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Distributed Power Loss Minimization in ResidentialMicro Grids: a Communications Perspective

Riccardo Bonetto†, Stefano Tomasin and Michele Rossi

Abstract—The constantly increasing number of power gen-eration devices based on renewables is calling for a transitionfrom the centralized control of electrical distribution gr idsto a distributed control scenario. In this context, distributedgenerators (DGs) are exploited to achieve other objectivesbeyondsupporting loads, such as the minimization of the power lossesalong the distribution lines.

The aim of this work is that of designing a full-fledged systemthat extends existing state of the art algorithms for the distributedminimization of power losses. We take into account practicalaspects such as the design of a communication and coordinationprotocol that is resilient to link failures and manages channelaccess, message delivery and DG coordination. Thus, we analyzethe performance of the resulting optimization and communicationscheme in terms of power loss reduction, reduction of aggregatepower demand, convergence rate and resilience to communicationlink failures.

After that, we discuss the results of a thorough simulationcampaign, obtained using topologies generated through a statis-tical approach that has been validated in previous research, byalso assessing the performance deviation with respect to localizedschemes, where the DGs are operated independently. Our resultsreveal that the convergence and stability performance of theselected algorithms vary greatly. However, configurationsexist forwhich convergence is possible within five to ten communicationsteps and, when just30% of the nodes are DGs, the aggregatepower demand is roughly halved. Also, some of the consideredapproaches are quite robust against link failures as they stillprovide gains with respect to the localized solutions for failurerates as high as50%.

I. I NTRODUCTION

The traditional centralized power distribution grid is nowa-days facing two important trends: the constantly increasingpower demand [1] and the worldwide diffusion of electricalpower generation devices based on renewables, e.g., photo-voltaic cells and wind turbines. While the former calls for radi-cal changes in the way the energy is generated and delivered tothe final users, we note that electrical power generation is stillmostly based on biofuels, fossil fuels and nuclear plants [2].On the other hand, the installed renewable sources do not seemto be exploited as they should be, as distributed generationdevices are mostly used to sell power to the energy provider byfully injecting it (through an inverter) into the grid or to fulfillthe owner’s power demand, without any interaction with thedistribution grid, other users, or the utility provider. The limitsof the fossil fuels fields, together with the inelastic nature ofthe demand of these goods, cause a steady price growth of

†Corresponding author: Riccardo Bonetto, e-mail: bonettor@de.unipd.it.All authors are with the Department of Information Engineering, University ofPadova, Via G. Gradenigo 6/B, 35131 Padova (PD), Italy (e-mail: bonettor,tomasin, rossi@dei.unipd.it).

the related electrical power. Moreover, nuclear-based energyproduction entails high costs and is subject to a diminishingpublic acceptance due to safety concerns. These facts areincreasingly motivating the shift from traditional centralizedand hierarchical power distribution grids towards smart anddistributed ones [3]–[5].

When communication and smart metering capabilities areadded to a power grid, local generators based on renewables(also called distributed generators, DGs) can be used toenhance the grid efficiency (in terms of power distribution,reactive power compensation and frequency stability) and torelieve electricity production plants from some of the powerload. In the last few years, several grid optimization techniqueshave been proposed [6]–[9], each exploiting some existingcommunication infrastructure and relying on online smartmetering procedures [10].

We stress that, a common approach of previous papershas been that of taking the communication infrastructure anda suitable DG coordination protocol for granted, assumingthat appropriate solutions are in place and that can be ex-ploited by the various algorithms. While this is certainly areasonable starting point for an initial design of distributedcontrol schemes, in the present paper, we aim at filling thegap between the electrical optimization techniques and thepeculiarities of a real communication network, such as thecoordination among the agents involved in the optimization.This is achieved through the definition and the subsequentimplementation of a suitable communication protocol, forpowerline communication (PLC) infrastructures [11], [12], thatmanages the channel access, the message delivery and therequired DG coordination procedures. Hence, this protocolisused in conjunction with four different power distributionlossminimization techniques [13]–[15] analyzing the performanceof the resulting optimization and communication system. Inparticular, we explore the system behavior in terms of powerloss reduction, reduction of the aggregate power demand tothe distribution network, convergence rate and resiliencetocommunication link failures. In addition, we have extendedthe current based surround control (CBSC) technique of [14]by improving its DG aggregation procedure, which allows fora further reduction of distribution power losses.

Finally, to obtain statistically valid results, as in [16] wehave used the smart grid topology generator proposed in [17],which is based on thesmall world model of [18]. Thus,we discuss the results of a thorough simulation campaign,obtained using the aforementioned topologies, by analyzingthe performance deviation with respect to localized schemes(where the DGs are operated independently).

