A Novel Method for Economic Dispatch with Across Neighborhood Search: A Case Study in a Provincial Power Grid, China

1Guizhou Key Laboratory of Intelligent Technology in Power System, College of Electrical Engineering, Guizhou University, Guiyang 550025, China 2State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China 3Guizhou Electric Power Grid Dispatching and Control Center, Guiyang 550002, China 4Key Laboratory of Network Control & Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education, Chongqing 400065, China


Introduction
Economic dispatch (ED), playing an important role in the power system operation and planning, has received significant attention in recent years.The purpose of ED is to schedule the committed generating unit outputs so as to simultaneously minimize the operating cost and meet the load demand of a power system while satisfying all the equality and inequality constraints [1].Traditionally, an approximate quadratic function is utilized to make the mathematical formulation of ED problem convex to reduce the computational difficulty.However, in practice, on one hand, the multi-valve steam turbines make the input-output curves of generators inherently present highly nonlinear characteristic.On the other hand, faults in the machines themselves or the associated auxiliaries prohibit generators from operating in some zones [2].Therefore, the solution space of ED problem with the presence of valve-point effects and prohibited operating zones is highly nonlinear and discontinuous, making the optimization hard to be tractable.In this context, conventional solution methods including linear 2 Complexity programming, Lagrange relaxation, nonlinear programming, quadratic programming, dynamic programming, and interior point method are likely to encounter dire difficulties and challenges mainly due to their heavy imposition of various restrictions such as continuity, convexity, and differentiability on the objective functions, and high sensitivity to the initial values of involved optimized variables.
The abovementioned metaheuristic methods have verified their efficacy in solving the ED problems.Regardless of the achieved promising results, the no free lunch theorem [57] indicates that there is no specific method which can be adopted as a gold standard for all kinds of optimization problems.Namely, there is no single universal superior method that, theoretically, always performs best in solving the ED problems.Therefore, there are still some possibilities to attempt new ones to provide more alternatives, which inspires the authors to apply a recently developed metaheuristic method named across neighborhood search (ANS) [58] to obtain high quality solutions for the ED problems.
As a simple yet versatile metaheuristic method, ANS is motivated by two common straightforward assumptions existing in different population-based algorithms: that searching around a superior solution has a higher probability to find another better solution and that high-quality solutions possess good solution components.In this context, ANS, following the law of parsimony, attempts to simultaneously search across the neighborhoods of multiple superior solutions to get as many potential good solution components as possible.The merits of ANS are its simple structure, ease of implementation, and strong robustness.In this paper, ANS is employed for the ED problems.The main contributions of this work are as follows.
(1) Four benchmark test cases with diverse complexities and characteristics are firstly used to verify the feasibility and effectiveness of ANS comprehensively.The superior performance of ANS is experimentally verified by comparing with four popular population-based algorithms and some recently proposed ED solution methods.
(2) The sensitivities of ANS to variations of population size and across-search degree are empirically investigated.
(3) ANS is finally applied to a practical provincial power grid of China.Its performance is further verified.In addition, the experimental results reflect the necessity and importance of the power construction policy of "replacing small power plants with large ones" in China.
The remainder of this paper is organized as follows.Section 2 briefly introduces the mathematical formulation of ED problems.In Section 3, ANS is described.Next, in Section 4, the flowchart of ANS in solving the ED problems is illustrated.In Section 5, four benchmark test cases are employed to verify ANS.ANS is then applied to a practical provincial power grid of China in Section 6.Finally, Section 7 is devoted to conclusions and future work.
The objective function of traditional ED problem is approximately formulated as follows [1,2,4]: where 푎  , 푏  , and 푐  are cost coefficients of the i-th generator.
In practice, modelling valve-point effects is necessary and can be formulated as follows [60]: where 푒  and 푓  are valve-point effects coefficients of the i-th generator and 푃 ,min is the minimum active power generation limit of the i-th generator (in MW).The total active generated power must be equal to the sum of the total system demand (푃  ) and the total transmission network loss (푃  ): where 푃  is commonly calculated using the following Bcoefficient method [9]: where 퐵  , 퐵 0 , and 퐵 00 are loss coefficients.

Generation Capacity Constraints.
The active power output of each generator should be within its minimum and maximum limits: where 푃 ,max is the maximum active power generation limit of the i-th generator (in MW).

Ramp Rate Limits
Constraints.The adjustment of active power output of each generator should be in an acceptable range: where 푃 pr  is the previous active power output of the i-th generator and 푈푅  and 퐷푅  are the up-ramp and down-ramp limits of the i-th generator, respectively.

