Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 710876 | https://doi.org/10.1155/2015/710876

Shengli Liao, Chuntian Cheng, Jing Wang, Zhongkai Feng, "A Hybrid Search Algorithm for Midterm Optimal Scheduling of Thermal Power Plants", Mathematical Problems in Engineering, vol. 2015, Article ID 710876, 13 pages, 2015. https://doi.org/10.1155/2015/710876

A Hybrid Search Algorithm for Midterm Optimal Scheduling of Thermal Power Plants

Academic Editor: Andrzej Swierniak
Received09 Sep 2014
Revised07 Jan 2015
Accepted26 Apr 2015
Published13 May 2015

Abstract

A hybrid search algorithm consisting of three stages is presented to solve the midterm schedule for thermal power plants (MTSFTPP) problem, where the primary objective is to achieve equal accumulated operating hours of installed capacity (EAOHIC) for all thermal power plants during the selected period. First, feasible spaces are produced and narrowed based on constraints on the number of units and power load factors. Second, an initial feasible solution is obtained by a heuristic method that considers operating times and boundary conditions. Finally, the progressive optimality algorithm (POA), which we refer to as the vertical search algorithm (VSA), is used to solve the MTSFTPP problem. A method for avoiding convergence to a local minimum, called the lateral search algorithm (LSA), is presented. The LSA provides an updated solution that is used as a new feasible starting point for the next search in the VSA. The combination of the LSA and the VSA is referred to as the hybrid search algorithm (HSA), which is simple and converges quickly to the global minimum. The results of two case studies show that the algorithm is very effective in solving the MTSFTPP problem accurately and in real time.

1. Introduction

Electric power generation in China mainly consists of coal-fired thermal power plants and hydroelectric power plants [1, 2]. At the end of 2013, China’s total installed generation capacity reached 12473.8 GW, with approximately 64.2% derived from coal-fired thermal power plants, 5% from thermal power using other fuels (e.g., oil and gas), 22.5% from hydropower, 6.1% from wind power, 1.2% from nuclear power, and the remainder from other sources [2]. However, coal-fired units are difficult to start up and shut down, and they respond slowly to system demand because preheating and cooling of the stream turbines is required. Moreover, startup and shutdown of a coal-fired unit require considerable amounts of fuel [3]. Therefore, thermal plants are usually scheduled to meet the base demand to ensure that the generation process proceeds smoothly and to minimize the startup and shutdown times of thermal units.

Over the past decade, extreme weather events such as storms, droughts, high temperatures, and damaging hail have occurred frequently in China, which has caused the demand on power systems to change substantially from day to day. These large deviations in demand over short periods cause difficulties in the scheduling of the power systems, a problem that is exacerbated by the large number of coal-fired thermal units in use. Consequently, the power dispatch centers in China must determine the optimal midterm schedule for thermal power plants (MTSFTPP), where “midterm” is a period of one to six months, to determine which units to start up and shut down or the boot capacity of every thermal plant each day before determining the short-term and daily schedules.

The midterm schedule is important for the economical operation of power systems. The objective of the scheduling problem is to find the optimal set of thermal generating units in a power system that will satisfy the system demand, operational restrictions, reliability constraints, and security requirements for each period considered. Unlike the short-term unit commitment problem associated with power systems, which involves determining a startup and shutdown schedule of units to meet the demand over a period of one day to one week, loads must be dispatched to each plant in the midterm to allow for maintenance and coordination with hydropower plants. Optimal midterm scheduling smoothes the demand on thermal plants, allowing them to operate more efficiently, and makes the best use of hydropower plants. In contrast with the short-term optimal scheduling of thermal power plants, there has been little research on the midterm problem.

The MTSFTPP problem is a variation of the unit commitment (UC) problem, which is a large-scale, nonlinear, nonconvex, mixed-integer combinatorial optimization problem. Various techniques [48] have been used to solve this computationally expensive problem [5]. Among these techniques are dynamic programming (DP) [912], branch-and-bound (BAB), Lagrangian relaxation (LR) [1315], integer programming (IP) [16, 17], and metaheuristic algorithms such as genetic algorithms (GA) [1824], particle swarm optimization (PSO), and neural networks [2527].

