G. R. Venkatakrishnan , R. Rengaraj and S. Salivahanan

1 Introduction

Real power economic dispatch (RPED) is one of the most important non - linear problem to be solved in the modern power system. The objective of the RPED problem is to allocate optimal real power generation to the existing thermal units without violating the constraints in the system. Conventional methods like lambda-iteration method, and so on are used to solve traditional RPED problem with assumptions many assumptions [Park, Lee, Shin et al. (2005); Sayah and Hamouda (2013)].

However, in practical, the nonlinearities and discontinuities like valve point loading, ramp rate limits and so on represent RPED problem as a non-smooth or non-convex optimization problem which makes it difficult for the traditional methods to obtain the global optimum[Park, Lee, Shin et al. (2005); Sayah and Hamouda (2013)]. Moreover considerable number of researchers has shown interest in developing an efficient algorithm in solving the RPED problem with nonlinearities [Mandal, Roy and Mandal (2014)]. Though the conventional methods have advantages like few control parameters and less computational time, it fails to reach global optima for the ELD problems with large dimensional and discrete search space [Nguyen and Vo (2015)].

According to No Free Lunch (NFL) theorem, there exist no meta heuristic optimization algorithm which is applicable in solving all real world optimization problems [Mirjalili,Mirjalili and Lewis (2014); Basu (2014)]. The development of numerous meta heuristic algorithms by various researchers around the world over the past two decades has successfully solved the ELD problem with superior convergence characteristics, high solution quality and robustness, eliminating most of the difficulties of classical methods[Mandal, Roy and Mandal (2014); Basu (2015)].

Grey Wolf Optimization (GWO) algorithm, a recent swarm intelligence algorithm is proposed to solve the non-convex optimization problem [Mirjalili, Mirjalili and Lewis(2014)]. The leadership and hunting behaviors of grey wolves in nature is incorporated in the algorithm and has superior exploration and exploitation ability. In solving real world problems, the GWO algorithm has the capability of providing higher quality solutions and good computational efficiency with few parameters and ease of implementation [Mandal,Roy and Mandal (2014); Mirjalili, Mirjalili and Lewis (2014)]. These properties have motivated few researchers to implement the GWO algorithm in solving problems like combined heat and power dispatch [Mandal, Roy and Mandal (2014)], hyper spectral band selection [Medjaheda, Ait Saadib, Benyettoua et al. (2015)], load frequency control [Guha,Roy and Banerjee (2015)], optimal reactive power dispatch [Sulaimana, Mustaffab,Mohameda et al. (2015)], power system stabilizer design [Shakarami and Faraji Davoudkhani (2015)], MPPT design [Mohanty, Subudhi and Ray (2016)], flow shop scheduling [Komakia and Kayvanfar (2015)], attribute reduction [Emarya, Yamany,Hassaniena et al. (2015)], feature selection [Emary, Zawbaa and Hassaniena (2015)],parameter estimation [Song, Tang, Zhao et al. (2015)] and automatic generation control[Sharma and Saikia (2015)]. In this paper, GWO algorithm is implemented to solve the RPED problem to validate its effectiveness over other meta heuristic algorithms. The simulation results show that this algorithm performs better than the other algorithms in terms of solution quality, convergence efficiency and robustness.

飞禽当中，可以吃洁净的，但以下皆不可食：鸢、秃鹫、黑雕，一切鹞隼，大小乌鸦；鸵鸟、夜莺、海鸥、鹗、猫头鹰之属；朱鹭、塘鹅、鸨、鸬鹚、鹳鹭一族；以及戴胜、蝙蝠。

Section 2 describes the formulation of ELD problem with constraints like ramp rate limits and so on. The detailed description of GWO algorithm is discussed in Section 3. Section 4 describes the implementation of GWO to the complex RPED problem. The numerical results and discussion of the GWO algorithm for different test systems are presented in Section 5 and conclusion is drawn in Section 6.

2 Formulation of the ELD problem

Minimization of the total cost in producing real power in a power system without violating constraints is the main aim of RPED [Sahoo, Dash, Prusty et al. (2015)]. In this paper,RPED problem without valve point loading is considered.

