Aggregation Points Planning for Software-DefinedNetwork Based Smart Grid Communications

Shaowei Wang ∗† and Xinxin Huang†

∗ State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210023, China† School of Electronic Science and Engineering, Nanjing University, Nanjing 210023, China

E-mail: wangsw@nju.edu.cn, huangxx@smail.nju.edu.cn

Abstract—Smart grid is characterized by a large number ofsmart meters (SMs) that exchange huge amounts of data withcontrol center, where an effective communication network isrequired to guarantee reliable data transmission. In this paper,we introduce software defined network (SDN) technology to thesmart grid, which decouples the control plane from the dataplane so as to satisfy the communication requirements in themart grid effectively. Aggregation points (APs) are employedin the data plane to process and forward data between SMsand control center. A general mathematical model is formulatedto plan the APs, where we try to minimize the total deploy-ment cost, including the opening expenditure of the APs, theconnection cost between SMs and APs, and the connection costbetween APs and control center. We present a 5-approximationalgorithm to address the generated NP-hard problem, whichyields performance-guaranteed solutions. Three representativescenarios are investigated to verity the efficiency of our proposal.Numerical results show that our proposed algorithm has greatadvantages over other heuristic ones.

I. INTRODUCTION

As an evolution of conventional power systems, smart gridemploys advanced information and communication technolo-gies to construct an intelligent bi-direction electricity and com-munication network [1]. That is, a smart grid not only deliverselectricity from suppliers to consumers, but also uses a two-way flow of data communications to exchange informationbetween consumers and suppliers to provide various services,such as integrating renewable distributed energy resources,applying demand side management and dealing with increasedenergy trade [2, 3].

Software defined networking (SDN), which decouples thecontrol plane from the data plane, is a promising method thatcan meet the requirements of the smart grid communication[4–9]. SDN can be applied for developing self-configuringintelligent electric devices for the substation communicationnetwork [10]. In [11], multicast transmission of measurementunit data is handled by an SDN enable network to adapt dif-ferent data rates requirements. Investigations in [12] indicatethat SDN can enhance data exchange and distribute resourcesin the smart grid efficiently. A multi-layer SDN for the smartgrid is proposed in [13], where the southbound applicationprogram interface is employed to the communication betweenthe data and the control plane.

In this paper, we introduce a novel SDN-based smart gridcommunication architecture, as shown in Fig.1. Smart meters(SMs) in the data plane gather energy consumption and control

APAP

SM

SM

SM

SM

SM

SMData Plane

Control Plane

Control center

Fig. 1. Architecture of SDN-based smart grid communication.

information from residential and industrial consumers [14].Recall that there are a huge number of SMs in the smartgrid, it is not practical to send these data to the controlcenter by deploying a direct link between each SM and thecontrol center. We suggest that multiple aggregation points(APs) should be installed in the data plane, and each ofthem serves its surrounding SMs to assist the informationexchanging between SMs and the control center. The datagenerated by SMs, such as demand response information, willbe sent to the control center via these APs [15, 16]. Meanwhile,the other way of data flow, such as the control informationfor load management and price broadcasting, can be also sentfrom the control center to SMs via these APs.

As can be seen from Fig.1, an AP receives data from itsserving SMs and send the data to the control center. Sincethe SMs always generate a large amount of data, the APneeds to compress the data before transmitting them for theconsideration of transmission efficiency. The compression ratiousually varies from 1/2 to 2/3, depending on the correlation ofdata [14]. In other words, the APs not only receive and forwardinformation, but also process the raw data originated from theSMs. Thus planing APs is an important task for designingsuch an SDN-based smart grid communication system. TheAPs planning includes the installation of APs, the physicallinks between each AP and its serving SMs, and the linksbetween the APs and the control center.

2

To reduce the capital expenditure of the smart grid com-munication system, APs should be planned in a cost-efficientway. Obviously, the key to achieve this goal is to minimize thenumber of APs while considering the links cost between APsand SMs, as well as the link cost between each AP and thecontrol center. Since power line communication (PLC) alwaysexists in the power systems, it is promising to use the PLCto provide the links between APs and SMs. As for the linkbetween APs and the control center, it could be wired andwireless communication as discussed in [17]. In this paper, weemploy the PLC to provide the communication link betweenAPs and SMs. The link cost will b discussed in detail in thefollowing section. To simplify analysis, the communicationcost between each AP and the control center is a given value.We try to minimize the sum cost of the installation of APs,the links between APs and SMs, and the links between APsand the control center.