Many methods such as DP, BAB, and IP suffer from the curse of dimensionality; that is, the solution space increases exponentially with the number of generating units. As a result, the computation time becomes unacceptable. LR, which decomposes the primal problem into a set of single plant or unit optimization subproblems that are easier to solve independently, has shown potential in solving the UC problem. However, this approach requires the conversion of the optimal dual solution into a feasible solution of the primal problem because of the duality gap, which represents the primary difficulty associated with LR. Metaheuristic algorithms have been widely used to solve the UC problem in recent years, but the quality of the solution depends on the choice of the control parameters, and choosing the control parameters of these algorithms is a very difficult task.

Essentially, the MTSFTPP problem is a multistage (in this context, the term “stage” refers to time) decision problem with unit constraints (type 1), thermal power load factor constraints (type 2), operating time constraints (type 3), and boundary condition constraints (type 4). Considering that types 1 and 2 are single-stage constraints and independent between any two stages, whereas type 4 constraints can be met by the initial solution, the optimization process mainly involves type 3 constraints, which are time constraints. This study investigates a hybrid search algorithm (HSA) to solve the MTSFTPP problem. The primary objective is to achieve equal accumulated operating hours of installed capacity [25] (EAOHIC) for all thermal power plants over a given period, with extra hours that are given to the plants with lower emissions, higher efficiency, and occasional unit maintenance. A more detailed description of the problem is provided in Section 2. Section 3 presents a detailed explanation of the HSA. A case study is presented in Section 4, and conclusions are presented in Section 5.

2. Problem Formulation

2.1. Objective Function

Equality, impartiality, and transparency are important in electric power dispatching. Based on power system security, stability, and economics, units should be treated equally in problems involving operations management and information disclosure. In many western countries, the driving force behind deregulation is the belief that competitive markets make the industry more efficient, leading to lower costs. In the case of China, similar to other developing countries, the incentive for restructuring is derived from the desire to provide equal opportunities to all participants. In China, the firms established by the government are responsible for power system operations, and the power systems are operated in an integrated fashion. Therefore, equality is usually the main concern in long-term and midterm scheduling in China. In this study, EAOHIC is the primary consideration in the MTSFTPP problem. Operating hours are accumulated from the beginning of the respective warm-up periods to the end of the computational period (Figure 1). The EAOHIC constraint for the MTSFTPP problem is formulated as follows: where , are the accumulated operating hours of installed capacity for plants , , respectively, in the selected period.

Although the concept captured by (1) is unambiguous and easily understood, the problem is very difficult to solve directly because the formulation is related to all plants and the objective cannot be evaluated by the computation of a single mathematical expression. Considering that the variance can be used to describe the degree of dispersion of data, let be the set of accumulated operating hours of the installed capacity for the plants, and let be the variance of the set . Equation (1) can be replaced by the following problem:where the variables are defined as follows: is the set of , the accumulated operating hours of the installed capacity of plant during the selected period, in hours (h); is the total number of plants; is the variance of set , in h2; is the average of set , in h; is the operating hours of installed capacity of plant accumulated during the warm-up period, in h; is the operating hours of installed capacity of plant accumulated during the computational period, in h; is the extra hours assigned to plant , in h; is the total number of steps in the computational period, where each step is one day; is the installed capacity of plant , in MW; is the average of the operating capacity of plant during the computational period, in MW; is the load factor; and is the operating capacity of plant in period , in MW.

2.2. Constraints

Due to the operational requirements, the minimization of the objective function is subject to the following types of constraints.

(1) Unit Constraints. Considerwhere and are the minimum and maximum numbers of units, respectively, in plant in period and is the number of active units in plant in period .

(2) Thermal Power Load Factor Constraints. Considerwhere is the operating capacity of the system in period , in MW, and and are the minimum and maximum load factors of thermal power, respectively, which are calculated from the long-term real-world operation of power grid. is the demand of system in period , in MW.

(3) Time Constraints. Midterm optimal thermal power plant scheduling only dispatches the system load to the plants and does not distribute the load among the units in each plant. Therefore, we transform the constraints on the response of a plant into minimum-time constraints at minimum or maximum output. Consider where is the duration of peak in the output of plant , is the minimum duration of the peak in the output of plant , is the duration of valley in the output of plant , and is the minimum duration of the valley in the output of plant .

