

prompter than the other two algorithms to find the feasible solutions. Additionally, the AIV of MBDE is equal to the Gmax , meaning that MBDE may not be able to find a feasible solution in the entire evolution process, which can be seen from its constraints violations in Section 5.8. Overall, EFDE performs the best among all presented DEs for the fuel cost minimization of four DEED cases. The results of three DE variants for the emission minimization of four DEED cases are presented in Table 2. According to the terms Emin , Emax , Emean , Emedian , and Estd , the performance of EFDE is better than that of the other two DEs for the emission minimization of four DEED cases. To be more specific, the Emax , Emean and Emedian obtained by EFDE is approximately equal to the Emin value of EFDE, and its standard deviation is equal to 0 for Case 1, indicating that EFDE can easily find the global best emission in each run. For the remaining cases, the Estd value of EFDE is far smaller than that of another two reported DEs, which demonstrates that the stability of EFDE is the best. The Emin values of SPS-DE and MBDE are even larger than the Emax value of EFDE, meaning that FEDE performs far

14

X. Shen et al. / Energy 186 (2019) 115801

better than SPS-DE and MBDE. Statistically, EFDE can produce better solutions than the other two DEs for all cases. In the computation cost, the ART of EFDE is the lowest among three DEs for all cases. EFDE saves 0.84%, 19.96%, 24.46% and 18.04% time cost compared to SPS-DE for the emission minimization of four cases, respectively. EFDE saves 2.75%, 4.50%, 11.13% and 38.67% time cost compared to MBDE for the emission minimization of four cases, respectively. Thus, it is seen that EFDE has higher computation efficiency compared to the other two DEs. According to the criterion AIV, the proposed constraint handling technique can help the three DEs to find the feasible solution quickly for cases 1, 3 and 4. For Case 2, the combining EFDE and constraint handling technique is faster than the other two methods to make the total constraint violation less than 0.0001. Based on the above observations, EFDE has desirable optimization performance in various DEED cases. According to the single-objective optimization results in Tables 1 and 2, the reason that EFDE is superior to SPS-DE and MBDE in most situations is that the operators of EFDE can improve its global search capacity and the convergence rate. Specifically, the introduced archive in EFDE provides more candidate solutions to enhance the global search of EFDE. The fitness-based mutation operation effectively balances the population diversity and convergence speed in the evolution process. The random-based F is beneficial to enhance the exploration ability of EFDE. The CR with learning mechanism is used to improve the quality of offspring. In addition, according to the criterions AIV, it can be concluded that the proposed constraint can help the infeasible individuals quickly find the feasible regions in most situations. Table 3 exhibits the results of three DE variants for the fuel cost and emission minimizations of four DEED cases. In terms of the ART, the three DE variants optimizing the cost and emission simultaneously are more time-consuming than those only optimizing the cost or emission in most cases, but the computation cost of EFDE is the lowest among three DEs for the fuel cost and emission minimizations of four DEED cases. Based on the values of AIV, the three DE variants can find the feasible solutions in the first generation for cases 1, 3 and 4. For Case 2, EFDE can help the infeasible solutions to move to the feasible regions with the less number of generation compared to the other two DE variants. Additionally, SPS-DE and MBDE can help population to finally reach the feasible regions for Case 2. Different from the cost and emission minimizations of Case 2, MBDE may not be able to find a feasible solution through the evolution process for the cost or emission minimizations. The terms Nimin and Nistd ði ¼ 1; 2; 3Þ are calculated based on the proposed NA (as Eq. (25)) for any case, and they are given by:

" Nimin ¼ u1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ,v " ,v u 3  u 3  2 2 uX uX C imin Eimin C imin t þ u2 Eimin t i¼1

i¼1

(38) " Nistd

¼

u1 C istd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ,v " ,v u 3  u 3  2 2 uX uX i i t t C std Eistd þ u2 Estd i¼1

i¼1

(39) where the C imin and Eimin obtained by the ith method denote the minimal values of costs and emissions in 20 runs for a problem, respectively. The C istd and Eistd obtained by the ith method denote the standard deviation values of costs and emissions in 20 runs for a problem, respectively. i ¼ 1, i ¼ 2 and i ¼ 3 represent, respectively, the indexes of the SPS-DE, MBDE and EFDE.

It is clearly seen that the N 3min value is smaller than the N1min and N2min

values for all test cases, which demonstrates that EFDE can produce the best compromise solution among all three DEs.

Furthermore, the N3std value is also smaller than the N 1std and N2std values for all test cases. Therefore, the stability of EFDE is better than that of the other two DEs for the four DEED problems. In short, EFDE can yield better compromise solutions compared to the other two DEs. According to the multi-objective optimization results in Table 3, it can be explained that the operators of EFDE are beneficial to accelerate the convergence and approach the best fuel cost and emission. Moreover, the proposed constraint can effectively help the infeasible compromise solutions find the feasible regions in all situations, and EFDE based on the adopted NA is competitive in obtaining the best compromise solution. 5.3. Convergence of three DE variants for the fuel cost or emission minimization The average convergence curves of three improved DEs for the fuel cost minimization of four DEED cases are plotted in Fig. 7. It is obviously seen that the convergence rates of EFDE are the fastest among all three DEs for the four selected cases. Meanwhile, the curves of EFDE can gradually reach a lower level compared to the other two DEs as evolution progresses. The convergence performance of SPS-DE is better than that of MBDE for all cases, but its average convergence curve is always above that of EFDE through the entire evolution process. MBDE performs the poorest in the convergence efficiency for all cases, and its average convergence curve always keeps a higher level than that of the other two DEs. In all, EFDE is more favorable to optimize the fuel cost of four DEED cases. Fig. 8 exhibits the average convergence curves of three improved DEs for the emission minimization of four DEED cases. Similar to the fuel cost minimization, EFDE converges faster than the two comparison algorithms for the four DEED cases. For cases 1, 3 and 4, the curves of EFDE and SPS-DE are close, and the curves of EFDE are slight better than those of SPS-DE. On the contrary, the curves of MBDE are obviously worse than those of EFDE and SPSDE. For Case 2, EFDE can rapidly converge to the feasible regions and its curve gradually reaches a low level compared to the other two DEs. Therefore, EFDE can produce lower emission while meeting various constraints of four DEED cases compared to its two competitors. 5.4. The evaluation of the improvement components of EFDE In this section, to evaluate the effect of each improvement component of EFDE, the EFDE excluding one of the three main improvement components is studied. These EFDE variants are denoted as follows. IC1: EFDE with the fixed crossover rate; IC2: EFDE excluding the archive setting and adopting the fixed mutation factor; IC3: EFDE without the fitness-based mutation. For the fuel cost minimization of four DEED cases, the three EFDE variants and EFDE are evaluated by the average fuel cost of 20 runs shown in Fig. 9. It can be seen from Fig. 9 that EFDE obtains the lowest mean fuel cost for the two 5-unit systems and one 10-unit system without transmission losses or POZs, and the average fuel cost of EFDE is slightly higher than that of IC2 for the 10-unit system considering transmission losses. Moreover, EFDE performs almost as good as IC3 for the 10-unit system considering transmission losses. It is

X. Shen et al. / Energy 186 (2019) 115801

15

Fig. 7. The average convergence curves of three improved DEs for the fuel cost minimization of four DEED cases.

clear that IC1 performs the worst among all algorithms, meaning that CR yielded by the proposed method is the most effective compared to the other improvement components of EFDE. For the two 5-unit systems and one 10-unit system without transmission losses or POZs, EFDE is better than IC2, which explains that the archive setting and random-based F are beneficial to improve the quality of solutions. EFDE is superior to IC3 for three out of four cases, thus the fitness-based mutation is also a good measure to enhance the global search of EFDE.