First, we give a brief survey of the related works. In[18], a general mathematical model is given to plan theAPs which locate in the neighborhood area network of thesmart grid. However, the channel characteristic of PLC isnot considered in this work. In [19], the placement of dataaggregation services in smart grid communication networkis investigated, where the authors developed a minimum-cost-forwarding-based asynchronous distributed algorithm tofind the optimal placement for the data aggregation servicetree with the minimal cost of in-network processing. Thecontroller placement problem in SDN is discussed in [20],where the objective is to determine the number and thelocations of controllers, as well as the connection cost amongthe controllers and the switches. The relay placement problemin smart grid is studied in [21], where the optimization task isto place the minimum number of relays so that all intelligentdevices in the smart grid can communicate with each othervia these relays under the constraint of the coverage rangeof each relay. A 4.5-approximation algorithm is introduced.The relay placement problem in [21] is different from theAPs planning problem investigated in this paper. The APs inthe data plane need not to communicate with each other. TheAPs in our considered SDN-based sart grid communicationsystem work as the base stations in the cellular systems formobile communications. In [22], an optimal placement of dataaggregation points in advanced metering infrastructures forsmart grid is proposed. The objective is to minimize the totalinstallation and transmission cost. It yields an integer programproblem and the authors proposed K-means algorithm to solveit efficiently.

In this paper, we investigate the APs planning problemin the SDN-based smart grid communication system, whichhas not been well discussed in the literature as far as theauthors have known. We show that the planning task leadsto a facility localization problem [23, 24] and introduce anefficient approximation algorithm to solve it. Three scenarioswith different SM densities are considered to verify ourproposed algorithm. Numerical results show our proposalperforms better than other heuristic methods, such as genetic

algorithm (GA) [25] and Tabu search (TS) [26]. Furthermore,our proposed algorithm is performance-guaranteed, sheddingsome insights on deploying a cost-efficient SDN-based smartgrid communication system.

The rest of this paper is organized as follows. In Section II,we illustrate the PLC channel characteristics and formulate theAPs planning problem in the SDN-based smart grid system.In Section III, we introduce a 5-approximation algorithm toaddress the formulated optimization task. Numerical resultsand discussions are given in Section IV. In Section V, weconclude this paper and point out future researches.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Consider an SDN-based smart grid communication systemwith M SMs, denoted by M = {1,2, . . . ,M}. There are Mc

control centers in the control plane. Here we only consider thecase of Mc = 1 to simplify analysis. Multiple APs are installedin the data plane to serve the SMs. That is, an SM must beconnected with one and only on AP. DenoteN = {1,2, . . . ,N}as a set of candidate sites where the APs can be installed. PLCis available for the communication links between the APs andthe SMs. Each AP is connected to one of the control centers bythe way of wired or wireless communication [17]. The numberof SMs is calculated as follows:

M = ρSMr2,

where r is the radius of the area of interest and ρSM is thedensity of SMs in the area. The value of ρSM is depend onthe area of interest being rural, urban or sub-urban [22]. Tomake the rest of this paper easy to follow, we give a summaryof the notations frequently used in this paper in Table I.

A. PLC Channel Model

PLC uses the existing power line infrastructure to providethe communication links between the APs and the SMs in theSDN-based smart grid [27, 28]. The channel characteristics ofPLC can be subdivided into transfer function Hf and additivenoise Nf . We assume that signal propagation only takes placealong a direct path between one AP and one SM. The lossof power line cable causes an attenuation A(f, disnm) thatincreases with frequency f and distance disnm between APn and SM m. B(f, disnm) is the delay portion which is alsorelated to f and disnm. Therefore, the transfer function Hf

can be expressed as

Hf = g ⋅A(f, disnm) ⋅B(f, disnm),

where g is a weighting factor representing the product of thereflection and transmission factor along the path. It is not morethan one,

g ≤ 1.

The attenuation of power line cable can be characterized by

A(f, disnm) = e(a0+a1f)⋅disnm ,

where a0 and a1 are cable parameters discussed in [29].

3

The delay τ of the path between AP n and SM m

τ = disnmεrv0

,

can be calculated from the dielectric constant εr of theinsulating material, the speed of light v0 and the distancedisnm between AP n and SM m. Therefore, the delay portionB(f, disnm) can be expressed as

B(f, disnm) = e−2πjf ⋅τ .

Thus the transfer function Hf can be expressed as

Hf = g ⋅ e(a0+a1f)⋅disnm ⋅ e−2πjf ⋅τ .

Another channel characteristic is the background noise ofPLC. The power spectral density of the background noise isa decreasing function of frequency. It can be calculated as

Nf = 10K−3.95⋅10−5f ,

where the value of K varies with the transmitter or receiverlocations [30]. Once an AP is selected, its location is alsofixed. Then the PLC channel noise is a constant value for agiven frequency.

B. Mathematical Model

The optimization objective of our APs planning problemis to install the minimum number of APs in the candidatesites N while considering the traffic demands of all the SMsand all the links cost discussed above. Assume that each APn ∈ N requires an installation cost of cAP

n . The connectioncost between an AP and the control center is P . When AP nis selected for opening, the connecting cost between SM mand AP n is cnm, which is related to the distance disnm:

cnm = ϕ(disnm).

We assume the connection cost cnm is linear to the distancedisnm for simplicity. However, tt is intuitive to extend ourresults to other convex cost functions.

Define xn as an index variable that indicates whether AP nis selected for opening or not:

xn = {1 if AP n is selected,0 otherwise.

Let ynm be an index variable to show whether SM m isconnected to AP n or not:

ynm = {1 if SM m is assigned to AP n,0 otherwise.

