Since wind power is integrated into the thermal power operation system, dynamic economic emission dispatch (DEED) has become a new challenge due to its uncertain characteristics. This paper proposes an adaptive grid based multi-objective Cauchy differential evolution (AGB-MOCDE) for solving stochastic DEED with wind power uncertainty. To properly deal with wind power uncertainty, some scenarios are generated to simulate those possible situations by dividing the uncertainty domain into different intervals, the probability of each interval can be calculated using the cumulative distribution function, and a stochastic DEED model can be formulated under different scenarios. For enhancing the optimization efficiency, Cauchy mutation operation is utilized to improve differential evolution by adjusting the population diversity during the population evolution process, and an adaptive grid is constructed for retaining diversity distribution of Pareto front. With consideration of large number of generated scenarios, the reduction mechanism is carried out to decrease the scenarios number with covariance relationships, which can greatly decrease the computational complexity. Moreover, the constraint-handling technique is also utilized to deal with the system load balance while considering transmission loss among thermal units and wind farms, all the constraint limits can be satisfied under the permitted accuracy. After the proposed method is simulated on three test systems, the obtained results reveal that in comparison with other alternatives, the proposed AGB-MOCDE can optimize the DEED problem while handling all constraint limits, and the optimal scheme of stochastic DEED can decrease the conservation of interval optimization, which can provide a more valuable optimal scheme for real-world applications.
Citation: Zhang H, Lei X, Wang C, Yue D, Xie X (2017) Adaptive grid based multi-objective Cauchy differential evolution for stochastic dynamic economic emission dispatch with wind power uncertainty. PLoS ONE 12(9): e0185454. https://doi.org/10.1371/journal.pone.0185454
Editor: Mauro Villarini, Universita degli Studi della Tuscia, ITALY
Received: May 3, 2017; Accepted: September 13, 2017; Published: September 29, 2017
Copyright: © 2017 Zhang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: This work is supported by the National natural fund (no. 61503199), the National natural science key fund (no. 61533010), the Jiangsu Province natural science fund (no. BK20150853), the Jiangsu Province high school natural science fund (no. 15KJB120009), the Ph.D. Programs Foundation of the Ministry of Education of China (no. 20110142110036), NUPTSF (no. NY214206), and the Open Fund (grant no. XJKY14019).
Competing interests: The authors have declared that no competing interests exist.
Dynamic economic dispatch (DED) plays a key role in power system operation: it assigns optimal output into power generators according to the system load requirement in a certain time period. The main goal of DED is to minimize economic cost while satisfying the output limits, ramp rate limits and transmission loss among different power generators. In comparison to static economic dispatch (ED), DED takes ramp rate limits into consideration, which makes solving DED problems very challenging. Generally, DED can be optimized via the static economic dispatch method by dividing the dispatch period into several intervals [1–3], and many optimization methods have been utilized to solve DED problems, such as evolutionary programming (EP) [4, 5], goal-attainment method , quadratic programming (QP) , and particle swarm optimization (PSO) .
With the increased concern regarding environmental protection, emission pollutants from thermal power generation have received increasing attention. To reduce emission pollutants, especially sulfur oxide and nitric oxide, the Clean Air Act Amendments were ordered to modify the design or operational strategy. However, modifying the design or replacing clean equipment can be quite expensive, and it can be merely taken as a long-term option. The dispatch method for reducing the emission pollutants can be an effective way in the short term and can take emission pollutants into consideration in the DED problem . DED with emission pollutants can be called dynamic economic emission dispatch (DEED), and it is generally considered a multi-objective optimization problem (MOP), in which both the fuel cost and emission rate can be minimized.
Recently, many alternatives have been proposed to solve the DED problem with emission pollutants; generally, three main research directions can be concluded: 1) Treating emission rate as a constraint limit during the optimization process ; 2) Converting the MOP into a single-objective optimization problem (SOP) with the weighting technique [11, 12]; 3) Optimizing fuel cost and emission rate simultaneously [13–19]. The first method can only optimize the emission rate to a certain degree, but it cannot minimize the emission rate at extreme effort. In the second method, the exact weight value can be difficult to obtain for real-world applications. The third method can obtain a set of Pareto optimal solutions for different situations without prior knowledge of objective weights; thus, it can be naturally taken as a relatively efficient way to solve the DEED problem. Multi-objective evolutionary algorithms (MOEAs) have also received great attention. Based on the Pareto optimal theory, MOEAs generate Pareto optimal solutions in a single simulation run. In literature , the non-dominated sorting genetic algorithm-II (NSGA-II) was used to optimize the DEED problem, which takes emission rate and fuel cost as competing and non-commensurable objectives. In literature , multi-objective differential evolution (MODE) was improved to solve the DEED problem and was implemented for a 10-thermal-unit system to verify its efficiency. In literature , multi-objective particle swarm optimization (MOPSO) was developed to solve the DEED problem.