In order to assess the quality of compromise solutions based on the proposed NA, the proposed NA is compared with the two NAs presented in the literature [72], which are marked as NA1 and NA2. To be more specific, the new objective functions based on the two reported NAs are expressed as follows. NA1:

(40)

NA2:

fa ¼ u1 ½ðCðxa Þ
Cmin Þ=ðCmax
Cmin þ nÞ þu2 ½ðEðxa Þ
Emin Þ=ðEmax
Emin þ nÞ

compared based on the term pi , and it is calculated by pi ¼ ðC imean
i i i NA i i C NA mean Þ=C mean þ ðEmean
Emean Þ=Emean . C mean and E mean represent the mean cost and emission based on the ith normalized approach,

5.5. Comparison among three normalized approaches

fa ¼ u1 ½Cðxa Þ=Cmax  þ u2 ½Eðxa Þ=Emax 

candidate solutions ðx1 ; x2 ; /; xA Þ, respectively. Emin and Emax represent the minimal and maximal emissions of candidate solutions ðx1 ;x2 ;/;xA Þ, respectively. n is very small number, which is to avoid the meaningless denominator. It is observed that the two reported NAs limit each objective function value of candidate solutions to the range ½0; 1, where the normalized cost and emission of candidate solutions are added to transfer into the singleobjective function. The three normalized approaches are

(41)

where Cmin and Cmax represent the minimal and maximal costs of

NA respectively. C NA mean and E mean represent the mean cost and emission based on NA, respectively. i ¼ 1 and i ¼ 2 denote the indexes of the NA1 and NA2, respectively. If the pi ði ¼ 1; 2Þ value is larger than 0, the compromise solution based on NA is better than that based on the normalized approach i. For the fuel cost and emission minimizations of four DEED cases, the pi ði ¼ 1; 2Þ values of EFDE based on NA1 and NA2 are shown in Fig. 10. It is clear that all p1 and p2 values are positive, indicating that the compromise solutions based on NA is better than those based on NA1 and NA2. Moreover, the p2 values are larger than the p1 values for all test DEED cases, meaning that NA is superior to NA2 more obviously compared to NA1. The reason that the proposed NA outperforms the other two NAs from the literature is that the fuel cost and emission of the currently

16

X. Shen et al. / Energy 186 (2019) 115801

Fig. 8. The average convergence curves of three improved DEs for the emission minimization of four DEED cases.

all candidate individuals are used to help a target individual limit its fuel cost and emission in the range ½0; 1 to construct the singleobjective function. Compared to NA, NA1 and NA2 only use Cmin , Cmax , Emin and Emax to form the single-objective function in the fuel cost and emission of all individuals.

5.6. The effects of Gf and l on the performance of EFDE The above selected four DEED cases are used to investigate the influence of different local parameters Gf and l on the performance of EFDE. The parameter Gf is set to 50%  Gmax , 60%  Gmax , 70%  Gmax , 80%  Gmax and 90%  Gmax , respectively. The parameter l is set to 0.6, 0.7, 0.8, 0.9 and 1, respectively. We adopt the variablecontrolling approach to analyze the influence of each parameter value on the performance of EFDE. In other words, only one factor is different in each experiment. Furthermore, to evaluate the compromise solutions based on different parameters values, Eq. (38) is used to obtain their comprehensive values. In Table 4, i ¼ 1, i ¼ 2, i ¼ 3, i ¼ 4 and i ¼ 5 denote the indexes of the Gf ¼ 50%  Gmax , Gf ¼ 60%  Gmax , Gf ¼ 70%  Gmax , Gf ¼ 80%  Gmax and Gf ¼ 90%  Gmax , respectively. It can be observed that the N5min value is the smallest among all N imin values for Case 2. For Case 1, case 3 and case 4, the N5min value is equal to or smaller than the other Nimin values. Therefore, it is suitable that Gf is set to 90%  Gmax .

In Table 5, i ¼ 1, i ¼ 2, i ¼ 3, i ¼ 4 and i ¼ 5 denote the indexes of the l ¼ 0:6, l ¼ 0:7, l ¼ 0:8, l ¼ 0:9 and l ¼ 1, respectively. It is clear that the N1min value is the smallest among all Nimin values for the first two cases, and the N 1min value is equal to or smaller than the other Nimin values for the remaining two cases. Therefore, we conclude that l ¼ 0:6 is the most appropriate as the ratio of the other individuals learning from the best individual. 5.7. The Pareto front based on the proposed NA To visualize the distribution of the Pareto front based on the proposed NA, the Pareto front based on the proposed NA is plotted in Fig. 11, where each point corresponds to a weight factor u1 value. To be more specific, u1 is decreased from 1 to 0, and the step of varying u1 is equal to 0.04. Since u2 is calculated by u2 ¼ 1
u1, u2 is gradually increased with the decreasing of u1 . The Pareto fronts of every cases contain 26 points, which are 26 potential schemes of producing the compromise solutions. The fuel cost and emission are two conflicting objectives, and their importance is determined by their weight factor in the new objective function (as Eq. (25)). About this characteristic, we can see from the Pareto front, where one objective function value reduces at the cost of increasing another objective function value. Furthermore, for the first case, it is shown that the distribution of the Pareto front is nonuniform when the u1 values are continuously taken from ½0:32; 0:68, and the distribution of the Pareto front is

X. Shen et al. / Energy 186 (2019) 115801

17

Fig. 9. The average fuel cost of three EFDE variants and EFDE for the fuel cost minimization of four DEED cases.

optimization problems is also popular and competitive. The more uniform the distribution of the Pareto front based on the NDA is, the better population diversity is. Different from the Pareto front based on the NDA, there is no conclusion that the nonuniform distribution of the Pareto front based on the NA is unreasonable, which need to further research. Meanwhile, similar researches have been ongoing, such as Fig. 12 of [73] and Fig. 7 of [74]. There may be many factors on the nonuniform distribution of the Pareto front, such as the nonlinearity of the objective functions, the strong-constraints, and the number of generating units. 5.8. Comparison between EFDE and the other approaches reported in the literature

Fig. 10. The pi values of EFDE based on NA1 and NA2 for the fuel cost and emission minimizations of four DEED cases.

also nonuniform when u1 lies in ½0:36; 0:64 for the second case. On the contrary, the distributions of the Pareto front are relatively uniform for the last two cases. In Ref. [73], the reported nondominated approach (NDA) of handling the multi-objective

To testify the optimization performance of EFDE for the singleobjective and multi-objective DEED cases, the results of EFDE are compared with those of the other approaches from the literature and the above presented two DEs. For the single-objective problems, not only the fuel cost or emission of these approaches are taken as the criteria to evaluate their performance, but also the corresponding constraints violations are considered as the important standard for checking the feasibility of a solution obtained. Because if a solution doesn't meet any of those constraints, the solution is infeasible, and the corresponding obtained fuel cost or emission are meaningless. For cases 1 and 3, these constraints violations are made up of the violations of power balance constraint with transmission losses Vpl , power generation limits Vg and ramp