SM m ∈M requests a traffic demand dm and AP n ∈ Nis equipped with a capacity of wn. The APs installed in thearea of interest should satisfy all the traffic demands of SMsin this area. Meanwhile, the traffic demands of SMs servedby an AP cannot exceed the capacity limitation of this AP.

TABLE INOTATIONS

Symbol SemanticsN Candidate sites set for placing APsM SMs setMc Number of control centerscAPn Installation cost of AP nxn Index variable showing that AP n is selected or notynm Index variable indicating whether SM m is assigned to AP n or notdm Traffic demand of SM m

disnm Distance between SM m and AP nρSM SM densitycnm Link cost between SM m and AP nP Link cost between AP and control centerwn Capacity of AP n

OPT Optimal valuec′n Per unit cost of distance to AP n

Nk Cluster centered around SM kk Center of Nk

C Set of cluster centerF Set of APs that could be selectedFm APs in F that fractionally serve SM mBm Set of unclustered APs

n∗(m) AP in Am nearest to mAm Set of APs not in Bm

Mathematically, our APs planning optimization problem canbe written as follows:

minxn,ynm

∑n∈N

cAPn xn + ∑

n∈N∑

m∈Mcnmynm + ∑

n∈NxnP

s.t. C1 ∶ ∑n∈N

∑m∈M

dmynm ≥ ∑m∈M

dm,

C2 ∶ ∑m∈M

dmynm ≤ wnxn,∀n ∈ N ,

C3 ∶∣Hf ∣≥ Q × ynm,∀n ∈ N ,∀m ∈M,

C4 ∶ xn ≥ ynm,∀n ∈N ,∀m ∈M,

C5 ∶ ∑n∈N

ynm = 1,∀m ∈M,

C6 ∶ xn ∈ {0,1},∀n ∈ N ,

C7 ∶ ynm ∈ {0,1},∀n ∈ N ,m ∈M,

(1)where

Q = Qa ⋅Qb.

Qa and Qb are the attenuation limit and the delay limit of thelinks between APs and SMs, respectively. The first term of theobjective function accounts for the installation cost of APs.The second term is the link costs between SMs and APs, andthe third term is the link costs between APs and the controlcenter. C1 guarantees that the total traffic demands of SMsshould be satisfied. C2 means that the sum traffic demandprovided by an AP cannot exceed its capacity limitation. C3

is the attenuation and relay requirements between APs andSMs. C4 means that an SM should be served by an AP thathas been selected to open. C5 and C7 ensure an SM can beserved by only one AP. C6 is intuitive.

4

III. PROPOSED ALGORITHM

Equation (1) defines an integer programming problem that isNP-hard. It falls into the general facility location problem class[23, 24]. The main difficulty of solving (1) lies in the integerconstraints which usually generate exponential complexity ifthe optimal solution could be worked out. An intuitive wayto tackle such kind of problems is the heuristic methodssuch as TS and GA, which can yield feasible solutionswith reasonable complexity. However, these methods cannotgive a provable gap between the produced solutions and theoptimum of the original problem. In this paper, we introducea 5-approximation algorithm to produce quality guaranteedsolutions. The relaxation form of (1) can be written as follows:

minxn,ynm

∑n∈N

cnxn + ∑n∈N

∑m∈M

cnmynm

s.t. C1 ∶ ∑n∈N

∑m∈M

dmynm ≥ ∑m∈M

dm,

C2 ∶ ∑m∈M

dmynm ≤ wnxn,∀n ∈ N ,

C3 ∶∣Hf ∣≥ Q × ynm,∀n ∈N ,∀m ∈M,

C4 ∶ xn ≥ ynm,∀n ∈ N ,∀m ∈M,

C5 ∶ ∑n∈N

ynm = 1,∀m ∈M,

C6 ∶ 0 ≤ xn ≤ 1,∀n ∈ N ,

C7 ∶ 0 ≤ ynm ≤ 1,∀n ∈N ,m ∈M.

(2)

where

cn = cAPn + P.

Equation (2) is the linear programming relaxation of (1), wherethe integrality constraints C6 and C7 of (1) are relaxed torational values between 0 and 1. The dual form of (2) can bewritten as follows:

max ∑m∈M

αm

s.t. C1 ∶ αm ≤ cnm + βnm + dmγn +Hfρn,

C2 ∶ ∑m∈M

βnm ≤ cn + δn −wnγn −Qρn,

C3 ∶ αm, βnm, γn, δn, ρn ≥ 0.

(3)

αm, βnm, γn, δn, ρn are dual variables. The original problem(2) is convex and is equivalent to the dual problem (3). We cantake αm as the total cost of SM m including the connectingcost and the deployment cost of the AP that serves SM m.That is, αm also includes part of the installation cost of thisAP.

