Source Rate Maximization for Smart Meter Mesh Networks ...yhe/my...

Source Rate Maximization for Smart Meter Mesh Networks Using Distributed Algorithm

Yifeng He and Ling Guan

Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada

Abstract

The two-way communications between the electricity customers and the utility company are important for smart grid. A higher source rate from the smart meters can lead to a quicker decision making at the utility company. However, the maximum source rate that can be supported in a smart meter mesh network depends on the location of the collector and the routing scheme. In this paper, we investigate the source rate maximization problem for a smart meter mesh network. We first optimize the placement of the collector by minimizing the mean squared distance between the collector and the smart meters. We next formulate the source rate maximization problem into a linear programming (LP). In order to solve the optimization problem in a distributed manner, we then convert the LP problem to a strictly convex optimization problem, which is solved with a distributed algorithm. In the simulations, we demonstrate that the proposed distributed algorithm can achieve the maximum source rate in a smart meter mesh network.

Keywords: smart grid, smart meter, wireless sensor networks, convex optimization

1. Introduction

The electric power systems throughout the world are undergoing a profound paradigm shift toward smart grid (SG). Smart grid is a term referring to next generation utility networks in which the electric power distribution and management is upgraded by incorporating advanced information and communications technologies for improved control, efficiency, reliability, and safety [1].

The two-way communications between the electricity customers and the utility company are important for smart grid. The two-way communications enable many new options and services such as automated meter data collection, outage management, dynamic rate structures, and demand response for load control [2]. The two-way communications are established on top of the smart grid communications infrastructure. Due to a large geographical coverage of the electric system, the smart grid communications infrastructure is typically a multitier network [3][4], which consists of: 1) the home area network (HAN), which provides access to in-home appliances, 2) the neighborhood area network (NAN), which connects smart meters to local collectors, and 3) the wide area network (WAN), which provides communications between collectors and the utility company. A smart grid communications network is illustrated in Figure 1.

Figure 1. Multi-tier network architecture for smart grid communications

This paper focuses on the investigation of NAN. The major function of the NAN is to gather the

smart meter readings from electricity customers to the collectors. Smart meters are typically equipped

Source Rate Maximization for Smart Meter Mesh Networks Using Distributed Algorithm Yifeng He, Ling Guan

Journal of Communications and Information Sciences (JCIS) Volume3, Number4, October 2013

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with IEEE 802.15.4 radios [5], which enable them to communicate with each other. The NAN can be formed in a tree topology or a mesh topology [2]. In a tree based NAN, the data is aggregated from the leaf nodes to the upper-level nodes, until reaching the root (e.g., the collector). In a mesh-based NAN, the smart meters establish ad hoc communications with their neighboring nodes, and forward the data toward the collector. One of the major drawbacks for the tree-based NAN is its poor resilience to node failures. A failed node in the tree-based NAN isolates its descendants from reaching the collector. The node in the mesh-based NAN may have multiple connections with its neighbors, thus more robust to node failures. Therefore we will investigate mesh based NANs in this paper.

In a smart meter mesh network, each node generates the data at its source rate. The utility company typically applies the same sampling rate to all the electricity customers in collecting the smart meter readings. Therefore, it is realistic to assume that all the smart meters in the NAN have the same source rate . The source rate can be changed by varying the sampling rate. For example, the source rate can be reduced by half if the meter readings are sampled from once per minute to once per two minutes. A higher source rate indicates a faster data acquisition, thus enabling the utility company to make a quicker decision. However, the maximum source rate that can be supported in a smart meter mesh network depends on the network topology, the routing scheme, and the allocation of resources (e.g., power and bandwidth). In a smart meter mesh network, the location of the collector has an impact on the maximum source rate. In addition, a better routing scheme can support a higher source rate. However, it is challenging to find the optimal routing scheme for a smart meter mesh network.