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To summarize, the main contributions of this paper are:

• a communication and coordination protocol designedto operate on smart micro grids in the presence ofsmart nodes (DGs) with communication capabilities. Thisprotocol ensures that the channel access is contentionfree, that the shortest length path is chosen in multi-hop communication scenarios and that only one nodecommunicates at any given time (thus avoiding collisionsand subsequent packet losses);

• a new design for CBSC [14] that permits a furtherreduction of the distribution power losses with respectto those obtained by the original protocol;

• a comparative study of the performance, in terms ofdistribution power loss reduction and convergence rate,of four selected control techniques for micro-grids [13]–[15], which constitute the state of the art in terms ofdistributed techniques for power loss minimization;

• a comparative study of the performance, in terms ofresilience to link failures, of the four selected controltechniques.

The rest of this work is structured as follows. In Section IIwe introduce the electrical grid model and the communicationnetwork, detailing the requisites that nodes equipped withPLC transceivers should meet in order to execute the con-trol algorithms. In Section III we briefly describe the fouroptimization techniques and, for each technique, we analyzethe related communication requirements. In Section IV wediscuss the way in which nodes can be aggregated in order toform optimization clusters and we propose a novel aggregationprocedure that results in a performance enhancement (in termsof distribution power loss reduction) with respect to thatproposed in [14]. In Section V we present a novel resilienttoken ring protocol, specifically designed for tree networks, sothat the nodes are able to access the communication channelin a contention free manner by making sure that only onenode performs the required control actions, at any given time.In Section VI we detail the simulation setup and we alsodescribe the procedure used to generate the test networks forour simulations. In Section VII we present and discuss thenumerical results, highlighting the most interesting features ofthe examined algorithms in terms of dependence on systemparameters such as the link failure probability and the numberof nodes that are equipped with smart functionalities. Finally,in Section VIII we draw our final considerations.

II. GRID MODEL

We consider a power micro grid modeled as a directed tree.The root of the tree represents the point of common coupling(PCC) and the other nodes represent loads, distributed gen-erators (DGs) and connection points. Loads are representedby complex impedances, the PCC is modeled as a voltagegenerator setting the voltage reference for the entire grid,while DGs are modeled as current generators. Fig. 1 showsan example of a power grid. Nodei is denoted by labelNi, load z and DGm are denoted respectively by Lz andGm and branch impedancej is denoted by Bj . Moreover,we assume that branches have constant impedance per meter

[14], [15]. Note that each DG has anassociated load. Thismodel, for example, a house (i.e., an aggregated load fed byphotovoltaic panels on the rooftop). In this paper we considerthe scenario where the DGs, besides feeding the respectiveassociated loads, are operated in order to reduce the powerdistribution losses through suitable control algorithms.FromFig. 1 we see that two main portions of the grid can beidentified: one connecting nodes PCC, N0, . . . ,N3 (see righthand side of Fig. 1) and the other one connecting nodesN4, . . . ,N8 (left hand side). These two portions are electricallyindependent and hence the DGs can be controlled separately.Generalizing this concept, if a power grid hasn ∈ N branchesexiting from the PCC node, then the correspondingn sub-grids can be controlled in parallel. The control algorithmsconsidered in this paper require a communication networkamong the controlled nodes, which is here assumed to bea powerline communication (PLC) infrastructure [19]. Nodesequipped with PLC transceivers are referred to as smart nodes(SNs) and suitable communication protocols are assumed toallow the communication between any pair of SNs, possiblyby appropriate routing of messages through intermediate SNs,as we detail shortly. We assume that SNs are also capableof measuring instantaneous electrical quantities (i.e., voltage,current and power) absorbed (by the loads) or injected (byDGs).

III. O PTIMIZATION METHODS

In this section, four different distribution losses minimiza-tion techniques are presented and the corresponding com-munication requirements are discussed. One algorithm onlyentails localized actions and, as such, does not require anycommunication among nodes. This scheme will be used as abenchmark to measure the improvements that can be achievedwhen communication capabilities are added to the grid.

A. Local Control

With the local control (LC) technique, [13] and [20], eachDG provides the reactive power absorbed by itsassociatedload. Assuming that only the reactive component injected bythe inverter is controllable by the optimization process, [13]and [20], LC only uses local informations available at theinverter: the active and reactive powers absorbed by the loadconnected to the same node. As an extension of LC, weconsider the case where both the active and the reactive powergenerated by the DG are the powers absorbed by itsassociatedload and we denote this control technique as extended LC(ELC). Since LC and ELC only use local information, theydo not require any communication among SNs.