(4) Boundary Condition Constraints. The planning for the current period depends on the previous states of the units. For the plant , considering the previous period as the maximum in and , the states of the units before the starting point affect the operation of the units through the minimum peak and valley time restrictions, as shown in Figure 1. Consider where and are the first duration of peak and valley in the output of plant , respectively.

Note that the boundary condition constraints are the same as the time constraints (type 3), the time constraints considering the time horizon in , while the boundary condition constraints considering in .

3. Solution Methodology

The MTSFTPP problem is a UC problem, and each approach to the problem has its disadvantages, as mentioned previously. Constraints of types 1 and 2 involve a single period and can be treated independently to reduce the search space, constraints of type 4 can be satisfied in the initial solution, and constraints of type 3 must be satisfied over multiple periods. This study investigates a novel method using three stages to solve the problem.

3.1. Producing a Feasible Solution Space Satisfying Constraint Types 1 and 2

Mathematically, the MTSFTPP problem, like the UC problem, suffers from the curse of dimensionality as the number of units and the computational period increases. Thus, it is difficult to solve this problem by considering a simple combination of units. For a power grid with five thermal plants, each having four units, the total number of unit combinations will reach for a horizon of one month, and the search will be computationally expensive. Fortunately, the actual constraints on the thermal power load factors and the number of units are beneficial because they reduce the search space when searching for feasible solutions. The specific procedure is as follows.

Step 1. For the thermal plants with different values of unit capacity, each plant will be represented by several “virtual plants” whose capacity depends on the number of units of a given type. For instance, for a thermal plant with 5 units, including 2 units with 300 MW of capacity and 3 units with 200 MW of capacity, the plant will be divided into 2 “virtual plants.” By initializing the plants in this manner, the final number of virtual plants will be greater than the number of actual plants and will be denoted as .

Step 2. Start from the first period; that is, let .

Step 3. Sort the combinations that meet the unit constraints in period ; that is, (), where is the number of combinations that meet constraint 1 (3) in period , , ().

Step 4. Select the combinations that satisfy the thermal power load factor constraints from in period ; that is, (), where is the number of combinations that satisfy constraints 1 and 2 ((3) and (4)) in period .

Step 5. If , set and return to Step 3. Otherwise, stop and retain the feasible solution space satisfying constraints 1 and 2 ((3) and (4)).

3.2. Generating an Initial Feasible Solution Using a Heuristic Method

At this point, we have obtained the feasible space satisfying constraints of types 1 and 2. Finding the optimum in the next stage requires a solution that satisfies all of the operational and system constraints. A heuristic method is proposed for this goal. The procedure is as follows.

Step 1. Obtain the operating data for days before the starting point. These data will be stored in an array and denoted as the scheme . The midterm schedule must satisfy the boundary constraints (type 4) between actual operation and future scheduling.

Step 2. Start from the first period; that is, let .

Step 3. Obtain all combinations for period () using the method described in Section 3.1.

Step 4. Choose a combination that satisfies constraint type 3 in (5) and that is suitable for the boundary constraints, type 4. For each subset () in combination space at period , check the type 3 and type 4 constraints. In particular, the type 4 constraints use the historical data to satisfy the minimum-time constraints at the starting points. Clearly, includes the historical data when . The search is stopped and the current subset appended to when a feasible subset is found for all subsets (). The subset number in the first period is stored to reduce the search in the next cycle. In this case, continue to the next period and set . Otherwise, return to the first period, start from the stored number, and repeat the previous procedure. An initial feasible solution will be generated for the optimization.

A flow chart and pseudocode for generating an initial feasible solution are shown in Figure 2 and Algorithm 1, respectively.

Heuristic for generating an initial feasible solution
Input actual operation data for d days before the start point
feasible solution space , ,
initial
Output initial feasible solution
for all periods from to
for all the feasible states from to
  if  
   
  end if
  else
   
  end else
  Define num to count, num = 0
  for all plants states from to
   if &
               num = num + 1
   end if
  end for
  if num = M
   Expand by appending to
    Store in
    if  
     
    end if
  end if
  if num < M &
   , ,
  end if
end for
end for

3.3. Hybrid Search Algorithm

Numerous operational and system constraints are imposed in the unit commitment problem. The aforementioned initial feasible solution meets only some constraints, and the optimization for unit commitment still must be performed. Sections 3.3.1 and 3.3.2 present a two-step procedure for optimization.