Since high penetration of wind power has increasingly become a great challenge to DEED due to its uncertain characteristics , great deviation may exist between forecasted wind output and actual wind output, most conventional deterministic methods cannot properly solve DEED with wind power uncertainty. In this paper, scenario technique is utilized on stochastic DEED model to simulate those possible situations caused by wind power uncertainty, and feasible domain of wind power output is divided into several intervals with different probabilities to generate those scenarios. For improving diversity distribution of obtained Pareto front, adaptive grid based multi-objective Cauchy differential evolution is proposed to optimize the stochastic DEED model, Cauchy mutation operation improves differential evolution by adjusting the population diversity, and adaptive grid mechanism retains the diversity distribution of obtained Pareto front. After the proposed AGB-MOCDE is implemented on three test systems, 5-thermal-unit and 10-thermal-unit systems are used to prove the efficiency of AGB-MOCDE and the thermal-wind power system is used to prove its optimization ability to deal with the stochastic DEED problem.
This paper is organized as follows: Section 2 presents the problem formulation of DEED with wind power uncertainty, some basic information is introduced in Section3, and the details of proposed AGB-MOCDE are substantially presented in Section 4. For testifying the efficiency of the proposed method, the implementation details of AGB-MOCDE for stochastic DEED with wind uncertainty are described in Section 5, and the case study and conclusion are presented in Section 6 and Section 7, respectively.
Due to the uncertainty of wind power generation, generated scenarios are taken to simulate those possible situations caused by wind power uncertainty . Scenario based technique has been an efficient method for solving stochastic optimization problem, probability distribution of each variable can be represented with a finite set of scenarios, which correspond with realizations of those random variables during the time span . The uncertainty domain of wind output can be divided into several intervals, each situation can be included in one interval, and the probability of each interval can be calculated combined with probability density function of wind power generation. The expected value of fuel cost and emission rate can be taken as two objectives, and the conventional DEED model can be developed into a stochastic DEED combined with generated scenarios.
1. Economic objective
Since it doesn’t exist economic cost for wind power generation in thermal-wind power system, fuel cost can be taken as the main economic cost. The fuel cost can be expressed by the summation of quadratic function and sinusoidal function of thermal output, which is caused by the sudden opening of the intake valve of a steam turbine. Generally, economic cost with the valve-point effect can be properly described as follows : (1) where T represents the total time period length, Nc is number of thermal units, Pci,t represents the output of the ith thermal unit in the tth time period, Pci,min is the minimum output of the ith thermal unit, and ai,bi,ci,di,hi are the cost coefficients of fuel cost at the ith thermal unit. In the stochastic model, combined with the probability density of wind power output, the expected value of economic cost can be expressed as follows: (2) where E(∙) represents the expected value of economic cost, ρs is the probability that the sth scenario occurs, is the output of the ith thermal unit in the tth time period in the sth scenario, and Ns is the number of generated scenarios.
2. Emission objective
The emission pollutant of thermal units represents another objective in the DEED problem, it mainly consists of sulfur dioxide and nitrogen oxides, and can be generally expressed with thermal output. Generally, emission pollutant caused by thermal units can be described as a combination of polynomial and exponential terms, which can be formulated as : (3) where αi,βi,γi,ζi,λi are the coefficients of emission rate at the ith thermal unit. Similarly, the expected value of the emission rate can be expressed as follows: (4)
(1) Power balance constraint [6, 25] (5) where is the output of the jth wind farm in the tth time period in the sth scenario, are system load requirement and transmission loss, respectively, in the tth time period in the sth scenario, and Nw is the number of wind farms. The transmission loss among different power generators is taken into consideration, which can be generally expressed as follows: (6) where Bij,B0i,B00 are the coefficients of transmission loss and is the output of the ith power generator in the tth time period in the sth scenario.