B. Current Based Surround Control

According the current based surround control (CBSC) [14],the grid is divided into clusters. Each cluster is composed ofa pair of DGs (GA and GB) such that the path connectingthe DGs only contains loads. Considering a single cluster, letIGAGB

be the current injected by node GA towards GB and,conversely, refer toIGBGA

as the current from GB to GA. The

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+L6

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N8 N7

N6

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+-

Fig. 1. Power micro grid example.

aim of CBSC is to find, for each cluster, the optimal currentsIGAGB

and IGBGA, as we describe next. According to [14],

fixing the initial DG, termed GA, we find all the possiblegenerators, termed Gh, that form a cluster with it, i.e., theportion of network between GA and Gh only contains loads.N (GA) is the set of indices of all generators Gh (includingthe PCC) found through this procedure. Hence, the optimalcurrent injected by GA that minimizes the distribution lossesis found as:

ICBSCGA

=∑

h∈N (GA)

IoptGAGh, (1)

where:

IoptGAGh=

1

RGA,Gh

∑

i∈L(GA,Gh)

IiRGh,Li(2)

and:

• RGA,Ghis the real part of the impedanceZGA,Gh

of thelines connecting DGs GA and Gh;

• RGh,Liis the real part of the (total) impedanceZGh,Li

ofthe lines connecting DG Gh and load Li;

• L(GA,Gh) is the set of indexes of the loads in cluster(GA,Gh);

• Ii, i ∈ L(GA,Gh) is the current absorbed by the loadLi.

A variant of CBSC provides that only the reactive currentinjected by the DGs is controlled in order to reduce distributionlosses, while the active current is regulated by other mecha-nisms, e.g., economic contracts or fully injected into the grid.In this case, the current injected by DG GA is 0+jIm(ICBSC

GA).

In order to operate CBSC, GA first builds a list of theclusters it belongs to. This list contains the setN (GA) and,∀h ∈ N (GA), the setL(GA,Gh). Moreover,∀h ∈ N (GA)and∀ i ∈ L(GA,Gh), GA estimates the resistancesRGhLi andRGAGh

. Once the list of clusters has been set up, Algorithm 1shows the actions taken by GA in order to estimate and injectICBSC

GA. Firstly GA creates and sends to Gh a special packet

denoted as DataGatheringPacket (i.e., see line 2). This packetis routed to its destination by the loads whose indexes arein L(GA,Gh). Once Gh receives the DataGatheringPacket, itsends back an acknowledgment which, according to line 3, is

stored in the Ack variable. Each load involved in the routingprocess adds to the acknowledgment its index and its actualcurrent demand. Function UpdateCluster(Ack) called in line 4builds the vector CurrentDemand containing the current de-mands stored in the Ack variable. Elements of CurrentDemandare indexed using the indexes ofL(GA,Gh), as shown inline 7. Once the optimum current has been computed, GA

injects ICBSC as dictated by the function InjectCurrent inline 12.

Algorithm 1 CBSC PseudocodeRequire: List of clusters

1. for all h ∈ N (GA) do2. SendDataGatheringPacket(h)3. Ack ← WaitForGatheringAck()4. CurrentDemand← UpdateCluster(Ack)5. I ← 06. for all i ∈ L(GA,Gh) do7. Ii ← CurrentDemand[i]8. I ← I + 1

RGAGh

IiRGhLi

9. end for10. ICBSC← ICBSC+ I11. end for12. InjectCurrent(ICBSC)

Since the voltage reference imposed by the PCC stabilizesthe grid, the current injected by the DGs does not influence theloads’ total current demand. Hence CBSC requires that eachDG runs Algorithm 1 only once in order to drive the grid’sstate towards the minimum distribution loss. Notably, whileCBSC has very fast convergence rates, this algorithm requiresthat each node is a SN, which means that each node musthave communication and metering capabilities (to measureor estimate the quantities required by the scheme). This maybe difficult to achieve in practice, especially when retrofittingexisting grids with old equipment.

C. Voltage Based Surround Control

The voltage based surround control (VBSC) algorithm [14]aims at reducing the communication requirements with respect

4

to CBSC. VBSC is based on the observation that losses areminimized when all DG voltages are as close as possible to thePCC voltage. Let GA be the generator performing the controlaction, then the voltage that GA should reach is:

UoptGA

=

∑

h∈N (GA)

RGAGh

|ZGAGh|2UGh

∑

h∈N (GA)

RGAGh

|ZGAGh|2

(3)

Given (3), the variation of the current injected by GA is:

∆IGA=

UoptGA− U0

GA

ZeqGA

, (4)

where U0GA

is the actual voltage of GA and ZeqGA

is theThevenin impedance of the whole grid as seen by GA. Asfor CBSC, if the active power is regulated by mechanismsother than power loss minimization, only the reactive currentcan be injected (see [14]). In this case, the variation of thecurrent injected by GA will be 0 + jIm(∆IGA

).

