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1 Florda State Unversty Lbrares Electronc Theses, Treatses and Dssertatons The Graduate School 2014 Consensus-Based Dstrbuted Control for Economc Dspatch Problem wth Comprehensve Constrants n a Smart Grd Janwu Cao Follow ths and addtonal works at the FSU Dgtal Lbrary. For more nformaton, please contact lb-r@fsu.edu

2 FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING CONSENSUS-BASED DISTRIBUTED CONTROL FOR ECONOMIC DISPATCH PROBLEM WITH COMPREHENSIVE CONSTRAINTS IN A SMART GRID By JIANWU CAO A Dssertaton submtted to the Department of Electrcal and Computer Engneerng n partal fulfllment of the requrements for the degree of Doctor of Phlosophy Degree Awarded: Fall Semester, 2014

3 Janwu Cao defended ths dssertaton on September 04, The members of the supervsory commttee were: Mng Yu Professor Drectng Dssertaton Anke Meyer-Baese Unversty Representatve Petru Andre Commttee Member Hu L Commttee Member The Graduate School has verfed and approved the above-named commttee members, and certfes that the dssertaton has been approved n accordance wth unversty requrements.

4 ACKNOWLEDGMENTS In the frst place, I would lke to express my sncere grattude to my academc supervsor Mng Yu PhD for hs contnuous supervson, advce, gudance and support from the very begnnng of my PhD study. Hs advce and support helped me get out of trouble and encouraged me to stay postve. Hs knowledge of communcaton theory and smart grd ncreased my understandng of ths specfc topc and helped me to produce ths work. He taught me to thnk, analyze and solve problems ndependently n a frendly manner. Dr. Yu helped me edt ths dssertaton and other publcatons. He also offers great advce and support for my lfe. Thank you for makng me a better researcher and a more mature person. I want to specally thank Hu L PhD for gvng me advce and drectons not only n my academc research but also n my lfe. Her advce always kept me reman enthusastc and ntatve. A great lesson can be learned from her every tme we had a dscusson. Furthermore, I d lke to thank Petru Andre PhD and Anke Meyer-Baese PhD for ther great advce and support. Thank you to my famly who are very supportve and patent for my further study. Everyone who contrbuted that was not specfcally named I thank you too.

5 TABLE OF CONTENTS LIST OF TABLES... v LIST OF FIGURES... v ABSTRACT... x 1. INTRODUCTION Background Motvaton Problem Statement STATE-OF-THE-ART ECONOMIC DISPATCH PROBLEM Conventonal Methods Decentralzed Methods Accuracy of EDP Optmzaton Methods GRAPH THEORY AND CONSENSUS ALGORITHM Graph Theory Consensus Algorthm Soluton to Economc Dspatch Problem Convergence Study of Consensus Algorthm DEFINITION OF COST FUNCTION AND CONSTRAINTS Cost Functon Defnton Constrants Defnton Nonlnearty of EDP Optmzaton ANALYTICAL RESULTS System Descrpton Case Studes Comparson wth Conventonal Methods SIMULATION TESTS PSCAD Smulaton Platform IEEE 14 Bus System Smulaton Results CONCLUSION AND FUTURE WORK Concluson Future Work...67 v

6 REFERENCES...69 BIOGRAPHICAL SKETCH...74 v

7 LIST OF TABLES 1 Percentage dfferences between consensus algorthm and optmal result Parameters of the fve generators Constrants of generators Lambda teraton parameters Partcle swarm optmzaton parameters Cost comparson of three methods Lne data of IEEE 14 bus system Bus data IEEE 14 bus system Cost coeffcents of generators Transformer tap settng IEEE 14 bus system...63 v

8 LIST OF FIGURES 1 Smart grd conceptual model A smple power system wth three unts A drawng of a graph Cost functon curve of a power generator A typcal generator operaton area Input-output curve of a thermal generator wth valve ponts Pecewse lnear cost curve of generaton unt Fve generators smart grd smulaton model Communcaton topology of fve generators: star topology Case study 1 ncremental cost of fve generators Case study 1 power generaton and power demand Case study 1 power msmatch of all generators Case study 2 ncremental cost of fve generators Case study 2 power generaton, power loss and power demand Communcaton topology of fve generators: loop connecton Case study 3 ncremental cost of fve generators Case study 3 power generaton, power loss and power demand Case study 4 ncremental cost for fve generators Case study 4 generator output Case study 4 power generatons, power loss and power demand Case study 5 ncremental cost of fve generators wth ε = 1e Case study 5 ncremental cost of fve generators wth ε = 10e v

9 23 Case study 5 ncremental cost of fve generators wth ε = 16e Case study 5 cost comparson of four dfferent convergence constants Case study 5 power comparson of four dfferent convergence constants Flow chart of lambda teraton EDP method Incremental cost comparson between lambda teraton and consensus algorthm Total cost between lambda teraton, PSO method and consensus algorthm IEEE 14 bus test system Power output of fve generators Power generaton and power demand Incremental cost of fve generators Total cost of fve generators...65 v

10 ABSTRACT Over the past few decades, the smart grd technology has been developed rapdly due to ts man features of more nvolvement of customers and abltes to accommodate all renewable energy and dstrbuted storages. More mportantly, t offers an mproved relablty, power qualty and self-healng capablty. However, there are many problems and challenges assocated the development of smart grd. For example, the economc dspatch problem (EDP) n a smart grd has become more complex and challengng due to specal characterstcs of smart grd. For example, one of the major characterstcs of smart grds s plug-and-play due to ts accommodaton of dstrbuted energy. Economc dspatch s the short-term determnaton of the optmal output of a number of electrcty generaton facltes, to meet the system load, at the lowest possble cost, subject to transmsson lne loss and generaton constrants. In short, EDP s an optmzaton problem and ts am s to reduce the total operaton cost. Varous mathematcal and optmzaton methods have been developed to solve EDP n power systems. Most of the conventonal methods collect global nformaton and process commands n a centralzed controller. In a smart grd, t s expensve and unrelable for these conventonal centralzed methods to acheve a mnmum cost when generatng a certan amount of power wthn certan power constrants. There are several reasons why t s not sutable to use centralzed methods for EDP n a smart grd. Frst of all, the centralzed controller requres a hgh level of connectvty to collect all the nformaton among power generators. A falure or error may mpar the effectveness of the centralzed controller. Secondly, the topologes of the smart grd and the communcaton network are lkely to be varable n a smart grd. Therefore, a small change n the smart grd may lead to x

11 reconfguraton of the centralzed algorthm. Thrdly, the centralzed controller s not able to accommodate the plug-and-play characterstc of smart grd. In ths work, we propose a dstrbuted controller based on consensus algorthm to solve the EDP n a smart grd. The consensus algorthm s based on graph theory n the area of communcaton. Compared wth the centralzed method, the dstrbuted algorthm features advantages of less nformaton requrement, robustness, and scalablty. In order to present a more practcal scenaro of EDP, a quadratc cost functon and comprehensve constrants are assumed n the problem defnton. It s assumed that the valve pont effect of the generaton unt s neglgble. Dfferent from the centralzed approach, the proposed algorthm enables each generator to collect the msmatch between power demand and power generatons n a dstrbuted manner. The msmatch power s used as a feedback for each generator to adjust ts power generaton. In order to mplement the consensus algorthm, the ncremental cost of each generator s selected as the consensus quantty and wll converge to a common value eventually. Smulaton results of dfferent case studes are provded to show the effectveness of the proposed algorthm. Effect of power constrants, communcaton topology and generator dynamc on the convergence and teraton speed of proposed algorthm s also examned. These case studes are smulated and analyzed n Matlab/Smulnk. The convergence speed and total generaton cost of proposed algorthm are also compared wth the conventonal algorthms such as lambda teraton method and partcle swarm optmzaton. The consensus algorthm has a better combned performance of convergence and total generaton cost compared to lambda teraton method and partcle swarm optmzaton. In order to valdate the consensus algorthm, an IEEE 14 bus system x

