Academic Editor: João P. S. Catalão Received: 10 February 2016; Accepted: 6 April 2016; Published: 14 April 2016

Transcription

1 eergies Article Load Cocetratio Factor Based Aalytical Method for Optimal Placemet of Multiple Distributio Geerators for Loss Miimizatio ad Voltage Profile Improvemet Mohsi Shahzad 1, *,, Ishtiaq Ahmad 1,, Wolfgag Gawlik 2, ad Peter Palesky 3, 1 Eergy Departmet, AIT Austria Istitute of Techology, Viea 1210, Austria; ishtiaq.ahmad.fl@ait.ac.at 2 Istitute of Eergy Systems ad Electrical Drives, TU Wie, Wie 1040, Austria; gawlik@ea.tuwie.ac.at 3 Departmet Electrical Sustaiable Eergy, TU Delft, Delft 2628 CD, The Netherlads; P.Palesky@tudelft.l * Correspodece: mohsi.shahzad.fl@ait.ac.at; Tel.: These authors cotributed equally to this work. Academic Editor: João P. S. Catalão Received: 10 February 2016; Accepted: 6 April 2016; Published: 14 April 2016 Abstract: This paper presets ovel separate methods for fidig optimal locatios, sizes of multiple distributed geerators (DGs) simultaeously ad operatioal power factor i order to miimize power loss ad improve the voltage profile i the distributio system. A load cocetratio factor (LCF) is itroduced to select the optimal locatio(s) for DG placemet. Exact loss formula based aalytical expressios are derived for calculatig the optimal sizes of ay umber of DGs simultaeously. Sice either optimizig the locatio or optimizig the size is doe iteratively, like existig methods do, the simulatio time is reduced cosiderably. The exhaustive method is used to fid the operatioal power factor, ad it is show with the results that the losses are further reduced ad voltage profile is improved by operatig the DGs at operatioal power factor. Results for power loss reductio ad voltage profile improvemet i IEEE 37 ad 119 ode radial distributio systems are preseted ad compared with the the loss sesitivity factor (LSF) method, improved aalytical (IA) ad exhaustive load flow method (ELF). The compariso for operatioal power factor ad other power factors is also preseted. Keywords: loss miimizatio; voltage profile improvemet; simultaeous optimal sizig; multiple distributed geeratores (DGs); load cocetratio factor (LCF), operatioal power factor; primary distributio system; loss sesitivity factor (LSF); improved aalytic (IA); exhaustive load flow (ELF) 1. Itroductio Distributed, reewable Geeratio (DGs) i covetioal power systems have received icreased iterest i research over the last decade due to its evirometally friedly ature ad potetial ecoomic beefits. The itegratio of DG(s) becomes the most ecoomical solutio to meet the icreased demad due to load growth i the covetioal system [1]. Techological advacemets i geerators, power electroics devices ad storage devices further fuel this tred. DGs cotribute i the applicatio of competitive eergy policies, diversificatio of eergy resources, reductio of o-peak operatig cost, deferral of etwork upgrades, lower losses ad lower trasmissio ad distributio costs, ad potetial icrease of service quality to the ed-customer [2]. Based o the techologies used for DGs, these ca be either reewable based or oreewable based, both havig their ow advatages ad disadvatages. The former beig evirometally Eergies 2016, 9, 287; doi: /e

2 Eergies 2016, 9, of 21 friedly is the preferable choice but these have a disadvatage of low efficiecy, high costs ad itermittecy because these techologies are ot matured eough. Improvemet i efficiecy ad reductio i cost is expected i the future due to techological developmets ad ew scietific methods of desigig [3]. Noreewable based DGs, e.g., iteral combustio egies or combustio turbies etc., are relatively more efficiet but are ot as evirometally friedly [4,5]. For geothermal, small hydro, ad combied cycles, combustio ad wid turbies with power electroics, DGs are cosidered as sychroous geerators. Moreover, DGs are modeled as power electroic coverters or static geerators for photovoltaic (PV) or fuel cell based plats ad droop-cotrol based techiques are used for cotrol (decetralised) of such DGs, as preseted i [6 8]. While providig evirometally friedly eergy ad while helpig to meet the icreasig load ecoomically, DGs are also prove to be beeficial i providig reductio i power losses, improvig voltage profile, ad improvig the power hadlig capability ad power quality of the system. Such beefits ca oly be exploited whe the locatio ad size of the DG(s) are properly chose. Voltage ad loadability ehacemet, reliability improvemet ad etwork upgrade deferral are also iflueced by the locatio ad sizig of DGs [9]. Similarly, DGs have bee itegrated i electricity markets for acillary services such as spiig reserve, reactive power support, loss compesatio, frequecy cotrol ad other fast respose services. O the other had, poorly plaed ad improperly operated DG uits ca lead to reverse power flows, excessive power losses ad overheatig of feeders [10]. Reverse power flow may be a problem because the covetioal cotrollig schemes ad operatioal procedures assume uidirectioal power flow ad hece bidirectioal (reverse) power flow ca lead to adverse impact o them [11]. Reverse power flow ca create problems such as voltage rise i distributio grids, leadig to the violatio of the requiremets set i stadard EN50160 [12]. Short circuit currets icreased to the damagig levels, protectio desesitizatio ad icorrect operatio of cotrol equipmet are some other expected problems due to reverse power flow [13]. The proposed method also works whe reverse power flow is allowed ad hadled properly i the system. I order to achieve oe or several of the aforemetioed beefits, locatio ad size of DG(s) have to be optimized. To get maximum advatages, this problem is usually approached by splittig it ito two parts, fidig the optimum locatio ad the sizig the DG at this optimum locatio [14]. Whe cosiderig the curret sceario of DG peetratio, it ca be referred to as the poppig up like mushrooms situatio because the ivestors ca make studies o their ow ecoomic beefits ad primary eergy availability for istallig a DG. If they abide by certai grid codes, these DGs must be coected to the system. Ufortuately, there are o detailed cosideratios of reducig power losses ad other techical aspects that might support the cosumer or distributio etwork operator (DNO). The major motivatio behid this study is to ivestigate the impact of a situatio which is cotrary to the curret sceario i.e., to provide the DNO with a certai level of cotrol for steerig up the directio of DG peetratio. I the curret situatio, loss reductio is ot cosidered. It is expected that the DNO will idetify the locatio(s) ad the desired geeratio patters from DGs so that the system performace is improved ad the grid reiforcemets ca be deferred. The proposed study tries to justify the eed of givig a DNO some cotrol over DG istallatio based o system improvemets. As this is ot i lie with the curret practice due to ubudlig rules, keepig i view the potetial beefits, the DNO ca offer beefits to the ivestors to istall DGs of desired sizes at studied locatios. Offerig beefits to the ivestors ca be justified agaist the ecoomic beefits a DNO gets because of reductio of cost for improvig voltage profile ad, maybe, savig because of reduced losses. The authors i [15] have obtaied the optimal locatio ad size for real power loss reductio to place a sigle DG. The importace of selectig the correct DG size ad locatio is also show. It is also agreed that real costraits may also be cosidered for choosig the locatio for DG placemet, but, oce the locatio is selected, the optimum size should ot be compromised to get maximum beefits. Authors i [16] have proved the impact of DG locatio ad sizig for differet loadig coditios ad showed that the losses also deped o the loadig coditios. The impact of DGs o