(4) The relationship between wind output and wind speed
The wind power output is closely related to wind speed; its relationship can be generally described as follows : (9) where is the wind speed of the jth wind farm in the tth time period in the sth scenario, vj,rate is the rated wind speed of the jth wind farm, vj,in,vj,out are the cut-in and cut-out wind speed, respectively, at the jth wind farm, and Pwj,max is the maximum output at the jth wind farm.
1. The description of the multi-objective optimization problem
The MOP generally has two or more competing objectives and cannot generate a single optimal solution; rather, it generates a set of non-dominated solutions in a single simulation run. In comparison to single-objective optimization, the fitness function cannot properly describe the dominance relationship among those individuals in an evolutionary population. For a given MOP, the MOP can generally be described as follows: (10) where F(∙) is the objective vector, which contains m objective functions, gj(∙) is the j th inequality constraint, hk(∙) is the k th equality constraint, and Sg,Sh are the number of inequality and equality constraints, respectively.
In comparison to single-objective optimization, multi-objective optimization utilizes the Pareto dominance relationship to decide which individual should be better. Because the objectives in the MOP are generally in conflict with each other, the relationship among different individuals cannot be easily described with a certain order. In the MOP, the Pareto dominance relationship with a partial order is utilized to describe the relationship among those individuals; the dominance order and Pareto optimal solution can be properly obtained using this dominance relationship.
The Pareto dominance relationship between x1 and x2 is that x1 dominates x2 and that x1 has higher Pareto dominance order than x2, which can be denoted as x1 ≻ x2.
Definition 2: Assume that no solutions can dominate x* in the solution set; x* is called the non-dominated solution or Pareto optimal solution in the solution set, which can be described as: ∀i = 1,2,…,n,¬xi ≻ x*.
2. Outline of differential evolution (DE)
DE is widely known as a simple yet powerful optimization algorithm due to its smaller number of parameters and population-based evolutionary strategy. The basic operations of DE include mutation, crossover and selection, the details about DE has been presented in literature . Combined with the above three basic operations, DE can guide individual vectors to move into the optimal solution. However, conventional DE cannot avoid a premature problem as other evolutionary algorithms can due to its fixed scaling parameter; it is apt to fall into local optima, especially for multi-modal functions. Here, the Cauchy mutation method is utilized to improve the differential evolution by checking population diversity, which can guide population evolution adaptively.
Interactive fuzzy satisfying-based adaptive grid-based multi-objective Cauchy differential evolution
For properly optimizing stochastic DEED problem, the AGB-MOCDE algorithm is proposed to improve the optimization efficiency as follows: (1) For avoiding the premature problem, adaptive Cauchy mutation operator is utilized to search global optimal solution by properly controlling population diversity during evolution process; (2) The adaptive grid based diversity maintenance strategy is proposed to properly control the diversity distribution, it makes obtained Pareto-optimal solutions evenly distributed, which can be convenient for decision-makers; (3) Interactive fuzzy satisfying method is utilized as a selection mechanism to screen the best optimal solution from those obtained nominated solutions, and the selected solution can be taken as the best scheme for DEED problem.
1. Adaptive Cauchy mutation-based multi-objective differential evolution
The essential reason for the premature problem is that the diversity of the population decreases during the evolution process . To avoid the premature problem, population diversity can be checked using the convergence metric C(P(g)), which denotes convergence progress as the number of generations increases . If the convergence metric C(P(g)) does not change obviously after h generations, which also means that ∇CP is smaller than a threshold ξCP ∈ [0.01,0.1], where ∇CP is defined as |C(P(g))−C(P(g−h))|/C(P(g)), check the population diversity. If the population diversity is less than a threshold ε, adaptive Cauchy mutation is carried out to update the current individuals and is expressed as follows: (13) where C(0,1) is a standard Cauchy variable, η ∈ [0.1,0.5] is a coefficient of Cauchy mutation, ε is the diversity threshold, and diversity(j) is the diversity value on the jth dimension, which is described as follows: (14) where NQ is the size of the archive set, is the average value of the jth dimension variables, and uj,lj are the upper and lower bounds, respectively, of the jth dimension variables. According to formulation (13), parameter η can adjust the search scale of the Cauchy mutation operation; the linear formulation in literature  is taken to control the mutation operation as population evolution proceeds. The linear formulation is presented as follows: (15) where Gmax is the maximum generation number. Combined with the above adaptive mutation operation, the differential evolution process can be properly controlled using population diversity, and it can adjust the search scale using adaptive scaling parameters, which can avoid the premature problem as population evolution proceeds.