12 wth the proposed algorthm s establshed n PSCAD/EMTDC and verfed by comparng wth the analytcal results. x

13 CHAPTER ONE INTRODUCTION 1.1 Background A smart grd s a modernzed electrc grd that uses analog or dgtal nformaton and communcaton technology to gather and act on nformaton, such as behavors of supplers and consumers, n an automated fashon to mprove the effcency, relablty, economcs, and sustanablty of the producton and dstrbuton of electrcty [1]. A typcal smart grd confguraton can be found n Fg. 1. Fg. 1: Smart grd conceptual model. The smart grd defnes several mportant objects: bulk generaton, transmsson, dstrbuton, customers, operatons, markets and servce provders. It shows all communcatons and power flows connectng each object. Each ndvdual s composed by smart grd elements that are connected to each other through both communcatons and electrc paths. The bulk generaton of the smart grd generates electrcty from renewable and nonrenewable energy resources n bulk quanttes. These resources can also be classfed as renewable, varable sources, such as solar and wnd energy; renewable, non-varable, such as bomass, 1

14 geothermal and pump storage; or non-renewable, non-varable, such as nuclear, coal and gas. Energy storage s also ncluded for later dstrbuton n bulk generaton. Dstrbuton element dstrbutes electrcty to and from the customers n the smart grd. The dstrbuton network connects the smart meters and all ntellgent feld devces, managng and controllng them through a two-way wreless or wrelne communcatons network. It may also connect to energy storage facltes and alternatve dstrbuted energy resources at the dstrbuton level. The customers of the smart grd s where the users of electrcty are connected to the electrc dstrbuton network through the smart meters. The smart meters control and manage the flow of electrcty to and from the customers and provde energy nformaton about energy usage. A customer may also generate, store and manage the use of energy as well as the connecton wth plug-n vehcles. The operaton of smart grd manages and controls the electrcty flow of all other objects n smart grd. It uses a two-way communcatons network to connect to substatons, customer network and other ntellgent feld devces. It provdes montorng, reportng, controllng and supervson status and mportant process nformaton and decsons. The market operates and coordnates all partcpants n electrcty market wthn the smart grd. It provdes the market management, sale, retal and trade of energy servces. It also handles energy nformaton operatons and nformaton exchange wth thrd-party servce provders. Fnally, the servce provder of the smart grd handles the thrd-party operatons among the objects. It may also manage other processes for the utltes, such as demand response programs, outage management and feld servces. 2

15 Due to ts advantages over conventonal power system, smart grd technology has developed and promsed around the world for many years. Together wth dstrbuted energy and storage technology, smart grd exhbts great benefts such as hgh relablty, mproved power qualty, flexblty, and real tme operaton. Development n power electroncs, control algorthms, communcaton technologes, and nformaton technque have played a crtcal part n the smart grd technology. However, ncreasng use of smart grd wll pose many challenges for the future grd. There are a lot of concerns assocated wth smart grd technology. For example, t becomes expensve and unrelable for a smart grd to solve economc dspatch problem (EDP) f we keep usng the conventonal method. Fundamentally, EDP s to solve an optmzaton problem and to reduce the total cost of power generators wth certan constrants. It s defned by US Energy Polcy Act of β005 as the operaton of generaton facltes to produce energy at the lowest cost to relably serve consumers, recognzng any operatonal lmts of generaton and transmsson facltes [2]. Many conventonal methods have been developed to solve the EDP n power system. All the conventonal methods requre global nformaton va a centralzed controller to acheve an optmal power generaton and a mnmum cost [4]. A smple power system s presented n Fg. 2 to llustrate the dea of EDP. Fg. 2. A smple power system wth three unts [3]. 3

16 As shown n Fg. 2, a central controller s requred to collect nformaton from all three generaton unts and dstrbuted loads. After solvng the dspatch problem n the controller, the commands of power generaton of each unt wll be sent out. However, the centralzed algorthm can be mpared due to falures or modelng errors [5]. Moreover, the smart grd topologes and communcaton network are lkely to change. For example, a new generaton unt may be added later to Bus 1. The central controller has to collect all nformaton agan ncludng the new generaton unt and assgns the power output correspondngly. Therefore, a slght change n the smart grd may lead to redesgn of the centralzed algorthm [6]. Also, collectng global nformaton from each generator may cause extra cost. It can be concluded that the centralzed controller s not sutable to solve the EDP n a smart grd. 1.2 Motvaton Economcs s one of the most promsng characterstcs of smart grd technology. Wth the development of smart grd technology, more and more renewable devces are connected to the smart grd system. For example, solar energy, wnd energy, and energy storage are all nterfacng wth smart grd. Most of them are plug-and-play devces. Therefore, economc dspatch becomes very mportant for all generaton unts n smart grd. In order to better solve EDP n a smart grd, a dstrbuted control algorthm s developed n ths work. The dstrbuted algorthm enables each generaton unt to control ts own power output n a dstrbuted manner. Compared wth the centralzed algorthm, the dstrbuted algorthm features the followng advantages. Frst of all, no centralzed controlled or global nformaton s needed by the dstrbuted algorthm. For example, n Fg. 2, no central controller s needed to gather all nformaton from generators and loads. Each generator wll have ts own generaton controller. Moreover, the dstrbuted algorthm s not 4

17 affected by the varaton of topologes. Therefore, the plug-and-play characterstcs of smart grd can be accommodated by the dstrbuted algorthm. The key to the dstrbuted algorthm n the smart grd s for all generated to reach a consensus [7]. There are many varables that can be used as the consensus. For example, n a smart grd, the consensus varables can be voltage, actve power, etc. The consensus protocol has been studed wdely n a varety of area for many years. It has been appled n system and control areas. It shows great advantages n dealng wth EDP n a smart grd. The benefts of consensus algorthm s presented n Chapter 2. In realty, the cost functon of a generaton unt s non-convex. However, t s approxmately represented by a convex equaton n most stuatons. Snce the economc dspatch has to meet certan generaton and transmsson constrants, comprehensve constrants are dscussed and mplemented n the proposed consensus algorthm. 1.3 Problem Statement As stated before, t s becomng more and more mportant for a smart grd to solve EDP n a dstrbuted manner. In ths work, a consensus-based dstrbuted controller for EDP n a smart grd s proposed. The dstrbuted algorthm exhbts great advantages than conventonal methods. Here are the problem statements of the proposed algorthm. Frst of all, a cost functon of EDP and generator constrants are defned. A convex cost functon s derved based on the non-convex functon of power generator. Practcally, the generators have to operate wthn certan operaton lmts. Meantme, the power balance between generaton and demand s also requred. For practce, the transmsson power loss s also taken nto consderaton for constrants. Based on grapy theory, the consensus algorthm s developed to solve EDP n the smart grd. A consensus varable n the smart grd s selected as the consensus that wll converge to the optmal value for each generator. The communcaton network can vary for 5

18 dfferent smart grds. Therefore, performance of proposed algorthm should be examned n dfferent communcaton topologes. The dynamcs of generators are also examned n ths work. In order to valdate the effectveness of the dstrbuted algorthm, t s compared wth conventonal EDP methods. The transents of the proposed algorthm s also consdered by ncludng the dynamcs of synchronous machnes. The performance s also valdated by testng n a practcal power system usng PSCAD/EMTDC. 6