The network protocols, network management and security issues for smart meter networks were presented in [2]. Optimal resource allocations for multi-hop wireless networks were reported in [6-12]. Joint optimization of the source rate and the routing scheme for a single video session over a wireless ad hoc network was presented in [6]. Joint optimization of the source rate, the routing scheme, and the power allocation for video multicasting over a wireless ad hoc network was presented in [7]. In this paper, we investigate the source rate maximization problem for multiple unicast sessions, which is different from a single unicast session [6] or a multicast session [7]. Joint optimization problem for multiple unicast sessions in wireless visual sensor networks was studied in [8], in which the objective is to seek an appropriate trade-off between maximum network lifetime and minimum video distortion. Our work in this paper has a different objective than that in [8]. In addition, the smart meter networks are powered by electricity, which are different from the battery-powered wireless sensor networks, in which the network lifetime is a major concern [9-12].

In this paper, we investigate the source rate maximization problem for a smart meter mesh network. We first optimize the placement of the collector by minimizing the mean squared distance between the collector and the smart meters. Given the network topology formed by the smart meters and the collector, we next formulate the source rate maximization problem, which maximizes the source rate under the link capacity constraints. Furthermore, we propose a distributed algorithm using dual decomposition techniques to solve the optimization problem.

The remainder of the paper is organized as follows. In Section 2, we optimize the placement of the collector by minimizing the mean squared distance between the collector and the smart meters. In Section 3, we propose a distributed algorithm to maximize the source rate by jointly optimizing the source rate and the routing scheme. The simulation results are presented in Section 4, and the conclusions are drawn in Section 5. 2. Placement of the Collector

In a smart meter mesh network, the locations of the smart meters are fixed. Based on the locations of the smart meters, we can optimize the location of the collector. In this paper, we assume that there is only one collector in the mesh network. However, the proposed algorithms can be easily extended to solve the source rate maximization problem for the mesh networks with multiple collectors. We denote the set of the smart meter nodes by V. The coordinate of node (∀ ∈ V) is denoted by ( , ). With the same resources (e.g., bandwidth and power), a smart meter node has a higher link capacity between the node and the collector if it is closer to the collector. Therefore, we determine the location of the collector by minimizing the mean


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squared distance between the collector and the smart meters. Mathematically, the optimization problem for collector placement is formulated as

minimize(,) 1|| (( −) +( −))∈

subjectto ≥ 0, ≥ 0, where (, ) is the coordinates of the collector, and |V| denotes the number of smart meters in the set V. The above optimization problem is a quadratic programming (QP). The objective function of the optimization problem is (, ) = || ∑ (( −) +( −))∈ . From the optimization problem

(1), we have (,) = || ∑ ( −)∈ = 0 and (,) = || ∑ ( −)∈ = 0, from which we can

find the optimal solution, given by ∗ = || ∑ ∈ and ∗ = || ∑ ∈ . 3. Source Rate Maximization Problem

In this section, we will first formulate the source rate maximization problem, and then solve the optimization problem with a distributed algorithm.

3.1. Problem Formulation

After determining the location of the collector, we can represent the network topology using

a network model. A smart meter mesh network can be modeled as a directed graph G = (N,L), where N is the set of nodes and L is the set of directed wireless links. Among the node set N, one node is the collector, denoted as t, while the other nodes belong to source-node set V. Thus, we have N = V ∪ {t}. Two nodes, nodes i and j, are connected by a link if they can directly communicate with each other.

The relationship between a node and its connected links is represented with a node-link incidence matrix A, whose elements are given by

= 1,iflinkisanoutgoinglinkfromnode,−1,iflinkisanincominglinkfromnode,0,otherwise.

All the source nodes will generate data traffic with a source rate R. We define session h as the traffic flow originating from the source node h (∀h ∈ V) to the collector. For each session, the flow conservation law holds at each node:

=∈ ,∀ℎ ∈ , ∀ ∈ , where is the link rate at link lfor session h, and is defined as = ,ifisthesourcenodeofsessionℎ,−,ifisthecollectorofsessionℎ,0,otherwise.