18

X. Shen et al. / Energy 186 (2019) 115801

Table 4 The effects of Gf on the performance of EFDE. Cases

Term

Gf ¼ 50%  Gmax

Gf ¼ 60%  Gmax

Gf ¼ 70%  Gmax

Gf ¼ 80%  Gmax

Gf ¼ 90%  Gmax

Case1

N imin N imin N imin N imin

0.447221

0.447218

0.447205

0.447218

0.447205

0.447045

0.447245

0.447303

0.447454

0.447018

0.447214

0.447213

0.447214

0.447214

0.447213

0.447214

0.447213

0.447213

0.447214

0.447213

Case2 Case3 Case4

Table 5 The effects of l on the performance of EFDE. Cases

Term

l ¼ 0:6

l ¼ 0:7

l ¼ 0:8

l ¼ 0:9

l ¼1

Case1

N imin N imin N imin N imin

0.447200

0.447238

0.447220

0.447206

0.447204

0.447040

0.447246

0.447316

0.447361

0.447102

0.447213

0.447214

0.447214

0.447213

0.447214

0.447213

0.447214

0.447214

0.447214

0.447213

Case2 Case3 Case4

rate limits Vr , which are added together to form the total constraint violation Vt ¼ Vpl þ Vg þ Vr . For Case 2, the total constraint violation is obtained by adding the violations of power balance constraint with transmission losses Vpl , power generation limits Vg , ramp rate limits Vr and POZs Vpoz . For case 4, the violation yielded by the equal constraint, that is the power balance constraint without

transmission losses, is represented as Vp , therefore, its total constraint violation is calculated by Vt ¼ Vp þ Vg þ Vr . To guarantee the satisfaction of solutions, all existing constraints violations should be equal to 0 or keep relatively small values. For the inequality constraints, its violation need to be exactly equal to 0. Otherwise, the corresponding solution is unsuitable. For the equality constraints, the smaller its constraint violation is, the better the solution is. For the multi-objective optimization problems, we give the fuel cost, emission and various constraints violations of the best compromise solution obtained by an approach. However, it is very difficult to confirm X solution obtained by X approach better than Y solution obtained by Y approach when one of the two objective function values of X solution is lower than that of Y solution while another of the two objective function values of X solution is higher than that of Y solution. Based on this cause, some extra comprehensive indexes are used to assess these compromise

Fig. 11. The effect of weight factor on the distribution of the Pareto front based on the proposed NA for all test cases.

X. Shen et al. / Energy 186 (2019) 115801

solutions of different methods. Here, the two comprehensive indexes from the literature are adopted, and they are, respectively, W i based on the weighted sum approach (WSA) [73] and mi based on the fuzzy-based approach (FA) [40]. Additionally, the two proposed comprehensive indexes are also used, and they are, respectively, the term N i based on the NA and pi based on the improvement percent (IP). The four comprehensive indexes are expressed as follows. WSA:

W i ¼ uC i þ ð1
uÞtEi ; i ¼ 1; 2; /; h

(42)

FA:

mij ¼

8 > > > > > > > > <

0;

If fj ðXi Þ  fjmax

fjmax
fj ðXi Þ ; If fjmin  fj ðXi Þ  fjmax > fjmax
fjmin > > > > > > 1; If fj ðXi Þ  fjmin > :

(43)

Xk

mij j¼1 mi ¼ Xh X k i¼1

(44)

mij j¼1

NA:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ,v " ,v u h   # u h   # uX uX 2 2 Ci Ei þ u2 E i t N i ¼ u1 C i t i¼1

(45)

i¼1

IP:

pi ¼

Ci
Ch Ci

þ

Ei
Eh Ei

; i ¼ 1; 2; /; h
1

(46)

where the weight factor u is equal to 0.5, which is the same as the u1 and u2 . h is the number of the selected comparison approaches. W i based on the WSA represents the term of the ith comparison approach for a case. C i and Ei represent the cost and emission obtained by the ith comparison approach for a case, which represent two objective function values of the best compromise solution. The factor t is calculated by t ¼ CðPmax Þ=EðPmax Þ, which denotes the ratio of the maximum generation cost to the maximum emission amount. mij represents the membership function of the jth objective of the ith solution. fj ðXi Þ represents the jth objective function value of the ith candidate solution. fjmax and fjmin represent the maximal and minimal function values of the j objective, respectively. Eq. (44) is used to select the best compromise solution among all candidate solutions. To be more specific, the higher mi is, the better corresponding solution is. Ni and pi represent the two criteria of the ith solution. The higher Ni is, the better corresponding solution is. For the criterion pi , if pi is positive, the solution obtained by EFDE is better than that obtained by the other approaches. C h and Eh represent the cost and emission of the best compromise solution obtained by EFDE for a case. In addition, to make a fair comparison, we amend some reported fuel cost or emission values from the literature, and they are denoted as C ia and Eia , respectively. In Table 6, the single-objective optimization results of different approaches for the 5-unit system with transmission losses are listed. If the single-objective optimization results include the fuel cost optimization and emission optimization results for an approach, the listed violations of the approach contain the violations for the

19

dynamic economic dispatch and dynamic emission dispatch. For example, the Vpl values of EFDE are denoted as 0.000011/0.000012, where the former is the Vpl value of EFDE for the dynamic economic dispatch, and the latter is the Vpl value of EFDE for the dynamic emission dispatch. For the dynamic economic dispatch, the reported fuel costs of SOA-SQP [76], HGABF [1], OGHS [80] and CMAES [81] are amended to compare fairly. In obtaining the fuel cost, SOA-SQP [76] seems to be better than EFDE. However, the Vr and Vpl values of SOA-SQP [76] are very large, and they are respectively equal to 606.1567 and 184.222910, which can't be ignored. Thus, the solution obtained by SOA-SQP [76] is infeasible. On the contrary, the constraints violations obtained by EFDE are satisfactory. The cost value of EFDE is the lowest among the cost values except the cost value of SOA-SQP [76]. All the available Vg values are equal to 0, and the Vr values are equal to 0 except the Vr value of SOA-SQP [76]. In Vpl , the two comparison DEs and EFDE perform better than the other approaches from the literature, indicating that the proposed constraint handling technique is very effective in handling the equality constraint. In addition, the Vpl values of SOA-SQP [76], OGHS [80] and CMAES [81] are far larger than those of the other approaches from the literature, and they can't be neglected. For the dynamic emission dispatch, EFDE also can find the least emission among all available emission. The emission obtained by NPAHS [41] is slightly larger than that obtained by EFDE, but EFDE has an advantage over NPAHS [41] for the constraint violations. Overall, EFDE is the most competitive approach for the dynamic economic or emission dispatch of Case 1. According to Table 7, all the available Vg and Vr values are exactly equal to 0. Furthermore, in view of the power balance constraint violation, the solutions of the last three DE variants are considered to be acceptable, and the Vpl values of the last three DE variants are lower than those of the other approaches. For the comprehensive performance of EFDE optimizing the fuel cost and emission, it can be seen from the four comprehensive indexes that EFDE outperforms the other approaches, and EFDE ranks first among all approaches based on each of the four indexes, and it is listed in the last row of Table 7. Therefore, EFDE can produce the best compromise solution. In Table 8, the results of five approaches optimizing the 5-unit system with transmission losses and POZs are presented. TVACIPSO [66] and LPSO_DVS [79] aren't used to optimize the 5-unit system with transmission losses and POZs for the dynamic emission dispatch. The Vpoz values of LPSO_DVS [79] and MBDE are relatively large, thus their solutions can't be accepted. All Vg and Vr values are equal to 0, which meet the requirement of the inequality constraints. The two single-objective function values of EFDE are the lowest, and equality and inequality constraint violations of EFDE are also satisfactory. It can be concluded that the performance of EFDE is the best among all approaches. According to Table 9, the constraints violations of the three DEs are equal to 0 or relatively small, which are quite reasonable. Moreover, both objective function values of the best compromise solution of EFDE are smaller than those of the other two approaches. On the other hand, EFDE ranks first according to the four comprehensive indexes, meaning that EFDE is superior to another two DEs for the dynamic economic emission dispatch of the 5-unit system with transmission losses and POZs. The results of different approaches for the dynamic economic or emission dispatch of the 10-unit system with transmission losses are reported in Table 10. For the dynamic economic dispatch, the Vg values of AIS [82] and IBFA [31] aren't equal to 0, which mean that the solutions obtained by the two approaches are infeasible. The two inequality constraints violations of EFDE are equal to 0 for the dynamic economic dispatch and dynamic emission dispatch, and

20

X. Shen et al. / Energy 186 (2019) 115801

Table 6 The results of different approaches for the dynamic economic or emission dispatch of the 5-unit system with transmission losses [38]. Approaches

