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 [23]: (1) where T represents the total time period length, N_c is number of thermal units, P_ci,t represents the output of the ith thermal unit in the tth time period, P_ci,min is the minimum output of the ith thermal unit, and a_i,b_i,c_i,d_i,h_i 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, ρ_s 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 N_s 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 [24]: (3) where α_i,β_i,γ_i,ζ_i,λ_i 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)

3. Constraints

(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 N_w 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 B_ij,B_0i,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.

(2) Output limits (7) where P_ci,max is the maximum output of the ith thermal unit.

(3) Ramp rate limits (8) where DR_ci,UR_ci are the down-ramp and up-ramp rates, respectively, of the ith thermal unit.

(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 [26]: (9) where is the wind speed of the jth wind farm in the tth time period in the sth scenario, v_j,rate is the rated wind speed of the jth wind farm, v_j,in,v_j,out are the cut-in and cut-out wind speed, respectively, at the jth wind farm, and P_wj,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, g_j(∙) is the j th inequality constraint, h_k(∙) is the k th equality constraint, and S_g,S_h 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.

Definition 1: Assume two feasible solutions x₁,x₂; the relationship between them exists as: (11)

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,¬x_i ≻ x^*.

Definition 3: The set of objective vectors obtained from Pareto optimal solutions is called the Pareto front, which can be described as follows: (12)

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 [19]. 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.

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 [27]. 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 [28]. If the convergence metric C(P^(g)) does not change obviously after h generations, which also means that ∇_CP is smaller than a threshold ξ_CP ∈ [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 N_Q is the size of the archive set, is the average value of the jth dimension variables, and u_j,l_j 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 [29] is taken to control the mutation operation as population evolution proceeds. The linear formulation is presented as follows: (15) where G_max 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.

3.2 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.

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Fig 1. Grid-based diversity maintenance mechanism of Pareto front.

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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:

1. Step.1: Add the newly generated individual into archive set, calculate the number of individuals in each grid , which satisfies and k_i ∈ N.
2. Step.2: Find the , and if there are N_grid grids that contain k_max individuals, and select all individuals in these grids.
3. Step 3: Calculate density degree of these individuals with following metrics:

(19)

1. 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 μ_rq 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.

1. The solution-encoding strategy

In the DEED problem, a set of thermal outputs with N_s scenarios is taken as the decision variable. In each scenario, there are T scheduling periods on the schedule horizon, and the output of N_c 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 [32]: (23) where v_j 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)

For a certain output interval [l_j,u_j], the probability of the output in this interval can be calculated as: (25)

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 [l_j,u_j], the interval division of wind power generation can be obtained as in Fig 2.

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Fig 2. Interval division of wind power generation.

https://doi.org/10.1371/journal.pone.0185454.g002

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:

1. Step 1: Initialize the reduced scenario set Scen_reduced = Scen, and the size of reduced scenarios set is , required scenarios number is , S_i represents the ith scenario in Scen_reduced, go to Step 2;
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…N_s), and covariance value set can be initialized with Cov_ij = Covariance(S_i,S_j), go to Step 3;
3. Step 3: Calculate the eigenvalues of covariance matrix Cov, and it can obtain eigenvalues, and go to Step 4;
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;
5. Step 5: Then , , and ρ_eigen,max1 = ρ_eigen,max1 + ρ_eigen,max2; and go to Step 6;
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 L_loss,t with formulation (6); then, the variation of the deviation can be found as follows: (28)

All thermal output is adjusted with the same increment of deviation, which means ℵ = ΔP_it = ΔP_jt(i ≠ j); the increment of deviation can be obtained as follows: (29)

According to the above increment of deviation, thermal output can be properly adjusted using the coupled constraint-handling technique in literature [19]. 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:

1. Step 1: According to the wind output domain of each wind farm in each time period, generate N_s 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;
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;
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;
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;
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;
6. Step 6: If g < g_max, g = g + 1; go to Step 3; otherwise, output all the non-dominated individuals in the archive set and go to Step 7;
7. Step 7: For a given reference membership value μ_rq, 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.

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 [33]. In Fig 3, 30 non-dominated solutions are produced by AGB-MOCDE method, and are shown with obtained schemes by MODE [34], 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 μ_rq is 0.75, scheme (14) can be selected as compromise scheme by AGB-MOCDE and scheme (16) is taken as compromise scheme by MODE.

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Fig 3. The obtained Pareto front of AGB-MOCDE and MODE.

https://doi.org/10.1371/journal.pone.0185454.g003

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Table 1. The memberships of non-dominated schemes by AGB-MOCDE and MODE for test system 1.

https://doi.org/10.1371/journal.pone.0185454.t001

In Table 2, SA [35], MSL [36], EP [37], PS [37], PSO [5] 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.

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Table 2. The comparisons among different optimization methods.

https://doi.org/10.1371/journal.pone.0185454.t002

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.

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Fig 4. The output process of compromise scheme by AGB-MOCDE in test system 1.

https://doi.org/10.1371/journal.pone.0185454.g004

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Table 3. The transmission loss of five thermal-unit system.

https://doi.org/10.1371/journal.pone.0185454.t003

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 N_Q is 30, and maximum generation number G_max 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 μ_rq 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 [14]. 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.

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Fig 5. Pareto front obtained by AGB-MOCDE and MODE.

https://doi.org/10.1371/journal.pone.0185454.g005

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Table 4. The membership of non-dominated schemes between AGB-MOCDE and MODE for test system 2.

https://doi.org/10.1371/journal.pone.0185454.t004

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 μ_rq 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 [14] and literature [38], 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 [14] and literature [38].

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Table 5. The comparison among different optimization methods for the DEED problem.

https://doi.org/10.1371/journal.pone.0185454.t005

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.

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Table 6. The output (MW) details of the compromise scheme obtained by AGB-MOCDE for test system 2.

https://doi.org/10.1371/journal.pone.0185454.t006

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 G_max is 1000, permitted constraint violation accuracy of output is set to 0.01MW, permitted total violation is set to 0.1MW, threshold parameter ξ_CP 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).

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Fig 6. Interval division of output domain for wind farm #1.

https://doi.org/10.1371/journal.pone.0185454.g006

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Fig 7. Interval division of output domain for wind farm #2.

https://doi.org/10.1371/journal.pone.0185454.g007

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Fig 8. Interval division of output domain for wind farm #3.

https://doi.org/10.1371/journal.pone.0185454.g008

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.

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Fig 9. The obtained optimal schemes in two extreme cases and in stochastic DEED.

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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.

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Table 7. The output (MW) details of the optimal scheme of AGB-MOCDE in extreme case 1.

https://doi.org/10.1371/journal.pone.0185454.t007

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.

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Table 8. The output (MW) details of the optimal scheme of AGB-MOCDE in extreme case 2.

https://doi.org/10.1371/journal.pone.0185454.t008

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 G_max is 1000, permitted constraint violation accuracy of output is set to 0.01MW, permitted total violation is set to 0.1MW, threshold parameter ξ_CP is set to 0.05, crossover rate is set to 0.4, and scenarios number is set to 50.

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Table 9. The output (MW) details of the optimal scheme of AGB-MOCDE in stochastic DEED.

https://doi.org/10.1371/journal.pone.0185454.t009

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.