19 CHAPTER TWO STATE-OF-THE-ART ECONOMIC DISPATCH PROBLEM 2.1 Conventonal Methods The economc dspatch s amng to supply load at a mnmum total cost. The EDP can be summarzed as the followng equaton. mn m k1 C k ( P k ) where m s the number of generaton unt, Pk s the power output of k-th generator, Ck s the generaton cost of k-th generator. There are varous numercal and optmzaton methods developed to solve the EDP n power system. The conventonal methods nclude lambda-teraton method [8], Lagrangan relaxaton method [9], gradent projecton method [10], and dynamc programmng [11]. In [8], the ncremental cost of each generator updates tself every teraton n order to meet the power balance. Lambda s the varable ntroduced n solvng constrant optmzaton problem and called a Lagrange multpler. Unlke usual teraton methods lke gauss sedel and newtonraphson, lambda teraton s dfferent. For example, n gauss sedel method, the next value of the unknown varable can be solved usng an equaton. However, n lambda teraton, the unknown varable gets ts next value based on ntuton. It wll be projected by nterpolatng the best possble value when a specfed msmatch has been reached. By usng Lagrange multpler, lambda s equal to the ncremental cost of each generator. However, ths method requres more computatons and takes a longer tme to process. And the result may not accurate due to a larger msmatch allowed. 7

20 The conventonal Lagrangan relaxaton method can be appled drectly nto a non-convex optmzaton problem by decomposng the non-convex space nto a small number of subsets [9]. In ths way, the assocated EDP s ether nfeasble or the one solved by the conventonal Lagrangan relaxaton method. The relaxaton s carred out so that the relaxed problem s decomposable to a number of problems correspondng to the perods n the dspatch horzon. These are solved smply by usng prorty lsts. The dual Lagrangan functon s optmzed usng subgradent optmzaton. If an overall soluton feasble n all constrants and suffcently close to a computed best lower bound s dscovered durng sub-gradent optmzaton. The cost wll be greatly reduced by the decomposed method among all feasble solutons. A gradent projecton method s also presented [10]. The gradent projecton s also usng ncremental cost of generator as the evaluaton crteron. The man convergence result s obtaned by defnng a projected gradent, and provng that the gradent projecton method forces the sequence of projected gradents to zero. A consequence of ths result s that f the gradent projecton method converges to a pont of a lnearly constraned problem. As an applcaton of EDP, a quadratc programmng algorthm that teratvely explore a subspace defned by the actve constrants s developed. Based on the look-ahead technque, EDP fnds the number of tme ntervals to guarantee the soluton optmzaton [11]. An effectve technque for fndng the optmal soluton va the nteror pont method based lnear programmng s descrbed. A detaled mathematcal dervaton of recursve dynamc programmng approach for the EDP s presented. It s noted that computaton efforts wll ncrease f the optmal look-ahead tme nterval s not used. Conventonal programmng based method depends on the sze of the dscrete capacty used. Wth a capacty step of one MW, the number of states at each stage s qute large for even a small system. 8

21 The methods above are assumed to have a convex cost functon. In order to handle a nonconvex cost functon, many optmzaton methods are developed, manly evolutonary programmng [12], dfferental evoluton [13], partcle swarm optmzaton [14], genetc algorthm [15], smulated annealng [16], and tabu search [17]. Evolutonary programs wth adaptatons based on scaled cost, as well as emprcal learnng rate, have been developed [12]. Evolutonary programmng s a stochastc optmzaton strategy by Lawrence J. Fogel. An ntally random populaton of ndvduals s created. Mutatons are then appled to each ndvdual to create new ndvduals. Mutatons vary n the severty of ther effect on the behavor of the ndvdual. The new ndvduals are then compared n a context to select whch should survve to form the new populaton. In smaller power system, all fast evolutonary programs perform much better than that wth Gaussan mutaton n terms of convergence rate, soluton tme, mnmum cost, and probablty of better solutons. As the sze of power system ncreases, mproved fast evolutonary program offers superor performance over other evolutonary programs. An mproved dfferental evoluton method to solve the EDP of generators consderng valvepont effects s proposed [13]. Dfferental evoluton s a populaton-based, stochastc functon optmzer usng vector dfferences for perturbng the populaton. Heurstc crossover technque and gene swap operator are ntroduced n ths approach to mprove the convergence characterstc of the dfferental evoluton algorthm. Consequently, t leads to a hgher probablty of gettng the global or near global soluton. Lke other evolutonary algorthms, the frst generaton s ntalzed randomly and further generatons evolve through the applcaton of certan evolutonary operator untl a stoppng crteron s reached. The optmzaton process s carred wth four basc operatons: ntalzaton, mutaton, crossover and selecton. 9

22 Partcle swarm optmzaton (PSO), one of the modern heurstc algorthms frst ntroduced by Kennedy and Eberhart, s used to solve EDP n power system [14]. PSO algorthm s motvated by socal behavor of organsms such as fsh schoolng and brd flockng. It provdes a populatonbased search procedure n whch ndvduals called partcles change ther postons wth tme. Compared wth genetc algorthms, PSO algorthm has been demonstrated to have superor features, ncludng hgh qualty soluton, stable converge characterstc, and good computaton effcency. Genetc algorthm (GA) nvented by Holland n the early 1970s s a stochastc global search method that mmcs the metaphor of natural bologcal evoluton. It s one of the advanced optmzaton method to solve the EDP [15]. The man objectve of GA method s to fnd strng wth a maxmum ftness that can be acheved by smply ncreasng the strng bt length. Smulated annealng algorthm s proposed to solve the optmal dspatch [16]. Smulated annealng s a random search technque for optmzaton that explots an analogy between the way n whch a metal cools and freezes nto a mnmum energy crystallne structure and the search for a mnmum n a more general system. Numercal studes for a sample test system are presented to demonstrate the performance and applcablty of the proposed method. A multple tabu search algorthm s presented to solve the dynamc EDP problem [17]. The multple tabu search algorthm ntroduces addtonal mechansm such as ntalzaton, adaptve searches, multple searches, crossover and restartng process. To show ts effcency, ths algorthm s appled to solve constraned EDP of power system wth 6 and 15 unts. And t s also compared wth conventonal approaches. 10

23 2.2 Decentralzed Methods All the conventonal methods requre global nformaton va a centralzed controller to acheve an optmal power generaton and a mnmum cost. However, the centralzed algorthm may cause a few problems n the smart grd. Frstly, the centralzed controller requres a hgh level of connectvty that may be affected by the falures and modelng errors. Secondly, the topologes of the smart grd and the communcaton network are lkely to be varable. Therefore, a small change n the smart grd may lead to reconfguraton of the centralzed algorthm. Thrdly, collectng global nformaton from each generator may cause extra cost. Therefore, t can be concluded that the centralzed controller s not sutable to solve the EDP n a smart grd. In order to solve the EDP n a smart grd, a dstrbuted control algorthm s employed. Compared wth the centralzed algorthm, the dstrbuted algorthm features advantages of less nformaton requrement, robustness, and scalablty. Frst of all, no centralzed controller or global nformaton s needed by the dstrbuted algorthm. Moreover, the dstrbuted algorthm s not affected by the varaton of topologes. Therefore, the plug-and-play characterstc of smart grd can be accommodated by the dstrbuted algorthm. The key to the dstrbuted algorthm n the smart grd s for all generators to reach a consensus. Consensus algorthm has been studed wdely for the past two decades. Its applcatons can be found n the area of system and control [18-19]. The man problem n a consensus algorthm s to reach agreement regardng certan quantty of nterest by usng local nformaton exchange [19]. The mult-agent system can be categorzed as ether formaton control problems wth applcatons to moble robots, unmanned ar vehcles, autonomous underwater vehcles, satelltes, arcraft, spacecraft, and automated hghway systems. Lately, consensus algorthm has been utlzed n smart grd assocated problems [20-21]. 11