2. Adaptive grid-based diversity maintenance strategy
Since archive set has a certain size, diversity maintenance strategy is needed when archive set is full. Here, an adaptive grid-based diversity maintenance strategy is proposed to improve diversity distribution of optimal individuals in each generation. Two-dimensional coordinate is divided into several small boxes, which are taken to measure the diversity distribution of Pareto optimal front.
3.1 Grid setting.
In the current archive set, the extreme values of optimal individuals can be found, it labels minimum value and maximum value of objective function F1 as , and minimum value and maximum value of objective function F2 as , and all those optimal individuals can be shown in sorted order in a two-dimensional coordinate. The grid division depends on the number of non-dominated solutions, and each square can be obtained: (16)
The represents the square length in F1,F2 objective direction, and feasible region can be divided into NQ × NQ pieces. Those optimal individuals scattered on those pieces need to keep certain distribution characteristics for ensuring diversity distribution of Pareto front. For each optimal individual, it satisfies: (17)
3.2 Grid based diversity maintenance mechanism of Pareto front.
Since archive set has a certain size, truncation mechanism must be taken to keep the elitism when the number of Pareto optimal individuals exceed the size of archive set. On the basis of above grid setting, the diversity distribution is taken to justify whether newly generated optimal solution is added into archive set when archive set is full. If generated optimal individuals B1,B2 are in the grids dominated by those individuals A1,A2 in archive set as it is shown in Fig 1, the newly generated solution can’t be added into the archive set.
If newly generated individual B3 and Pareto optimal individual A3 are in the same grid, diversity distribution of Pareto optimal front is taken to judge which one should be replaced by the other one. The main procedures are presented as follows:
- Step.1: Add the newly generated individual into archive set, calculate the number of individuals in each grid , which satisfies and ki ∈ N.
- Step.2: Find the , and if there are Ngrid grids that contain kmax individuals, and select all individuals in these grids.
- Step 3: Calculate density degree of these individuals with following metrics:
- Step.4: Select the individual that has the largest density, and delete this individual from the archive set, then archive set can be properly maintained to certain extent. Especially when newly generated individual has better minimum value in objective in F1 or F2, this new individual can be directly added into archive set and delete one individual that has the largest density.
3. Interactive fuzzy satisfying method as a selection mechanism
Due to the uncertain features of wind power and a decision-maker’s judgment, objective values can be naturally considered fuzzy or imprecise, which brings a challenge to the conventional selection mechanism in multi-objective optimization. This paper utilizes the interactive fuzzy satisfying method to decide which individual vector in the archive set is the best optimal individual vector. The satisfaction of the individual vector in the archive set can be described using membership functions, which can be expressed as follows [30, 31]: (20) where are the minimum and maximum value, respectively, of the qth objective function. In the decision-making process, desirable levels of the membership function or reference membership value μrq need to be specified; the best optimal solution can be selected by solving the mini-max problem, as follows: (21)
In the selection operation, formulation (24) is taken to decide which individual is the best individual that satisfies the decision maker’s requirement. The fuzzy satisfying method takes the uncertainty of environmental circumstances and subjective consciousness into consideration, being able to obtain the best optimal solution, which can be a better fit for real-world applications.
Implementation of fuzzy satisfying method-based multi-objective differential evolution for a stochastic DEED problem
Because DEED with wind power uncertainty has various coupled constraints with randomness characteristics, the implementation of the proposed algorithm has a great impact on the efficiency of solving the DEED problem. The maximum and minimum output of wind power can be obtained according to its probability density function (PDF), and the output domain can be equally divided into several intervals. Simultaneously, the system load balance constraint with transmission loss connects the thermal units and wind farms together; its constraint handling technique plays an important role in solving the DEED problem. Here, the coupled constraint-handling technique simplifies the system load balance constraint, which ensures that the proposed MODE can be properly implemented for the DEED problem with wind power uncertainty.