Single-objective optimization Ci

DE-SQP [13] PSO-SQP [13] PSO [36] PS [37] MOHDE-SAT [39] DE [75] HS_MD [2] SOA-SQP [76] NPAHS [41] HGABF [1] HIGA [77] BBPSO with DCS [78] LPSO_DVS [79] OGHS [80] CMAES [81] SPS-DE MBDE EFDE

=C ia

Constraints violations Ei

=Eia


/

/
19094/18192/17884/
/

/

/
17853/
/

/

/

/

/

/
17855.693833/18688.822728/17852.958340/-

43161/43263/47852/46530/46478/43213/43210.95/40701.42/41923.32 43072.99/41574.80/43845.76 43125.365/43222.7/43125.51661/31549.3/54454.17 43526/45543.91 43952.118875/48319.635111/43047.851168/-

Vg

Vr

Vpl

0/0/
/
0/
/
0/0/0/0/0 0/0/0/0/0/0/0/0 0/0 0/0

0/0/
/
0/
/
0/0/606.1567/0/0 0/0/0/0/0/0/0/0 0/0 0/0

0.001527/0.001282/
/
0.032466/
/
0.011571/0.097577/184.222910/0.125939/0.001719 0.995436/0.07364/0.014702/0.000631/170.288864/193.178605/0.000013/0.000012 0.000013/0.000012 0.000011/0.000012

Table 7 The results of different approaches for the dynamic economic emission dispatch of the 5-unit system with transmission losses [38]. Approaches

DE-SQP [13] PSO-SQP [13] PSO [36] PS [37] MOHDE-SAT [39] NPAHS [41] SPS-DE MBDE EFDE Rank

Multi-objective optimization

Constraints violations

Terms

Ci

Vg

Vr

Vpl

Ni

pi

Wi

mi

0 0 0 0 0 0 0 0 0 e

0 0 0 0 0 0 0 0 0 e

0.001449 0.001178 0.001417 0.088777 0.022764 0.002145 0.000009 0.000011 0.000010 e

0.330123 0.331822 0.357763 0.336320 0.329332 0.324088 0.323857 0.341390 0.322383 1

0.043263 0.052773 0.197590 0.082610 0.039055 0.010371 0.009005 0.106443 e 1

35860.315216 36014.752674 39462.041430 37111.883110 36626.660601 35547.934873 35688.772403 38082.800954 35423.563443 1

0.117841 0.109689 0 0.097452 0.133027 0.150012 0.153319 0.080054 0.158607 1

=C ia

Ei

44450/44542/50893/47911/48214/45196/45918.406478/50501.977074/45242.515604/
/


=Eia

19616/19772/20163/18927/18011/18630/18312.978085/18460.066994/18417.618036/
/


Table 8 The results of different approaches for the dynamic economic or emission dispatch of the 5-unit system with transmission losses and POZs [38,66]. Approaches

Single-objective optimization

TVAC_IPSO [66] LPSO_DVS [79] SPS-DE MBDE EFDE

Constraints violations

C i =C ia

Ei =Eia

Vg

Vr

Vpoz

Vpl

40126.2/50803.015612 51293.513/50268.396417/50946.600138/43431.576531/-


/

/
23015.751097/20082.454549/17858.900947/-

0/0/0/0 0/0 0/0

0/0/0/0 0/0 0/0

0/98.033/0/0 0.801419/0.795572 0/0

0.009715/0.000730/0.000010/0.000010 0.000013/0.000012 0.000008/0.000010

Table 9 The results of different approaches for the dynamic economic emission dispatch of the 5-unit system with transmission losses and POZs [38,66]. Approaches

SPS-DE MBDE EFDE Rank

Multi-objective optimization

Constraints violations

C i =C ia

Ei =Eia

Vg

Vr

Vpoz

Vpl

Ni

pi

Wi

mi

50778.771075/51675.584137/45406.143765/
/


22957.670589/19866.674934/18381.022610/
/


0 0 0 e

0 0 0 e

0 0 0 e

0.000013 0.000007 0.000011 e

0.620376 0.582074 0.524482 1

0.305156 0.196104 e 1

41347.535777 39647.354416 35479.939607 1

0.050754 0.239631 0.709615 1

its Vpl values are the lowest among all presented Vpl values. In terms of the objective function values, the fuel cost and emission of EFDE are the smallest, which suggest that the global search capability of EFDE is better than that of the other methods. Given the best fuel cost, best emission and corresponding constraints violations, EFDE

Terms

performs the best among all methods. In Table 11, there is an approach whose inequality constraints violations aren't equal to 0, and it is HCRO [43]. Hence the best compromise solution obtained by HCRO [43] is undesirable. For the power balance constraint, the Vpl value of EFDE is the lowest, and it

X. Shen et al. / Energy 186 (2019) 115801

21

Table 10 The results of different approaches for the dynamic economic or emission dispatch of the 10-unit system with transmission losses [10]. Approaches

Single-objective optimization Ci

DE-SQP [13] PSO-SQP [13] RCGA [10] AIS [82] IBFA [31] MOHDE-SAT [39] MAMODE [42] HCRO [43] SPS-DE MBDE EFDE

=C ia

Constraints violations Ei

2465900/2466800/2516800/2519700/2481733.2570/2484676.3172 2508195/2492451/2479931.38/2479962.21 2469980.185316/2601963.956486/2463022.017655/-

=Eia


/

/
304120/
/
295833.0308/295596.0157 296069/295244/298456.27/298451.65 292417.029736/328468.581102/291816.137769/-

Vg

Vr

Vpl

0/0/0/0 4.56780/37.9288/0
/

/
0/2.3976 0/0 0/0 0/0

0/0/0/0 0/0/0
/

/
0/10.7834 0/0 0/0 0/0

0.008293/0.010245/0.001488/0.001282 32.284669/43.360406/14.211320
/

/
0.286255/0.001314 0.000012/0.000017 0.000016/0.000015 0.000009/0.000013

Table 11 The results of different approaches for the dynamic economic emission dispatch of the 10-unit system with transmission losses [10]. Approaches

Multi-objective optimization Ci

DE-SQP [13] PSO-SQP [13] NSGA-II [10] IBFA [31] MOHDE-SAT [39] MAMODE [42] PSOAWL GIDN [15] HCRO [43] SPS-DE MBDE EFDE Rank

=C ia

Ei

2468800/2470100/2522600/2517116.746/2527929/2514113/2514117 2549000/2517076.39/2517067.36 2535138.720482/2622443.184124/2532717.411389/
/


=Eia

315640/315070/309940/299036.7059/297760/302742/302743 294780/299065.50/299060.47 295225.923964/315793.968856/294992.589715/
/


Constraints violations

Terms

Vg

Vr

Vpl

Ni

pi

Wi

mi

0 0 0 0 0 0 0 0.0014 0 0 0 e

0 0 0 0 0 0 0 0.0523 0 0 0 e

0.005116 0.001786 0.001698 0.00145 0.074859 0.121606 0.0009 0.107508 0.000013 0.000015 0.000010 e

0.304026 0.303821 0.304409 0.298670 0.298682 0.300331 0.298461 0.298679 0.297855 0.313274 0.297594 1

0.039524 0.038373 0.044216 0.007326 0.007400 0.018202 0.005667 0.007385 0.001745 0.100085 e 1

1899721.352397 1899169.878658 1914606.615010 1888882.494105 1891597.518279 1895194.928174 1895851.629260 1888907.892232 1889860.929733 1976867.487506 1888158.441901 1