3 Eergies 2016, 9, of 21 load supply cost has bee discussed i [17], which uses evolutioary programmig (EP). The high losses idex has bee utilized to improve the performace of the proposed algorithm. Referece [18] proposed a particle swarm optimizatio (PSO) based techique for optimal DG placemet ad sizig. Alog with reductio i real power losses, impact of various load models ad techical issues such as power factor, voltage profile, lie loadig ad total power itake from grid have also bee discussed. The local PSO based method for optimal sittig of DGs i order to miimize losses i etworks with reverse power flow is discussed i [19]. Optimal sittig is advatageous as it gives the DG sizes ad locatio i such a way that the load is fed locally. To address the issue of optimum locatio ad size separately, the use of suitable optimizatio techiques for each part is also used. Diversificatio of optimizatio variables ad hece the improvemet i results are the mai advatages expected from such approaches. To get the maximum out of such approaches, the selectio of suitable methods for the respective part of the problem is importat. For example, the authors i [20] have applied geetic algorithm (GA) for selectig the optimum bus, as it is a iteger-based optimizatio algorithm while, for optimizig the size of DG at give cadidate locatios provided by GA, PSO is used. Despite beig good i meetig the desired performace, it is oteworthy that such approaches are time cosumig compared to the case where sigle optimizatio is applied throughout. Aother such approach is preseted i [21] where authors have used Discrete PSO ad GA, but these were ot used idividually for each sub-part. Istead, both algorithms have bee mixed, where GA is used for mutatio ad crossover to elimiate the chaces of gettig stuck at local miimum. A few other examples of recet research iclude Kalma filter algorithm [22], EP [23], chace costraied programmig [24] ad PSO [25]. Alog with heuristic ad meta-heuristic methods, there has bee various research about fidig the optimum sizes ad locatio aalytically. I [20], the optimal locatio of a sigle DG with uity power factor has bee doe for radial as well as meshed etworks. This study is performed with the objective of loss miimizatio, but the optimum size is ot cosidered. Referece [26] presets a aalytical approach based o exact loss formula for optimizig the size ad locatio of DG. This is a cosiderably fast method but with a disadvatage that it ca oly compute the size of sigle a DG capable of deliverig real power oly. This problem is addressed i [27] where the optimum size of a DG capable of deliverig both active ad reactive power, ad the optimal power factor of idividual DGs at optimum locatio has bee calculated. This work is exteded for placig multiple DGs i [28], where a iterative procedure is used to select the optimum bus i the system. Despite beig able to fid the exact size, usig iterative methods for fidig the optimum locatio makes this approach slower. Aother importat poit i this method is that it eglects the impact of ocomig geerators whe calculatig the size of a DG. This avoids the method from fidig the best solutio because it does ot iclude the variatios of loss coefficiets (alpha ad beta i exact loss formula) [29]. I additio, i iterative techiques, the sizes of ocomig DG(s) are calculated based o the already istalled DG(s), which limits the rage for the optimal sizes of the ocomig DGs. A recet example of aalytical method based optimal sittig is discussed i [30] where loss miimizatio is doe by reducig the lie currets after placig DGs at various locatios. Moreover, based o the foud optimal sizes, the combiatio of DG types is also suggested. The ucertaities of the DGs ca also be cosidered for optimum locatios ad sizig as they itroduce disturbaces i the power etworks i terms of large curret trasiets, voltage fluctuatios, waveform distortios, ad log voltage variatio etc. I [31], three step GA ad decisio theory based procedure is proposed to obtai best geeratio size ad locatio for DG cosiderig uavoidable ucertaities. A robust plaig methodology to improve the accuracy of Mote Carlo simulatios ested i a evolutioary algorithm is formulated i [32] to assess the trade-offs i techical ad ecoomic aspects. Geetic algorithm embedded with poit estimate method (GA-PEM) based approach has bee proposed for solvig optimizatio problem of probabilistic power flow (PPF) modeled ucertaity uder chace costraied programmig for o-determiistic load growth ad odispatchable reewable resources i [33]. A Mote Carlo simulatio-embedded