通过对传统伦理学的反思和对现代技术现实境况的考察，约纳斯形成了对技术时代伦理氛围的基本认识。在传统社会，技术的影响范围极其有限，因而人的伦理行为遵循此时此地的原则；而在现代社会，由于科学技术的影响超越时空 , 因此人类应实行远距离的“责任”伦理。[29]至此，约纳斯试图将“责任”维度重新置入伦理学理论之中，通过阐发一种“未来责任”的理念，构建适应“技术时代”需要的“未来伦理学”。

2.1 RPED problem with smooth cost function

The objective of the RPED problem with smooth cost function is given by

Step 2: Initialization of GWO parameters i.e. population size N and select the stopping criteria.

吴铁成一见到戴笠，就批评他说：“雨农啊，这几天，山城政界搞‘吼’了，都是你惹的祸啊。你那样搞法，自认为是忠于领袖和国家，那是你个人的想法，不一定是大家的想法。你给党国、给领袖帮了倒忙。你们做特务、情报工作的，要准确无误嘛。黄炎培虽然可恨，但他爱国和坚决抗日的态度，是众所周知的。说他家藏有日伪人员，没有哪个会相信的。下面有这样的情报来，作为局长，你应慎重地研判一下，不能糊里糊涂地下令叫部下去乱搞。雨农，你知道，我过去也做过半个情报人员的公安局长。我那时处理这类问题非常慎重。这一回你恰巧碰到天不怕地不怕的黄炎培头上，所以闹得你下不了台。以后，你一定要吸取这次的教训。”

whererepresents total fuel cost of all the thermal units present in the system ($/hr),N is the total number of thermal units existing in the system andrepresents fuel cost of thethermal unit ($/hr) andrepresents power generated by thethermal unit (MW).In general, the fuel cost functionthermal unit is expressed in quadratic polynomial as

where aq,bqandare the cost coefficients ofthermal unit.

The different practical constraints to which the above minimization problem is subjected are power balance or demand constraint, generator output limits, prohibited operating zones and ramp rate limits.

2.2 Power balance or demand constraint

The sum of individual power generated from each thermal unit existing in the system must be equal to the sum of transmission loss and total demand of the system which is represented as

1.3.1 株数计算法 每一块田中某种杂草的株数=同一块田9点样方中该杂草的株数之和/9；同一类型田块中某杂草的株数=相同类型每一块田中这种杂草的株数之和/该类型田块数。

where D is the total demand of the system (MW) and PLossis the transmission loss of the system (MW).

The transmission loss of the system PLosswhich is a function of power generated by each unit is given by

whereandare the loss coefficients or B-coefficients.

2.3 Real power generating limits

The power generated Pqfrom each thermal unit must lie within its permissible limits which is represented as

where Pq,minand Pq,maxare the minimum and maximum generation of thegenerator(MW) respectively.

2.4 Ramp rate limit

Theoretically in RPED problems it is assumed that the output from thermal units is adjusted linearly. But in practical, this assumption is not plausible as the operating limits of each generators are restricted by their corresponding up-rate limit URq, down-rate limit DRqand previous hour generationsHence, forgenerating unit,

“嘿嘿，太平本是将军定，哪个将军见太平？本将军平定天下，功高劳苦，想不到功劳越大，越是不能安享太平。”将军苦笑之余，颓然问道，“沉渊楼什么时候也肯杀忠臣良将了？”

Therefore, using the ramp rate limits the real power generating constraints given in Eq. (5)can be modified as

2.5 Prohibited operating zones

In the characteristic curve of the thermal units, due to some non-linear behavior existing in shaft bearing or faults in the machines or its associated auxiliary equipment, some thermal unit might have prohibited operating zones which is to be avoided. The input-output characteristics of a generator with POZ is shown in Fig. 1 [Subbaraj, Rengaraj and Salivahanan (2009)]

Figure 1: Characteristic curve of thermal unit with POZs

Therefore, the operating constraint of the qthunit with POZ is

whereis the index of POZ ofthermal unit,is the total number of POZ exist forgenerator andandare the lower and upper limit ofPOZ of thethermal unit (MW) respectively

3 Grey Wolf Optimization (GWO) algorithm

GWO is a very recent optimization algorithm inspired by gray wolves and is developed in 2014 [Mirjalili, Mirjalili and Lewis (2014)]. The algorithm imitates the hunting and the social hierarchy behaviors of grey wolves. In addition, to the advantages of meta heuristic algorithms the GWO algorithm requires no specific input parameters to be initialized. Also,the GWO algorithm is straight forward, free from computational complexity and can be easily implemented in any programming languages [Guha, Roy and Banerjee (2015)]. The interesting fact of grey wolves is that it possesses social dominant hierarchy as shown in Fig. 2 and this hierarchy is used in GWO. The leader wolf or alpha（α）wolf takes decisions like hunting, searching, time to wake and so on. The beta （β）wolf supports alpha （α）wolf in decision making and the delta（δ）wolf follows the alpha （α）and beta （β）wolves. The wolves which do not come under these category are called as omega （ω）wolves and are used basically as a scapegoat [Medjaheda, Ait Saadib, Benyettoua et al. (2015); Sharma and Saikia (2015)].