3.3.1. Vertical Search Algorithm

Over the past few decades, the progressive optimality algorithm [28] (POA) has been shown to have great advantages over classical optimization methods and has been one of the most widely used techniques for hydroelectric generator scheduling and water resources problems [2931]. The POA attempts to find the best solution in a given decision space based on Bellman’s Principle and is free of a particular model structure. The advantages of the POA over other optimization techniques are that it can decompose a multistate decision problem into several nonlinear programming subproblems to reduce the dimensionality. In fact, the optimal scheduling of thermal power plants is a multistate and multistage problem, and hence it is suitable for the POA. The POA is defined as follows [27]:where is the state in period , is the objective function in period , and and are the decision variables in periods and , respectively. For the optimal scheduling of thermal power plants, the problem can be mathematically stated as follows:where and are functions that give the operating hours of installed capacity when the status is and and are the objective functions for period . With this formulation, we choose from with the status in the other periods fixed to obtain the optimal objective function value. A diagram of the operating of the POA is shown in Figure 3, and pseudocode for the algorithm is shown in Algorithm 2.

VSA
Input initial feasible solution ,
actual operation data for d days before the start point
feasible solution space , ,
initial value
Output optimized solution
for   to
for all periods from to
  for all the feasible states from to
   Define num to count, num = 0
   for all plants states from to
    if   &
     num = num + 1
    end if
   end for
   if num = M
    Calculate
    if  
     Replace using ;
    end if
   end if
  end for
end for
 Obtain new solution
If  I = R
  Obtain optimized solution
end if
end for

The aforementioned search is a sequential procedure, which we refer to as the vertical search algorithm (VSA), as distinguished from the following search algorithm.

3.3.2. Lateral Search Algorithm

The solution obtained using the VSA is clearly affected by the starting point (i.e., initial guess) and can converge to a local minimum. Hence, another search strategy is presented to improve the solution by enlarging the search space. The search addresses the objective that all plants have equal or approximately equal operating hours of installed capacity. To attain this goal, the plants whose operating hours are greater than or less than the average value will be adjusted. The plant with the greatest bias from the average is selected for adjustment. Generally, we choose the periods of maximum or minimum plant output to perform the adjustments. This state involves situations in which the system load allows for a change in the number of active units. Another state occurs in periods of maximum or minimum output during which a change in the number of active units in other periods is not possible. For the first case, the adjustment is started from the first period among the periods of maximum or minimum output. The increase or decrease in the number of units depends on the difference from the average and the constraints. If the plant’s operating hours of installed capacity are greater than the average, we decrease the number of units for selected periods. This adjustment is a trial procedure because of the complex constraints and because the starting period for the adjustment is related to the constraints. For the second case, the changes in the maximum/minimum output periods will cause a change in the unit status in subsequent periods due to the minimum-time constraints. The adjustment features two aspects: satisfying all of the constraints in the maximum- or minimum-output periods and satisfying the constraints in subsequent periods. In either case, a new feasible solution will be produced. Consequently, an updated optimal solution can be obtained using the VSA. Herein, we refer to the search procedure as the lateral search algorithm (LSA). An optimal solution that satisfies the constraints can be obtained using the VSA and LSA alternately. A flowchart for the hybrid search algorithm that combines the VSA and LSA is shown in Figure 4.

4. Case Study

The proposed method was applied to the MTSFTPP problem for an actual power system, the Yunnan Power Grid (YPG) in China. Currently, it is used as the primary tool to determine the midterm optimal scheduling of thermal power plants by the operators of YPG. Yunnan is among the richest in hydropower resources of all of the provinces in China. Its potential hydropower capacity is 103.6 GW, which contributes 15.3% of the national capacity and ranks third in the nation. The exploitable hydropower capacity is 90 GW, which contributes 17.9% and ranks second in the nation. By the end of 2013, the total installed capacity of the YPG had reached 26.1 GW, with hydropower capacity accounting for 16.7 GW and thermal capacity accounting for 8.935 GW. Proper scheduling of thermal power plants can make the best use of hydropower systems and improve system security.

