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Grey Wolf Optimizer to Real Power Dispatch with Non-Linear Constraints

2018-06-07 06:46:57VenkatakrishnanRengarajandSalivahanan
關(guān)鍵詞:水周株數(shù)鱷魚

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.

飛禽當(dāng)中,可以吃潔凈的,但以下皆不可食:鳶、禿鷲、黑雕,一切鷂隼,大小烏鴉;鴕鳥、夜鶯、海鷗、鶚、貓頭鷹之屬;朱鷺、塘鵝、鴇、鸕鶿、鸛鷺一族;以及戴勝、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.

通過對(duì)傳統(tǒng)倫理學(xué)的反思和對(duì)現(xiàn)代技術(shù)現(xiàn)實(shí)境況的考察,約納斯形成了對(duì)技術(shù)時(shí)代倫理氛圍的基本認(rèn)識(shí)。在傳統(tǒng)社會(huì),技術(shù)的影響范圍極其有限,因而人的倫理行為遵循此時(shí)此地的原則;而在現(xiàn)代社會(huì),由于科學(xué)技術(shù)的影響超越時(shí)空 , 因此人類應(yīng)實(shí)行遠(yuǎn)距離的“責(zé)任”倫理。[29]至此,約納斯試圖將“責(zé)任”維度重新置入倫理學(xué)理論之中,通過闡發(fā)一種“未來責(zé)任”的理念,構(gòu)建適應(yīng)“技術(shù)時(shí)代”需要的“未來倫理學(xué)”。

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.

吳鐵成一見到戴笠,就批評(píng)他說:“雨農(nóng)啊,這幾天,山城政界搞‘吼’了,都是你惹的禍啊。你那樣搞法,自認(rèn)為是忠于領(lǐng)袖和國家,那是你個(gè)人的想法,不一定是大家的想法。你給黨國、給領(lǐng)袖幫了倒忙。你們做特務(wù)、情報(bào)工作的,要準(zhǔn)確無誤嘛。黃炎培雖然可恨,但他愛國和堅(jiān)決抗日的態(tài)度,是眾所周知的。說他家藏有日偽人員,沒有哪個(gè)會(huì)相信的。下面有這樣的情報(bào)來,作為局長,你應(yīng)慎重地研判一下,不能糊里糊涂地下令叫部下去亂搞。雨農(nóng),你知道,我過去也做過半個(gè)情報(bào)人員的公安局長。我那時(shí)處理這類問題非常慎重。這一回你恰巧碰到天不怕地不怕的黃炎培頭上,所以鬧得你下不了臺(tái)。以后,你一定要吸取這次的教訓(xùn)?!?/p>

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 株數(shù)計(jì)算法 每一塊田中某種雜草的株數(shù)=同一塊田9點(diǎn)樣方中該雜草的株數(shù)之和/9;同一類型田塊中某雜草的株數(shù)=相同類型每一塊田中這種雜草的株數(shù)之和/該類型田塊數(shù)。

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,太平本是將軍定,哪個(gè)將軍見太平?本將軍平定天下,功高勞苦,想不到功勞越大,越是不能安享太平?!睂④娍嘈χ?,頹然問道,“沉淵樓什么時(shí)候也肯殺忠臣良將了?”

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.

為大力宣傳水法,普及水法律知識(shí),促進(jìn)水法規(guī)的貫徹實(shí)施,水利部于1988年6月確定每年的7月1日至7日為“中國水周”,集中開展水法規(guī)宣傳活動(dòng)??紤]到“世界水日”與“中國水周”的主旨和內(nèi)容基本相同,從1994年開始,水利部將“中國水周”的時(shí)間調(diào)整到每年的3月22日至28日。兩項(xiàng)活動(dòng)時(shí)間的重合,加大了水法規(guī)宣傳活動(dòng)的力度。

3.1.5 Searching the prey (Exploration)

第五,剛才說到口號(hào)多了,朗誦多了,概念性的話多了點(diǎn),戲劇必須是靠情節(jié)和人物性格來說話。通過情節(jié),通過人物性格,通過人物思想發(fā)展、情感發(fā)展、心理發(fā)展、行動(dòng)發(fā)展來表達(dá),而不是靠喊口號(hào),盡管詞寫得很好,但是所有的人物和情節(jié)全部斷掉了,還是要按照藝術(shù)形式來。

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.

數(shù)字測繪檔案工作的數(shù)據(jù)安全風(fēng)險(xiǎn)大,歸檔時(shí)很多數(shù)據(jù)沒有背景信息,信息不完整,而且歸檔數(shù)據(jù)組織混亂、歸檔內(nèi)容不完整等情況普遍,使得數(shù)據(jù)的完整性和有效性安全性都不高。雖然保管部門已經(jīng)進(jìn)行了一定改革,但是每年讀取電子文件和處理設(shè)備登記更新工作還存在一些問題,影響載體和新設(shè)備的兼容性。在數(shù)字測繪檔案的管理工作很少會(huì)檢查數(shù)據(jù)軟盤,也沒有加大人力和財(cái)力支持進(jìn)行有效的數(shù)據(jù)抽查。

綜上所述:點(diǎn)P為任意△ABC內(nèi)的一點(diǎn),∠A=α,∠B=β,∠C=γ,α+β+γ=180°.點(diǎn)P到△ABC三個(gè)頂點(diǎn)A、B、C的距離分別為a、b、c,且滿足條件asinα+csinγ>bsinβ、bsinβ+csinγ>asinα、asinα+bsinβ>csinγ.則△ABC的面積可以表示為:

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.

這一環(huán)節(jié),由學(xué)生喜聞樂見的鱷魚吃肉情境引入,兩條鱷魚比嘴大小,同樣的一條鱷魚嘴張開的角度隨其張開的大小而變化,直觀呈現(xiàn)和感受角的大小與其兩邊長短無關(guān),而與其張開的程度有關(guān)。

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

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

許沁走了,葛局長安靜了下來,開始擔(dān)憂了。想到許沁勝券在握的語氣,不禁更擔(dān)憂了。稅務(wù)局長哪有屁股干凈的?不只許沁懂,地球人都懂。但是,幫許沁是絕對(duì)不可能的。葛局長對(duì)許沁恨之入骨,絕不肯做違背個(gè)人意志的事情。可是,如果不幫許沁,她必定要在那段錄音上做文章,問題就鬧大了。許沁有句話說得沒錯(cuò),即使鉆戒還了,他也是有前科的人。有前科的人,就會(huì)進(jìn)入辦案人員的視線。打鐵還需自身硬,葛局長知道自身并不硬。許沁現(xiàn)在就像是定時(shí)炸彈,讓他感受到了巨大威脅。

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

仆人來不及查看釣竿,就發(fā)現(xiàn)地上有一支小巧的銅笛,沾著鮮血。確切地講,那不是笛子,只不過模樣像笛子,小得多,也短得多。就是這玩意,剛才砰的一下從釣竿握把底端射出,貫通李霸崖身體,洞穿其肝臟。

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.

從一例非法用地案看設(shè)施農(nóng)用地的認(rèn)定(葉隆生) ........................................................................................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

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