A special case of (2), for which there is only one SM,plays an important role for designing our proposed algorithm.Assume that only one SM k ∈M exists, the special case can

TABLE IIGREEDY ALGORITHM FOR THE SINGLE SM CASE

Algorithm: Greedy Algorithm

1: Initialize gn = vn = 0;2: for n in increasing order of ( cn

wn+ c

′n);

3: while Dk ≠ 04: gn =min(wn,Dk);5: vn = gn

wn;

6: Dk = Dk − gn;7: end while8: end for9: return (g, v)

be described as follows:

minvn,gn

∑ncnvn +∑

nc′

ngn

s.t. C1 ∶ ∑n∈Lk

gn ≥Dk,

C2 ∶ gn ≤ wnvn,∀n ∈ Lk,

C3 ∶∣Hf ∣≥ Qgn,∀n ∈ Lk,

C4 ∶ vn ≤ 1,∀n ∈ Lk,

C5 ∶ vn, gn ≥ 0,∀n ∈ Lk,

(4)

where Lk is the set of APs that fractionally serves SM k. Dk

is the total traffic demand served by these APs, and c′

n is thecost of connecting SM k with AP n; vn indicates if AP n isopen or not; gn is the traffic demand assigned to AP n. Setvn = gn

wn, and we can obtain a feasible solution with no greater

cost. Therefore, we rewrite (4) into the following form:

mingn∑n( cnwn+ c

′

n)gn

s.t. C1 ∶ ∑n∈N

gn ≥Dk,∀n ∈ Lk,

C2 ∶ gn ≤ wn,∀n ∈ Lk,

C3 ∶ gn ≥ 0,∀n ∈ Lk.

(5)

We propose a Greedy algorithm for the single SM problem asshown in Table II, by which an optimal solution to (5) can beobtained. The solution generated the greedy algorithm has thefollowing property: at most one of the vn’s is not zero; if itexists, it falls into the interval 0 ≤ vn ≤ 1.

By decomposing (2) into a collection of single SM case thatcan be solved independently, we can obtain the optimal frac-tional solution of (2). Then we introduce an efficient roundingtechnique to yield a feasible solution to (1) with qualityguarantee. Stack x and y into a vector x = {x1, x2, ..., xN}and a vector y = {y11, y12, ..., yNM}, respectively. Let (x, y)be the optimal solution to (2). Denote F = {n ∶ xn > 0} asthe set of APs in (x, y), and Fm = {n ∶ n ∈ F, ynm > 0}as the set of APs in F that fractionally serve SM m. Ourproposed approximation algorithm can be divided into twostep: clustering and rounding.

5

TABLE IIIPROPOSED ALGORITHM FOR APS PLANNING PROBLEM

Step 1: Clustering

Clustering:2: while S ≠ ∅ do3: for m in increasing order of αm

4: Bm = {n ∈ Fm ∶ n ∉ ⋃k∈C Nk, cnm ≤mink∈C cnk};5: S = S ∖m;6: while Bm ≠ ∅7: k =m;8: Nk = Bm;9: C = C ∪ {k};

10: end while11: end for12: end while13: U = F −⋃k∈C Nk;14: for n ∈ U do15: if k = argmink∈C cnk

16: Nk = Nk ∪ {n};17: end if18: end for

Step 1: Clustering. Partition the APs with xn > 0 intoclusters and each of them can be centered around an SM. Wecall it as the cluster center. Denote the cluster centered aroundSM k as Nk. The cluster Nk consists of SM k, the set of APsassigned to it, and the fractional demands served by these APs,that is, ∑n∈Nk

∑m ynm. At the clustering step, the followingtwo properties are held:

(1) ∑n∈Nkxn ≥ 1

2;

(2) if the APs in Nk serve SM m, the center k is not toofar away from SM m.

Let C be the set of current cluster centers, which is initiallyempty. For each SM m ∉ C, Bm represents the set ofunclustered APs that are closer to it than to any other clusteredcenter, i.e.,

Bm = {m ∈ Fm ∶ n ∈ ⋃k∈C

Nk, cnm ≤mink∈C

cnk}.

To find all cluster centers, m ∈M is ordered in the increasingof αm and the cluster Nk can be formed by Bm. The procedureof clustering is described in Table III.

Step 2: Rounding. In this step, we decide which AP will befully opened in each cluster. For each cluster obtained in thefirst step, it can be tackled as the single SM problem. There-fore, the cluster Nk can be obtained by the greedy algorithmshown in Table II. Then we have Lk = {n ∈ Nk ∶ xn < 1} andDk = ∑n∈Lk

∑m dmynm, where the tuple of (g(k)n , v(k)n ) is an

optimal solution produced by the algorithm in Table II. While0 < v(k)n < 1, all APs are fully opened in Lk, i.e., xn = 1 forn ∈ Lk. Piecing together the solutions for all clusters, xn’s areassigned by {0,1}. Once all APs have been fixed, each SM isassigned to the closest AP around it.

Lemma 4.1 and 4.2 show that the cost cnk is bounded.