24 For example, a consensus-based algorthm s appled to solve EDP n a smart grd [20]. Effectve dstrbuted control algorthms could be embedded n dstrbuted controllers to properly allocate electrcal power among connected generators. By selectng the ncremental cost of each generator as the consensus varable, the algorthm s able to solve the conventonal centralzed control problem n a dstrbuted manner. To satsfy the power balance, the msmatch between demand and total generatons s fed back to consensus algorthm so that the ncremental cost wll converge to the optmal pont. The communcaton topology among generators are assumed to be undrected that the nformaton exchange s bdrectonal. However, the assumpton s not practcal snce the communcaton may not be symmetrc n real stuatons [4]. Moreover, the algorthm s not completely dstrbuted snce a leader has to be selected to collect all power output from each generator n order to calculate the power msmatch. Smlarly, a decentralzed algorthm of self-organzng dynamc agents equpped wth consensus protocol s proposed to solve EDP n a smart grd [21]. Modern trends n economc dspatch analyss are orented towards the deployment of computng archtectures that move away from the tradtonal centralzed paradgms to archtectures dstrbuted n the feld wth an ncreasng pervason of cooperatve smart enttes. The soluton of the economc dspatch problem s obtaned by the agents network by exchangng and processng local nformaton accordng to a dstrbuted consensus protocol. Due to ths feature, each agent can compute the most mportant varables characterzng the global power system operaton wthout the need for a centralzed center. The effectveness of proposed algorthm s proved on 118 and 300 IEEE test networks. However, these results are obtaned by assumng some smplfcaton on the actve power loss modelng. And no generator dynamcs are consdered when evaluatng the convergence rate. 12

25 A load dspatch model for the system consstng of both thermal generators and wnd turbnes s developed [25]. The stochastc wnd power s ncluded n the model as a constrant. It s shown that under certan condtons, the presented model has a set of closed-form solutons. The avalablty of closed-form solutons s helpful to gan more fundamental nsghts, such as the mpact of a partcular parameter on the optmal soluton. Furthermore, the probablty dstrbuton and the average of solutons are derved. Ths s called the wat-and-see approach n the dscplne of stochastc programmng. In order to acheve an optmal network operaton, t requres a contnuous real tme montorng, control and economc dspatch by means of a smart dstrbuton management system [26]. The system counts on hgh performance algorthms and real tme nformaton systems. Three very effcent algorthms are presented, whch are: load estmaton, ac power flow, and optmal reconfguraton of loss mnmzaton. Furthermore, these algorthms are tested n many real and large scale dstrbuton networks for several utltes. The prerequste power generaton and load nformaton for decson makng s dscovered by each dstrbuted generaton unt va a multagent coordnaton wth guaranteed convergence [27]. To avod a slow convergence speed whch potentally ncreases the generaton cost because of the tme-varyng nature of dstrbuted generaton, a heterogeneous wreless network archtecture for smart grd s presented. Low cost short range wreless communcaton devces are used to establsh an ad hoc network as a basc nformaton exchange, whle auxlary dual-mode devces wth cellular communcaton capabltes are optonally actvated to mprove the convergence speed. Two multagent coordnaton schemes are proposed for the sngle-stage and herarchcal operaton modes, respectvely. The optmal number of actvated cellular communcaton devces s obtaned based on the tradeoff between communcaton and generaton cost. 13

26 An addtonal constrant s mplemented to ensure the stable slanded operaton of a smart grd [28]. A detaled formulaton accordng to the power sharng prncple regardng the dstrbuted generaton s presented. A test system wth 15 unts s developed for numercal smulaton takng nto account the source type and part load performance. The effect of varous parameters on the cost s also analyzed. An ntellgent economc operaton of smart grd envronment facltatng an advanced quantum evolutonary method s presented [29]. Wnd generators, PV generaton and thermal generators are ncluded n the model. An ntellgent quantum nspred evolutonary algorthm s proposed and appled to perform the ntellgent economc schedulng operaton concernng schedulng and dspatchng. The quantum algorthm features ntellgent operators such as sophstcated rotaton operator, dfferental operator, etc. The method s tested on a hypothetcal power system wth 10 unts, smlar number of hybrd electrc vehcles, smlar number of solar energy and wnd energy. An approach based on the smultaneous perturbaton technque s proposed to deal wth the equalty and nequalty constrants n the economc dspatch problem [30]. The effects of cap-andtrade polces, energy storage, and transmsson le flow lmts n economc dspatch are dscussed. The Lagrangan relaxaton ncludes all the system constrants n the objectve functon wth Lagrange multplers. In ths paper, we have constrants across multple tme perods, such as water resources and rampng constrants. In order to reduce computaton tme, the nequalty constrants are not ncluded n the optmzaton problem. Instead, we check f the soluton satsfes the nequalty constrants after converge. If not, then the volated nequalty constrants are ncluded nto the Lagrangan as bndng equalty constrants wth respectve Lagrangan multplers. Then the optmzaton contnues wth the new bndng constrants enforced. 14

27 2.3 Accuracy of EDP Optmzaton Methods There are many crtera to evaluate the performance of EDP optmzaton methods. For example, the crtera could nclude the convergence speed, computatonal complexty, total generaton cost, etc. One of the most mportant crtera s total generaton cost because the purpose of optmzaton s to reduce the cost of the smart grd. There are a lot of work that have been done to evaluate the qualty of EDP optmzaton methods, especally consensus-based methods. A modfed partcle swarm optmzaton algorthm s proposed to mprove the performance of partcle swarm optmzaton for economc dspatch problem n power systems [32]. The modfed PSO method shows a hgher probablty of achevng better solutons among the exstng methods. A comparson between lambda teraton method and ncremental cost consensus method s carred out [6]. The comparson s based on 6-generator system, 15-generator-system, and 110- generator system, respectvely. The solutons of both methods shown that the consensus algorthm has almost the same cost as the lambda teraton method n 6-generator and 15-generator systems. The results of lambda teraton method can be treated as the optmal values that consensus algorthm tres to reach. To be specfc, the percentage dfferences of the correspondng costs s 0.014% and 0.067%, respectvely. The two methods have the same generaton cost for the 110- generator system when transmsson loss s neglected. It means that the dfference between them s 0%. A smlar consensus based on aucton s proposed to optmze the EDP n the smart grd [5]. In ths work, dfferent scenaros of power system are tested to evaluate the qualty of proposed aucton-based algorthm. The consensus algorthm s compared to most of the centralzed algorthms n 10-generator, 15-generator, and 40-generator systems. As compared to the best 15

28 optmal results, the percentage dfferences are 0.022%, 0.094%, and 0.302%, respectvely. Although, the consensus algorthm does not have the best optmzaton results, the dfference between the optmal result and the consensus algorthm s neglgble. It s noted that most of the centralzed algorthms take many trals to obtan the optmal result. However, the consensus algorthm only need 1 sngle tral. For the heurstc, the results are not determnstc due to specal characterstcs of heurstc methods. The optmal results vary every tral for heurstc methods. In contrast to the heurstc methods, gven the ntal parameters, the results of consensus algorthm are always the same for every tral. It s noted that the percentage dfference between consensus algorthm and real optmal result ncreases as the number of power unts n the system ncreases. The percentage dfference between consensus algorthm and real optmal results s summarzed n the followng table. TABLE 1 Percentage dfferences between consensus algorthm and optmal result No. of Unts Percentage Dfference % % % % It can be seen from Table 1 that percentage dfferences go up from 0.014% n 6-generator to % n 40-generator. Although the dfferences go up, they are stll less than 1% of the optmal soluton. It shows that the consensus has a hgh percentage to acheve the optmal soluton of economc dspatch problem. 16