4 Eergies 2016, 9, of 21 geetic-algorithm-based approach is employed i [24] for optimally sittig ad sizig of DGs. The chace costraied programmig (CCP) model is developed of cost miimizatio with security limitatios as costraits, cosiderig ucertaity of stochastic power productio. For choosig the optimum bus umber, various methods have bee explored. Loss sesitivity factor (LSF), exhaustive load flow (ELF), improved aalytical (IA) method, which is a iterative techique, gradiet ad secod order methods, ad GA based selectio of optimum locatio(s) are just a few methods for choosig optimum locatios i order to reduce losses [27,28,34]. All these methods provide the locatio selectio as a builti feature; therefore, oe eeds to implemet a whole algorithm i order to fid the locatios oly. Cost of istallatio ad operatio is always a importat factor, which depeds mostly o the size ad type of DG to be istalled. Selectig DGs with high rated output ca be a simple solutio to the problem of DG sizig. Ultimately, the (active or reactive) power curtailmet ca be doe to make the sizes optimal. Such a approach has multiple disadvatages, icludig high startig istallatio cost, less efficiecy of eergy geeratio due to curtailmet ad loger period of retur o ivestmet. Although such costs are ot cosidered i this work, it is easily uderstood that eedlessly high sizes will lead to uecoomical situatio. Loss miimizatio, beig the utmost objective of this work, eeds optimally sized DGs at optimal locatios. Therefore, cost is ot cosidered as primary cocer. I additio, for very high sized DGs, the chaces of curtailmet are also very high as compared to optimal sizes (to miimize loss). Aother importat poit is that, at high DG sizes, the loss variatio is more rapid ad ca be higher tha the base case (without ay DG) [26]. All the methods cited before help to improve the voltage as a additioal advatage, but it is ot always cosidered as a primary objective. Moreover, maitaiig the bus voltages withi the permissible limits is amog fudametal requiremets of etwork operator. I additio, the voltage problem occurs mostly due to the high power demad by the load. Fially, the voltage problem is a cosequece of the high load cocetratio i some area of the etwork. Keepig i view such importat poits, a method for selectig a optimal locatio, which should ot oly reduce the losses but also improve voltage; hece, the stability of the system, is highly eeded. Aother aspect of this work is to split the optimal locatio selectio from that of optimal size selectio. It is expected that such effort ca expad the versatility i a way that, i future, more factors (i additio to loss reductio or voltage profile improvemet) ca be developed ad locatio selectio ca be made more practical. Aother advatage of this approach would be the liberty of choosig either locatio or size idepedet of the each other. This importat advatage esures the opeess of the method. The fuctioality of the method ca be ehaced due to the idepedet ature of both variables. It is easy to iclude costrait(s) specific to oe part oly, ad, hece, the other part remais the same. For istace, a ivestor ca have iterest i istallig DG(s) at some particular locatio(s) due to differet reasos. Similarly, based o some other factor(s) (tha loss reductio ad Voltage Profile Improvemet (VPI) of iterest for DNO, the locatios ca be made fixed. I such a sceario, the method ca be implemeted for fidig optimal sizes oly. Moreover, other factors, such as for ecoomic beefits or istallatio ad/or operatioal costs, ca be formulated ad combied with a load cocetratio factor (LCF) to fid the optimal locatios based o other objectives. I this paper, the cocept of loss i a power system is used so that DGs are placed ear the load, which is more realistic as it ca decrease the power trasmissio ad hece the losses. Additioally, supplyig heavily loaded area of etwork locally helps to reduce voltage problems. Based o this, a ew idea of LCF is itroduced. Cosiderig the desired umber of DGs, the cadidate bus locatios are selected, for which the sizes are calculated simultaeously with aalytical expressios, a limited (to three DGs oly) versio of which has already bee published i [35]. These aalytical expressios are based o the exact loss formula. Moreover, the operatioal power factor is also foud for further ehacig the capability of preseted method i reducig losses ad improvig voltage profile. Although relatively fast ad accurate, the preseted aalytical approach bears the limitatios like iability to icorporate the bus voltage limitatios, thermal limits of lies ad other such parameters.

5 Eergies 2016, 9, of 21 It is worth metioig that the other factors, alog with LCF, ca be formulated to further ehace the selectio criteria of optimal locatio. The accuracy ad effectiveess of the proposed method is compared with already developed aalytical methods i.e., the LSF method, improved aalytical (IA) method ad exhaustive load flow (ELF) methods. Major cotributios of this work are: 1. Load cocetratio factor as a method of choosig optimal locatio. 2. Geeralized aalytical expressios for N DGs with improved mathematical represetatio i compariso to [35]. 3. Exhaustive method of fidig operatioal power factor for DG operatio. The rest of the paper is structured as follows. A brief problem formulatio is give i Sectio 2. Sectio 3 presets the LCF-based method for selectig the optimal locatio(s) for DGs. Sectio 4 details the derivatio of aalytical expressios for fidig the optimum sizes of multiple DGs simultaeously while cosiderig the iterdepedecies. Explaatio of operatioal power factor is also icluded i this sectio. A brief descriptio of LSF, IA ad ELF methods is also preseted i Sectio 5. Experimetal use cases, results ad their compariso with LSF, IA ad ELF methods, alog with discussio, are preseted i Sectio 6. The iovatios ad coclusio are summarized i Sectio Problem Formulatio 2.1. Distributio System Power Losses Based o the real ad reactive power ijectios at the buses, total real power loss i a N bus system is give as [36]: where P L = α ij = j=1 [α ij (P i P j + Q i Q j ) + β ij (Q i P j P i Q j )], (1) R ij V i Vj cos(δ i δ j ); β ij = R ij V i Vj si(δ i δ j ), P i, P j : Active power ijectios at the ith ad jth buses, respectively; Q i, Q j : Reactive power ijectios at the ith ad jth buses, respectively; V i δ i,v i δ j : Complex voltages at the ith ad jth buses, respectively; R ij + jx ij : ijth elemet of impedace matrix [Zbus]; : Total umber of buses i the system. Active power loss miimizatio problem is defied as: Subject to the followig power, voltage ad lie flow costraits: Miimize(P L ). (2) N P DGdg P Di, dg=1 N Q DGdg Q Di, dg=1 V mi <V i < V max, I i < I max, P DGdg ad Q DGdg : real ad reactive power output from DGs; : total umber of DGs to be istalled; P Di ad Q Di : the active ad reactive power demads o ith bus; N: the total umber of buses i the system; V mi ad V max : the lower ad upper bus voltage bouds; V i : the ith bus voltage; I i : flow o