Suppose that Time Division Multiple Access (TDMA) is applied at the Medium Access

Control (MAC) layer to resolve the link interferences. In addition, we assume that the transmission power at each link, denoted by P, is constant, and the fraction of time allocated to a link, denoted by τ, is also constant. The number of the links in the mesh network is denoted by ∣L∣. Therefore we can set τ= 1/∣L∣. The link capacity at link is modeled by

= log 1 + ,∀ ∈

(1)

(2)

(3)

(4)

(5)


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where is the channel bandwidth, ℎ is the channel gain at link , and is the noise power spectral density.

The aggregate link rate at link is given by = ∑ ∈ . The aggregate link rate at each link should be no larger than the link capacity. This constraint is expressed by ∑ ≤∈ , ∀ ∈ .

The source rate maximization problem can be described as: given the topology of a smart meter mesh network, to maximize the common source rate by jointly optimizing the source rate and the routing scheme, subject to the link capacity constraints. Mathematically, the problem is formulated as follows.

maximize(,) subjectto ∑ =∈ ,∀ℎ ∈ , ∀ ∈ , ∑ ≤ ∈ ,∀ ∈ , ≥ 0,∀ℎ ∈ , ∀ ∈ , ≥ 0.

In the optimization problem (6), the optimization variables are the common source rate and the link rate matrix f. The constraints, ∑ =∈ , ∀ℎ ∈ , ∀ ∈ , represent the flow conservation law at each node for each session. The constraints, ∑ ≤ ∈ , ∀ ∈ , represent the link capacity constraint at each link. The routing scheme is actually determined by the link rates for each session. Optimizing the routing scheme is equivalent to optimizing the link rates for each session.

The problem in (6) is a LP. It can be solved in a centralized way using the simplex method or the interior point method [13]. However, imposing the entire computational tasks on a central node is inefficient and unreliable in smart meter mesh networks. In the following subsection, we will develop a distributed algorithm to solve the optimization problem.

3.2. Distributed Algorithm

We will use dual decomposition techniques [14] to develop a distributed algorithm for the

optimization problem. The objective function in the optimization problem (6) is not strictly convex with respect to variables R and f. Therefore, the corresponding dual function is non-differentiable, and the optimal values of R and f are not immediately available. We change the objective from maximizing R to minimizing (−) and add a quadratic regularization term for the source rate and each link rate, respectively. The optimization problem (6) is then approximated to the following:

minimize(,) − + + ∑ ∑ ∈∈ subjectto ∑ =∈ ,∀ℎ ∈ , ∀ ∈ , ∑ ≤ ∈ ,∀ ∈ , ≥ 0,∀ℎ ∈ , ∀ ∈ , ≥ 0.

where δ (δ > 0) is the regularization factor. When the regularization factor δ is close to 0, the objective value in the optimization problem (7) will be close to that in the optimization problem (6).

In the problem (7), the objective function is strictly convex, the inequality constraint function is convex, and the equality constraint function is linear. Therefore, it is a convex optimization problem [15]. In addition, there exists a strictly feasible solution that satisfies the inequality constraint in the optimization problem (7). In other words, the Slater’s condition is satisfied, and the strong duality holds [15]. Thus, we can obtain the optimal solution indirectly by first solving the corresponding dual problem [15]. The dual-based approach leads to an efficient distributed algorithm [14].

We introduce dual variables (∀ℎ ∈ , ∀ ∈ ) and (∀ ∈ ) to formulate the Lagrangian corresponding to the primal problem (7) as below:

(6)

(7)


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(, , , ) = − + + ∑ ∑ ∈∈ +∑ ∑ (∑ −∈ )∈∈ +∑ (∑ − ∈ )∈ = − − ∑ ∑ ∈∈ +∑ ∑ + (∑ ∈ + )∈∈ −∑ ∈

The Lagrange dual function (, ) is the minimum value of the Lagrangian over primal

variables (R, f), and it is given by (, ) = min(,){(, , , )}.

The Lagrange dual problem corresponding to the primal problem (7) is then given by maximize(,) (, ) subjectto ≥ 0,∀ ∈ .

The objective function in the Lagrange dual problem is a concave and differentiable function.

Therefore we can use subgradient method [16] to find the maximum of the objective function. If the step size ()(() > 0) at the iteration follows a non-summable diminishing rule: lim→ () = 0, ()

= ∞, the subgradient method is guaranteed to converge to the optimal value [16].