0.075596 0.076996 0.069674 0.111290 0.110568 0.099519 0.110919 0.111229 0.116097 0 0.118112 1

is equal to 0.00001. The quality of the best compromise solutions obtained by the other approaches is verified by the four criteria. Obviously, EFDE is the most excellent approach in yielding the compromise solution for the dynamic economic emission dispatch of the 10-unit system with transmission losses. In Table 12, the Vg , Vr and Vpl values of the three DEs are small enough for the dynamic economic emission dispatch of the 10-unit

system without transmission losses or POZs, which indicate that the three DEs succeed in producing the reasonable solutions. According to the objective function values C i and Ei , EFDE is the most efficient in achieving the relatively low fuel cost and emission. Table 13 reports the results of four approaches for the dynamic economic emission dispatch of the 10-unit system without transmission losses or POZs. The inequality constraints violations of the

Table 12 The results of different approaches for the dynamic economic or emission dispatch of the 10-unit system without transmission losses or POZs [10,39]. Approaches

Single-objective optimization Ci

SPS-DE MBDE EFDE

=C ia

Constraints violations Ei

2343764.799204/2482843.791884/2337199.608281/-

=Eia

261246.350897/297235.425431/260700.953361/-

Vg

Vr

Vp

0/0 0/0 0/0

0/0 0/0 0/0

0.000011/0.000009 0.000014/0.000019 0.000009/0.000009

Table 13 The results of different approaches for the dynamic economic emission dispatch of the 10-unit system without transmission losses or POZs [10,39]. Approaches

MOHDE-SAT [39] SPS-DE MBDE EFDE Rank

Multi-objective optimization

Constraints violations

Terms

Ci

Vg

Vr

Vp

Ni

pi

Wi

mi

0 0 0 0 e

0 0 0 0 e

0.019 0.000009 0.000015 0.000009 e

0.495282 0.494384 0.515606 0.494326 1

0.003844 0.000229 0.082041 e 1

1764776.560057 1767331.606276 1829239.133310 1766768.447572 2

0.342418 0.327239 0 0.330343 2

=C ia

2406934/2424096.277256/2475942.800021/2422486.555346/
/


Ei

=Eia

266295/263436.117144/280507.677480/263550.784888/
/


22

X. Shen et al. / Energy 186 (2019) 115801

four approaches meet the requirement. In addition, the Vpl values of the three DEs using the proposed constraint handling technique lie in a relatively low level compared to that of MOHDE-SAT [39]. In terms of the two proposed evaluation criteria, EFDE owns the lowest Ni , and all the pi values of the other approaches are larger than 0, which mean that EFDE wins the other three approaches in producing the best compromise solution. For the two reported criteria, the best compromise solution obtained by EFDE is worse than that obtained by MOHDE-SAT [39], but it is better than those obtained by SPS-DE and MBDE. Thus, EFDE ranks second according to the W i and mi . In all, EFDE dominates the other approaches based on Ni and pi , but it loses to the second approach based on W i and mi . As can be seen from Tables 6e13, the performance of EFDE is better than that of the two DEs and different approaches from the literature in most situations, which furthermore explains that the operators of EFDE, the proposed constraint handling mechanism, and NA are competitive. To investigate output of each generating units with respect to EFDE, the best generation schedules obtained by EFDE for all four DEED cases are presented in Tables 14e22, where the transmission losses of each time period and percentage of losses (loss/load) for the DEED cases with transmission losses are also reported. As can be seen from Tables 14 and 16, for the single-objective and multi-objective optimization of the 5-unit system with transmission losses, output of each generating unit at each time period is under control, ramp rate of each unit is also properly satisfied. Not only the proposed method is used to minimize the fuel cost and emission, but also is also to reduce the generating transmission losses, which can save cost. the total losses don't exceed 2% of load demand, which may be considered as an acceptable percentage in power plant [37]. For the single-objective and multi-objective optimization, the best generation schedules and loss of the 5-unit system with transmission losses and POZs are shown in Tables 15 and 16 Output of each generating unit at each time period properly satisfies ramp rate and POZs. In addition, we find that the P3 and P4 values are

larger than the other three generating unit output values at each time period, which mean that the third and fourth generating units bear more power generation to meet the load demand. For the 5unit system with transmission losses and POZs, the percentage of losses doesn't exceed 2%, and it is equal to or lower than the percentage of losses of the 5-unit system with transmission losses. For the 10-unit system with transmission losses, it can be seen from Tables 17e19 that the dispatch results meet the actual load demand, transmission losses and existing constraints. When the fuel cost is taken as the only optimization objective, the total losses and percentage of losses obtained by EFDE are, respectively, equal to 1290.505107 (MW) and 3.24%, which are lower than those obtained by EFDE for dynamic emission dispatch and dynamic economic emission dispatch. In addition, for the single-objective and multi-objective optimization of the 10-unit system with transmission losses, the percentage of losses obtained by EFDE doesn't exceed 5% of load demand, which may be considered as a reasonable percentage in power plant [39]. Different from the first three cases, case 4 neglects the transmission losses, thus the loss term isn't reported in Tables 20e22 Output of each generating unit at each time period properly satisfies the load demand and ramp rate. It is worth noting that the ninth and tenth generating units output their maximum power limits at each time period for dynamic emission dispatch and dynamic economic emission dispatch, which is to reduce emission. However, the emission is reduced at the cost of increasing the fuel cost.

6. Conclusions To minimize the fuel cost and emission of the DEED problems, we have presented an efficient fitness-based differential evolution algorithm (EFDE), a constraint handling mechanism (CHM) and a multi-objective optimization method for the DEED problems. The innovations of this paper are listed as follows: (1) An archive is constructed, which contains the information of the current

Table 14 The best generation schedules of the 5-unit system with transmission losses obtained by EFDE for dynamic economic or emission dispatch [38]. Hour

Dynamic economic dispatch P1

P2

1 20.304592 98.591922 2 10.000000 97.684802 3 10.000000 98.048520 4 10.001026 98.591031 5 10.000000 96.470963 6 10.000000 98.242542 7 10.000000 72.690823 8 12.551281 98.539348 9 42.551281 103.985181 10 63.923073 98.573779 11 75.000000 103.254695 12 74.994943 124.499550 13 64.335373 98.458126 14 46.197352 101.013796 15 16.197352 98.492147 16 10.000000 78.466751 17 10.000000 88.245090 18 10.001288 98.580353 19 12.500511 98.736588 20 42.236726 120.381048 21 38.933931 98.630433 22 10.056437 98.627435 23 10.000000 98.336414 24 10.000000 80.741398 Total losses Percentage of losses

Dynamic emission dispatch

P3

P4

P5

Loss

P1

P2

P3

P4

P5

Loss

30.000000 66.852855 106.852855 112.598252 109.798002 112.683252 112.586791 112.679784 114.285955 112.741201 112.783939 112.747527 112.652620 113.288378 112.352161 112.536343 112.528145 112.778253 112.656676 112.675311 112.924274 112.676442 112.673684 112.609726

125.120068 124.911101 125.111561 175.056580 208.709940 209.792584 209.567736 209.891321 209.848959 209.797751 210.173715 209.853784 209.735242 210.020572 206.675078 156.675078 124.754417 165.039003 209.813659 209.833556 209.837778 161.988297 124.873598 124.673168 194.618882 1.33%

139.800241 139.675411 139.767345 139.767904 139.793271 185.272438 229.614359 229.596992 229.516925 229.523406 229.831585 229.623326 229.377233 229.658417 229.519348 229.520004 229.155768 229.551513 229.551830 229.533367 229.575315 229.521978 187.021895 139.464140

3.816823 4.124170 4.780281 6.014793 6.772176 7.990816 8.459708 9.258725 10.188301 10.559210 11.043934 11.719129 10.558594 10.178515 9.236085 7.198175 6.683420 7.950411 9.259263 10.660008 9.901731 7.870590 5.905590 4.488432

54.678619 58.067127 63.526150 71.120494 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 70.703330 61.883257

58.235516 62.383464 69.080288 78.429737 83.268787 93.462902 97.206739 103.116738 111.476694 115.394434 119.921261 125.000000 115.394404 111.476701 103.116713 87.718978 83.268787 93.462928 103.116729 115.394397 108.704980 92.842968 77.915026 67.062902