6 Eergies 2016, 9, of 21 the ith lie; I max : the maximum loadig limit of ith lie. I this work, lie loadig is take i form lie currets. The optimal sizes of DGs are calculated by derivative based miimizatio techiques Methodological Steps The proposed method cosists of two major parts: i. Optimal locatio selectio; ii. Simultaeous optimal sizig. Optimal locatio selectio method is based o the LCF, whereas the mathematical expressios are derived for fidig the optimal size(s) simultaeously. I part of the optimal size calculatio, the method for operatioal power factor selectio is also explaied. I the ext sectios, both of these parts are explaied i detail. It is worth metioig that exact sizes of the DGs for the purpose of loss miimizatio are foud by this method. For the purpose of icorporatig operatioal costraits such as voltage limits or lie flow limits, the algorithm checks the system coditios after optimally placig the DGs at respective locatios. If certai costraits are ot met, the method adjusts itself as explaied i Sectio Optimal Locatio Selectio 3.1. Basic Idea Behid Loss Reductio with DGs This sectio elaborates o the selectio of optimum locatio(s) of DG(s) i order to miimize the active power loss. Covetioally, power plats were meat to produce huge amouts of bulk power which is the delivered to the loads via log trasmissio lies, resultig i high power loss. Nowadays, distributed geeratios ca be placed ear the loads i order to supply local demad, leadig to multiple beefits e.g., reductio i stress o covetioal power plats, trasmissio losses, ad stress o the trasmissio system. Hece, buses with higher load cocetratio are cosidered as the optimal locatios. Cosiderig power losses, the cadidates locatio selectio is very importat. Ideally, the trasmissio losses ca be reduced to almost zero if the DG is coected at every bus i order to serve its local load but this is seldom feasible. Hece, i this study, the cadidate locatio has bee selected based o cocetratio of load beig faced by a certai bus. For every bus i the system, its LCF is calculated ad buses are arraged i descedig order with respect to their LCF. The, based o the desired umber of DGs to be placed, the buses with topmost LCF will be chose as cadidate locatios Types of Buses To uderstad the idea of LCF preseted i this paper, the buses i typical IEEE bus systems are classified ito two categories i.e., the directly coected buses ad the loaded buses. For ay bus i i the system, a directly coected bus is coected directly to i ad it must ot pass through ay other bus i.e., o other bus comes i betwee it ad bus i. From this defiitio, it is obvious that, for bus i, the umber of directly coected buses will be the same as the lies coected to it. For the sake of simplicity ad easy mathematical formulatio, the set C i is defied as the set of the directly coected buses to bus i ad bus i itself. For the bus itself, the lie legth is zero, hece the lie loss. Loaded buses are the load carryig buses. For a bus without ay load, a zero load ca be cosidered i order to get geeralized defiitios ad mathematical expressios. For the IEEE 37 ode system give i [37], all the buses except 1, 3, 4, 6, 7, 8, 12, 16, 18, 25, 29 ad 34 are categorized as loaded buses. For bus 22; 21, 23 ad 24 are directly coected buses ad set C22 will be give as {21, 23, 24}. Similarly, for bus 10, C10 will be give as {8}.

7 LCF (MW) LCF (MW) Eergies 2016, 9, of Load Cocetratio Factor (LCF) The LCF of bus i i the system is the sum of loads coected to each member bus of set C i ad the losses across the lies coectig bus i to the members of set C i. Mathematically, LCF i = j C i P Dj, (3) where P Dj is the power demad i.e., load coected at bus j. As metioed above, if there is o load coected to ay bus, its P Dj will be take as zero Selectig the Optimal Locatios It is worth metioig here that the system may require multiple DGs; hece, there ca be differet approaches for selectig the optimum locatio. The most commo practice i the research is either fidig the optimum locatios for the whole system to miimize loss or dividig the whole system ito regios ad fidig a optimum with respect to the specific regio. Although the first approach ca guaratee the maximum reductio i power loss, it ca lead to a o-suitable geographical locatio. Alog with the possibility of improved loss miimizatio, the later approach ca also help to fid out the optimum locatio with respect to availability of geographical space if regios are decided strategically Selectio of Optimal Locatios i the Systems Uder Study Based o these steps, the LCF i are give i Figures 1 ad 2 for IEEE 37 ad IEEE 119 ode systems, respectively. Ideally, the buses with topmost LCF should have bee cosidered for DG placemet, but, due to the high lie flows i some lies, the losses o them icrease. If the DGs are placed o buses coected to such lies, the losses may icrease. The lie flows for both etworks uder study (without DGs) are give i the Figures 3 ad 4. Aother importat cosideratio was avoidig the DG placemet o directly coected buses, which is agai to reduce the chaces of icreased lie flows ad voltage violatios. Therefore, buses (12, 18, 22, 32) ad (43, 52, 74, 82, 115) are cosidered i 37 ad 119 ode systems, respectively. The case of various regios has ot bee cosidered i this work. If it is desired to make regios, the same strategy ca be applied for optimal locatio selectio i every regio idividually Bus Number Figure 1. Load cocetratio factors for 37 ode system Bus Number Figure 2. Load cocetratio factor for 119 ode system.

8 Lie Flow (ka) Lie Flow (ka) Eergies 2016, 9, of Lie Number Figure 3. Lie flows for 37 ode system Lie Number Figure 4. Lie flows for 119 ode system. 4. Simultaeous Optimal Sizig This sectio elaborates o the mathematical steps take to formulate expressios for fidig optimal DG sizes simultaeously for give specific bus umbers. The geeralized method for optimal sizig of N DGs simultaeously is explaied. To check the accuracy, effectiveess ad time for fidig a exact solutio, the LSF, IA ad ELF methods are used. The results are validated by usig the slightly modified IEEE 37 ad 119 ode radial systems with system data give i [37,38]. The ode 1 is always cosidered as a referece ad voltage regulated ode. The method proposed here ot oly gives expressio for optimal sizig but also a method for fidig operatioal power factors which ca esure the least losses. The extesive mathematical steps take to fialize the expressios for calculatig the size of DGs at N buses simultaeously are first itroduced followed by steps for fidig the operatioal power factor. The complete set of steps take to implemet these methods is also give Aalytical Expressios where For a DG, reactive power ijectio is give as [27]: Q DGi = ap DGi, (4) a = (sig) ta(cos 1 (PF DG )), ad sig = +1 or 1 for DG ijectig or cosumig reactive power. PF DG is the power factor of the DG. Active ad reactive power ijectios at the bus where the DG is istalled are give, i terms of active ad reactive power demads P Di ad Q Di, as: P i = P DGi P Di, (5) Q i = Q DGi Q Di = ap DGi Q Di. (6)