TABLE IVTHE PROPOSED ALGORITHM FOR APS PLANNING PROBLEM

Step 2: Rounding

Rounding:19: for each Nk

20: Lk = {n ∈ Nk ∶ xn < 1};21: Dk = ∑n∈Lk

∑m ynm;

22: find (g(k)n , v(k)n ) by greedy algorithm in Table I and

calculate the value O∗k;23: end for24: for each Nk

25: while 0 < vkn < 126: xn = 1;27: end while28: end for

Lemma 4.1. Consider SM m and AP n ∈ Am. If AP n isassigned to the cluster Nk, then cnk ≤ αm.

Proof: If SM m is a cluster center, we can get cnm ≤αm with the analysis of complementary slackness. It can beverified as follows. Denoe C

′as the set of cluster centers when

SM m is removed from S. Since n ∉ Bm, we have

cnm > mink′∈C′

cnk′ .

Let C′′

be the set of cluster centers when AP n is assigned toNk. So k is the cluster center closest to n among all clustercenters in C

′′, that is:

cnk = mink′∈C′′

cnk′ .

Hence,

cnk ≤ αm.

Lemma 4.2. Consider SM m and AP n ∈ Fm ∖ Am. Ifm ∈ C, then cnk ≤ cnm; otherwise, cnk ≤ cnm+cn∗(m)m+αm.

Proof: If SM m is a cluster center, when it is removedfrom S, we can construct the cluster Nk that is equal to thecurrent set Bm, which is also Fm ∖Am. Then we can simplyget cnk ≤ cnm.

Suppose m ∉ C. Let k′∈ C

′be the cluster center if AP

n∗(m) is removed from Bm, we have

cn∗(m)k′ < cn∗(m)m.

Since

αk′ ≤ αm,

we also have

cn∗(m)k′ ≤ αm.

6

Using Lemma 4.1., we have

cnk =mink′∈C′′ cnk′′

≤ cnm + cn∗(m)m + cn∗(m)k′

≤ cnm + cn∗(m)m + αm.

(6)

Let k(n) ∈ C denote the cluster where AP n is assigned, wecan obtain that n ∈ Nk(n). (g

(k)n , v

(k)n ) construct the optimal

solution to (9) and the value is O∗k . Define R1 = {n ∶ xn < 1}and R2 = {n ∶ xn = 1}, we can obtain the following Lemma4.3 which shows that the optimal solution to the single SMcase problem is comparable to the OPT .

Lemma 4.3. The optimal value O∗k ≤∑n∈Lk

cnvn + ∑m∑n∈Lkcnkynm, and ∑k∈C O∗k ≤

∑R1cnvn +∑m∑R1

cnkynm.

Proof: Setvn = xn,

andgn =∑

m

dmynm,

for all n ∈ Lk. The AP cost is at most

∑n∈Lk

cnvn = ∑n∈Lk

cnvn.

The second bound follows from the first since the clusters Nk

are disjoint. Then we can prove the lemma.Let x be the 0−1 vector indicating which AP is open. That

is, xn = 1 if n is open, and 0 otherwise. Let x(k) be the portionof x consisting of the APs in Lk, we can show that the costof the solution is bounded by aggregating the bounds obtainedfor each partial solution.

Lemma 4.4. ∑n∈Lkcnx

(k)n +∑m∑n∈Lk

cnmy(k)nm is at most

O∗k+2∑n∈Nkcnxn+∑m∑n∈Lk

cnmynm+∑m∑n∈Lkcnkynm.

Proof: First, it is intuitive that∑n∈Nkxn ≥ ∑n∈Nk

ynk ≥ 12

since Nk is generated with this property in the first step. Sinceall APs have the same cost, the cost of the APs in Nk is

c ≤ c ⋅ 2 ∑n∈Nk

xn = 2 ∑n∈Nk

cnxn.

Therefore, the AP cost ∑n∈Lkcnx

(k)n is at most

∑n∈Lk

cnv(k)n + 2 ∑

n∈Nk

cnxn.

The service cost ∑m∑n∈Lkcnmy

(k)nm can be

∑m∑

n∈Lk

cnky(k)nm +∑

m∑

n∈Lk

cmky(k)nm.

So the service cost is

∑m∑

n∈Lk

cnmynm ≤ ∑n∈Lk

cny(k)n +∑

m∑

n∈Lk

(cnm + cnk)ynm.

(7)

Recall that O∗k = ∑n∈Lk(cnv(k)n +cng(k)n ), the lemma is proved.

Next, we prove a bound of the opening APs cost and therelated connecting cost.

Lemma 4.5. The sum of opening APs cost and connectingcost is at most ∑m∑R2

αmynm −∑n δn.

Proof: For AP n that δn > 0, we have xn = 1. ConsiderEquation (5-7), we have,

∑mαmynm = ∑

mcnmynm +∑

mβnmynm +∑

mγnynm

= ∑mcnmynm +∑

mβnmxn +wnγnxn

= ∑mcnmynm + cn + δn.

(8)

∑m βnm +wnγn = cn + δn. So Lemma 4.5 can be proved bysum all n with xn = 1.

Finally, we can prove our main results that is shown in thefollowing Lemma 4.6 using these bounds.

Lemma 4.6. The total cost of the solution returned byAlgorithm 2 is at most 5 ⋅OPT .