116.571661 121.851433 130.220851 141.551593 147.240109 158.804304 162.903452 169.210355 175.000000 175.000000 175.000000 175.000000 175.000000 175.000000 169.210365 152.359888 147.240115 158.804296 169.210356 175.000000 174.990095 158.117748 140.939280 127.720696

110.598165 117.981854 129.750170 145.801668 153.905825 170.449191 176.326333 185.370351 197.650638 203.192097 209.428719 217.410084 203.192106 197.650623 185.370374 161.220814 153.905831 170.449199 185.370344 203.192101 193.649203 169.465245 144.930988 126.226595 188.134935 1.29%

73.363996 78.601569 87.063872 98.890122 105.016028 117.938442 122.686739 130.183812 140.785986 145.750313 151.483661 159.061972 145.750335 140.785995 130.183804 110.655733 105.016017 117.938416 130.183826 145.750345 137.272920 117.152240 98.238752 84.513874

3.447958 3.885448 4.641329 5.793614 6.430750 7.654839 8.123263 8.881256 9.913318 10.336844 10.833641 11.472056 10.336844 9.913318 8.881256 6.955413 6.430750 7.654839 8.881256 10.336844 9.617198 7.578200 5.727376 4.407324

X. Shen et al. / Energy 186 (2019) 115801

23

Table 15 The best generation schedules of the 5-unit system with transmission losses and POZs obtained by EFDE for dynamic economic or emission dispatch [38,66]. Hour

Dynamic economic dispatch P1

P2

1 13.987808 99.104188 2 10.000000 101.172160 3 10.012587 97.688403 4 13.309081 98.426706 5 10.000000 91.983030 6 38.957836 100.394045 7 12.596586 97.262092 8 12.891578 98.847487 9 42.891578 103.894374 10 64.580882 98.438710 11 75.000000 102.117027 12 75.000000 124.844469 13 66.936705 97.096401 14 47.706148 99.519960 15 17.761327 95.909330 16 10.000000 90.130554 17 10.000000 96.721868 18 10.000000 100.905162 19 13.707139 98.769424 20 42.290613 117.205027 21 38.932605 98.309505 22 12.253764 100.181608 23 10.000000 96.570422 24 10.000000 77.376203 Total losses Percentage of losses

Dynamic emission dispatch

P3

P4

P5

Loss

P1

P2

P3

P4

P5

Loss

30.000000 30.444133 30.000000 70.000000 110.000000 113.539213 111.444609 112.426107 113.016082 112.305245 113.533066 112.684572 113.235360 113.075944 112.353131 100.891122 110.328234 114.288474 111.674444 115.104580 113.841214 112.648900 72.648900 32.648900

40.757296 69.871364 119.871364 124.737988 124.316674 133.287633 183.263346 209.280607 210.961816 209.842067 211.681720 209.129705 209.271924 210.295228 208.045792 158.045792 121.769537 159.620051 209.620051 210.946983 209.316338 159.320930 124.341037 122.861154 196.583635 1.35%

230.288712 228.046334 222.694264 229.757466 228.398220 229.680668 229.873192 229.812220 229.432985 229.393610 228.707003 230.060960 228.006094 229.576241 229.157117 228.198438 225.883021 231.135439 229.487563 229.091049 229.494968 228.456324 229.594754 225.083970

4.138005 4.533991 5.266618 6.231241 6.697924 7.859395 8.439826 9.257999 10.196835 10.560514 11.038816 11.719706 10.546484 10.173521 9.226698 7.265905 6.702660 7.949126 9.258621 10.638253 9.894631 7.861526 6.155113 4.970227

54.684146 60.001286 65.202848 70.946129 75.000000 74.999989 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 74.999991 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 75.000000 70.428659 62.445070

58.303245 62.083351 71.083026 78.358852 79.998106 90.586383 96.064660 103.140749 111.409602 115.335129 120.054271 125.000000 115.466936 111.462204 103.047656 90.002527 79.998830 90.632899 103.097784 115.393000 108.698805 95.702182 77.508324 67.692624

116.619984 121.322774 124.989843 141.245087 148.232813 155.666453 161.695588 169.158292 175.000000 174.999867 175.000000 175.000000 175.000000 175.000000 169.196259 151.804070 148.089295 155.796964 169.248754 175.000000 174.948073 161.274452 140.503553 124.997103

110.522814 117.267505 133.380559 145.236703 155.314477 180.003863 180.002764 185.374877 197.682945 203.199865 209.353613 217.350885 203.141634 197.602677 185.427561 159.996808 155.221002 180.001422 185.324022 203.246443 193.678740 159.999569 144.280053 127.289906 188.168803 1.29%

73.317849 78.210750 84.996904 100.004669 105.880025 114.417400 121.367081 130.207420 140.820600 145.801754 151.426076 159.120947 145.728409 140.848211 130.209697 110.155870 106.115928 114.243041 130.210493 145.697587 137.291637 120.589431 100.003173 84.984655

3.448039 3.885667 4.653181 5.791440 6.425420 7.674088 8.130092 8.881338 9.913147 10.336615 10.833961 11.471832 10.336979 9.913092 8.881164 6.959275 6.425055 7.674326 8.881052 10.337031 9.617255 7.565634 5.723762 4.409358

Table 16 The best generation schedules of two 5-unit systems obtained by EFDE for dynamic economic emission dispatch [38,66]. Hour

The 5-unit system with transmission losses P1

P2

1 27.511899 98.533105 2 52.905800 98.540568 3 68.826607 98.539669 4 74.999694 98.530845 5 74.999873 98.539018 6 75.000000 98.540864 7 73.602046 98.526026 8 75.000000 98.541941 9 75.000000 100.309597 10 74.999885 114.787508 11 75.000000 125.000000 12 75.000000 98.723236 13 75.000000 111.692596 14 75.000000 100.332793 15 75.000000 98.540564 16 74.999846 98.541372 17 75.000000 98.540086 18 74.999206 98.540912 19 74.998838 98.538596 20 75.000000 114.770441 21 74.999559 98.539582 22 75.000000 98.540127 23 75.000000 98.534806 24 59.727132 98.536378 Total losses Percentage of losses

The 5-unit system with transmission losses and POZs

P3

P4

P5

Loss

P1

P2

P3

P4

P5

Loss

112.671937 112.671835 112.673688 112.662138 112.639780 113.867792 112.674019 139.853104 175.000000 175.000000 175.000000 175.000000 175.000000 175.000000 139.857760 112.721212 112.745449 114.045743 139.873159 175.000000 166.528539 126.528740 112.674056 112.673508

124.907200 124.908011 124.907714 124.886596 138.541040 188.541040 209.761269 209.815956 209.844465 209.814987 209.847510 209.817273 209.822147 209.818362 209.815511 161.005183 138.433677 188.432937 209.815483 209.820649 209.813891 172.799565 124.907766 124.908318 189.654780 1.3%

50.000000 50.000000 74.780789 124.757475 139.759541 139.829594 139.753103 139.762869 139.760645 139.757305 146.011604 192.825773 142.831202 139.763533 139.760020 139.760172 139.759705 139.759612 139.747745 139.768540 139.756277 139.760046 121.654601 71.654601

3.624142 4.026214 4.728468 5.836747 6.479253 7.779290 8.316463 8.973870 9.914708 10.359685 10.859114 11.366282 10.345944 9.914688 8.973856 7.027785 6.478917 7.778410 8.973821 10.359630 9.637848 7.628478 5.771230 4.499937

30.043858 52.851250 69.049495 75.000000 75.000000 75.000000 74.122084 75.000000 74.999423 75.000000 75.000000 75.000000 74.990051 75.000000 74.980513 75.000000 74.994707 75.000000 75.000000 74.998918 75.000000 69.526160 75.000000 66.179849