29 It s also noted that a more accurate optmal soluton may need a bgger teraton steps to acheve the optmal value. The effect of teraton on the performance of EDP algorthms wll be analyzed later. Some mght argue that the consensus algorthm may not work because renewable energy s non-dspatched. Renewable energy source s non-dspatchable due to ts fluctuatng nature, lke wnd power and solar power. However, wth the nteracton of energy storage, the renewable energy can be treated as dspatchable sources. Combned wth energy storage, the renewable energy can generate relatvely constant power. Therefore, the consensus algorthm can be appled to renewable energy system and also smart grds. 17

30 CHAPTER THREE GRAPH THEORY AND CONSENSUS ALGORITHM 3.1 Graph Theory In ths secton, basc graph theory s ntroduced. In mathematcs and computer scence, graph theory s the study of graphs, whch are mathematcal structures used to model parwse relatons between objects [31]. An example of graph s shown n Fg. 3. Fg. 3. A drawng of a graph [31]. A graph s composed of nodes and edges. Edges are lnes that connect dfferent nodes. A graph may be undrected, meanng that there s no dstncton between the two nodes assocated wth each edge. Graphs can be used to model many types of relatons and processes n physcal, bologcal, socal and nformaton systems. Many practcal problems can be represented by graphs. In computer scence, graphs are used to represent networks of communcaton, data organzaton, computatonal devces, etc. A graph G s used to model the power system components and the way they exchange nformaton based on communcaton theory. Let 18

31 G = (V, E, A) (1) Where V s a set of elements called nodes, E s a set of pars of dstnct nodes called edges, and A = [aj] R n*n s the adjacency matrx. A drected grapy s a graph where the edges have drectons assocated wth them. In a smart grd, nodes represent the buses of the power system, the edges represent the transmsson lnes between the buses, and the adjacency matrx represent the edge weghts. A drected edge from to j s denoted by a par (, j) E. The par means that generator j can receve nformaton from generator. The n-neghbor of the -th generator s denoted by: N = {j V (j, ) E} (2) Lkewse, the out-neghbor of the -th generator s denoted by: N = {j V (, j) E} (3) Practcally, a generator can receve nformaton from ts n-neghbor, and send nformaton to ts out-neghbor. The n-degree and out-degree of node s denoted as: d N and d N (4) A drected graph s strongly connected f there exsts a connecton between any par of two nodes. It s noted that d 0 and d 0 n a strongly connected graph. 3.2 Consensus Algorthm Two matrces P, Q R n*n assocated wth a strongly connected graph G = (V, E, A) are defned as follows: 1 j N p, j d, j V (5) 0 otherwse Lkewse, 19

32 1 j N j q, j d, j V (6) j 0 otherwse From the defntons of matrces P and Q, t can be verfed that the summaton of row elements of P s equal to one, and the summaton of column elements of Q s equal to one. It s free to choose the weghts of P and Q. And P and Q have to satsfy the followng assgnments, p,j > 0 f j N, p,j = 0 otherwse, q,j > 0 f N j, q,j = 0 otherwse. The convergence rate s not affected by the weghts. Consderng the followng two dscrete-tme systems: ( k 1) p, ( k) (7) jn j j ' ( k 1) q ( k) (8) jn, j ' j where ξ(k) and ξ (k) are state varables assocated wth node n graph G at tme step k. n a smart grd, the state varable represents a physcal quantty such as the output power, ncremental cost, power msmatch, etc. Equatons (7) and (8) have the same structures but wth two dfferent sets of weghts. They can be composed n the followng format: ( k 1) P ( k) (9) ' ' ( k 1) Q ( k) (10) where ξ(k) and ξ (k) are the column stack vectors of ξ(k) and ξ (k). In order to nvestgate the behavors of (9) and (10), the followng theorem s ntroduced. Theorem 1[19]: If A R N*N s a nonnegatve and prmtve matrx, then 1 lm ( ( A) A) xy 0 (11) k where Ax = ρ(a)x, y T A = ρ(a) y T, x > 0, y > 0, x T y = 1, and ρ(a) s the specal radus of A. k T 20

33 The symbol > means that all the entres n a matrx or vector are larger than zero. Based on the defntons of P and Q, both P and Q are nonnegatve and stochastc. Therefore, ρ(p) = ρ(q) = 1. They are derved from a strongly connected graph, and ther dagonal elements are postve. Then P N-1 > 0 and Q N-1 > 0,.e., P and Q are prmtve. Accordng to theorem 1, we can obtan the followng two propertes. lm P k k T T 1 where 0 and 1 1 (12) lm Q 1 where 0 and 1 1 (13) k k T T From equatons (12) and (13), we can get lm ( k) (0) for equaton (7), and k T lm k '( k) N ' 1 (0), where s the -th element of. In equaton (7), all state varables converge to a common value, whch depends on the communcaton topology and ntal condtons. Equaton (7) represents the consensus algorthm for the frst order dscrete-tme system. Whle n equaton (8), the state varables do not converge to a common value. But the summaton of all state varables s constant,.e., N 1 N 1 ' ' ( k) (0), k. 3.3 Soluton to Economc Dspatch Problem The communcaton topology among generators s a strongly connected grapy G as descrbed n 3.1. At the begnnng, we are not consderng power generaton constrants. Accordng to the ncremental cost crteron, all generators operate at optmal ncremental cost. The ncremental cost of -th generator s derved as: dc ( P ) 2 dp P where P s the output power of generator, C s the cost of generator,, are the cost coeffcents of generator. Therefore, the optmal ncremental cost of generators wll be: (14) 21

34 * * 2 P (15) And the optmal output of generator I can be obtaned from equaton (15). * * P (16) Let us denote (k) the estmaton of optmal ncremental cost of -th generator, P(k) the estmated optmal power output of -th generator, and ΔP(k) the estmated local power msmatch between power demand and total power generatons. And we have the followng ntalzatons: 2 P (0) (0) any P (0) any D P f (0) N 0 feasble feasble N 0, value value P (0), otherwse (17) where D s the total power demand, 0. N 0 s the vertces set that can receve nformaton from vertex Now we can derve the consensus algorthm: ( k 1) p, ( k) P ( k) (18) jn j j P ( k 1) ( k 1) / 2 / / 2 / (19) P ( k 1) q, P ( k) ( P ( k 1) P ( k)) (20) jn j j where ε s a small postve constant, and k s the tme step. q,j s the element of matrx Q. The teraton processes stated n equatons (18), (19), and (20) only need local nformaton. For example, the teraton of generator n equaton (20) only needs nformaton from ts nneghbor set N. Hence, the process s a dstrbuted algorthm. 22

35 In order to analyze the propertes and convergence of proposed consensus algorthm, equatons (18), (19), and (20) are wrtten n the followng format: ( k 1) P( k) P( k) (21) P ( k 1) B( k 1) (22) P( k 1) QP ( k) ( P( k 1) P( k)) (23) where P, ΔP, α, λ are the column stack vectors of P, ΔP, -/2/,, respectvely. And B=dag ([1/2/1, 1/2/2,, 1/2/N]). It can be seen that equaton (23) preserves the summaton of P + ΔP. Multplyng equaton (23) by 1 T, and we can get By equaton (13), we can have T T T 1 P( k 1) 1 QP( k) 1 ( P( k 1) P( k)) (24) T T T 1 P( k 1) 1 P( k 1) 1 P( k) 1 P( k) (25) T T Therefore, 1 P( k) 1 P( k) s a constant for all k. Snce P ( 0) (0) = D, 1 T P( k) wll be T P the actual msmatch between demand and total power generatons. It s notced that ths msmatch s collected by a dstrbuted manner nstead of a centralzed controller. Observed from equaton (21), the term εδp(k) provdes a feedback to the ncremental cost so that (k) wll converge to the optmal value. Then we have another theorem statng f constant ε s small enough, them the algorthm s stable. And all varables wll converge to the soluton to the EDP. Theorem 2: for k, V * * ( k), P ( k) P, P ( k) 0 (26) 23