Prohibited Operating Zones Constraints.
Generators should avoid operating in prohibited zones [2]: where 푝푧  is the number of prohibited operating zones of the i-th generator and 푃  , and 푃  , are the lower bound and upper bound of the k-th prohibited zone of the i-th generator, respectively.

Across Neighborhood Search
ANS is a very young population-based algorithm proposed by Wu [58] in 2016.ANS, following the law of parsimony and showing good performance compared with other methods [61], attempts to simultaneously search across the neighborhoods of multiple superior solutions to achieve as many potential good solution components as possible.At the same time, it needs to dynamically maintain and update the superior solutions to guarantee the advancement and convergence of the population.The main difference between ANS and other population-based algorithms is that other algorithms mainly utilize some operations such as crossover and mutation to generate new solutions, whereas ANS directly searches across the neighborhoods of multiple superior solutions to produce new solutions.
Like other population-based algorithms, ANS starts with a population of 푝푠 individuals  = [푋 1 , 푋 2 , . . ., 푋  ] representing the potential solutions.Each individual 푋  = [푥 ,1 , 푥 ,2 , . . ., 푥 , ] (푖 = 1, 2, . . ., 푝푠) consists of 퐷 variables and it is initialized as where 푑 = 1, 2, . . ., 퐷, rand(0,1) is a uniformly distributed random real number in (0,1), and 푙  and 푢  are the lower bound and upper bound of the d-th dimension, respectively.In ANS, a group of individuals collaboratively navigate through the search space for obtaining the optimal solution.Each individual searches across the neighborhoods of multiple superior solutions.These superior solutions, being archived in a collection  = (푆 1 , 푆 2 , . . ., 푆  ) where 푐 is the cardinality and is generally set to be the population size 푝푠, are directly derived from the individuals' best positions found so far.The searching strategy is as follows: where 푂 is a pool used to record the randomly selected 푛 (푛 is called across-search degree, 1 ≤ 푛 ≤ 퐷) dimensions for individual 푖, 푟 ( ̸ = 푖) is a randomly superior solution selected from the superior solution collection , and 퐺(0, 휎 2 ) is a Gaussian random value with mean zero and standard deviation 휎 which is usually set to be 0.5.
It can be seen from ( 10) that each individual, on one hand, searches across the neighborhood of the individual's best position achieved so far.On the other hand, it also simultaneously searches across the neighborhoods of other individuals' best positions found so far.After updating individual 푋  , 푖 = 1, 2, . . ., 푝푠, its own superior solution 푆  will be replaced by 푋  if 푋  has a better fitness value.
The main procedure of ANS is given in Algorithm 1.It can be seen that the individuals are guided by multiple superior solutions and the structure, following the law of parsimony, is very simple, making the implementation easy.

Implementing ANS for Solving ED Problem
The flowchart of implementing ANS for solving ED problem is depicted in Figure 1.The main steps are as follows.
Step 3. Evaluate the fitness for each individual.
Step 4. Replace the superior solution 푆  with 푋   if 푋   has a better fitness value.
Step 5. Update the position of each individual.
Step 6. Go to Step 2 to handle constraints if the stopping condition is not satisfied.Otherwise, stop the ANS and output the obtained results.In this study, the maximum number of fitness evaluations (Max FEs) is used as the stopping condition.

Experimental Results on Benchmark Test Cases
In this section, we employ four benchmark test cases with different characteristics to verify the proposed ED method.These cases are described as follows.
Case I.A 13-generator system considering valve-point effects [62].No Update each individual using (10) Output the results, including the optimal dispatching schedule and the corresponding cost End Case II.A 15-generator system considering ramp rate limits, prohibited operating zones, and transmission network losses [63].
In order to validate the feasibility and effectiveness of the proposed ED method, four popular population-based algorithms, i.e., biogeography-based optimization (BBO) [64], competitive swarm optimizer (CSO) [65], differential evolution (DE) [66], and particle swarm optimization (PSO) [67], are employed for comparison.Their involved parameters are listed in Table 1.For the following experiments, 푝푠 and Max FEs are, respectively, set to be 10 and 1 × 10 3 × 푁  for the test cases without considering the valve-point effects, whereas they are 40 and 1 × 10 4 × 푁  for the test cases with the valvepoint effects unless a change is mentioned.50 independent runs are conducted to eliminate contingency.All experiments are executed in MATLAB 2017b.Case II is a canonical traditional ED problem with a smooth and continuous solution space.Although the solution space of Case III is discontinued by the prohibited operating zones, its objective function is also quadratic.Therefore, both cases are typical multiconstraint unimodal optimization problems which have high requirement on the solution methods' local searching ability.It can be observed from Tables 3 and 4 that ANS outperforms BBO, CSO, DE, and PSO on both cases in terms of minimum, maximum, and mean costs.In addition, ANS is also better than other reported solution methods.The comparisons demonstrate that ANS is able to search local range meticulously and thereby possesses good exploitation ability.
Cases I and IV, whose solution spaces are highly nonconvex due to the valve-point effects, are multiconstraint multimodal optimization problems.The number of local minima increases at an exponential rate with the problem scale.They demand a lot on the solution methods' global searching ability.The experimental results tabulated in Tables 2 and 5 consistently indicate that ANS is significantly better than BBO, CSO, DE, and PSO in both cases in terms of minimum, maximum, and mean costs.Furthermore, the larger the system scale, the more significant the superiority.Additionally, ANS also can achieve better or highly competitive results compared with other reported solution methods.The comparisons fully conclude that ANS is with the capability of breaking away from the local minima and locating the global or near-global range.Namely, ANS has good exploration ability.
The obtained optimal dispatching schedules for these four cases are presented in Tables 6-9, respectively.
In conclusion, the abovementioned comparison results sufficiently demonstrate that ANS is able to achieve a strong equilibrium between the local exploitation and global exploration.