1. The solution-encoding strategy
In the DEED problem, a set of thermal outputs with Ns scenarios is taken as the decision variable. In each scenario, there are T scheduling periods on the schedule horizon, and the output of Nc thermal units needs to be properly assigned in each scheduling period; the decision variable can be encoded as follows: (22)
2. The probability of the output interval under wind power uncertainty
The integration of wind power generation brings strong randomness into the DEED problem, which presents a great challenge for optimization methods. The randomness characteristics of wind power generation need to be analyzed with its probability density function and cumulative distribution function, which can be generally described based on the wind speed in literature : (23) where vj is the wind speed of the jth wind farm; k,c are the scaling parameters. Combined with the relationship presented in formula (9), the cumulative distribution function for wind power output can be obtained as: (24)
3. The division interval of the wind power output
Regarding the randomness of wind power generation, the feasible domain of wind power output can be divided into seven intervals, with different probabilities for each interval. The feasible domain can be equally divided, and the probability of each interval can be properly calculated using the method in Section 4.2. For a given feasible domain [lj,uj], the interval division of wind power generation can be obtained as in Fig 2.
Because scenarios of the thermal wind-power system simulate those possible wind outputs during the entire time period, the binary parameter of wind power generation exists in each interval; the interval index interval is selected when interval = 1; otherwise, the interval index interval is not selected. In each generated scenario, the probability of the generated scenario can be described by the Bayesian probability formulation: (26) where represents the binary parameter of the output interval of wind power generation in the jth wind farm in the tth time period in the sth scenario.
4. The reduction of scenarios number
Since a lot of scenarios need to be generated to simulate the power generation process in the power system operation, it makes large dimensionality for high computational complexity. Here, a reduction of scenarios method is proposed to screen out those similar scenarios for simulating more power generation process with less scenarios, a covariance based method is utilized to find covariance between every two scenarios, and screen out one similar scenario with minimal covariance, which can retain the efficiency of the scenario based method. The procedures of covariance based scenarios reduction method can be presented as follows:
- Step 1: Initialize the reduced scenario set Scenreduced = Scen, and the size of reduced scenarios set is , required scenarios number is , Si represents the ith scenario in Scenreduced, go to Step 2;
- Step 2: Define the covariance value set , which can be considered as the covariance matrix. Calculate the covariance between every two scenarios (i,j ∈ 1,2…Ns), and covariance value set can be initialized with Covij = Covariance(Si,Sj), go to Step 3;
- Step 3: Calculate the eigenvalues of covariance matrix Cov, and it can obtain eigenvalues, and go to Step 4;
- Step 4: Select the maximum eigenvalue and eigenvector , and compare the eigenvector with each scenario, and find two most similar scenarios and , and delete one , and go to Step 5;
- Step 5: Then , , and ρeigen,max1 = ρeigen,max1 + ρeigen,max2; and go to Step 6;
- Step 6: If , go to Step 2. Otherwise, complete the scenarios reduction procedures and exit.
5. The constraint-handling method for system load balance
The proper constraint-handling technique can improve the efficiency of the proposed algorithm, especially when the system load balance constraint is properly handled. Here, transmission loss among different power generators is taken into consideration, which increases the difficulty of handling the system load balance constraint. A constraint technique is utilized to decrease the deviation of the system load balance constraint, which can be calculated as follows: (27)
Replace transmission loss Lloss,t with formulation (6); then, the variation of the deviation can be found as follows: (28)
According to the above increment of deviation, thermal output can be properly adjusted using the coupled constraint-handling technique in literature . The system load balance constraint can be properly handled using the coarse adjustment and fine tuning technique, and the total constraint violation is utilized to ensure the feasibility of those individuals in the evolutionary population.
6. The flowchart of implementation for the DEED problem with wind power uncertainty
The DEED with wind power uncertainty is a complex-coupled optimization problem; all the procedures must be substantial and intact. The interval of wind power output can be equally divided into seven intervals in each scenario, and the probability of each interval can be calculated by solving formulation (25) using the probability density function; the obtained probability of each interval can be input into the DEED model. Then, the proposed MODE with the fuzzy satisfying method can be used to solve the DEED model with the constraint handling technique. The procedure details can be described as follows:
- Step 1: According to the wind output domain of each wind farm in each time period, generate Ns scenarios in the thermal wind-power system, divide the uncertainty domain of each wind output into seven intervals, calculate the probability of each interval, and input it into the DEED model; go to Step 2;
- Step 2: Combined with DEED under wind power uncertainty, take thermal output as the decision variable, initialize all the parameters and the evolutionary population, and set the generation number g = 0; go to Step 3;
- Step 3: Check the population diversity diversity (j); if diversity (j) < ε, adaptive Cauchy mutation is taken to update the individual vector; then, go to Step 4;
- Step 4: Check the feasibility of individuals in the evolutionary population and use the constraint-handling technique to deal with those infeasible individuals; go to Step 5;
- Step 5: Apply the crossover and selection operations to the evolutionary population, screen out those non-dominated individuals and store them into the archive set via the archive retention strategy; go to Step 6;
- Step 6: If g < gmax, g = g + 1; go to Step 3; otherwise, output all the non-dominated individuals in the archive set and go to Step 7;
- Step 7: For a given reference membership value μrq, select the optimal individual that has the best satisfying value from the archive set; the selected optimal individual vector is the optimal solution of the DEED problem.