Figure 2: Hierarchy of grey wolves

In addition, the group hunting another social behavior is considered in the algorithm. The three stages by which the grey wolf attacks the prey are explained in Muro et al. [Muro,Escobedo, Spector et al. (2011)] and is modeled as follows.

3.1 Modeling of GWO

3.1.1 Social hierarchy

Step 4: The fitness of each population is calculated using. After sorting the fitness value in descending order, the minimal fitness value is saved as alphanext minimal as betaand third minimal as deltagrey wolves as given in Eq. (18).

3.1.2 Encircling of prey

The comparison of statistical data of GWO algorithm with the results obtained using different algorithm is given in Tab. 2. The results presented in Tab. 2 suggest that GWO algorithm has the capability of attaining global minimum value for the ELD problems. To move further, the GWO algorithm is applied to large sized problems to assess the efficiency of the algorithm.

Where the current iteration in the problem is represented asis the position of the prey,indicates the position of grey wolf atindicates the position of grey wolf at t+1andandare the coefficient vectors which are computed using Eq. (12)and Eq. (13) respectively.

Wheredecreases from 2 to 0 linearly as the iteration increases and rand is the random vectors betweensuch that A gets values within

The total demand is 2630 MW. Thermal units 2, 5, 6 and 12 have prohibited operating zones. The best fuel cost reported in Basu [Basu (2016)] is 32,548.17 $/hr. The best fuel cost obtained by GWO algorithm for SYS2 is 32,548.13 $/hr. The optimal power generation obtained using GWO algorithm is given in Tab. 3. The comparison of statistical results obtained using GWO algorithm and other algorithms are summarized in Tab. 4.

The results obtained using GWO algorithm for SYS3 is compared with the previously obtained results using various algorithms and is summarized in Tab. 6. The statistical data for these algorithms are obtained from Moradi-Dalvand et al. [Moradi-Dalvand,Mohammadi-Ivatloo, Najafi et al. (2015)]. The authors Moradi-Dalvand et al. [Moradi-Dalvand, Mohammadi-Ivatloo, Najafi et al. (2015)] suggest that the reported results in algorithms SOA, CGPSO and CMSFLA do not satisfy the system constraints.

Whereare the position of first, second and third best fitness value,is determined using Eq. (10) ,are determined using Eqs. (12)and (13) ,and 3are the updated position ofbased on position of alpha,beta and grey wolves respectively.

3.1.4 Attacking prey (Exploitation)

During this phase,value gets reduced which reduces the fluctuation of. Sinceis a vector whose value is in the range ofthe position of grey wolf will be towards the position of prey in the next generation.

为大力宣传水法，普及水法律知识，促进水法规的贯彻实施，水利部于1988年6月确定每年的7月1日至7日为“中国水周”，集中开展水法规宣传活动。考虑到“世界水日”与“中国水周”的主旨和内容基本相同，从1994年开始，水利部将“中国水周”的时间调整到每年的3月22日至28日。两项活动时间的重合，加大了水法规宣传活动的力度。

3.1.5 Searching the prey (Exploration)

第五，刚才说到口号多了，朗诵多了，概念性的话多了点，戏剧必须是靠情节和人物性格来说话。通过情节，通过人物性格，通过人物思想发展、情感发展、心理发展、行动发展来表达，而不是靠喊口号，尽管词写得很好，但是所有的人物和情节全部断掉了，还是要按照艺术形式来。

Theand δwolves diverges to search the prey and then converges to attack it. All other grey wolves search the prey with respect to above three wolves. This process of searching the prey emphasizes the exploration capability of grey wolves to search globally. Figure 3 represents the flowchart of GWO algorithm.

4 Implementation of GWO algorithm to RPED (GWO-RPED)

The implementation of GWO algorithm to solve RPED complex problem is described as follows:

Step 1: For the chosen test system, read the input data to compute the total fuel cost of the system.