95.968650 98.506939 98.396260 98.614840 98.535293 98.739060 98.236245 98.370326 100.263550 114.622508 125.000000 125.000000 114.748384 100.285572 98.488925 98.516033 98.476689 98.545476 98.520577 114.620012 100.191232 98.529465 98.565162 97.866985

112.699648 112.648958 112.642985 112.613598 112.841055 113.959750 112.659935 140.164632 175.000000 175.000000 175.000000 174.976578 175.000000 174.997632 140.018644 114.001733 112.675240 113.834525 140.025238 175.000000 164.842652 124.842652 112.947486 112.610969

124.898236 124.894519 124.904204 124.883631 138.362628 188.310383 209.662722 209.773696 209.830022 209.842581 210.819077 209.815470 209.849191 209.853008 209.727034 159.747283 138.609412 188.609412 209.687638 209.944276 209.828155 180.012420 130.109748 124.696736 189.765851 1.3%

50.000000 50.124174 74.735292 124.724988 139.739639 139.769662 139.634388 139.663916 139.821479 139.893955 145.043910 166.654080 139.772014 139.778440 139.757898 139.756947 139.723003 139.790209 139.739504 139.796248 139.782435 139.737992 116.152897 66.152897

3.610391 4.025840 4.728236 5.837057 6.478615 7.778855 8.315374 8.972570 9.914474 10.359044 10.862987 11.446128 10.359640 9.914651 8.973014 7.021995 6.479052 7.779622 8.972956 10.359455 9.644474 7.648690 5.775294 4.507437

population and a previous one. The archive aims to enhance the population diversity. (2) Either of the two mutation strategies that have different characteristics is chosen based on the population similarity, which is beneficial to balance the local and global search. (3) The crossover rate of each individual is updated based on the quality of the current individual, which is helpful to improve the quality of offspring. (4) A constraint handling mechanism is

proposed to obtain feasible solutions. (5) A multi-objective optimization method, that is a normalized approach (NA), is adopted to better ease the conflict between the fuel cost and emission. CHM, EFDE and NA have together been applied to four DEED cases. The obtained results reveal that CHM can better satisfy various constraints compared to other approaches, EFDE is more efficient than other approaches in minimizing the fuel cost and

24

X. Shen et al. / Energy 186 (2019) 115801

Table 17 The best generation schedules of the 10-unit system with transmission losses obtained by EFDE for dynamic economic dispatch [10]. Hour

P1

1 150.000000 2 150.000000 3 150.000000 4 150.000000 5 150.000000 6 150.000000 7 150.000000 8 177.018295 9 257.018295 10 288.962773 11 368.962773 12 344.530716 13 338.415489 14 258.415489 15 178.415489 16 150.000000 17 150.000000 18 150.000000 19 228.849272 20 308.849244 21 265.028935 22 185.028935 23 150.000000 24 150.000000 Total losses Percentage of losses

P2

P3

P4

P5

P6

P7

P8

P9

P10

Loss

135.000000 135.000000 135.000000 135.000000 135.000000 135.000000 188.594576 229.532927 309.532927 384.619669 396.897082 470.000000 390.001073 310.001073 230.001073 150.001073 135.000000 150.034554 230.034554 310.034554 301.518653 221.518653 141.518653 135.000000

75.378010 96.196515 170.332919 204.249366 265.277307 323.176173 340.000000 339.999999 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 299.190990 297.397879 306.387004 299.489818 340.000000 340.000000 264.310138 184.310138 114.973381

120.415252 126.558313 176.558313 226.558313 241.433321 291.433321 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 250.000013 243.209880 293.209880 300.000000 300.000000 300.000000 250.000000 200.000000 180.830074

172.733107 222.599655 222.599660 222.599520 222.599896 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 242.999999 242.999999 222.599720 243.000000 243.000000 243.000000 243.000000 222.919438 222.244232 222.599642

122.449868 122.474444 122.449834 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 126.442580 160.000000 160.000000 160.000000 160.000000 160.000000 122.425253 122.449831 1290.505107 3.24%

129.590445 129.590445 129.590451 129.590635 129.590449 129.999988 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 129.999996 130.000000 129.590457 130.000000 130.000000 130.000000 130.000000 129.590522 129.590441 129.590441

120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 119.999998 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000

20.000000 20.000000 20.000000 50.000000 52.057072 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 52.057069 52.057068 80.000000 80.000000 80.000000 80.000000 80.000000 50.634983 20.634983

10.000000 10.000000 40.000000 43.421246 43.421198 43.421200 43.422544 54.855767 55.000000 55.000000 55.000000 55.000000 55.000000 53.155292 53.003289 43.421206 43.421177 43.421238 43.421227 55.000000 55.000000 43.421206 43.271642 13.271642

19.566683 22.419372 28.531177 35.419079 39.379243 48.030682 53.017120 58.406988 70.551222 79.582442 87.859855 92.530716 84.416562 70.571853 58.419846 43.670349 39.718759 48.052676 58.794871 74.883798 70.547588 48.788892 31.995342 25.349994

Table 18 The best generation schedules of the 10-unit system with transmission losses obtained by EFDE for dynamic emission dispatch [10]. Hour

P1

1 150.000000 2 150.000000 3 165.310193 4 200.866972 5 218.290805 6 254.749344 7 274.229871 8 292.367510 9 325.170264 10 348.644430 11 382.809466 12 407.113227 13 364.081322 14 325.236559 15 293.412242 16 233.731747 17 218.290790 18 254.749325 19 290.056735 20 336.686118 21 333.452351 22 253.452351 23 176.048255 24 150.567236 Total losses Percentage of losses

P2

P3

P4

P5

P6

P7

P8

P9

P10

Loss

135.000000 137.112030 165.622749 201.258920 218.710114 255.192540 274.665372 292.782224 325.527503 348.960803 383.086773 407.373380 364.376293 325.596191 293.816100 234.158678 218.710051 255.192542 290.556175 337.026531 333.949859 253.949859 176.379611 150.830255

90.875090 101.490541 120.297425 145.267910 158.514017 190.011126 210.048823 231.559543 278.445350 316.325974 340.000000 340.000000 340.000000 278.551136 232.881216 171.111219 158.514052 190.011135 228.663958 296.898039 282.545831 202.545831 127.670659 110.460383

90.939229 101.573947 120.413034 145.399230 158.649607 190.156302 210.202130 231.716691 278.590937 300.000000 300.000000 300.000000 300.000000 278.352532 228.352532 178.352532 158.649692 190.156311 239.073354 289.073354 258.087990 208.087990 158.087990 110.563497

129.600369 144.801263 171.235712 204.843486 221.698744 243.000000 243.000000 242.999998 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 236.930834 221.698702 243.000000 243.000000 243.000000 243.000000 214.974203 181.281196 157.542841

129.703897 144.934820 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 157.662493 1315.298636 3.3%

97.311411 108.800668 129.060855 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 118.477896

97.267997 108.743487 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 118.407953

80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000

55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000

19.697994 22.456756 28.939969 36.636519 40.863288 50.109313 55.146195 60.425966 71.734053 79.931207 87.896239 92.486607 84.457615 71.736419 60.462090 45.285011 40.863286 50.109313 60.350221 75.684043 72.036031 50.010234 32.467711 25.512555

X. Shen et al. / Energy 186 (2019) 115801

25

Table 19 The best generation schedules of the 10-unit system with transmission losses obtained by EFDE for dynamic economic emission dispatch [10]. Hour

P1

1 150.000000 2 150.000000 3 150.000000 4 150.000000 5 162.126651 6 226.624211 7 230.734300 8 254.917581 9 303.245342 10 334.385407 11 379.872633 12 406.041570 13 361.327929 14 303.246723 15 255.023039 16 206.329455 17 162.126277 18 226.624226 19 253.598556 20 314.978259 21 306.893129 22 226.893129 23 150.000000 24 150.000000 Total losses Percentage of losses