9 Eergies 2016, 9, of 21 P L = By substitutig Equatios (5) ad (6) i Equatio (1), the active power loss becomes: j=1 [α ij {(P DGi P Di )P j + (ap DGi Q Di )Q j )} + β ij {(ap DGi Q Di )P j (P DGi P Di )Q j }]. (7) It ca be proved that the miimum of active power loss of the system ca be foud if partial derivative of Equatio (7) with respect to ijected real power from the DG at the ith bus computed equal to zero. Hece, Equatio (7) ca be writte as: P L = 2 P DGi j=1 [α ij (P j + aq j ) + β ij (ap j Q j )] = 0. (8) As metioed earlier, the method for calculatig the optimum size of N DGs is preseted here. Suppose that the DGs are to be placed at bus umber x 1, x 2,..., x, where x caot exceed the total umber of buses i.e., N, Equatio (8) ca be re-writte as: P L P DGx1 = α x1 x 1 (P x1 + aq x1 ) + β x1 x 1 (ap x1 Q x1 ) + α x1 x 2 (P x2 + aq x2 ) + β x1 x 2 (ap x2 Q x2 ) α x1 x (P x + aq x ) + β x1 x (ap x Q x ) + a j=1,j =1,2... j=1,j =1,2... (α x1 x j Q j + β x1 x j P j ) = 0 (α x1 x j P j β x1 x j Q j )+ P L P DGx2 = α x2 x 1 (P x1 + aq x1 ) + β x2 x 1 (ap x1 Q x1 ) + α x2 x 2 (P x2 + aq x2 ) + β x2 x 2 (ap x2 Q x2 ) α x2 x (P x + aq x ) + β x1 x (ap x Q x ) + a j=1,j =1,2... j=1,j =1,2... (α x2 x j Q j + β x2 x j P j ) = 0 (α x2 x j P j β x2 x j Q j )+. P L P DGx = α x x 1 (P x1 + aq x1 ) + β x x 1 (ap x1 Q x1 ) + α x x 2 (P x2 + aq x2 ) + β x x 2 (ap x2 Q x2 ) α x x (P x + aq x ) + β x x (ap x Q x ) + a j=1,j =1,2... j=1,j =1,2... (α x x j P j β x x j Q j )+ (α x x j Q j + β x x j P j ) = 0. (9) X x1 = j=1,j =x 1,x 2...x (α x1 x j P j β x1 x j Q j ) + a j=1,j =x 1,x 2...x (α x1 x j Q j + β x1 x j P j ) X x2 = j=1,j =x 1,x 2...x (α x2 x j P j β x2 x j Q j ) + a j=1,j =x 1,x 2...x (α x2 x j Q j + β x2 x j P j ) Let. X x = j=1,j =x 1,x 2...x (α x x j P j β x x j Q j ) + a j=1,j =x 1,x 2...x (α x x j Q j + β x x j P j ).

10 Eergies 2016, 9, of 21 I additio, β ii = 0, α ij = α ji ad β ij = β ji. Thus, by substitutig Equatios (5) ad (6) i set of Equatio (9), ad arragig for P DGxi, (1 + a 2 ) (1 + a 2 ) (P DGxi α x1 x i ) (P DGxi α x2 x i ) [P Dxi (α x1 x i + β x1 x i a) + Q Dxi (α x1 x i a β x1 x i )] + X x1 = 0, [P Dxi (α x2 x i + β x2 x i a) + Q Dxi (α x2 x i a β x2 x i )] + X x1 = 0, (1 + a 2 ) (P DGxi α x x i ). [P Dxi (α x x i + β x x i a) + Q Dxi (α x x i a β x x i )] + X x1 = 0. (10) Writig i the form of matrices to solve usig Cramer s rule: AX = B, (11) where α x1 x 1 α x1 x 2... α x1 x P DG1 B x1 A = (1 + a 2 α x2 x ) 1 α x2 x 2... α x2 x......, X = P DG2., B = B x2.., α x x 1 α x x 2... α x x P DG B x ad B x1 = B x2 = [P Dxi (α x1 x i + β x1 x i a) + Q Dxi (α x1 x i a β x1 x i )] X x1, [P Dxi (α x2 x i + β x2 x i a) + Q Dxi (α x2 x i a β x2 x i )] X x2, B x =. [P Dxi (α x x i + β x x i a) + Q Dxi (α x x i a β x x i )] X x. (12) For N DGs to be istalled, the order of matrices A, B ad X is, 1 ad 1, respectively. It is importat to metio that the method fids the optimal active power output desired from DGs to miimize the losses. The reactive power output ca be calculated correspodig to the desired power factor usig Equatio (4). I additio, i this method, the DG sizes deped upo the active ad reactive power, whether geeratio or load, already coected to the system. This meas that ay equipmet or system (storage or geeratio, etc.) that ca be represeted i the form of active ad reactive power ca be used by takig care of respective sig otatio i.e., positive for geeratios ad egative for loads. I this way, the method ca be used to iclude ay type of existig geeratios as has bee doe i [35] Operatioal Power Factor To make these expressios useable for practical cases, the power factor demaded from a DG is also icorporated. I improved aalytical (IA) method, combied load power factor (CLPF) is

11 Eergies 2016, 9, of 21 cosidered as the optimal power factor. This approach of fidig the optimal power factor has produced good results, but the CLPF caot be useful for the systems with higher reactive power demad such as IEEE 119 ode radial system cosidered i this work. CLPF for this system was about 0.79, which is ot preferred i most of the distributio grid codes. For a geerator, power factor ear uity is preferred but, sometimes, it is ot possible i practical systems to keep power factor ear uity. Therefore, a rage of cosidered power factor i this work is take to be O this groud, the suggested power factor is amed as the operatioal power factor. As already metioed, the proposed method is cosiderably fast with respect to the other cotemporary methods, a exhaustive approach for fidig the operatioal power factor is suggested i this work. Startig from power factor of 0.90, the method foud optimal DG sizes ad losses with a step size of 0.01 util either a uity power factor is reached or active power loss starts icreasig the those for power factor of previous step. This helped i fidig operatioal power factor without sigificat icrease i the total simulatio time. Although the primary objective was active power loss miimizatio, the voltage profile is also improved sigificatly i compariso to the other stated methods. Although the method stops operatig after it fids that the losses are icreased for a power factor as compared to the previous step, the results of the complete rage are preseted i Sectio 6.2. The operatioal power factor for the 37 ode ad 119 ode radial systems is 0.97 ad 0.90, respectively Algorithm for Optimal Sizes Calculatio Figure 5 illustrates the flow chart of the computatioal steps eeded to fid the results based o proposed aalytical expressios. The power of the slack bus is kept positive i order to limit the chaces of icreasig the losses. 1. Eter the base case etwork. 2. Eter the desired umber of DGs to be placed. 3. Ru base case load flow ad calculate losses usig Equatio (1). 4. For all the buses i a regio, calculate the LCF i ad arrage the buses i descedig order with respect to this. 5. Choose N buses for placig N DGs. The iitial set of bus umbers will cotai the bus(es) with highest LCF i each regio. 6. Based o iput data i step (4), fid the optimal size of DGs usig the expressio give i Equatio (11). 7. Calculate operatioal power factor of DG usig exhaustive method. 8. Stop if: a. The sum of power of DGs to be istalled is less tha the total power demad plus losses. b. The bus voltages are withi a permissible limit. c. The lies are ot overloaded. Else, look for ew locatios by usig these steps. i. Try to chage oly oe bus i the regio from the set of bus umbers chose i step (4) i.e., for placig DGs, ( 1) buses will remai the same. ii. The most suitable cadidate for the chaged bus will be the oe which has the least differece i LCF from the LCF of a previously selected bus, while stayig i the same regio. iii. I case of a clash betwee ay two or more regios for the selectio of the secod highest LCF bus, priority will be give to the bus which carries the highest load. Go to step (6) 9. Place sized DGs i the system ad calculate losses usig Equatio (1). 10. To check for better sizes, optimal sizes i earest proximity ca also be checked. 11. To check for eve better solutios, the ext cadidate buses i the list of LCF ca also be checked, but experimets showed that this leads to zero or egligible improvemets.