Proof: The proof is referred to Appendix A.

IV. NUMERICAL RESULTS AND DISCUSSIONS

As can be observed from Lemma 4.6, our proposed algo-rithm is approximation algorithm that can provide the worstperformance guarantee for the APs planning problem in theSDN-based smart gird. In this section, we verify the efficiencyof the algorithm in three representative scenarios: urban, sub-urban and rural. We compare the performance of the proposedalgorithm with other heuristic methods that can be employedto address NP-hard problems with reasonable complexity: GAand TS. We will show that our proposal outperforms othersfor all considered scenarios. The parameters of the GA andthe TS employed in this work is summarized in Table V.

Consider a certain area served by an SDN-based smartgrid communication system, where all the SMs and thecandidate sites to install the APs are uniformly distributedwithin an area of 1 by 1 kilometer for urban, suburban andrural environments. The capacity wn of AP n is distributeduniformly within (600,900). The traffic demand dm of SMm is distributed uniformly within (20,30). The number ofSMs, the control center and the candidate sites to installAPs are summarized as Table VI for different scenarios. Thecommunication technology in data plane is PLC and thefrequency of PLC channel is 15M. The weighting factor gis 1. The attenuation limit Qa and delay limit Qb of thelinks between APs and SMs are 15 and 20, respectively.a0 = 6.5 ⋅10−3 and a1 = 2.46 ⋅10−9, which have been proposedin [29]. Without loss of generality, we assume the installationcost of each AP is the same.

Numerical results yielded by our proposed algorithm aregiven in Fig.2-4, where the black circles represent the can-didate sites for installing APs and the red squares are thelocations of SMs. Fig.2, Fig.3 and Fig.4 correspond to urban,

7

TABLE VPARAMETERS OF GA AND TS

Parameters of GA Parameters of TS

Population size 100 Tabu size (active) 3

Crossover probability 0.7 Tabu size (candidate) 7

Mutation probability 0.3 Iterations 300

TABLE VIPARAMETERS OF THE SYSTEM IN DIFFERENT SCENARIOS

Scenario ρSM

(perkm2)N Mc SMs per AP

Urban 1000 500 1 34

Suburban 500 500 1 22

Rural 10 500 1 2

suburban and rural areas, respectively. Table VI shows theaverage number of SMs served by an AP in different scenarios.In Fig.2-4, the sub-figure on the left shows the distribution ofSMs and the candidate sites for deploying APs. And the rightone shows the selected APs by our proposed algorithm, aswell as the links between the SMs and the AP serving them.

Since the SMs are sparse in the rural area, we can observefrom Fig.2 that there are on average much fewer SMs servedby an AP as compared to the suburban and urban scenariosshown in Fig.3-4, where on average 22 and 34 SMs areconnected to one AP. Moreover, from Fig.2-4 we can see thatour proposed algorithm can produce load balanced planningresults for all considered cased. In other words, our proposedalgorithm can fully exploit the capacity potential of each AP,leading to the minimal number of required APs for servinggiven number of SMs. So we can conclude conservatively thatour proposal is cost-efficient.

To verify the promising performance of our proposed algo-rithm, we also compare it with other representative heuristic

Fig. 5. Average deployment cost as the number of SMs M changes. N = 500,cAPn = 9.3,∀n ∈ N , P = 10, cnm = 0.001disnm,∀n ∈ N ,m ∈M.

methods that can tackle our formulated problem. Fig.5 showsthe average costs as a function of the number of SMs, whichare produced by our proposed algorithm, the GA and the TSdiscussed above. The cost of AP n is set to cAP

n = 9.3 andthe connection cost between AP n and the control center isset to P = 5. The connection cost between AP n and SM mfor PLC is cnm = 0.001disnm, where disnm is the distancebetween the SM and the serving AP. As can be seen fromthe Fig.5, our proposed algorithm outperforms the GA andthe TS remarkably. The cost of our proposal is about 20%lower than that of the TS. Obviously, when the number of SMsincreases, the total deployment cost is also increases. Recallthat our proposed algorithm has the worst case performanceguarantee which is difficult even impossible for other heuristicmethods to achieve such a performance. It is promising forapplications in the SDN-based smart grid communicationswhere scalability and reliability are essential prerequisites.

V. CONCLUSION

In this paper, we studied the SDN-based smart grid com-munications, where the data plane and the control controlare decoupled to provide flexible two-way flow of data com-munications for the smart grid. The APs planning in dataplane plays an important role for such an SDN-based smartgrid communication network. We formulated a general APsplacement problem and introduce a 5-approximation algorithmto address it. Our proposed algorithm can yield solutionswith the worst case performance guarantee. Moreover, itoutperforms other representative heuristic methods for allconsidered scenarios as verified by numerical results. In futurework, hybrid networking model should be investigated for thecommunication links between APs and SMs, e.g., both wiredand wireless technology can be selected for these links inpractical environment, depending on the density of SMs, trafficrequirements of SMs, as well as the installation and connectioncost between APs and SMs.