Algorithm 2 VBSC PseudocodeRequire: List of neighborsN (GA)Require: GA feeding associated load with currentIVBSC

Require: ImpedanceZGA,Gh∀h ∈ N (GA)

1. U0GA← GetMyVoltage()

2. for all h ∈ N (GA) do3. SendVoltageGatheringPacket(h)4. Ack ← WaitForGatheringAck()5. UGh

← UpdateNeighborVoltage(Ack)

6. Unum ← Unum +real(ZGA,Gh

)

abs(ZGA,Gh)2UGh

7. Uden ← Uden +real(ZGA,Gh

)

abs(ZGA,Gh)2UGh

8. end for9. Zeq

GA← MeasureEquivalentImpedance()

10. UoptGh←

Unum

Uden

11. IVBSC ← IVBSC +Uopt

Gh− U0

GA

ZeqGA

12. InjectCurrent(IVBSC)

Note that the update of the current according to (4) changesthe voltages of all the other nodes, including the value ofUGh

, ∀ h ∈ N (GA). Therefore, the optimum voltage isobtained through multiple control actions that gradually drivetowards zero the absolute voltage difference between the DGsand the PCC. In order to operate VBSC, GA must know theindexes of the neighboring DGs (N (GA)) and the impedanceof the path connecting GA and Gh, ∀h ∈ N (GA). Algorithm 2shows the procedure that GA executes everytime it performsthe control action. According to line 1, GA firstly stores itsvoltage, then for each DG whose index is inN (GA) it sendsa VoltageGatheringPacket (i.e., see line 3). Once Gh receivesthe VoltageGatheringPacket, it measures its current voltageand sends it back to GA as an acknowledgment. Once theacknowledgment is received and the neighbor’s voltage has

been updated (see line 4 and line 5), GA computes the numer-ator and denominator of (3) corresponding to neighbor Gh.Once this operation has been performed for all neighboringDGs, the equivalent Thevenin impedance seen by GA (seeline 9) is measured and the current step is computed (seeline 10 and line 11). Neglecting the information about theloads, this algorithm requires that only the DGs are SNs,considerably reducing the communication requirements withrespect to CBSC.

D. Distributed Optimal Reactive Power Flow Control

The distributed optimal reactive power flow control(DORPF) algorithm, proposed in [15], assumes that only thereactive power is controlled for distributed loss minimization.DORPF groups the DGs into (possibly overlapping) clustersand, for each cluster, a portion of the full optimization problemis solved using an approximated representation of the grid.DORPF is based on a distributed linearization of the optimalreactive power flow problem which is not reported here for thesake of conciseness (see [15]). The most effective clusteringstrategy appears to be that of [14], used also for CBSC.Optimizing the reactive power injection requires the generatorsin a cluster to estimate the PCC’s voltage, the line impedance,and the neighbor’s voltage. In order to get this informationonly distributed generators need to be SNs and hence, thecommunication requirements of this algorithm are similar tothose of VBSC.

IV. CLUSTERING

As discussed in the previous sections, the ability to buildclusters of distributed generators is an essential featureofall the considered distributed algorithms. In this sectionwedescribe an online procedure to build clusters in a distributedfashion, by also describing a novel approach that extends thework of [14], and makes the optimization more robust tocertain topologies.

In [14] and [15] pairs of generators (including the PCC)such that the path connecting them only includes loads (and nogenerators) are defined as clusters. According to this definition,considering Fig. 1, four clusters can be identified as shown inTab. I.

TABLE IGENERATOR PAIRS ASSOCIATED WITH THE CLUSTERS INFIG. 1.

Cluster Number Generator 1 Generator 2

1 PCC G02 G0 G33 PCC G54 G5 G8

For VBSC and DORPF only distributed generators arerequired to be SNs, hence the clustering process is reducedto a neighbor discovery process. The CBSC algorithm, on thecontrary, requires detailed informations about the loads (whichare SNs too) along each cluster’s path. Let GA be the DGperforming the clustering procedure shown in Algorithm 3,then to obtain this information, onceN (GA) has been setup, GA sends a special information gathering packet (called

5

BuildClusterPacket, see line 2) to all DGs Gh : h ∈ N (GA).Each load Lk : k ∈ L(GA,Gh) appends to this packet itscurrent demand, the impedance of the lines connecting it toGA and its identifierk and forwards it to the next node inthe path between GA and Gh. This procedure is repeated forall nodesk in the path, until the BuildClusterPacket reachesthe destination, as shown in Algorithm 4 (see line 2, line 3and line 4). Gh stores the received load current demands andimpedances from GA in the clusters table on the positioncorresponding to the sender’s identifier and then sends backanacknowledgment with piggybacked the loads’ current demandsand line impedances. Once the clusters table has been set up,changes in loads current demands can be dynamically updatedby the loads.