12 Eergies 2016, 9, of 21 Start Eter Number of DG(s) Ru Base Case Load Flow ad Calculate Losses Calculate LCF i ad Idetity Optimal Locatio(s) Calculate Optimal DG Size(s) Fid Operatioal Power Factor for DG(s) Place at Respective Locatio(s) Ru Load Flow ad Calculate Losses Select Next Optimal Locatio(s) i LCF i List Start Eter Number of DG(s) ad Bus Locatio(s) Ru Base Case Load Flow ad Calculate Losses Calculate Optimal DG Size(s) Place at Respective Locatio(s) Ru Load Flow ad Calculate Losses Check for other Stoppig Criteria Yes Sca earest Optimal Size(s) for Improvemet No Stop ad Prit Results Ed Stop ad Prit Results Ed 16 Figure 5. Flowchart of optimal placemet. 5. Comparative Studies For the purpose of compariso ad validatio, the followig methods have bee implemeted Loss Sesitivity Factor The LSF method [26] for selectio of optimum locatio is based o the priciple of liearizatio of the origial exact loss curve aroud a iitial operatig poit. I this way, the size of the solutio space is reduced. Based o Equatio (1), the LSF of the ith bus is give as: LSF i = P L / P i = 2 j=1 (α ij P j β ij Q j ). (13) The LSFs for all buses i the systems are calculated to arrage them i descedig order. The top 20% of buses i the LSF list are selected for iterative placemet ad sizig of the DGs. For each iteratio, the losses are calculated by placig the DGs from zero up to the maximum, which is chaged i small steps. The size with least losses is selected. For placig the ext DG, LSFs are calculated agai ad the DG is placed usig the very same method Improved Aalytical Method The improved aalytical (IA) method, proposed i [28], calculates the optimum sizes of multiple DGs iteratively. This advaced techique is preseted i the latest literature ad is used here for compariso ad validatio. The iteratios for fidig the optimum locatios have bee removed ad oly the optimum DG sizes are calculated for respective buses to reduce the computatio time.

13 Eergies 2016, 9, of Exhaustive Load Flow Method This method is cosidered to be the most suitable ad accurate method i literature for optimizig the locatio ad sizig of DGs. It is preseted i [39]. This method places oe DG at a time o a bus, ad icreases its output power from zero to some specified maximum value i steps. For every iterative step, it fids the active power losses. At the ed of this scaig of every bus, it stores the size of the DG correspodig to the least losses. The same practice cotiues util the specified stoppig criteria is reached, which is usually either the sum of all DG outputs to be equal or greater tha total demad i the etwork or the maximum umber of DGs. 6. Results 6.1. Experimets/Use Cases Two differet systems, IEEE 37 ad 119 odes, give i Figures 6 ad 7, have bee cosidered for testig the proposed methodology. Details of total active power demads ad losses without DG are summarized i Table 1. Our software is developed i Pytho coupled with DIgSILENT PowerFactory (DSPF) through Applicatio Programmig Iterface (API). The DSPF is a widely used commercial tool for studyig the power systems. Its fuctioality is expaded further by couplig it with Pytho, which helped to program the complex equatios very easily. Further details about this ca be foud i [40]. Aother possibility of couplig DSPF with MATLAB is give i [41]. The software is maily operated from a Pytho commad lie eviromet, which computes the base case losses, fids optimum locatios for DG placemet based o LCF, the computes 11 the optimum sizes of multiple DGs. Fially, the losses are calculated after placig these DGs. The Newto Raphso method was used to solve power flow of the systems. A stadard PC with Itel 12 Xeo Processor W3505 ad 12 GB RAM is used for simulatio. To geeralize, the DGs with the ability 13 of ijectig both14 active ad reactive power are cosidered. The operatioal power factor ad power factor of 0.95 are 15 used for calculatig the DG sizes at static demad. I additio, the upper ad lower 18 bus voltage limits are kept betwee ±6% of omial i.e., 1.06 ad 0.94 p.u. [42]. Based o the etwork data used, the 16 I max is take to be 0.8 ka. The size of the DG ca vary betwee some kilowatts to 300 MW [43] but 17 depeds upo the capacity of the power system ad ca be cosidered maximum up to the total power demad plus the losses [9] Exteral Grid Figure 6. IEEE 37 ode etwork.

14 Eergies 2016, 9, of Exteral Grid Figure 7. IEEE 119 ode etwork. Table 1. Base case system data. Parameters 37 Node System 119 Node System Active Power Demad 4.98 MW MW Reactive Power Demad 1.35 MVar MVar Active Power Loss without DGs kw kw Mi Voltage i Network p.u p.u Max Voltage i Network p.u p.u 6.2. Operatioal Power Factor Test Results The DGs cosidered throughout this work are capable of ijectig both real ad reactive power. Hece, the impact of the power factor is also importat. For results show i Figures 8a ad 9a, it ca be clearly see that the operatio beyod the operatioal power factor icreases the losses, with either of the methods discussed here. The LSF method appeared to be the least efficiet method, whereas the ELF ad Mohsi method (MM) methods produced early the same as well as the best results. The IA method also produced good results i compariso to the LSF but was outperformed by the ELF ad MM methods. Similar treds of loss reductio is observed for both test systems studied i this work.