With the subgradient method, the dual variable at the ( + 1)iteration is updated by () =() − () () − ()∈ ,∀ℎ ∈ , ∀ ∈ , () = max 0, () − () () − ()∈ ,∀ ∈ .

The step size that we use in our algorithm is: () = (/), where ω is a positive real number and q is a positive integer. It follows non-summable diminishing rule.

Given the dual variables at the iteration, we can calculate the primal variables R and f. The source rate R is given by

() = arg min() − − ()∈∈ . The link rate at link l for session h is given by () = arg min() + ()∈ + (),∀ℎ ∈ , ∀ ∈ .

The proposed algorithm is performed in a distributed manner. First, the collector is

responsible for updating the source rate using the dual variables of the source nodes. Second, each source node is responsible for updating the link rate of each of its outgoing links using the dual variable of the outgoing link and the dual variable of the node and the corresponding downstream node.

(8)

(9)

(10)

(10)

(12)

(13)

(11)

(14)

(15)


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4. Simulations

We simulate a smart meter mesh network in a square region of 150m-by-150m. The smart meters are randomly placed in the square region, and the location of the collector is determined by minimizing the mean squared distance between the collector and the smart meters. Each node has a maximum transmission range of 50 m. The bandwidth of the wireless channel is W= 1MHz, the noise power spectral density is =10 /. The transmission power at each link is set to 500 mW. The channel gain at link l is given by ℎ =10/, where is the distance from the transmitting node to the receiving node of the link. In the distributed optimization, we set the regularization factor δ = 0.1. In the default setting, the smart meter mesh network consists of 30 source nodes and 1 collector.

(a) (b)

Figure 2. Iterations of the optimization variables: (a) the source rate, and (b) the aggregate link rates Figure 2 shows the iterations of the optimization variables in a smart meter mesh network

with 30 source nodes and 1 collector. The iteration of the source rate is shown in Figure 2(a). The source rate is converged to 45.896 Kbps within 1000 iterations. The maximum source rate obtained from the proposed distributed algorithm is almost the same to that obtained from the centralized algorithm. By solving the optimization problem (7), we obtain the link rate at link l for session h. The aggregate link rate at link l is given by =∑ ∈ . The iterations of the aggregate link rates are shown in Figure 2(b). All the aggregate link rates converge to the optimal values within 1000 iterations.

(a) (b)

Figure 3. Illustration of the routing flows: (a) for a single session, and (b) for all sessions The computed routing flows are shown in Figure 3. The thickness of an edge is proportional

to the amount of link rate at the corresponding wireless link. Figure 3(a) illustrates the routing flows for the session from node 25 to the collector. Figure 3(b) illustrates the routing flows for

0 200 400 600 800 10000

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40

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Iteration No.

Sou

rce

rate

[Kbp

s]

0 200 400 600 800 10000

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Iteration No.

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rega

te li

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te [K

bps]

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X [m]

Y [m

]

Collector

Node 25

0 50 100 1500

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X [m]

Y [m

]

Collector


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all sessions. The proposed algorithm disperses the traffic into multiple paths to fully utilize the link bandwidth, thus maximizing the source rate. The nodes close to the collector have a heavier burden in forwarding the traffic than the nodes far away from the collector.

Figure 4. Comparison of maximum source rate with different location of the collector

The location of the collector has a great impact to maximum source rate in a smart meter

mesh network. The proposed method places the collector at the location which minimizes the mean squared distance between the collector and the smart meters. The comparison of maximum source rate with different location of the collector is shown in Figure 4, from which we can see that the proposed location (74.4, 71.0) achieves a higher maximum source rate compared to the other random locations.

Figure 5. Comparison of maximum source rate with different number of nodes

We vary the number of the nodes in the square region of 150m-by-150m and compare

maximum source rate in Figure 5. We keep the channel bandwidth and the transmission power unchanged. When the number of the nodes in the same region is increased, the average link capacity is reduced, which causes a reduction of maximum source rate, as shown in Figure 5.