The thermal power system in the YPG consists of 9 thermal power plants with a total installed capacity of 8.935 GW and 27 units. The dispatching center in the YPG controls the thermal power system. Table 1 lists the basic data for these plants. The HSA was implemented in Java and executed on a PC with an Intel Core 2 Duo 2.93 GHz CPU and 2 GB of memory. An actual schedule for 2013 in the YPG was used to test the validity and the computational efficiency of the proposed method.


Plant numberUnitsCapacity (MW)
(number capacity, MW)

14 6002400
26 3001800
34 3001200
42 200 + 2 3001000
52 300600
62 300600
72 300600
82 300600
91 135135

Two typical months, May and September, were selected to demonstrate the practicality and the efficiency of the method. May represents the beginning of the flood season, when the hydropower plants can operate at their peak and the demand on the thermal system can be decreased. Conversely, September represents the beginning of the dry season, when the hydropower system has the lowest output and the demand on the thermal system increases.

According to the real-world operating experience and users’ actual demands, the basic parameters for the HSA are set as follows:(i)minimum load factor ;(ii)maximum load factor ;(iii)initial load factor ;(iv)actual operating hours of installed capacity in the warm-up period ();(v)extra hours ();(vi)minimum number of units (, ), and for ;(vii)maximum number of units (, ), where is the number of available units for plant in period ;(viii)minimum duration of peak in plant output days ();(ix)minimum duration of valley in plant output days ();(x)considering previous periods days ().

The average cost time of 10 runs for May is 9764 ms and for September is 10241 ms, and the calculation efficiency can be met the actual demand in midterm scheduling.

In a comparison of the results obtained by the HSA with those obtained by the heuristic, the VSA and the LSA are shown in Table 2. As indicated by the table, the results improve progressively in going from the heuristic to the HSA for most of the evaluation criteria, including the average, the max-min difference, and the objective value, for both May and September. It should be noted that the most improved solutions were achieved by the VSA, which demonstrated that the VSA has the capability to find the optimal solution. The LSA attempts to improve the objective function value and avoid convergence to a local minimum. As shown in Table 2, Plant 1 was changed in May by the LSA. Although the max-min difference remained the same, the objective function was diminished from 50.06 to 36.4. There is little difference in the results obtained between September and May. Due to the adjustments to the generating units in the valley periods by the LSA, the max-min difference was greater, but the objective function decreased from 10.81 to 7.17. Equal values for all three evaluation criteria were attained for May and September using the HSA, which indicates that the HSA has the capability to find the global minimum.


MaySeptember
HeuristicVSALSAHSAHeuristicVSALSAHSA

Operating hours of installed capacity (h)
 Plant 1475.2484.8499.2499.2264288288288
 Plant 2499.2499.2499.2499.2304284.8284.8288
 Plant 3475.2499.2499.2499.2244.8278.4292.8288
 Plant 4499.2499.2499.2499.2268.8282.2282.2288
 Plant 5489.6499.2499.2499.2288288288288
 Plant 6499.2499.2499.2499.2288288288288
 Plant 7499.2499.2499.2499.2288288288288
 Plant 8489.6499.2499.2499.2288288288288
 Plant 9480480480499.2268.8288288288

Average (h)489.6495.5497.1499.2278.04285.9287.5288
Max-min difference (h)2419.219.2059.29.610.60
Objective value (h2)97.2850.0636.40282.6110.817.170

The generation requirements for the 9 plants in May are shown in Figure 5 and the details of results are listed in Table 3. Meanwhile, the generation requirements and details in September are shown in Figure 6 and Table 4. From Tables 3 and 4, it can be seen that the boot capacity result of each plant can meet all constraints. In Plant 2 in Table 3, for example, the days of first peak in plant output are 8 (from 7th to 14th) which is bigger than minimum duration of peak in plant output days, while the days of first valley in plant output are 4 (from 17th to 20th) which is bigger than minimum duration of valley in plant output days. Figure 5 shows that the outputs of most of the plants by the end of May were reduced to take advantage of the hydropower system. Plant 4 appears to have been an exception to this trend in May and could not satisfy the peak time constraint by the end of May, but this result is misleading. Table 1 shows that Plant 4 has two types of turbine units, 2 units of 200 MW capacity and 2 units of 300 MW capacity. As described in Section 3.1, Plant 4 was divided into two “virtual plants” for computational purposes. Table 3 and Figure 7 show the generation requirements for “virtual plants” 1 and 2 and demonstrate that these plants exhibit the same performance as the other plants. Conversely, the outputs of all of the plants increased from the beginning of September. Figures 5 and 6 demonstrate that the MTSFTPP results for May and September are reasonable.