Algorithm 3 Basic Clustering Pseudocode, Generator sideRequire: Nodes are synchronized to central clockRequire: List of neighboring generators N

1. for all Neighbor n in Ndo2. SendBuildClusterPacket(CurrentTime);3. WaitForAck();4. if ReceivedAck()then5. D←SetImpVector(Ack.Impedances);6. PW←Ack.PowerDemand;7. UpdateClusterTable(n, D, Pw);8. end if9. end for

Algorithm 4 Basic Clustering Pseudocode, Load sideRequire: Nodes are synchronized to central clock

1. if BuildClusterPacketRx()then2. d←EstimateImpFromSource();3. UpdatePacket(GetPowerDemand(), d);4. SendPacket(Packet);5. end if

Since the CBSC requires that all the nodes are SNs, theoptimization process can be improved. The clustering ob-tained through Algorithms 3 and 4 considers only couples ofneighboring DGs, thus leaving out of the optimization processportions of the network ending with a leaf node with a singleconnected load. In Fig. 1 two portions of the network areisolated: one made by edge B1 and node N1 and the othermade by edge B6 and node N6. Since the leaf nodes have justone neighbor, they are able to determine their position in thenetwork and hence to communicate to the nearest distributedgenerator the presence of a portion of the network that wouldnot be optimized using the standard clustering approach. Oncea DG learns of such a portion of the network, which can beachieved through a simple probing procedure, it considers it asa special cluster and fully feeds such portion of the grid. Thisclustering procedure (called enhanced clustering, EC) permitsto enhance the performance of CBSC, with respect to thestandard clustering scheme, when an appropriate number ofdistributed generators is accounted for. As an example, Fig. 2shows the performance in terms of dissipated power when EC

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Optimization Steps

No LeavesLeaves

Fig. 2. Dissipated powervs optimization steps for CBSC for standardclustering and the proposed clustering technique.

is used. For this plot, CBSC has been executed on the topologyof Fig. 1 using the parameters of Tab. II. In Section VII-E,EC is further investigated for a higher number of topologiesand system parameters.

TABLE IIPARAMETERS FOR THE NETWORK OFFIG. 1.

Active Generators G0 G5 G8Impedance [Ω/m] (0.8 + j0.8)10−3

Branch Lengths [km]

B0 0.1B1 0.023B2 0.045B3 0.026B4 0.035B5 0.067B6 0.032B7 0.012B8 0.066

V. COMMUNICATION PROCEDURES

The update of the injected current by DGs alters the oper-ating points of all other grid nodes. To ensure convergence ofthe distributed algorithms of Section III, an iterative approachto update the injected currents has been proposed in [14].Nevertheless, in that paper communications aspects, such asthe design of protocols, the selection of the routing pathsand possible approaches to make the control resilient to linkfailures are not discussed. In this section, we fill this gapby proposing a failure resilient token ring protocol for treenetworks. This protocol is then used in conjunction withthe clustering algorithm of Section IV, obtaining the finaldistributed algorithms that will be evaluated in Section VII.

A. Token Ring Protocol

As suggested in [14], we have implemented the selectedalgorithms by passing the control to a single node at a time.This is achieved through a round robin approach, wherebya token is utilized to assign the control to a certain node ata certain instant. To make sure that a single node owns the

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Fig. 3. Token ring network example. The arrows indicate the token’s path.

PCC

N0

N1

N2

N3

N4

N5

N6

N7

N8

Fig. 4. Example of the token’s path in the communication network relatedto the power grid of Fig. 1.

token (and hence performs the control actions as dictated bythe selected algorithm) we exploit a token ring communicationprotocol.

The token ring protocol (known as protocol IEEE 802.5)has originally been developed for networks whose nodes areconnected in a ring fashion. Hence, a special packet calledtoken is circulated in the network and the noden receivingthe token has the right to transmit packets while all the othernodes remain silent, unless they receive a specific request fromnoden. Once the token’s owner has completed its operations,it sends the token to the next neighbor, selected according toa certain schedule. Fig. 3 shows an example of a token ringnetwork. Solid lines connecting the nodes numbered from1to 8 represent the actual communication links between them,while the counterclockwise pointed lines represent the token’spath in the network.

B. Token Ring Protocol for Tree Networks

To adapt the standard token ring protocol to the treetopology treated in this work, a new token’s owner selectionprocedure is devised. LetN ≥ 0 be the number of SNs and letthe current token’s owner identifier bei ∈ N with i ≤ N , thenthe next token’s owner is attained asj = (i + 1) mod N .This owner selection rule ensures that when SNi releases thetoken, all the other SNs will receive it beforei owns it again,thus ensuring fairness in the communication process. Fig. 4shows an example of the token path on the power grid of Fig. 1

assuming that all nodes are SNs (i.e.,N = 9). Nodes receivethe token on the basis of their identifiers (on the contrary, inregular token ring networks the token exchange is based on theactual physical position of the nodes). It is also worth notingthat by correctly setting the nodes’ identifiers, the token’s pathcan be forced into a depth first path search on the tree, asillustrated in Fig. 4. Note that this minimizes the number ofjumps of the token between non-adjacent nodes.