Experimental results and discussion
In this section, the proposed AGB-MOCDE is implemented for three test systems, the whole scheduling time period is 24h with each hour an interval. Test system 1 consists of five thermal units with minimizing fuel cost and emission rate, and non-linear power generation loss is taken into consideration combing with valve point effect. Test system 2 consists of 10 thermal units with the valve-point effect, considering transmission loss among the thermal units, it mainly demonstrates the efficiency of the proposed multi-objective optimization method, and scenario and interval division methods are not required in this test system. Test system 3 consists of 10 thermal units and 3 wind farms with the valve-point effect, considering transmission loss among the thermal units and wind farms, it does not merely verify the efficiency of the proposed multi-objective optimization method, and it also shows its ability to deal with wind power uncertainty.
1. Test system 1
The main goal of this test system is to properly assign the output of five thermal units to minimize fuel cost and emission rate simultaneously, all data details about five thermal units are presented in literature . In Fig 3, 30 non-dominated solutions are produced by AGB-MOCDE method, and are shown with obtained schemes by MODE , which is a Pareto dominance based algorithm with DE/rand/1/bin strategy. It can be seen that all non-dominated solutions by AGB-MOCDE are more evenly distributed than that in MODE due to its adaptive grid-based mechanism, and has better convergence ability than MODE.
For further analysis on efficiency of obtained Pareto front, compromise schemes represents those obtained Pareto fronts with calculating fuzzy satisfying membership of those non-dominated schemes, which are presented in Table 1. According to formula (21), if it is assumed that reference membership value μrq is 0.75, scheme (14) can be selected as compromise scheme by AGB-MOCDE and scheme (16) is taken as compromise scheme by MODE.
In Table 2, SA , MSL , EP , PS , PSO  and MODE are taken to compare those obtained results for minimum fuel cost, minimum emission rate and compromise results. In comparison to other alternatives for solving this five-unit problem, it can be found that AGB-MOCDE can obtain better minimum fuel cost and minimum emission rate than other methods, and obtained compromise scheme dominates all optimal schemes by other alternatives, which means that AGB-MOCDE can properly optimize fuel cost and emission rate simultaneously for solving DEED problem, and it can also retain extreme objective value for completely analysis on obtained Pareto front.
Then, further analysis is taken on the output process of compromise scheme obtained by AGB-MOCDE, the output process is shown in Fig 4. The output of each thermal unit at each time period can properly satisfy those constraint limits, and system load balance at each time period is also properly satisfied with considering transmission loss of five thermal-unit, which has been presented in Table 3. It can be seen that transmission loss period can’t exceed 2% of system load at each time, and nonlinear transmission loss has been properly controlled within permitted accuracy.
According to those obtained results in this test system, the proposed AGB-MOCDE can properly solve DEED problem of five-thermal-unit power system with satisfying those constraint limits. The comparison with MODE reveals that adaptive grid mechanism can improve the diversity distribution of Pareto front, and the comparison with other alternatives on fuel cost and emission rate demonstrates that Cauchy mutation operator can promotes the optimal efficiency of differential evolution. After checking those constraint limits, all the constraint limits are properly satisfied and transmission loss is also controlled in permitted accuracy. Combined with above results, it can be found that the proposed AGB-MOCDE can be taken as a viable alternative for solving DEED problem.
The main parameter settings in this test system are presented as follows: The population size is set to 60, the size of archive set NQ is 30, and maximum generation number Gmax is 1000. Since wind power is not taken into consideration, the scenarios method is not used in this test system. The permitted total violation accuracy is set to 0.1, and the permitted output violation is set to 0.01, and coarse adjustment number is set to 5, and fining tuning number is set to 10. The reference membership value μrq is set to 0.75, and the initial square length represents the square length are set to 200 and 20.