P2

P3

P4

P5

P6

P7

P8

P9

P10

Loss

135.000000 135.000000 135.000000 168.475676 214.647295 222.266463 252.550300 284.760030 309.532925 342.145975 386.013567 408.440986 367.120716 309.532925 284.828099 222.266218 214.646556 222.266465 283.894107 324.527486 309.496140 229.496140 149.496140 135.000000

81.310613 94.270155 146.801784 175.592558 185.199737 213.157298 243.674794 260.869654 297.177886 337.070597 340.000000 340.000000 340.000000 297.396901 260.918945 185.199740 185.199735 213.157296 260.203946 319.635915 309.815090 229.815090 149.815090 98.185656

94.435861 116.553419 130.682251 180.060222 180.830487 227.229946 241.246278 247.182952 297.182952 300.000000 300.000000 300.000000 300.000000 296.962761 246.962761 196.962822 180.830486 227.229932 250.000801 300.000000 281.017360 231.017360 181.017360 131.017360

122.866555 129.031832 179.031832 222.599650 232.325651 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 232.326756 243.000000 243.000000 243.000000 243.000000 215.532491 189.019090 172.732962

122.449847 127.932435 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 137.903974 1303.18686 3.27%

94.686770 124.686770 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 129.590967 129.594358 129.590441

120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000

80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000

55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000

19.749645 22.474611 28.515867 35.728106 40.129821 49.277918 54.205673 59.730217 71.139105 79.601979 87.886200 92.482556 84.448645 71.139310 59.732844 44.758234 40.129811 49.277918 59.697411 75.141661 71.221719 49.345178 31.942038 25.430393

Table 20 The best generation schedules of the 10-unit system without transmission losses or POZs obtained by EFDE for dynamic economic dispatch [10,39]. Hour

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

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

150.000000 150.000000 150.000000 150.000000 150.000000 150.000000 150.000000 150.000000 226.624211 284.467075 288.467075 325.200612 303.248422 226.624211 150.000000 150.000000 150.000000 150.000000 161.706107 241.706107 226.624210 150.000000 150.000000 150.000000

135.000000 135.000000 135.000000 135.000000 135.000000 135.000000 135.000000 209.578801 269.375789 309.532925 389.532925 396.799388 340.751578 269.375789 209.578801 135.000000 135.000000 142.293893 222.293893 302.293893 269.375790 189.375790 135.000000 135.000000

73.000000 79.712213 147.712863 185.132002 226.079966 296.136007 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 340.000000 291.331641 226.085905 297.400656 315.578800 340.000000 340.000000 269.081980 189.081980 143.395812

103.226622 120.463967 170.463967 211.622257 241.251636 291.251636 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 300.000000 250.000000 241.245732 290.235946 300.000000 300.000000 300.000000 250.000000 200.000000 180.830489

172.733100 222.599652 222.599650 222.598234 222.599655 222.600710 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 222.599651 222.599656 243.000000 243.000000 243.000000 243.000000 236.530556 222.470743 172.733100

122.449836 122.450497 122.449849 159.999997 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 122.435334 122.449855

129.590441 129.590441 129.590441 129.590441 129.590476 129.590448 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 129.590441 129.590441 129.590904 130.000000 130.000000 130.000000 129.590476 129.590441 129.590442

120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000

20.000000 20.183229 50.183229 52.057068 52.057068 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 52.057068 52.057068 52.057403 80.000000 80.000000 80.000000 80.000000 50.000303 20.000303

10.000000 10.000000 10.000000 40.000000 43.421199 43.421198 44.000000 43.421199 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 43.421199 43.421198 43.421198 43.421198 43.421199 55.000000 55.000000 43.421198 13.421198 10.000000

26

X. Shen et al. / Energy 186 (2019) 115801

Table 21 The best generation schedules of the 10-unit system without transmission losses or POZs obtained by EFDE for dynamic emission dispatch [10,39]. Hour

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

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

150.000000 150.000002 160.346835 193.610439 210.429005 242.973362 262.135798 280.063762 312.249995 331.508703 350.597995 362.718478 341.280490 312.249992 280.121561 226.777620 210.428980 242.973349 278.990356 322.638371 318.125000 239.856652 170.608816 150.000000

135.000000 135.000000 160.346842 193.610430 210.428950 242.973370 262.135779 280.063767 312.249982 331.508698 350.597992 362.718462 341.280504 312.250000 280.122300 226.761771 210.428979 242.973376 278.991785 322.635257 318.170831 239.813293 170.564283 147.033024

88.045271 98.542798 116.052748 138.651375 150.852976 177.026645 194.864210 213.936232 255.750020 285.491303 316.804014 336.563060 301.439009 255.750003 214.021114 163.487750 150.852960 177.026632 212.724405 271.432919 260.437182 180.437182 122.934029 107.375483

88.045257 98.542806 116.052734 138.651332 150.852945 177.026623 194.864213 213.936245 255.750003 285.491296 300.000000 300.000000 300.000000 255.750006 213.735025 163.735687 150.852961 177.026643 217.293453 267.293453 239.266987 189.266987 139.266987 107.375475

125.631489 140.739139 165.550855 196.476424 212.436123 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 242.999998 243.000000 243.000000 228.237173 212.436119 243.000000 243.000000 243.000000 243.000000 233.625885 183.625885 153.326414

125.631490 140.739122 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 153.326386

94.323240 105.718060 124.649986 130.000000 130.000000 130.000000 130.000000 129.999994 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 115.281616

94.323253 105.718074 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 115.281602

80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000

55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000

Table 22 The best generation schedules of the 10-unit system without transmission losses or POZs obtained by EFDE for dynamic economic emission dispatch [10,39]. Hour

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

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

150.000000 150.000000 150.000000 150.000000 154.508715 226.624210 226.624211 239.020912 289.424454 308.821292 338.125618 358.437318 326.155648 289.424305 240.763649 176.207958 154.509349 226.624210 239.020989 300.534085 301.889260 221.889260 150.000000 150.000000

135.000000 135.000000 135.000000 152.349980 201.511636 222.266463 229.643884 265.144422 304.017889 317.671428 344.517098 363.562682 333.687680 304.017997 267.820862 219.115830 201.511339 222.266463 265.143633 309.532925 302.266256 222.266256 142.266256 135.000000

85.363093 90.569046 135.072312 166.569838 176.179381 196.748856 222.026128 242.588937 275.550520 307.507280 335.357283 340.000000 324.156672 275.550536 243.939015 185.199737 176.179421 196.748862 242.589647 287.332702 280.339079 200.339079 134.825930 95.306970

102.232282 110.599785 120.604141 169.480485 180.200617 194.360471 235.705777 241.245730 267.007137 300.000000 300.000000 300.000000 300.000000 267.007162 235.476475 185.476475 180.200241 194.360465 241.245730 286.600288 251.505406 201.505406 151.505406 115.954068

122.866550 122.866550 172.733106 222.599697 222.599650 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 243.000000 242.999999 222.599650 243.000000 243.000000 243.000000 243.000000 237.000000 208.811966 172.733099

122.450428 122.877047 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 160.000000 130.415422

93.087572 123.087572 129.590441 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 130.000000 129.590443 129.590441

90.000075 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000 120.000000

80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000 80.000000

55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000 55.000000

emission, and the proposed NA is more competitive than the other two normalized approaches in balancing the fuel cost and emission. In view of the excellent performance of our proposed method, it is expected to be promising for solving the other DEED problems such as DEED considering wind power [34], DEED considering plug-in electric vehicles charging [35] and DEED with demand side management [83]. In addition, EFDE has an underlying shortcoming. Several local parameters are contained in EFDE, and these local parameters values need extra optimization trials to find the most suitable ones. Our further work will focus on reducing the local parameters of the algorithm.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 61403174 and 61573172), Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17 1575).

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