36 Next, we wll prove the theorem 2. Replace P n equaton (βγ) wth λ by usng equatons (β1) and (22), we can get P( k 1) ( QB) P( k) B( I P) ( k) (27) where I s the dentty matrx of approprate dmenson. Combne (21) and (27) together, we can have the followng matrx equatons: ( k 1) P P( k 1) B( I P) I ( K) Q B P( k) (28) Let us defne M P B( I P) 0 0 I Q, and. 0 B The system matrx n equaton (28) can be seen as M perturbed by εδ. The egenvalues of M s the unon of the egenvalues of P and Q. So M has two egenvalues θ1 = θ2 = 1, and the rest egenvalues le n the open unt dsk on the complex plane. Denote vectors u1, u2 and follows: T v 1, T v 2 as U 0 I (29) where N 1 1/ 2 /, and V T T 1 B T 1 0 T T (30) whch are two lnearly ndependent egenvectors of M. and V T U = 1. When ε s small the varaton of θ1 and θ2 perturbed by εδ can be represented by egenvalues of V T ΔU, and 24

37 V T U 0 T 0 T (31) It s found out that θ1 does not change wth ε, and when ε > 0, θ2 becomes smaller. Snce egenvalues contnuously depend on the entres of a matrx, the rest of egenvalues of M + εδ contnuously depend on ε. Hence, f we choose ε smaller, we can guarantee that the rest egenvalues wll le n the open unt dsk. As k, ( k) 1 converges to span P( k) 0 That s P 0. The demand constrant s satsfed n equaton (23). From equaton (21), the ncremental cost wll converge to a common value. Therefore, the ncremental cost crteron s satsfed. 3.4 Convergence Study of Consensus Algorthm Convergence speed s one of the most mportant factors to evaluate the effectveness of proposed EDP methods. The convergence performance of tradtonal methods s dscussed n lterature [37-42]. A real-parameter quantum evolutonary algorthm from evolutonary algorthm and quantum computng s presented to solve non-lnear EDP [37]. Although heurstcs whch employ hstory of better solutons obtaned n the search process such as genetc algorthm and evolutonary algorthm may have better convergence of soluton. However, problems of slow or premature convergence reman. It s proved n [γ7] that real-parameter quantum evolutonary algorthm has better convergence and qualty of soluton than other evolutonary algorthms. And t requres a 25

38 relatvely smaller populaton sze. It s also noted that too small a convergence constant as stated n equaton (18) wll result n a slow convergence speed. Moreover, premature convergence of genetc algorthm s accompaned by a very hgh probablty of trap nto a local optmzaton. In order to solve ths problem, a bacteral foragng optmzaton technque s provded to deal wth EDP optmzaton [38]. In the bacteral foragng method, a gradual or sudden changes n the locaton may occur because of consumpton of nutrents or some other nfluence. Ths may cause the elmnaton of a set of bactera or dsperse them to a new envronment. Consequently, ths wll reduce the chances of convergence at local optmal locaton. It s also mentoned that a small convergence constant may cause slow convergence. Whle a large one may cause the optmal go past the optmal pont wthout stoppng. The convergence characterstcs of the bacteral foragng method s evaluated by measurng the ftness functon wth respect to the number of teratons. It s found out that the bacteral foragng method has a better convergence than other evolutonary algorthms. Taguch method that requres less computaton s developed to solve the EDP wth a nonsmooth cost functon [39]. The taguch method s employed that nvolves the use of orthogonal arrays n estmatng the gradent of the cost functon. A convergence stoppng crteron of 0.01 s proposed to evaluate the convergence of the taguch method. It means that the mprovement of the cost functon from one cycle to the next s less than Such a choce of the convergence crteron has been shown to be a good choce n ths partcular method. The taguch method s sx tmes faster than the fast evolutonary programmng and ten tmes faster than classcal evolutonary programmng when appled to the economc dspatch problem wth 40 generaton unts. However, one dsadvantage of taguch method s that t s very senstve to the choce of ntal values of parameters. 26

39 Due to slow convergence suffered from generc algorthms and evolutonary algorthms, a bogeography-based optmzaton s presented [40]. Bogeography deals wth the geographcal dstrbuton of bologcal speces. Mathematcal models of bogeography descrbe how a speces arses, mgrates from one habtat to another and gets wped out. It has some common features wth other bology-based methods, such as genetc algorthm, partcle swarm optmzaton. Compared to other bology-based methods, t requres fewer computatonal steps per teraton. Therefore, the bogeography-based optmzaton wll have a faster convergence. Also, the method has a hgher probablty to reach global optmal n a consstent manner. A multobjectve evolutonary algorthm for EDP s presented [41]. Ths approach employs a dversty-preservng mechansm to overcome the premature convergence and search bas problems. A herarchcal clusterng algorthm s also mposed to provde the decson maker wth a representatve and manageable Pareto-optmal set. Moreover, fuzzy set theory s employed to extract the best compromse nondomnated soluton. Several optmzaton runs of the proposed approach have been carred out on a standard test system. The results demonstrate the capabltes of the proposed approach to generate well-dstrbuted Pareto-optmal solutons of the multobjectve economc dspatch problem n a sngle run. Evolutonary programs wth adaptons based on scaled cost, as well as emprcal convergence constant, have been developed and examned on three test cases of a power system [42]. In smaller problems, all fast evolutonary programs perform much better than evolutonary programs wth Gaussan mutaton n terms of convergence rate, soluton tme, mnmum cost, and probablty of attanng better solutons. As the sze of the problem ncreases, as n large-scale modern power systems, mproved fast evolutonary programs offer superor performance over all other evolutonary programs wth both types of adaptatons. Though the soluton tme of mproved fast 27

40 evolutonary programs n case of larger systems wth more nonlneartes s slghtly hgher than fast evolutonary programs, t offers a hgher convergence rate and better soluton qualty as compared to fast evolutonary programs. The effect of consensus-based algorthms on convergence soluton of economc dspatch s also examned [43-45]. A novel energy plannng approach nvolvng the actual wnd energy as well as the energy traded wth the man grd s ntroduced [43]. A sample average approxmaton approach wth convergence guarantee s effcently utlzed to deal wth the nvolved multdmensonal ntegral n the expectaton functon. Wth the attractve advantages of beng computatonally effcent and reslent to communcaton outages, decentralzed schedulng over the Mcrogrd communcatons network s developed based on the alternatng drecton method of multplers. Some objectve values of the teraton can be even smaller than the optmal value due to the constrant volaton. However, for the day-ahead energy plannng problem, alternatng drecton method of multplers outperforms alternatve dstrbuted solvers thanks to ts fast convergence. An analyss of the dstrbuted average consensus algorthm n networks wth stochastc communcaton falures s presented [44]. It s shown that the problem can be formulated as a lnear system wth multplcatve nose. For systems wth no addtve nose, t s shown that the convergence rate of the consensus algorthm can be characterzed by the spectral radus of a Lyapunov-lke matrx recurson, and they have developed expressons for the multplcatve decay factor n the asymptotc lmts of small falure probablty and large networks. For systems wth addtve nose, t s shown that the steady-state total devaton from average s gven by the soluton of a Lyapunov-lke equaton. Usng ths method, these second order statstcs for varous network topologes can be computed as a functon of lnk falure probablty. These computatons ndcate 28