数字测绘档案工作的数据安全风险大，归档时很多数据没有背景信息，信息不完整，而且归档数据组织混乱、归档内容不完整等情况普遍，使得数据的完整性和有效性安全性都不高。虽然保管部门已经进行了一定改革，但是每年读取电子文件和处理设备登记更新工作还存在一些问题，影响载体和新设备的兼容性。在数字测绘档案的管理工作很少会检查数据软盘，也没有加大人力和财力支持进行有效的数据抽查。

综上所述:点P为任意△ABC内的一点，∠A=α,∠B=β,∠C=γ，α+β+γ=180°.点P到△ABC三个顶点A、B、C的距离分别为a、b、c,且满足条件asinα+csinγ>bsinβ、bsinβ+csinγ>asinα、asinα+bsinβ>csinγ.则△ABC的面积可以表示为：

Step 3: Select the number of design variables, D and initialize the design variables i.e. the real power outputs for each generating units in the chosen system. In accordance to the population size, the design variable is generated randomly using Eq. (17).

Figure 3: Flowchart of GWO

where

Therefore, the matrix of D×N is initialized using Eq. (17).

For modeling the GWO algorithm, the wolves are classified based on the fitness value of the problem. The best solution is considered as αwolf, followed by β,δand ωwolves.

Step 5: The individual population corresponding to,andare saved asandrespectively.

这一环节，由学生喜闻乐见的鳄鱼吃肉情境引入，两条鳄鱼比嘴大小，同样的一条鳄鱼嘴张开的角度随其张开的大小而变化，直观呈现和感受角的大小与其两边长短无关，而与其张开的程度有关。

Step 6: Determineandusing Eqs. (12) and (13).

Step 7: Update the position of each grey wolf in the population using Eqs. (14)-(16).

许沁走了，葛局长安静了下来，开始担忧了。想到许沁胜券在握的语气，不禁更担忧了。税务局长哪有屁股干净的？不只许沁懂，地球人都懂。但是，帮许沁是绝对不可能的。葛局长对许沁恨之入骨，绝不肯做违背个人意志的事情。可是，如果不帮许沁，她必定要在那段录音上做文章，问题就闹大了。许沁有句话说得没错，即使钻戒还了，他也是有前科的人。有前科的人，就会进入办案人员的视线。打铁还需自身硬，葛局长知道自身并不硬。许沁现在就像是定时炸弹，让他感受到了巨大威胁。

Step 8: Select the termination criterion

Step 4 to step 7 will be repeated till the termination criteria is reached by the algorithm.

5 Results and discussions

In this section, the performance of the algorithm in solving various complex RPED problems with 6, 15, 20 and 40 thermal unit is discussed. The different constraints considered for these test systems are ramp rate limits, POZ and individual generator limits.The GWO algorithm for different test system has been implemented in MATLAB 2013a on Intel (R) Core (TM) i7-3517U CPU 2.40GHz with 8G-RAM. Simulation results obtained are compared with the results reported in the recent literatures in terms of solution quality.

5.1 RPED problem with POZ and transmission line loss

For RPED problem with POZ and transmission line loss characteristics, the GWO algorithm has been implemented on (i) 6 generating unit and (ii) 15 generating unit for comparison. The performance of GWO is compared with the results obtained in the recent literatures

5.1.1 Test system 1: 6 unit system

Initially, the GWO algorithm is applied to a small test system comprising of 6 generating unit with load demand of 1263 MW which is referred as SYS1. The system coefficients and loss coefficients are listed in Gaing [Gaing (2003)]. The transmission loss, ramp rate limits and POZ are considered for this test system. All the six generating units have two sets of prohibited operating zones. The minimum fuel cost reported in recent literature [Mandal, Roy and Mandal(2013)] is 15,443.06 $/hr. The best fuel cost obtained by GWO algorithm is 15,443 $/hr. The result obtained using GWO indicates that the algorithm attains the global solution with reasonable computational time. The optimal power generation and its corresponding minimum cost obtained using GWO algorithm is given in Tab. 1.

The encircling behavior of grey wolf around the prey is modeled mathematically using Eq.(10) and Eq. (11). Using these equations, a grey wolf updates its position within solution space around the prey.

Table 1: Optimal power for SYS1 using GWO

5.1.2 Test system 2: 15 thermal unit system

Here, 15 thermal unit system is considered to demonstrate goodness of the GWO algorithm in solving this convex RPED problems including all the constraints and is referred as SYS2 in this paper. The system coefficients and loss coefficients for SYS2 are listed in Gaing[Gaing (2003)]. The transmission loss, ramp rate limits and POZ are considered.