APPENDIX APROOF OF THEOREM 4.6

To bound the total cost, it suffices to give a fractionalassignment znm such that x, y are feasible solutions to (2)and have the cost at most 5 ⋅OPT . Set ynm = ynm for everyAP n with xn = 1 = xn. Then

∑m∑R2

cnxn +∑m∑R2

cnmynm =∑m∑R2

αmynm −∑n

δn. (9)

(x, y) satisfies ∑m ynm = ∑R1ynm + ∑R2

znm. Since theclusters Nk are disjoint, by Lemma 4.5 we have,

∑R1

cnxn +∑m∑R1

cnmynm

≤ ∑k∈C

O∗k + 2∑m∑R1

cnmynm +∑m∑R1

cnmynm

≤ 3∑ncnxn +∑

m∑R1

cnmynm + 2∑m∑R1

cnk(n)ynm.

(10)

For any SM m and AP n ∈ Fm, if n ∈ Am, then we havecnk(n) ≤ αm; otherwise, cnk ≤ cnm + αm for m ∈ C, and

8

(a) Rural area (b) Selected APs and the connected SMs in rural area

Fig. 2. APs placement for rural area

(a) Suburban area (b) Selected APs and the connected SMs in suburban area

Fig. 3. APs placement for suburban area

(a) Urban area (b) Selected APs and the connected SMs in urban area

Fig. 4. APs placement for urban area

9

cnk ≤ cnm + cn∗(m)m + αm for m ∉ C. Plugging this in theabove expression,

∑R1

cnxn +∑m∑R1

cnmynm

≤ 3∑ncnxn +∑

m∑R1

cnmynm + 2∑m∑R1

αmynm

+ 2∑m∑

n∶xn<1cnmynm + ∑

m∉C2cn∗(m)m ∑

R1

ynm.

(11)

For m ∉ C, denote n ∈ Am as R3,

cn∗(m)m =minR3

cnm ≤∑R3

cnmynm

∑R3

ynm< 2∑

R3

cnmynm. (12)

This implies that

∑R1

cnxn +∑m∑R1

cnmynm

≤ 3∑ncnxn +∑

m∑R2

cnmynm + 2∑m∑R2

αmynm

+ 2∑m∑R1

cnmynm + 2 ∑m∉C∑R3

cnmynm

≤ 2∑m∑R1

αmynm + 3(∑ncnxn +∑

m∑ncnmynm).

(13)

Combine (12) and (13), we obtain that

Ctotal ≤ (∑m∑R2

αmynm −∑nδn) + 2∑

m∑R2

αmynm

+3(∑ncnxn +∑

m∑ncnmynm)

≤ 2(∑m∑R2

αmynm −∑nδn

+∑m∑R1

αmynm) + 3 ⋅OPT1

= 5 ⋅OPT1.

(14)

ACKNOWLEDGEMENT

This work was partially supported by JiangsuSF(BK20151389), the Fundamental Research Funds forthe Central Universities (021014380013) and the openresearch fund of National Mobile Communications ResearchLaboratory, Southeast University (2016D08).

REFERENCES

[1] X. Fang, S. Misra, G. Xue, and D. Yang, “Smart grid - the new andimproved power grid: A survey,” IEEE Commun. Surv. Tut., vol. 14,no. 4, pp. 944–980, Dec. 2012.

[2] Z. Fan, P. Kulkarni, S. Gormus, and Efthymiou, “Smart grid communi-cations: Overview of research challenges, solutions, and standardizationactivities,” IEEE Commun. Surv. Tut., vol. 15, no. 1, pp. 21–38, Jan.2013.

[3] M. H. U. Ahmed, M. G. R. Alam, R. Kamal, C. S. Hong, and S. Lee,“Smart grid cooperative communication with smart relay,” IEEE Comm.and Netw., vol. 14, no. 6, pp. 640–652, Dec. 2012.

[4] A. Gelberger, N. Yemini, and R. Giladi, “Performance analysis ofsoftware-defined networking (SDN),” in Proc. IEEE MASCOTS’13, Aug.2013, pp. 389–393.

[5] T.-C. Yen and C.-S. Su, “An SDN-based cloud computing architectureand its mathematical model,” in Proc. ISEEE’14, vol. 3, Apr. 2014, pp.1728–1731.

[6] P. Xiao, W. Qu, H. Qi, Z. Li, and Y. Xu, “The SDN controller placementproblem for WAN,” in Proc. IEEE ICC’14, Oct. 2014, pp. 220–224.

[7] M. Obadia, M. Bouet, J.-L. Rougier, and L. Iannone, “A greedy approachfor minimizing SDN control overhead,” in Proc. IEEE NetSoft’15, Apr.2015, pp. 1–5.

[8] Y. Jimenez, C. Cervello-Pastor, and A. Garcia, “On the controllerplacement for designing a distributed SDN control layer,” in Proc. IFIPNetworking Conference’14, June 2014, pp. 1–9.

[9] H. Farhad, H. Lee, and A. Nakao, “Data plane programmability in SDN,”in Proc. IEEE ICNP’14, Oct. 2014, pp. 583–588.