15 Active Power Loss (kw) Voltage Variatio (p.u) Active Power Loss (kw) Voltage Variatio (p.u) Eergies 2016, 9, of 21 Aother importat result that ca be draw from the graphs give i Figures 8b ad 9b is about the voltage stability of the system at operatioal power factor. Not oly is the voltage profile improved (as give i Sectio 6.4), but the voltage variatio is also reduced. Voltage variatio is give as the differece betwee the maximum ad miimum voltage appearig i the etwork (at ay bus) for a specific power factor. For the MM method, the voltage variatio is the miimum over most of the cosidered rage of power factors, which clearly highlights the usefuless of the preseted method. Based o the results of this sectio, it is prove that the power factor selectio is also a importat poit which must be properly addressed while optimizig the size ad locatio for placemet of DGs MM IA LSF ELF Power Factor (a) Power Factor vs. Active Power Loss Power Factor (b) Power Factor vs. Voltage Variatio Figure 8. Operatioal power factor for 37 ode system ELF LSF IA MM Power Factor (a) Power Factor vs. Active Power Loss Power Factor (b) Power Factor vs. Voltage Variatio Figure 9. Operatioal power factor for 119 ode system Active Power Loss Miimizatio Results The sizes of the DGs were calculated for the give umber of DGs. It is ever cosidered to be the oly possibility of, for example, four DGs i the case of IEEE 37 ode system. Moreover, the sizes of DGs foud here are to esure the least possible losses, ad, subsequetly, the voltage profile improvemet. After DG istallatio, i operatioal phase, icreasig or decreasig the sizes beyod the calculated sizes is possible. Cosequetly, o-optimal sizes will be resulted; hece, the losses will vary. I this case study, it was cosidered that the total power supplied by DG is ot more tha total demad [28]. This is doe because supplyig power to the exteral grid may icrease the losses due to icreased lie flows. As i real etworks, the reverse power flow ca happe, ad this costrait ca be altered to allow a certai amout of power to be delivered to the exteral grid. The sizes of DGs ad losses will be differet Bus System The simulatio results with the proposed methodology ad other metioed techiques have bee summarized i Table 2 for the IEEE 37 ode system. The losses for o DG case ad after

16 Eergies 2016, 9, of 21 placemet of optimally sized DGs with various techiques alog with computatio time are show i Table 2. Table 2. DG placemet by differet methods for IEEE 37 ode system at operatioal power factor. Case Method Istalled DG Schedule (MW) DG (MW) Ploss (kw) Loss Red (%) Time (s) No DG Total Real Load = MW DGs LSF IA ELF MM Bus Size Bus Size Bus Size Bus Size From these results, the proposed method proves its superiority both i terms of simulatio time ad loss reductio. The IA method appears to be the earest i terms of both of these objectives while the LSF method is ot oly slow i fidig the optimum size but also gives poor loss reductio. For the 37 ode system, loss reductios of 46.60%, 86.52%, 96.33% ad 95.92% is observed for LSF, the IA method, the ELF method ad our proposed method (MM), respectively. It is worth metioig that the ELF method produced slightly better results i compariso to the method preseted i this work because it scas all the buses iteratively for every possible size. Optimal placemet of DG is a plaig phase issue where the computatioal time of a few miutes may ot be very importat but is preseted here for the sake of compariso amog differet methods. Comparig the computatioal time for placemet of four DGs with differet techiques discussed here shows the advatage of our proposed methodology. The earest competitor i terms of simulatio time take is the IA method, which took about 33% more time tha the proposed method. Similarly, the LSF method, beig iterative over the limited umber of buses, took a cosiderably high amout of time (about more tha five times higher) for placig four DGs. Despite beig slightly better at reducig the losses, the ELF method is highly demadig, as ca be see from the simulatio time take by it. The time take by the ELF method to reach the fial solutio is approximately 33 times more tha that of the MM method. It is importat to ote that the ELF method could be made faster by makig the step size of DG size bigger, which would have ultimately adversely affected the optimal sizes ad hece the loss reductio. Hece, the proposed method appears to be the best i respect to both the loss reductio ad computatioal time. Exteral ifeed for this system was reported to be 5.40 MVA i the base case, which was reduced to 1.46 MVA after optimal placemet of DGs Bus System Table 3 presets the simulatio results for the IEEE 119 ode system i the same fashio as they were doe for the 37 ode system. It ca be see that the proposed methodology could fid the optimum sizes for reductio of losses i the system cosiderably faster. I additio, the sizes foud depict the effectiveess of the proposed aalytic expressio for simultaeous sizig of DGs because the loss reductio is higher amog all the preseted techiques except the ELF method. The LSF method is proved to be the least efficiet i terms of simulatio time ad optimum sizes that ca reduce losses to the lowest level. Loss reductio with the LSF method, the IA method, the ELF method ad the proposed method are 65.94%, 76.04%, 81.20% ad 78.82%, respectively. The ELF method is proved better i loss reductio but at the cost of very high computatio time.

17 Eergies 2016, 9, of 21 Table 3. DG placemet by differet methods for IEEE 119 ode system at operatioal power factor. Case Method Istalled DG Schedule (MW) DG (MW) Ploss (kw) Loss Red (%) Time (s) No DG Total Real Load = MW DGs LSF IA ELF MM Bus Size Bus Size Bus Size Bus Size The proposed methodology ca fid the optimum size for five DGs i the 119 ode system approximately 2.5 times faster tha by the IA method. The LSF method, beig a iterative techique for fidig the optimum size, cosumes 14.4 times more time as compared to the proposed method. Although slightly better i loss reductio, the ELF method cosumed approximately 72 times more time tha the proposed method. Based o these importat observatios, the proposed method outperformed the other methods give here. As per most of the grid code requiremets ad prefereces, the most desired power factor is The results of which are summarized i the Table 4. These results show the same treds i loss reductio ad simulatio time as were give i Table 3. Slightly higher losses are observed because of the fact that this is ot a optimal power factor for this system. This also speaks about the eed for optimizig the power factor. I this system, exteral ifeed of MVA i the base case ad MVA after optimal placemet of DGs was recorded. Table 4. DG placemet by differet methods for IEEE 119 ode system at 0.95 power factor. Case Method Istalled DG Schedule (MW) DG (MW) Ploss (kw) Loss Red (%) Time (s) No DG Total Real Load = MW DGs LSF IA ELF MM Bus Size Bus Size Bus Size Bus Size Voltage Profile Improvemet Results A additioal advatage of the optimal sittig ad sizig of DGs with the proposed method is depicted clearly i Figures 10 ad 11. These graphs were take whe the system was operatig at the operatioal power factor proposed i Sectio 4.2. As metioed i the Table 1 ad ca also be see i these figures, the miimum voltage i either etwork is about 0.87 p.u. approximately, whereas the maximum voltage is p.u., which appeared at the ode where the exteral grid is coected ad is a voltage regulated ode. After DG placemet, all the methods improved the voltage i the