41 that there s a relatonshp between the network topology, the algorthm parameter, and the probablty of falure that s more complex than ntuton suggests. The selecton of samplng rate s very mportant for the overall system convergence performance n dstrbuted consensus algorthm [45]. A proper samplng rate has to be hgh enough to satsfy the samplng theorem, yet not so hgh as to ncur unnecessary cost or nstablty. Smulaton results show the convergence performance under dfferent selectons of samplng rate and dfferent tme constants of dynamcs. A mathematcal formulaton of the system conssts of both consensus and dynamcs s provded n ths work. The same formulaton methodology can be appled to a system of a dscrete consensus system wth second-order, thrd-order, ffth-order, etc. The relatonshp between the samplng rate and the convergence performance s analyzed. Overall, there are several factors that can affect the convergence speed of consensus algorthm. Usually, the convergence can be guaranteed by usng the consensus algorthm. However, the optmal soluton may not be the deal soluton due to the power constrants such as prohbted operatng zones, power lmts, etc. 29

42 CHAPTER FOUR DEFINITION OF COST FUNCTION AND CONSTRAINTS 4.1 Cost Functon Defnton In ths secton, the EDP wth transmsson losses and generator constrants n a smart grd s defned. The objectve of EDP s to mnmze the total cost of power generaton. A typcal cost functon curve of a power generator s llustrated n Fg. 4. Fg. 4. Cost functon curve of a power generator. A quadratc cost functon of -th generator s gven as follows: C ( P) P P (32) 2 Then the total cost of operaton for all generators s calculated as: C t m 1 C ( P) P P (33) 2 30

43 where m s the number of power generators, P s the output power of generator, α,, are the cost coeffcents of generator. The purpose of EDP s to mnmze Ct. The total transmsson loss s a functon of generator power outputs that can be represented usng B coeffcents [23]. P l m m PB P m j j 1 j 1 1 B0 P B00 (34) where Bj, B0, B00 are transmsson loss coeffcents. 4.2 Constrants Defnton The EDP has to be solved under several generaton and transmsson constrants. Therefore, the generators have to operate under several constrants. The constrants are categorzed nto the followng groups. Power balance Frst of all, n order to meet the power demand, the power.generatons have to be equal to the power demand plus the power transmsson loss. The power loss s calculated usng equaton (34). m 1 P P d P l (35) where Pd, Pl are power demand and power loss, respectvely. Ramp rate lmt The constrants of ramp rate lmts for generaton changes are gven. When generaton ncreases, P P 0 UR (36) and when generaton decreases, P P 0 DR (37) 31

44 where P, P 0 are the current and prevous output power, respectvely. UR and DR are the up ramp and down ramp lmts of -th generator. In dscrete-tme smulaton systems, the UR can be defned as the ncrease lmt between two tme steps. Smlarly, DR can be defned as the decrease lmt between two tme steps. Generaton lmt Each generator has to satsfy ts own generaton lmts. When selectng a power generator, there are nherent lmts on the actve and reactve power whch can be delvered. P mn P P (38) max where P mn, P max are the mnmum and maxmum output of -th generator, respectvely. Ths property can be llustrated n the followng fgure. Fg. 5. A typcal generator operaton area. Prohbted operaton zone 32

45 A typcal thermal unt wth many valve ponts can generate prohbted operaton zones. In practce, a generator has to avod operatng n those prohbted zones. The expresson of the operaton zones can be found as follows: mn l P P P, 1 u l P, j 1 P P, j P P P, j = β, γ,, n (39) u max, n where n s the number of prohbted zones of generaton. It s shown n Fg. 6 that the nput-output performance curve for a typcal thermal unt wth many valve ponts. These valve ponts generate many prohbted zones. Fg. 6. Input-output curve of a thermal generator wth valve ponts. For example, there are fve valve ponts n Fg. 6. They are two dfferent curves when there are several valve ponts for a generaton unt. Lne flow constrant 33

46 carres. The transmsson lne power flow should reman less than the maxmum capacty that a lne P P (40) max Lne, k Lne, k where PLne,k s the power flow of lne k, max P Lne,k s the maxmum capacty of lne k. There are some other constrants that are not consdered n ths work. For example, the constrant of energy storage capacty s not taken nto account. Also, the securty of the power system s not consdered n our model. However, they are not the man constrants of our proposed model. 4.3 Nonlnearty of EDP Optmzaton The EDP optmzaton can be summarzed as the soluton to the quadratc cost functon stated n equaton (32). However, ths equaton s a smplfed verson of the practcal cost functon of the generaton unts. The real-world cost functon of generaton unts wth valve pont loadng s gven n (41) [33]. 2 mn C ( P ) P P e sn( f ( P P )) (41) where P s the output power of generator, α,, are the cost coeffcents of generator, e and f are ntroduced to model the valve pont loadng. The generaton cost of generator wth valve pont s llustrated n Fg. 6. Large steam turbne generators wll have a number of steam admsson valves that are opened n sequence to obtan ever-ncreasng output of the unt. As the unt loadng ncreases, the nput to the unt ncreases and the ncremental heat rate decreases between the openng ponts for any two valves [34]. However, when a valve s frst opened, the throttlng losses ncreases rapdly and the ncremental heat rate rses suddenly. Ths s the valve pont that leads to a non-smooth, non-convex characterstc. 34

47 If the cost functon s approxmately represented by a quadratc functon as n (32), the problem can be solved by usng pecewse lnear programmng method, quadratc programmng method [35], [36]. However, none of these methods may be able to fnd the global optmal soluton and most of them get stuck at a local optmum. Practcally, the nput-output characterstc of modern generaton unts are hghly non-lnear n nature due to valve pont, ramp rate lmts, and prohbted operaton zones. In order to smplfy the optmzaton process, a lnear programmng approach usng pecewse lnear cost curves can be appled to solve economc dspatch. The process of lnear programmng method can be demonstrated n the followng fgures. The orgnal cost curve s plotted n Fg. 4, whch s a nonlnear curve. Fg. 7. Pecewse lnear cost curve of generaton unt. As shown n Fg. 7, the nonlnear curve of generaton unts s dvded nto a seres of straghtlne segments. P1, P2 and P3 are power ncrements, respectvely. The slopes of each one of the 35

48 lne segments are denoted by s1, s2, and s3. Then the ncrement n cost functon to each correspondng lne segment s gven by C s Therefore, the cost functon can be approxmated usng the lne segments, and that approxmaton can be mproved to nay desred level by ncreasng the number of lne segments used. However, the quadratc cost functon s stll a nonlnear problem because of the equalty and nequalty constrants. The equalty constrant s represented by the power balance equaton as shown n (35). The nequalty constrants are represented by generaton lmt, ramp rate lmt, prohbted operaton zone, and transmsson lne lmt. Wthout these constrants, the quadratc cost functon can be treated as a lnear program f usng the lnear programmng method. k P k 36

49 CHAPTER FIVE ANALYTICAL RESULTS 5.1 System Descrpton In ths secton, analytcal studes are carred out to evaluate the performance of the proposed consensus-based dstrbuted algorthm. A smart grd system of fve generators s developed usng Matlab/Smulnk n dscrete tme step. The smulaton models can be dvded nto three man layers: power layer, communcaton layer, and control layer. The system of fve generators are modeled n the power layer. The smplfed synchronous generator n Matlab s used to model the generator dynamc n ths layer. Communcaton network s bult up n the communcaton layer. Communcaton delay s also consdered n ths layer. Each generator s connected to other generators by power transmsson lnes and communcaton sgnals. The fve-unt smart grd system s shown n Fg. 8. Power Msmatch G3 G2 G4 G1 G5 G2 G4 G1 G5 G3 Communcaton layer Consensusbased Control algorthm Control layer Power layer Incremental Cost Fg. 8. Fve generators smart grd smulaton model. 37