Convergence Property.
The convergence curves of the mean costs for the four cases are plotted in Figures 2-5, respectively.It can be seen that although CSO and DE are slightly faster than ANS at the very beginning, both methods are quickly trapped into local search later and thus suffer from prematurity.The convergence speed of BBO  stable optimal result in each trial.In addition, through careful observation, we can see that the standard deviation values of Cases I and IV are bigger than those of Cases II and III.This is because the valve-point effects make the ED problems exhibit highly multimodal characteristic and the solution methods are more likely trapped into different local minima in different trials.Therefore, the valve-point effects are more challenging for solution methods.

Statistical Analysis.
A nonparametric statistical test called Wilcoxon's rank sum test for independent samples is employed to compare the significance differences between ANS and its competitors.The results based on the Wilcoxon's rank sum test at a 0.05 confidence level are summarized in Table 10.The mark " †" symbolizes that ANS is statistically better than its competitor.It can be seen that ANS performs significantly better than BBO, CSO, DE, and PSO in all cases, meaning that ANS is capable of obtaining overall higher quality of the final solutions than the other four popular population-based algorithms.The following can be seen: (i) For the unimodal ED problems, i.e., Cases II and III, the smaller the population size, the better the performance ANS yields.(ii) For the multimodal ED problems, i.e., Cases I and IV, the bigger or smaller the population size, the worse the performance ANS obtains, and ANS can achieve the best results in 푝푠 = 40.The reason might be that, for the unimodal ED problems, it is relatively easy for ANS to find the correct searching direction and there is no need for exploring the entire solution space.Therefore, a small population can swarm towards the global optimum easily.While for the multimodal ED problems, on one hand, a small population tends to converge very fast before fully exploring the entire solution space, thus resulting in prematurity.On the other hand, although a large population can increase the population diversity significantly, the distribution of individuals is sparse and the probability of finding the correct searching direction is sharply reduced.In addition, a large population size will consume a large number of fitness evaluations in each iteration, which is not proper for computationally expensive problems.In general, for the traditional ED problems without considering the valve-point effects, a small population size is recommended, whereas for the nonconvex ED problems, it is safe to set a moderate population size.

Influence of the Across-Search Degree.
The across-search degree 푛 in ANS is utilized to control the amount of information deriving from other superior solutions.In this subsection, an experiment is conducted to investigate the influence of 푛 on the performance of ANS.The experimental  results in different across-search degrees are presented in Figure 7.It can be observed that the best values of 푛 for all cases are all 1.Besides, for the unimodal ED problems, the bigger the value 푛 is, almost the worse the performance of ANS gets.For the multimodal ED problems, a temperate value of 푛 will considerably deteriorate ANS though a bigger value is also not good for ANS.The reason might be that, for both unimodal and multimodal ED problems, bigger values of 푛 will damage individuals' solution components vastly in each iteration, which is not conducive to the consistency of convergence.In addition, for the multimodal ED problems, although a more bigger value of 푛 can fully recombine each individual to maintain the population diversity, it will make individuals trap into different local optima frequently and thus slow down the convergence considerably.Generally, the recommendation value of 푛 for different ED problems is 1.