18 Bus Voltage (p.u) at pf = 0.90 Bus Voltage (p.u) at pf = 0.97 Eergies 2016, 9, of 21 etwork; however, the voltage improvemet is varyig. It is already show i Figures 8b ad 9b that voltage variatio (the differece betwee maximum ad miimum voltages appearig i the etwork) is the lowest with the method proposed here. For the 37 ode etwork, the miimum voltage observed after optimal DG placemet was p.u. with the ELF method, whereas with the proposed method (MM), the voltage was p.u. Moreover, the highest voltage observed with ELF method of optimal DG placemet was p.u., ad, with the proposed method, it was p.u. The IA method also brought the miimum voltage to the permissible limit of p.u. ad maximum voltage to p.u. Although the voltage improvemet by IA method is withi the allowed bad, it is closer to the miimum level, which makes it proe to fallig below with small variatio i loads or geeratios. The LSF method was uable to icrease the miimum voltage to the permissible level. From Figure 11, it is clear that the proposed method outperformed the other methods where the miimum voltage was p.u. ad maximum voltage was p.u. The ELF ad IA methods also improved miimum voltage to p.u. each, but the maximum voltage was p.u. ad p.u., respectively. For the LSF method, miimum voltage was agai out of the permissible bouds ad appeared to be p.u., whereas the maximum voltage was p.u. As a fial commet, it is said that, for a etwork, it is ot oly required to keep maximum ad miimum voltages withi the limits, but the differece betwee these two should also be kept closer to zero to make the system more stable with respect to the voltage Bus Number No DG ELF IA LSF MM Figure 10. Voltage profile for 37 ode system with differet methods (pf = 0.97) Bus Number No DG ELF IA LSF MM Figure 11. Voltage profile for 119 ode system with differet methods (pf = 0.90). 7. Coclusios This paper explais a ovel method for fidig the optimal bus umbers for placig multiple DGs. The method is based o load cocetratio faced directly by the bus or through directly coected buses. Based o the optimal locatios selected, aalytical expressios for the optimal sizes of sigle or multiple DGs are derived for a large scale distributio system. The geeralized steps to

19 Eergies 2016, 9, of 21 derive the aalytic expressios for simultaeous optimum sizig, which also iclude the impact of all DGs o each other, are preseted. The exhaustive method for selectig operatioal power factor required from a DG is also icorporated, resultig i the versatility of the derived expressios to be used for DGs with both active ad reactive power geeratio capabilities. Moreover, the LSF, IA ad ELF methods have bee preseted. The proposed methodology for simultaeous optimal sizig of multiple DGs is cofirmed to be effective ad efficiet i compariso with LSF ad IA both i terms of miimizig the losses ad simulatio time. For the ELF method, the results of the preseted methods are comparable i terms of loss miimizatio but superior for voltage profile improvemet ad simulatio time. The proposed methodology is the geeral set of steps that ca be applied for ay umber of DGs. I this work, the cases of oly four DGs i a IEEE 37 ode ad five DGs i a IEEE 119 bus systems are discussed. Aother importat coclusio is the eed for optimizig the power factor that a DG must provide withi the limit of grid code requiremets. It is observed that the DGs with optimal power factor ca reduce the losses further if kept withi the limits. Hece, to achieve the maximum beefits from DG(s), their locatio, size ad power factor must be properly chose. The study presets a ovel, useful ad highly competitive method for plaig ad decisio support before implemetig ad coectig DGs to the electric power grid. It makes uused ad ormally wasted optimizatio potetial accessible ad reduces losses ad trasmissio ad distributio system stress. Author Cotributios: All authors cotributed equally to this work. Coflicts of Iterest: The authors declare o coflict of iterest. Abbreviatios The followig abbreviatios are used i this mauscript: CLPF: Combied Load Power Factor DSPF: DIgSILENT PowerFactory ELF: Exhaustive Load Flow IA: Improved Aalytical LSF: Loss Sesitivity Factor LCF: Load Cocetratio Factor MM: Mohsi s Method pf: Power Factor VPI: Voltage Profile Improvemet Refereces 1. Sigh, D.; Sigh, D.; Verma, K. GA based eergy loss miimizatio approach for optimal sizig & placemet of distributed geeratio. It. J. Kowl. Based Itell. Eg. Syst. 2008, 12, Georgilakis, P.; Hatziargyriou, N. Optimal distributed geeratio placemet i power distributio etworks: Models, methods, ad future research. IEEE Tras. Power Syst. 2013, 28, Taylor, M.; Daiel, K.; Ilas, A.; So, E. Reewable Power Geeratio Costs i 2014; Iteratioal Reewable Eergy Agecy: Masdar City, Abu Dhabi, UAE, Li, W.; Joos, G.; Belager, J. Real-time simulatio of a wid turbie geerator coupled with a battery supercapacitor eergy storage system. IEEE Tras. Id. Electro. 2010, 57, Pigazo, A.; Liserre, M.; Mastromauro, R.; Moreo, V.; Dell Aquila, A. Wavelet-based isladig detectio i grid-coected PV systems. IEEE Tras. Id. Electro. 2009, 56, Yao, W.; Che, M.; Matas, J.; Guerrero, J.; Mig Qia, Z. Desig ad aalysis of the droop cotrol method for parallel iverters cosiderig the impact of the complex impedace o the power sharig. IEEE Tras. Id. Electro. 2011, 58,

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