C. Lost Token Recovery

The circulating token can be lost for various reasons as,for example, external electromagnetic interference, communi-cation link failures and device failures on the token’s path.In order to recover from these events, the token receptionmust be acknowledged to the sender. Nodei releasing thetoken remains the owner until it successfully receives theacknowledgment sent by nodej. If after a timeout timeT(also dependent on the network size and on the transmissionrate) no acknowledgment is received, then the token exchangeprocedure is repeated until either the token reception is ac-knowledged or the maximum number of transmission attemptsis exceeded. In the latter case, we skip the next node in thepath and start a new exchange procedure with the followingnodej′ = (j + 1) modN .

D. Handling Disconnected Portions of the Network

When a portion of the network gets disconnected froma communication perspective, the nodes therein first haveto discover that such an event has occurred. To this end,each SN utilizes the following approach. A timeout timer isreset at every node, every time the token is received (thetimeout period must be the same for all the SNs and mustbe dispatched by a coordinator). If the timeout timer of agiven SN counts down to zero, then the SN promotes itselfas the new coordinator for the disconnected portion of thenetwork and -from a communication standpoint- starts actingas the coordinator. It performs a new neighbor discovery, andeach node involved in such process updates its routing tablesaccording to the new information it receives. Once this processgets to completion, the obtained subtree will be optimizedindependently, using as a reference voltage the PCC’s voltageestimated (or measured) by the new coordinator before thefailure.

VI. SIMULATIONS SETUP

Numerical results comparing the performance of the con-sidered algorithms have been obtained over a large set ofrandomly generated grids. Instead of relying on standard testfeeders as proposed in [21], we adopted the approach used in[16]. Using the generator from [17], we generated more than athousand power distribution grids and averaged the numericalresults obtained by testing the optimization techniques takeninto account on every single grid. The impact of communica-tion and clustering protocols has also been assessed.

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Fig. 5. Regular ring lattice with eight nodes.

1

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Fig. 6. Example of small world network generated from the regular ringlattice of Fig. 5.

A. Random Grid Generation

For a meaningful performance evaluation, we have consid-ered a large number of networks, which have been obtainedusing a network generator that accounts for the theoreticaland experimental results of [18]. In fact, power micro gridscan be modeled as directed graphs (with the orientation ofthe edges determined by the direction of the active current).These graphs are included in the class of small-world networkswhich are connected graphs characterized by a large numberof vertices with sparse connections and fill the gap betweencompletely random graphs and regular graphs. With thisapproach, we can generate numerous synthetic networks byfixing their relevant parameters such as the number of nodes,the number of generators, the depth of the tree, etc., by makingsure that the generated networks have statistical propertiesresembling those of real power grids.

In order to build a small-world network we start from aring lattice. Then for each edgee, one of its endpoints isreplaced with probabilityp by another node, chosen uniformlyat random among all other nodes. Fig. 5 shows an exampleof regular ring lattice with8 nodes. Instead, Fig. 6 shows asmall-world graph generated from the regular lattice of Fig. 5when edgesea andeb connecting nodes(6, 7) and (4, 6) arerewired to links(4, 7) and (4, 8), respectively.

B. Electrical Parameters Setup

We assume that the generated grids operate in steadystate and that they are single phased electrical networkswhose distribution lines have a constant specific impedanceof (0.08 + j0.08)10−3 Ω/m. The phase voltage at the PCCis set at230 V and the voltage drops along the distributionlines are neglected in determining the loads’ instantaneouspower demand, as assumed in [14]. DGs with associated loadsautomatically feed them with the required current. Moreover,we do not assume limitations on the maximum current thatcan be injected by the DGs.

C. Communication Assumptions

From a communication standpoint, a first assumption isthat a routing protocol connecting each pair of SNs in thegrid exists. A second assumption is that no packet is lost orcorrupted along a communication path unless at least one ofthe links in the path is broken (which is accounted for usingan i.i.d. process with a certain probability).

In order to obtain statistically relevant results, the optimiza-tion methods treated in Section III and the communicationprocedures treated in Section V and Section IV have beentested over a large number of grids and the correspondingcommunication network conditions. Tab. III summarizes thefixed parameters considered in the simulations.

TABLE IIIGRID PARAMETERS.

Parameter ValueNumber of Nodes 30Line Impedance (0.08 + j0.08)10−3 Ω/m

Average Line Length 30 mRewiring Probability 50%

VII. N UMERICAL RESULTS

In this section, we show the numerical results obtainedconsidering the simulation setup described in Section VI.