[10] A. Cahn, J. Hoyos, M. Hulse, and E. Keller, “Software-defined energycommunication networks: From substation automation to future smartgrids,” in Proc. IEEE SmartGridComm’13, Oct. 2013, pp. 558–563.

[11] A. Goodney, S. Kumar, A. Ravi, and Y. H. Cho, “Efficient pmu network-ing with software defined networks,” in Proc. IEEE SmartGridComm’13,Oct. 2013, pp. 378–383.

[12] J. Zhang, B.-C. Seet, T.-T. Lie, and C. H. Foh, “Opportunities forsoftware-defined networking in smart grid,” in Proc. ICICS’2013, Dec.2013, pp. 1–5.

[13] N. Dorsch, F. Kurtz, H. Georg, C. Hagerling, and C. Wietfeld, “Software-defined networking for smart grid communications: Applications, chal-lenges and advantages,” in Proc. IEEE SmartGridComm’14, Nov. 2014,pp. 422–427.

[14] M. Ringwelski, C. Renner, A. Reinhardt, A. Weigel, and V. Turau, “Thehitchhiker’s guide to choosing the compression algorithm for your smartmeter data,” in Proc. IEEE ENERGYCON’12, Sep. 2012, pp. 935–940.

[15] R. Yu, Y. Zhang, and Y. Chen, “Hybrid spectrum access in cognitiveneighborhood area networks in the smart grid,” in Proc. IEEE WCNC’12,Apr. 2012, pp. 1478–1483.

[16] D. Niyato, L. Xiao, and P. Wang, “Machine-to-machine communicationsfor home energy management system in smart grid,” IEEE Commun.Mag., vol. 49, no. 4, pp. 53–59, Apr. 2011.

[17] Z. Fadlullah, M. Fouda, N. Kato, A. Takeuchi, N. Iwasaki, and Y. Noza-ki, “Toward intelligent machine-to-machine communications in smartgrid,” IEEE Commun. Mag., vol. 49, no. 4, pp. 60–65, Apr. 2011.

[18] X. Huang and S. Wang, “Aggregation points planning in smart gridcommunication system,” IEEE Commun. Lett., vol. 19, no. 8, pp. 1315–1318, Aug. 2015.

[19] Z. Lu and Y. Wen, “Distributed algorithm for tree-structured dataaggregation service placement in smart grid,” IEEE Syst. J., vol. 8, no. 2,pp. 553–561, June 2014.

[20] S. Lange, S. Gebert, T. Zinner, P. Tran-Gia, D. Hock, M. Jarschel, andM. Hoffmann, “Optimal model for the controller placement problem insoftware defined networks,” IEEE Trans. Net. Serv. Manag., vol. 12,no. 1, pp. 4–17, Mar. 2015.

[21] W.-L. Wang and Q. Wu, “Relay placement problem in smart griddeployment,” in Proc. LNCS ISoLA’10, Oct. 2010, pp. 411–424.

[22] F. Aalamifar, G. Shirazi, M. Noori, and L. Lampe, “Cost-efficient dataaggregation point placement for advanced metering infrastructure,” inProc. IEEE SmartGridComm’14, Nov. 2014, pp. 344–349.

[23] K. Jain and V. V. Vazirani, “Primal-dual approximation algorithmsfor metric facility location and k-median problems,” in Proc. IEEEFOCS’99, pp. 2–13.

[24] R. Levi, D. B. Shmoys, and C. Swamy, “LP-based approximationalgorithms for capacitated facility location,” Math. Program., vol. 131,no. 1-2, pp. 365–379, Feb. 2012.

[25] N. Weicker, G. Szabo, K. Weicker, and P. Widmayer, “Evolutionarymultiobjective optimization for base station transmitter placement withfrequency assignment,” IEEE Trans. Evol. Comput., vol. 7, no. 2, pp.189–203, Apr. 2003.

[26] C. Lee and H. Kang, “Cell planning with capacity expansion in mobilecommunications: a tabu search approach,” IEEE Trans. Veh. Technol.,vol. 49, no. 5, pp. 1678–1691, Sep. 2000.

[27] I. Stievano, F. Canavero, W. Garcia Valverde, L. Guerrieri, and P. Bis-aglia, “Multipath modeling of automotive power line communicationchannels,” IEEE Trans. Indus. Inform., vol. 10, no. 2, pp. 1381–1391,May 2014.

[28] G. Bumiller, L. Lampe, and H. Hrasnica, “Power line communicationnetworks for large-scale control and automation systems,” IEEE Com-mun. Mag., vol. 48, no. 4, pp. 106–113, Apr. 2010.

[29] M. Zimmermann and K. Dostert, “A multipath model for the powerlinechannel,” IEEE Trans. on Commun., vol. 50, no. 4, pp. 553–559, Apr.2002.

[30] O. Hooijen, “A channel model for the residential power circuit usedas a digital communications medium,” IEEE Trans. Electrom. Comp.,vol. 40, no. 4, pp. 331–336, Nov. 1998.