50 The communcaton topology for ths dagram s shown n Fg. 9. G1 G2 G5 G3 G4 Fg. 9. Communcaton topology of fve generators: star topology. It s usng star topology n Fg. 9. There are bdrectonal communcatons between G1 and the rest four generators. Accordng to equatons (5) and (6), matrces P and Q can be derved as: 1/ 5 1/ 2 P 1/ 2 1/ 2 1/ 2 1/ 5 1/ / 5 0 1/ / / 2 0 1/ / 2 1/5 1/5 Q 1/5 1/5 1/5 1/ 2 1/ / 2 0 1/ / / 2 0 1/ / 2 It s apparent that summaton of row elements of P and column elements of Q s equal to one, respectvely. The parameters of fve generators are shown n Table 2. And constrants are also 38

51 lsted n Table 3. The load demand for ths model s 850 MW. Dfferent case studes are carred out n ths work. Ther performance are shown n the next secton. TABLE 2 Parameters of the fve generators Gen TABLE 3 Constrants of generators Gen. UR DR Prohbted zones [90 110] [ ] [90 110] [90 110] [90 110] In Table 2, power lmts and cost coeffcents of all generators are presented. In Table 3, up rate lmt, down rate lmt, and prohbted zones of all generators are presented. 5.2 Case Studes Case Study 1: Wthout Generator Constrants In ths case study, no generator constrants or transmsson lne loss s consdered. The convergence rate ε s equal to As dscussed before, a small convergence rate wll guarantee the stablty of the smulaton. The smulaton s carred out at a dscrete tme step of s. The 39

52 generator power output, ncremental cost of all generators, total power generaton, power msmatch, and total generaton cost are plotted n the followng fgures. 9 Incremental Cost Incremental Cost ($/MWh) IC1 IC2 8.2 IC3 IC4 IC Tme (S) Fg. 10. Case study 1 ncremental cost of fve generators. Fg. 11. Case study 1 power generaton and power demand. 40

53 Fg. 12. Case study 1 power msmatch of all generators. Fgure 10 shows that ncremental cost of fve generators all converge to the optmal value,.e., * = 8.682$/MWh. The correspondng power output of generators are: 243.9MW, 214.4MW, 73.86MW, 243.9MW, and 73.86MW. The comparson between power demand and total power generaton s gven n Fg. 11. It can be seen that the optmal power generaton s acheved at 0.04s. The local power msmatch of each generator wll converge to zero as shown n Fg. 12. Case Study 2: Wth Generator Constrants Fg. 13. Case study 2 ncremental cost of fve generators. 41

54 Fg. 14. Case study 2 power generaton, power loss and power demand. In order to calculate the transmsson power loss, the transmsson power loss coeffcents are gven n the followng B j 1.0e 5* , B o 1.0e 2*[ ; ;0.7047;0.2161; ], B oo

55 In ths case study, the generator s constrants and transmsson lne loss are consdered for a more practcal stuaton. For example, the generator should satsfy ts operaton constrants ncludng operaton lmts, UR/DR rates, and prohbted zones. The ncremental costs and power generatons are presented n Fg. 13 and Fg. 14, respectvely. It can be seen that generator 1 reaches ts lmt at sec. and ts ncremental cost settles at 1 = $/MWh. However, n order to satsfy power balance, the other four generators have to generate more power. And the correspondng ncremental cost wll ncrease as ndcated by Fg. 13. The new optmal ncremental cost for the four generators * s equal to $/MWh. The optmal power output are: 200MW, 232.5MW, 81.12MW, 266.3MW, and 81.12MW. The power generaton and power loss are MW and 11.04MW, respectvely. Case Study 3: Wth a Dfferent Communcaton Topology In ths study, a dfferent communcaton topology s presented n order to test the capablty of the proposed algorthm. Dfferent from Fg. 9, a new topology s provded n Fg. 15. G1 G2 G5 G3 G4 Fg. 15. Communcaton topology of fve generators: loop connecton. Smlarly, the correspondng matrces P and Q are derved as follows. 43

56 1/ 3 1/ 3 P 0 0 1/ 3 1/ 3 1/ 3 1/ / 3 1/ 3 1/ / 3 1/ 3 1/ 3 1/ / 3 1/ 3 1/ 3 1/ 3 Q 0 0 1/ 3 1/ 3 1/ 3 1/ / 3 1/ 3 1/ / 3 1/ 3 1/ 3 1/ / 3 1/ 3 The ncremental costs are plotted n Fg. 16. Fg. 16. Case study 3 ncremental cost of fve generators. It s observed that ths topology has a slghtly hgher ncremental cost than topology n case study 1. The optmal ncremental cost for the four generators s $/MWh. The power output of generators are also shown n Fg

57 Comparng wth case study 2, t takes slghtly more tme for the generators n case study 3 to reach the optmal generatons. It s also requrng more generatons from the generators. The power output of fve generators are: 200MW, 239.2MW, 83.86MW, 274.8MW, and 83.81MW. The correspondng power generaton and power loss are: MW and 31.67MW, respectvely. It can be concluded that dfferent topologes can have dfferent effects on the consensus algorthm. Fg. 17. Case study 3 power generaton, power loss and power demand. Case Study 4: Wth Generator Dynamcs 45

58 Fg. 18. Case study 4 ncremental cost for fve generators. In ths case study, the dynamcs of generators are taken nto consderaton. The characterstcs of generators are modeled usng the smplfed synchronous generator from Matlab/Smulnk. The smulaton results are shown n the Fg. 18, Fg. 19 and Fg. 20. Fg. 19. Case study 4 generator output. 46

59 Fg. 20. Case study 4 power generaton, power loss and power demand. It s notced that t s takng a longer tme for generators to reach consensus due to the nerta of the generator. After about 2s, the generaton reach the same consensus as that n case study 2. And there s a large transent n power generatons. The optmal ncremental cost of all generators s $/MWh. And the correspondng power generatons are: 200MW, 232.5MW, 81.12MW, 266.3MW, and 81.12MW, respectvely. Lkewse, the smlar dynamcs are exhbted n the power generatons. The generator power output are presented n Fg. 19. And the power balance s valdated n Fg. 20. The power generaton and power loss are: MW and 31.67MW, respectvely. Case Study 5: Wth Dfferent Convergence Constants It s notced that dfferent convergence constant ε has dfferent effect on convergence speed of consensus algorthm. And t s been dscussed that a relatvely larger convergence constant causes a better convergence soluton. However, a very large convergence constant wll cause the 47

60 optmzaton to be unstable. Three dfferent convergence constants are presented and compared wth ε = 5e-4. Fg. 21. Case study 5 ncremental cost of fve generators wth ε = 1e-4. Fg. shows the ncremental cost of fve generators wth ε = 1e-4. It s shown that wth a smaller convergence constant, the consensus algorthm can stll obtan the optmal soluton. However, t takes more than 0.2s to acheve the optmal pont. Fg. 22. Case study 5 ncremental cost of fve generators wth ε = 10e-4. 48

61 When the convergence constant s ncreased to 10e-4, t takes less tme for the generator to acheve the consensus pont. However, the oscllaton of ncremental cost becomes larger as the constant goes up. Fg. 23. Case study 5 ncremental cost of fve generators wth ε = 16e-4. Fg. 24. Case study 5 cost comparson of four dfferent convergence constants. 49