Application to a Practical Provincial Power Grid of China
In the previous section, the feasibility and effectiveness of ANS in solving ED problems are comprehensively validated on four benchmark test cases.In this section, ANS is applied 10 Complexity   ANS is the most robust method among the five methods.In addition, the convergence curves in Figure 8 show that CSO converges the fastest in the early phase, followed by DE, ANS, PSO, and BBO.However, both CSO and DE stagnate quickly and are surpassed by ANS after about 7500 FEs.BBO and PSO are very slow during the whole evolutionary process.ANS is able to converge to the optimal solution.In order to learn the operation performance of each generator, the detailed load rate and coal consumption values are presented in Figure 9.It is worth noting that the 150MW generators (#1∼#2) are combined heat and power (CHP) generators whose peak regulation depth must not exceed 30%; namely, the adjusting range is from 70% to 100% of the rated capacity.For the other generators, their peak regulation depths are all 50%.It can be seen from Figure 9 that the operation performance indicators of different generators vary from one to another.The mean values of load rate and coal consumption of various capacity generators are summarized in Table 12.It is obvious that, except the 150MW generator, the larger the rated capacity of the generator, the higher the load rate, and the lower the coal consumption.For the 150MW generator, the load rate equals the allowed minimum   limit, meaning that its operation performance is poor.In addition, although its load rate is greater than those of the 300MW and 600MW generators, its coal consumption is significantly higher, which indicates that its energy conservation is inefficient.The abovementioned results fully reflect the necessity and importance of the power construction policy of "replacing small power plants with large ones" in China.

Conclusions and Future Work
In this paper, a simple yet very young and efficient population-based algorithm named across neighborhood search (ANS) is applied to ED problems.Four benchmark test cases with diverse complexities and characteristics are firstly employed to comprehensively verify the feasibility and effectiveness of ANS.The experimental results demonstrate that ANS is able to effectively coordinate the local exploitation and global exploration.It significantly outperforms four popular population-based algorithms (BBO, DE, CSO, and PSO) in terms of the final solution quality, convergence speed, robustness, and statistics.ANS is also capable of achieving better or competitive results compared with some recently proposed ED solution methods.In addition, the sensitivities of ANS to variations of population size and across-search degree are investigated.The experimental results indicate that a small population size is recommended for the traditional ED problems, whereas a moderate population size is relatively safe for the nonconvex ED problems with valve-point effects.For both convex and nonconvex ED problems, the acrosssearch degree with value 1 is appropriate.In addition to the benchmark test cases, ANS is further applied to a practical provincial power grid of China.The experimental results strongly verify ANS once again and fully reflect the necessity and importance of the power construction policy of "replacing small power plants with large ones" in China.In conclusion, ANS can be used as an efficient and reliable alternative for the ED problems.
ANS is a young and promising population-based algorithm.In future work, we will employ some advanced learning strategies such as orthogonal learning and oppositional learning to further enhance its performance and then apply it to solve other power system optimization problems such as combined economic and emission dispatch and optimal power flow.

Start
Read system data Set parameters for ANS Generate a random initial population Initialize the iteration counter t = 1 Handle the quality and inequality constraints for each individual Evaluate the fitness for each individual Is the updated individual better than its own superior solution?No Yes Yes Update the superior solution Is the stopping condition satisfied?

Figure 1 :
Figure 1: Flowchart of implementing ANS for solving ED problem.

Figure 8 :Figure 9 :
Figure 8: Convergence curves for the practical provincial power grid of China.

Table 1 :
Parameter settings of different algorithms.
deviation results provided in Tables 2-5 clearly illustrate that the recorded values of ANS are significantly smaller than those of BBO, CSO, DE, and PSO.Moreover, they are also very competitive with those of other recently proposed ED problem solution methods.The comparisons indicate that ANS has strong robustness and it can achieve a relatively

Table 6 :
Best output power for 13-generator system (Case III).
a TP denotes the total power output.b TC denotes the total generation cost.

Table 7 :
Best output power for 15-generator system (Case II).
a TL denotes the total transmission network loss.

Table 8 :
Best output power for 38-generator system (Case III).It is worth pointing out that the ED objective of this system is minimization of the total coal consumption rather than the total generation cost.The main reasons are twofold.On one hand, the price of coal fluctuates frequently.On the other hand, utilizing the coal consumption instead of coal cost to measure the generation efficiency of generators is more intuitive and reasonable.The objective function is a canonical traditional ED problem with a smooth and continuous solution space.The experimental results are tabulated in Table11.It can be seen that ANS is better than the other involved methods in terms of the minimum, maximum, and mean values of the total coal consumption.The statistical results based on Wilcoxon's rank sum test also further confirm the conclusion.With respect to the standard deviation, the value of ANS is significantly smaller than those of compared methods, indicating that

Table 9 :
Best output power for 40-generator system (Case IV).

Table 10 :
Statistical analysis results based on Wilcoxon's rank sum test.

Table 11 :
Comparison of simulation results (t/h).

Table 12 :
Operation performance of various capacity generators.