5. Conclusions

In this paper, we investigate the source rate maximization problem for smart meter mesh networks. We first determine the location of the collector by minimizing the mean squared distance between the collector and the smart meters. Based on the network topology formed by the smart meters and the collector, we next formulate the source rate maximization problem into

1 2 3 40

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Location of collector (x,y) [m]

Max

imum

sou

rce

rate

[Kbp

s]

(112.0, 54.0) (49.0, 113.0) (96.0, 84.0) (74.4, 71.0)

Proposed

21 31 41 510

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Number of nodes

Max

imum

sou

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[Kbp

s]


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a LP problem, which maximizes the source rate under the link capacity constraints. In order to develop a distributed solution, we convert the LP problem to a strictly convex optimization problem. We then develop a distributed algorithm using dual decomposition techniques to solve the strictly convex optimization problem. In the simulations, we demonstrate that the maximum source rate can be obtained by the proposed distributed algorithm. 6. References

[1] F. Li, W. Qiao, H. Sun, H. Wan, J. Wang, Y. Xia, Z. Xu, and P. Zhang, “Smart Transmission Grid:

Vision and Framework,” IEEE Transactions on Smart Grid, vol. 1, no. 2, pp. 168-177, 2010. [2] C. Bennett and D. Highfill, “Networking AMI Smart Meters,” in Proc. of IEEE Energy 2030

Conference, pp. 1-8, Sep. 2008. [3] A. Ghassemi, S. Bavarian, and L. Lampe, “Cognitive Radio for Smart Grid Communications,” in

Proc. of IEEE International Conference on Smart Grid Communications (SmartGridComm), pp. 297-302, Oct. 2010.

[4] P.P. Parikh, M.G. Kanabar, and T.S. Sidhu, “Opportunities and challenges of wireless communication technologies for smart grid applications,” in Proc. of IEEE Power and Energy Society General Meeting, Jul. 2010.

[5] IEEE Standard 802.15.4, “Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LR-WPANs),” Sep. 2006.

[6] Y. He, I. Lee, and L. Guan, “Optimized multi-path routing using dual decomposition for wireless video streaming,” in Proc. of IEEE International Symposium on Circuits and Systems (ISCAS), pp. 977-980, May 2007.

[7] Y. He, I. Lee, and L. Guan, “Dptimized video multicasting over wireless ad hoc networks using distributed algorithm,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 19, no. 6, 796-807, Jun. 2009.

[8] J. Zou, H. Xiong, C. Li, R. Zhang, and Z. He, “Lifetime and Distortion Optimization With Joint Source/Channel Rate Adaptation and Network Coding-Based Error Control in Wireless Video Sensor Networks,” IEEE Transactions on Vehicular Technology, vol. 60, no. 3, pp. 1182-1194, Mar. 2011.

[9] R. Madan, S. Lall, “Distributed algorithms for maximum lifetime routing in wireless sensor networks,” IEEE Transactions on Wireless Communications, vol. 5, no. 8, pp. 2185-2193, Aug. 2006.

[10] F. Liu, C.Y. Tsui, and Y.J. Zhang, “Joint Routing and Sleep Scheduling for Lifetime Maximization of Wireless Sensor Networks,” IEEE Transactions on Wireless Communications, vol. 9, no. 7, pp. 2258-2267, Jul. 2010.

[11] Y. He, I. Lee, and L. Guan, “Distributed algorithms for network lifetime maximization in wireless visual sensor networks,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 19, no. 5, pp. 704-718, May 2009.

[12] C. Li, J. Zhou, H. Xiong, and C.W. Chen, “Joint Coding/Routing Optimization for Distributed Video Sources in Wireless Visual Sensor Networks,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 21, no. 2, pp. 141-155, Feb. 2011.

[13] R. J. Vanderbei, Linear programming: Foundations and Extensions, 2nd New York: Springer, 2001.

[14] D. Palomar and M. Chiang, “A tutorial on decomposition methods and distributed network resource allocation,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 8, pp. 1439-1451, Aug. 2006.

[15] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. [16] D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and Optimization, Athena

Scientific, 2003.


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