2. Test system 2
In this test system, thermal power output is taken as a decision variable, its main goal is to minimize the fuel cost and emission rate simultaneously, and all details on the data can be found in literature . After the proposed AGB-MOCDE is implemented for the thermal power system, the Pareto front and membership value of non-dominated schemes can be properly obtained, as shown in Fig 5 and Table 4. The Pareto front is obtained using 30 non-dominated schemes; all schemes are evenly distributed on the Pareto front. In comparison to MODE, AGB-MOCDE has better results, while all non-dominated schemes have a wider extreme edge and better diversity distribution.
According to the 30 obtained non-dominated schemes in Fig 5, the minimum cost of AGB-MOCDE is $22437 less than that of MODE, and the minimum emission rate of AGB-MOCDE is 1172 lb less than that of MODE. Among the 30 non-dominated schemes, approximately 20 AGB-MOCDE schemes have a lower fuel cost and a higher emission rate than those of MODE; the obtained results reveal that AGB-MOCDE has higher efficiency than MODE when solving the DEED problem.
For further analysis of the output process of the thermal units, a compromise scheme can be selected from the non-dominated schemes with the fuzzy satisfying method. The reference value μrq is set to 0.75; it can then be calculated that scheme (16) of AGB-MOCDE is taken as the compromise scheme and scheme (16) of MODE is taken as the compromise scheme. In comparison to different results obtained in literature  and literature , it can be found in Table 5 that AGB-MOCDE has both lower minimum fuel cost and minimum emission rate and also has a relatively good compromise scheme. AGB-MOCDE integrates the Cauchy mutation operation into the MODE, which needs to verify the population diversity and calculate the convergence metric. The computational time is longer than that for MODE but still shorter than those for the alternatives in literature  and literature .
The output process of the compromise scheme of AGB-MOCDE is presented in Table 6. All the output of thermal units is properly controlled in the feasible domain; the output limits, ramp rate limits and system load balance are properly satisfied in each time period. According to calculation of the transmission loss in each time period, the transmission loss does not exceed 5% of the system load, which reveals that transmission loss is also controlled properly. Thermal units 8 and 9 maintain the maximum output due to their low power capacity, while thermal units 5, 6 and 7 almost maintain the maximum output during the entire time period.
The main parameter settings are presented as follows: The size of archive set is set to 30, size of evolutionary population is set to 100, and maximum generation number Gmax is 1000, permitted constraint violation accuracy of output is set to 0.01MW, permitted total violation is set to 0.1MW, threshold parameter ξCP is set to 0.04, crossover rate is set to 0.3. According to the above comparison and analysis, it can be found that the proposed AGB-MOCDE has both better convergence ability and diversity distribution when solving the DEED problem. In comparison to MODE and other alternatives, AGB-MOCDE can produce a set of non-dominated schemes that have a wider extreme edge and a more evenly distributed diversity distribution. In the output process, the output of each thermal unit at each time period is properly controlled within the feasible domain; system load balance with transmission loss is also properly dealt with using the proposed constraint handling technique.
3. Test system 3
Since wind power is integrated into the DEED problem, the scenario-based method is utilized to simulate those possible outputs of wind power generation. The uncertainty domain of wind power in each time period is equally divided into several levels, which represent those possible situations caused by wind power generation. In this test system, the uncertainty domain of wind power output in three wind farms is divided into 7 intervals with 8 levels, which are shown in Figs 6–8. Among these divided intervals, interval 1 represents extreme case 1 of the worst situation for wind power generation, while interval 7 represents extreme case 2 of the best situation for wind power generation. Combined with the probability density function of wind power generation, the probability of each interval can be calculated using formulation (25), and the probability of the generated scenario can be calculated using formulation (26).
According to different intervals of wind power output, the proposed AGB-MOCDE is implemented on thermal-wind power system with optimizing fuel cost and emission rate simultaneously, and 30 non-dominated schemes can be properly obtained in the archive set. Extreme case 1 and extreme case 2 are taken for comparison with the Pareto front of stochastic DEED, the obtained Pareto fronts are shown in Fig 9. Since less wind power generation occurs in extreme case 1, more thermal power generation is needed to meet the system load requirement, which also leads more fuel cost and emission rate. Similarly, case 3 has less fuel cost and emission rate due to more wind power generation. The stochastic DEED is the situation between these two extreme cases, the obtained fuel cost and emission rate are range in the interval between these two extreme cases, which can be seen in Fig 9.