Table 2: Comparison of various algorithms with GWO for SYS1

3.1.3 Hunting

仆人来不及查看钓竿，就发现地上有一支小巧的铜笛，沾着鲜血。确切地讲，那不是笛子，只不过模样像笛子，小得多，也短得多。就是这玩意，刚才砰的一下从钓竿握把底端射出，贯通李霸崖身体，洞穿其肝脏。

Table 3: Optimal power for SYS2 using GWO

It can be inferred from Tab. 3 and Tab. 4 that the best result has been obtained using GWO algorithm without violating any system constraints.

Table 4: Comparison of various algorithms with GWO for SYS2

5.1.3 Test system 3: 20 generating unit system-ELD problem with transmission losses

For this variant of ELD, a test system with 20 thermal unit is adopted for evaluating using GWO algorithm and is referred as SYS3. In this system, POZ is not considered. The demand to be met by SYS3 is 2500 MW. The system data and the transmission loss coefficients are considered from Su et al. [Su and Lin (2000)]. The authors in Moradi-Dalvand et al. [Moradi-Dalvand, Mohammadi-Ivatloo, Najafi et al. (2015)] have reached a optimum value of 62,456.63 $/hr. The exploration and exploitation of GWO algorithm has converged the system to reach a better optimum value of 62,454.27 $/hr without violating any system constraints. Tab. 5 provides the optimal power for each unit in the test system obtained using GWO.

Table 5: Optimal power for SYS3 using GWO

Though all the grey wolves can recognize the prey's location, α,βand δgrey wolves have more knowledge about the location. Therefore, the positions of these wolves are saved and force the other wolves to update their position using Eq. (14) through Eq. (16).

Table 6: Comparison of various algorithms with GWO for SYS3

5.1.4 Test system 4: 40 thermal unit system-RPED problem with POZ

A complex system with 40 thermal unit with POZ is considered here and is referred as SYS4. The transmission line losses are neglected. The total demand for SYS4 is 7000 MW.The test system data is available in Chen et al. [Chen and Chang (1995)]. The optimal generation schedule for the test system using GWO algorithm is presented in Tab. 7. The minimum fuel cost achieved by GWO is 99722.99 $/hr. In the recent literature, the minimum fuel cost achieved for the 40 unit system is 100767.68 $/hr in Balamurugan et al.[Balamurugan and Subramanian (2008)].

Table 7: Optimal power for SYS4 using GWO

Tab. 8 summarizes the statistical cost achieved by various algorithms for 40 unit system with prohibited operating zone over the decade. It can be observed from Tab. 7 that the GWO algorithm results in a better solution when compared to others and it reveals the capability of algorithm to produce the global optimal cost from a large solution space which has large local optima.

从一例非法用地案看设施农用地的认定（叶隆生） ........................................................................................8-49

Table 8: Comparison of various algorithms with GWO for SYS4

5.2 Result analysis

5.2.1 Parameter selection

According to many research experts, the efficiency of stochastic search algorithms (such as GA, PSO, DE, etc.) depends on user defined parameters. Using parameter tuning,testing and evaluating different combinations of parameters, the optimal parameter values of an algorithm are obtained for a specific test system [Barisal and Prusty (2015); Amjady and Sharifzadeh (2010)]. In GWO, the parameter which affects the convergence and search capability of the algorithm is the number of grey wolf population. An optimal choice of population size is necessary as the other values makes an algorithm slow, computationally inefficient and leads to local minima than to the global minima. The optimal population size directly depends on problem dimension and complexity to achieve optimum value for the problem [Chaturvedia, Panditb and Srivastava (2009); Roy, Roy and Chakrabartic(2013)]. The numerical values presented in Tabs. 2, 4, 6 and 8 summarizes that GWO algorithm provides the optimal fuel cost when compared to the recent literatures. In addition, the performance of GWO algorithm is demonstrated by executing 50 test runs for different population sizes and the obtained solutions are presented in Tab. 9. The population size selected for different test systems is indicated in Tab. 9.

Table 9: Effect of population size on different test systems

5.2.2 Convergence characteristics

The convergence characteristic of GWO algorithm for SYS1,SYS2, SYS3 and SYS4 discussed in the previous sections is presented in Fig. 4. The search agents in GWO explores solution space and determine the optimal solution quickly and since it has good search mechanism, the algorithm attains the optimal solution within 100 iterations for small test system and within an ad