A. Dissipated Power

Figs. 7 and 8 show the average dissipated power over onehundred random network realizations where both active andreactive current injection is allowed and where only reactivecurrent injection is allowed, respectively. For each network,30% of the nodes are DGs. For all algorithms, the startingpoint of the iterative optimization procedure provides thatthe DGs do not inject any current. When both active andreactive current is injected, only ELC, CBSC and VBSCare considered. A first important result that can be noticedis that both CBSC and VBSC achieve a considerably lowerpower distribution loss than ELC, due to their exploitationof communication capabilities. With reference to Fig. 7, whileELC reduces the power loss by more than7 kW with respect tothe starting point, CBSC further reduces losses by over2 kWin less than ten iterations. VBSC exhibits a slower convergencerate while reducing power loss by nearly1 kW with respectto ELC. When only reactive current is controlled (Fig. 8), LC

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5 10 15 20 25 30 35 40

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ELCCBSCVBSC

Fig. 7. Dissipated powervs optimization steps for ELC, CBSC and VBSC.30% of nodes are DGs.

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5 10 15 20 25 30 35 40

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sipa

ted

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er [k

W]

Optimization Steps

ELCLC

CBSCVBSC

DORPF

Fig. 8. Dissipated powervs optimization steps for LC, ELC, CBSC, VBSCand DORPF.30% of nodes are DGs, reactive current injection.

reduces the power losses by nearly1.75 kW with respect tothe starting point. It is worth noting that all the distributedalgorithms exploiting communication capabilities still allowthe reduction of the power loss by (up to)0.5 kW with respectto ELC. Overall, CBSC outperforms all other approaches,relying on a complete knowledge of the clusters, since eachDG has a complete knowledge of the branches connecting itto other neighboring DGs, and hence can compute the exactamount of power that is needed in each branch to minimizethe loss.

B. Convergence Time

As shown in Figs. 7 and 8, all algorithms after a certainnumber of iterations converge to a minimum dissipated power.In Fig. 9, we show the number of control steps that arerequired, for each algorithm, so that its performance fallswithin 5% of the associated minimum power loss. This numberof steps is then plotted against the dissipated power by varying,as a free parameter, the percentage of DGs from10% to 95%of the nodes. A first important result is that CBSC, VBSC

1

10

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103

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Con

verg

ence

Tim

e [S

teps

]

Dissipated Power [kW]

10%95% Percentage of DGs

LCCBSCVBSC

DORPF

Fig. 9. Optimization stepsvs dissipated power for ELC, CBSC, VBSC andDORPF. DGs are from10% to 95% of nodes, reactive current injection.

and DORPF guarantee that the dissipated power is comparableto that of LC even when only10% of the nodes are DGs.Moreover, the aforementioned algorithms permit to achieveapower loss very close to zero when the DGs are about70%of the nodes (or more). CBSC, once again, exhibits the fastestconvergence rate and the lowest power loss at each point. It isremarkable that this algorithm always converges within a few(at most twenty for the considered networks) iterations andthat this number weakly depends on the percentage of DGs.When only the distributed generators (DGs) are SNs, optionsreduce to DORPF and VBSC, as explained in Section III. Inthis case, we note that DORPF ensures the best convergencetime for the same dissipated power performance. Nevertheless,from Fig. 10, we note that when the specific line impedancegrows, the performance gap between DORPF and VBSCincreases leading to a higher dissipated power, up to25%, forDORPF with respect to VBSC for a specific line impedanceof (0.08+ j0.08)10−3 Ω/m. Thus, DORPF appears to be lessrobust for an increasing line impedance and this fact shouldbe carefully evaluated for practical implementations of thisalgorithm. In particular, when the specific line impedance iswell known and ensures that the voltage drops along the powerlines are less than3% of the PCC’s voltage, DORPF canbe successfully used. On the contrary, when the specific lineimpedance is not known in advance or the voltage drops arenot within the3% range (as it may occur in rural or isolatedareas), optimization should be performed through VBSC sincethis technique exhibits a higher robustness with respect tothegrid parameters.

C. Resilience to Link Failures

In the two previous sections we assumed a communicationnetwork with error free links. In this section, instead, thealgorithms’ resilience to link failures is considered. In thefollowing results, broken links are chosen uniformly at randomamong all links according to a given percentage. Fig. 11 showsthe dissipated power as a function of the percentage of brokenlinks; we note that despite high percentages of broken links,

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(3+j3)e-05 (7+j7)e-05 (1.1+j1.1)e-04 (1.5+j1.5)e-04

Dis

sipa

ted

Pow

er [k

W]

Line Impedance [Ω/m]

VBSCDORPF

Fig. 10. Dissipated powervs specific line impedance for VBSC and DORPF.15 nodes,50% rewiring probability,30% of nodes are DGs, reactive currentinjection.

2.3

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0 10 20 30 40 50 60 70 80 90

Dis

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W]

Broken Links [%]

LCCBCVBC

DORPF

Fig. 11. Percentage of broken linksvs dissipated power for ELC, CBSC,VBSC and DORPF.30% of nodes are DGs.