In extreme case 1, wind power output occurs in the poorest situation of wind resources, which is also the worst case in the thermal-wind power system. After fuzzy satisfying method is utilized to select a compromise scheme from the above non-dominated schemes, the thermal output details of the compromise scheme are as shown in Table 7. It can be observed that the output of each thermal unit in each time period is properly controlled in the feasible domain, while output limits, ramp rate limits and system load balance are properly handled. The transmission loss in each time period cannot exceed 5% of the system load requirement, which also reveals that system load balance is properly dealt with by the transmission loss among thermal units and wind farms.
In extreme case 2, wind farms make full use of their wind turbines when the wind speed is at the maximum level. Since system load requirement in each time period is certain, the thermal units bear relative low output in comparison to other cases, and thermal output details of the compromise scheme are shown in Table 8. The output of each thermal unit in each time period is properly controlled in the feasible domain, and system load balance at each time period is properly satisfied. In comparison to those obtained results in extreme case 1, thermal output and transmission loss are smaller than that in extreme case 1 at most time.
Combined with the probabilities of different output intervals, the stochastic DEED model in Section 2 can be optimized by using AGB-MOCDE, the obtained Pareto front is shown in Fig 9. After these scenarios are generated in this test system, obtained optimal schemes can be taken as best schemes for most possibilities. The compromise scheme can be properly selected from archive set with fuzzy satisfying method, the output details of compromise scheme are shown in Table 9. In comparison to the above two extreme cases, the thermal output is within thermal output interval of extreme case 1 and extreme case 2, transmission loss can also be controlled properly, and thermal output at each time period satisfies all the constraint limits. In a real-world application, compromise scheme in stochastic DEED can be taken as the optimal scheme for guiding the output assignment in the thermal-wind power system. The main parameter settings in this test system are presented as follows: The size of archive set is set to 30, size of evolutionary population is set to 100, and maximum generation number Gmax is 1000, permitted constraint violation accuracy of output is set to 0.01MW, permitted total violation is set to 0.1MW, threshold parameter ξCP is set to 0.05, crossover rate is set to 0.4, and scenarios number is set to 50.
DEED with wind uncertainty in different intervals can represent possible situations in a real-world application, the optimal scheme in extreme case 1 can guide the DEED, especially when wind power is generated under the weakest wind speed at the three wind farms, and the optimal scheme in extreme case 2 can deal with the situation in which wind power is generated under abundant wind resources for power generation. The optimal schemes obtained for the above two extreme cases may cause the conservation problem in the stochastic DEED problem, the proposed AGB-MOCDE with scenario-based method generates the most probable optimal scheme with considering probability density function of wind power generation, which can be considered the optimal scheme that can better fit real-world application.
Since wind power generation is taken into consideration in the DEED problem, stochastic characteristics of wind power brings a great challenge for optimizing stochastic DEED problem. For properly tackling with above problem, this paper proposes an adaptive grid-based multi-objective Cauchy differential evolution combining with scenario-based technique to solve the DEED problem. The improved scenario-based technique simulates those possible situations according to different levels of stochastic domain, which divides the output domain of the wind power into several intervals. In comparison to other alternatives for solving DEED problem, the proposed method has following innovations: (1) For properly improving convergence ability and diversity distribution of Pareto front, an adaptive grid-based mechanism and Cauchy mutation operator are interpolated into multi-objective differential evolution, which mainly manipulates Pareto-optimal solution with grid-based technique and improves optimization efficiency with considering population diversity; (2) For reducing computational complexity of generated scenarios, reduction mechanism of scenario-based method is improved with covariance among different scenarios; (3) Since emission rate and economic cost are taken into consideration simultaneously, Pareto dominance-based multi-objective evolutionary algorithm is utilized to optimize two objectives in single optimization run. In this mode, a set of Pareto optimal schemes are produced instead of single optimal solution, which can provide more viable choices for decision makers. In three test systems, the obtained optimal schemes represent different possible situations in DEED, the conservation of the obtained optimal scheme can be decreased according to the probability density function, and the compromise scheme of stochastic DEED can be taken as the optimal scheme in real-world applications.
This work is supported by the National natural fund (NO. 61503199), the National natural science key fund (NO. 61533010), the Jiangsu Province natural science fund (NO. BK20150853), the Jiangsu Province high school natural science fund (NO. 15KJB120009), the Ph.D. Programs Foundation of the Ministry of Education of China (NO. 20110142110036), NUPTSF (NO. NY214206), and the Open Fund (Grant NO. XJKY14019).
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