DatePlant 1Plant 2 Plant 3Plant 4Plant 5Plant 6Plant 7Plant 8Plant 9
Virtual plant 1Virtual plant 2Total

1st2400150012004006001000300300300300135
2nd2400150012004006001000600300300300135
3rd24001500900200600800600600600300135
4th24001500900200600800600600600600135
5th24001500900200600800600600600600135
6th24001500120040060010006006006006000
7th24001800120040060010006006006006000
8th24001800120040060010006006006006000
9th2400180012004006001000600600600600135
10th240018001200400300700600300300600135

11th240018001200400300700600300300600135
12th240018001200400300700600300300600135
13th1800180012004006001000600600600600135
14th180018009004006001000600600600600135
15th180015009004006001000600600600600135
16th240012009004006001000300600600600135
17th24009001200200600800300600600600135
18th24009001200200600800300600600300135
19th24009001200200600800300600600300135
20th24009001200400300700600300300300135

21st240012001200400300700600300300300135
22nd240012001200400300700600300300300135
23rd240012001200400300700600300300600135
24th120018003004006001000600600600600135
25th60018003004006001000600600600600135
26th60018003004006001000600600600600135
27th6001800900200600800300600600600135
28th120018009000600600300600600600135
29th120018009000600600300600600600135
30th1200180090006006003006006003000
31st24009009004003007003003003003000


DatePlant 1Plant 2Plant 3Plant 4Plant 5Plant 6Plant 7Plant 8Plant 9

1st600300300300300300300300135
2nd6003003006003003003003000
3rd6003006006003003003003000
4th6003006006003003003003000
5th6006006006003003003003000
6th600600600600300300300300135
7th1200600600600300300300300135
8th1200600600600300300300300135
9th1200600600600300300300300135
10th1200900600600300300300300135

11th1200900600600300300300300135
12th1800900600600300300300300135
13th18009006006003003003003000
14th18009006006003003003003000
15th18009006006003003003003000
16th18009006006003003003003000
17th18009006006003003003003000
18th18009006006003003003003000
19th18009006006003003003003000
20th1800900600600300300300300135

21st1200900600600300300300300135
22nd6001200600600300300300300135
23rd6001500600600300300300300135
24th6001500600600300300300300135
25th12001500600600300300300300135
26th12001500600600300300300300135
27th120015006006003003003003000
28th120015006006003003003003000
29th120015006006003003003003000
30th120030012009003003003003000

Figures 8 and 9 show the generation and capacity requirements in May and September, respectively. It can be observed that the generation requirements are between maximum and minimum capacity.

5. Conclusions

A model for the MTSFTPP problem was proposed. The model includes three stages. The feasible spaces for the MTSFTPP problem are first produced by narrowing the search space based on constraints 1 and 2. Next, an initial feasible solution for the MTSFTPP problem is obtained using virtual plants and considering constraints 3 and 4. The POA is used to solve the MTSFTPP problem, and the optimal search is referred to as the VSA. To avoid convergence to a local minimum, the LSA is used, and the updated solution is used as a new feasible starting point for the next search with the VSA. Thus, the HSA is produced by combining the LSA with the VSA. A complete framework for solving the MTSFTPP problem was established and demonstrated using a case study. The results showed that the proposed method can efficiently produce satisfactory solutions with practical requirements.

A novel method focusing on solving strategies for the MTSFTPP problem was proposed, in which the thermal system load was given after the other electric sources were optimized. However, the joint modeling of thermal power system with other source systems in midterm schedule can be considered as a research basis for future studies.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This study is supported by the Major International Joint Research Project (no. 51210014), the National Natural Science Foundation of China (no. 51209029), and the Fundamental Research Funds for the Central Universities (no. DUT14QY15).

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Copyright © 2015 Shengli Liao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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