CBSC and VBSC achieve a lower dissipated power than ELC.On the contrary, when the percentage of broken links exceeds25%, DORPF performs worse than ELC. We recall that, whenonly DGs are SNs only VBSC and DORPF can be used.VBSC exhibits a considerably higher degree of resilience tolink failures with respect to DORPF. The higher resilience tolink failures, together with the independence from the specificline impedance discussed in the previous section, make VBSCthe best algorithm when little information is available aboutthe grid or link failures are frequent events.

D. PCC Workload

Fig. 12 shows the average PCC’s workload as a function ofthe percentage of DGs in the grid. A first noticeable result isthat at least40 kW are saved when using LC. When ELC isused, it permits to save at least60 kW. Distributed optimizationtechniques appear to be useful when the number of DGs isbetween10% and50% of the total number of nodes. In thisrange, the distributed optimization techniques allow to saveup to 15 kW with respect to ELC. When, instead, more than

0

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ower

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Distributed Generators [%]

ELCLC

CBSCVBSC

DORPF

50

60

70

30 35

50

60

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30 35

Fig. 12. Percentage of DGs over the total number of nodesvs PCC workloadfor LC, ELC, CBSC, VBSC and DORPF.

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10 20 30 40 50 60 70 80 90

PC

C P

ower

Del

iver

y [k

W]

Broken Links [%]

ELCCBSCVBSC

DORPF

Fig. 13. Percentage of broken linksvs PCC workload for ELC, CBSC,VBSC and DORPF. 30% of nodes are DGs.

50% of the nodes are DGs, the gain in terms of power losswith respect to the local control technique may not be worththe communication infrastructure needed by the distributedoptimization techniques.

Fig. 13 shows the PCC workload as a function of thepercentage of broken links in the communication network.We note that when the percentage of broken links is in therange of10%-50%, CBSC or VBSC outperforms the localizedapproach (ELC), guaranteeing a noticeable gain with respectto ELC (which, as expected, is insensitive to link failures).However, when the percentage of broken links exceeds50%of the total number of links, the gain with respect to ELC ismodest and may not motivate a distributed approach. Thus,when link failures can be detected, a sensible solution couldbe that of switching between distributed control (CBSC orVBSC) and ELC as a function of the percentage of brokenlinks in the network.

E. Impact of the EC procedure

In Fig. 14 we show the relative gain (expressed as apercentage), in terms of PCC workload reduction, obtained

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30

40

50

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90

10 20 30 40 50 60 70 80 90

Gai

n [%

]

Distributed Generators [%]

Gain [%]

Fig. 14. PCC power delivery gain obtained using ECvs percentage of DGs.Reactive current injection.

using CBSC together with EC as opposed to using CBSC inconjunction with the standard clustering technique of Fig.12.In this plot we vary the percentage of nodes that are DGsfrom 10% to 95%. Notably, when the distributed generatorsare between50% and 95% of the nodes, the gain rangesfrom 15% to 85%. However, in practice the cases wheremore than75% of the nodes are DGs may be unlikely and,besides that, although the gain in this case is high in terms ofpercentage, the PCC workload reduction in terms of absolutevalue is rather small (see Fig. 12). On the one hand, when DGsare between40% and75% of the nodes, the PCC workloadreduction ranges between3 and 8 kW. Note also that whenthe percentage of DGs is between20% and 35%, standardclustering performs slightly better than EC. The highest gapis in this case is obtained when30% of the nodes are DGs,where standard clustering permits a5% gain (i.e., about3 kW)with respect to EC. This is due to the fact that when thepercentage of DGs is small, the special clusters optimized byEC have bigger length on average with respect to the casewhere the percentage of DGs is higher (i.e., above35%).Hence, a higher fraction of the power fed by the DGs to theleaf nodes is wasted along the distribution lines. The PCCtries to compensate for the power dissipated on these branchesinjecting more reactive power and, in turn, the total power lossslightly grows.

VIII. C ONCLUSIONS

In this paper, we have presented a communication-orienteddesign of a full-fledged system for the minimization of powerlosses in micro-grids, where communication and power lossoptimization algorithms are jointly implemented. We havestudied the performance of such system in terms of powerloss reduction and convergence time when the communicationnetwork provides error free communications. We have thenexplored the resilience of selected optimization techniqueswhen communication links are subject to failures. To test theimpact that the communication protocol has on the optimiza-tion process, we have defined a novel contention free tokenring protocol, specifically designed for tree topologies, that

allows the SNs to communicate without interfering with oneanother. We have also devised a new aggregation procedurethat permits, when the leaf nodes are SNs, a further reductionof the dissipated power that is achieved by the best algorithm.The results that we have discussed in this paper highlightthe differences among the selected optimization techniquesfor power loss minimization. Their performance in terms ofconvergence time, resilience to link failures and to the specificline impedance, largely vary. However, configurations exist forwhich convergence is achieved within only ten communicationsteps. Also, the aggregate power demand of the micro-grid canbe roughly halved with just30% of the users being SNs.

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