Open AccessArticle

On the Probabilistic Deployment of Smart Grid Networks in TV White Space

Angela Sara Cacciapuoti

^1,2,*

Marcello Caleffi

^1,2 and

Luigi Paura

^1,2

Department of Electrical Engineering and Information Technologies, University of Naples Federico II, Naples 80138, Italy

Multimedia Communications Laboratory, CNIT, Naples 80126, Italy

Author to whom correspondence should be addressed.

Sensors 2016, 16(5), 671; https://doi.org/10.3390/s16050671

Submission received: 29 February 2016 / Revised: 4 May 2016 / Accepted: 6 May 2016 / Published: 10 May 2016

(This article belongs to the Section Sensor Networks)

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Figure 1
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Versions Notes

Abstract

To accommodate the rapidly increasing demand for wireless broadband communications in Smart Grid (SG) networks, research efforts are currently ongoing to enable the SG networks to utilize the TV spectrum according to the Cognitive Radio paradigm. To this aim, in this letter, we develop an analytical framework for the optimal deployment of multiple closely-located SG Neighborhood Area Networks (NANs) concurrently using the same TV spectrum. The objective is to derive the optimal values for both the number of NANs and their coverage. More specifically, regarding the number of NANs, we derive the optimal closed-form expression, i.e., the closed-form expression that assures the deployment of the maximum number of NANs in the considered region satisfying a given collision constraint on the transmissions of the NANs. Regarding the NAN coverage, we derive the optimal closed-form expression, i.e., the closed-form expression of the NAN transmission range that assures the maximum coverage of each NAN in the considered region satisfying the given collision constraint. All the theoretical results are derived by adopting a stochastic approach. Finally, numerical results validate the theoretical analysis.

Keywords:

white space; smart grid; cognitive radio; sensor network

1. Introduction

To accommodate the rapidly increasing demand for wireless broadband communications in Smart Grid (SG) scenarios, research efforts [1] are currently ongoing to enable the SG Networks (SGNs) to utilize the channels of the TV spectrum temporarily unused by the licensed users, referred to as incumbents, according to the Cognitive Radio paradigm [2,3,4,5,6]. The vacant channels are known as TV White Space (TVWS) channels.

SG communications over TVWS channels are conditioned by regulators on the ability of the SGNs to avoid harmful interference toward incumbents. To this aim, the existing rulings [7] obviate the spectrum sensing as the mechanism for the SGNs to determine the TVWS availability. Instead, they require the SGNs to periodically access to a geolocated database, referred to as White Space DataBase (WSDB) [8,9,10], for acquiring the list of TVWS channels.

Let us consider the typical SG scenario shown in Figure 1, where several smart meters connected to a gateway constitute a Home Area Network (HAN), and multiple gateways connected to a Data Aggregate Unit (DAU) constitute a Neighborhood Area Network (NAN) [1,11,12] (please note that the typical wireless technologies and the corresponding values of the coverage ranges for HANs and NANs are reported in [12], along with further details on their communication features). The gateway is responsible for transmitting the meter data periodically collected within its HAN to the DAU via a TVWS channel, once declared available by the WSDB. However, experimental studies have shown that the number of TVWS channels is significantly limited in urban areas [13]. Hence, it is likely that multiple closely-located NANs are authorized to use the same TVWS channel and their transmissions may collide, leading so to a performance degradation. Moreover, since so far there are no regulatory requirements for the coexistence among NANs operating in TVWS spectrum, such a performance degradation can be severe. The described scenario of multiple closely-located NANs sharing the same TVWS channel has been considered for the first time in [14], where the sensing duration that maximizes the achievable data rate at a given NAN has been determined.

Differently from [14], in this letter, we develop an analytical framework for the optimal deployment of multiple closely-located NANs using the same TVWS channel with the objective to derive the optimal values for both the number of NANs and their coverage. More specifically, regarding the number of NANs, we derive the optimal closed-form expression, i.e., the closed-form expression that allows to evaluate the maximum number of NANs in the considered network region satisfying a given collision constraint on the transmissions of NANs.

Regarding the NAN coverage, we derive the optimal closed-form expression, i.e., the closed-form expression of the NAN transmission range that assures the maximum coverage of each NAN in the considered network region satisfying the given collision constraint.

All theoretical results are derived by adopting a stochastic approach and by considering the specificity of the NAN traffic demands. Finally, numerical results validate the theoretical analysis.

2. System Model

In a geographic region of area

A

, K NANs are deployed. The HAN gateways of each NAN are responsible for transmitting the meter data periodically collected within their HANs to the DAU via the same TVWS channel declared available by the WSDB, as depicted in Figure 1. We denote with

G_{i}

the number of HAN gateways belonging to the i-th NAN and with

R_{TX}^{i}

their transmission range. The traffic of the i-th NAN is modeled thought the parameters

P_{a}^{i}

and

P_{in}^{i} \overset{▵}{=} 1 - P_{a}^{i}

denoting the probabilities of an arbitrary HAN gateway belonging to the i-th NAN being active (i.e., having meter data collected within its HAN to transmit to the DAU) and inactive, respectively.

3. Optimal Deployment Analysis

At a certain time instant t, the transmissions of two arbitrary HAN gateways, say

H_{ℓ}^{i}

and

H_{m}^{j}

belonging to the i-th and j-th NANs, respectively, collide if and only if the HAN gateways are both active and their distance

d_{H_{ℓ}^{i} H_{m}^{j}} (t)

(denoted in the following as

d_{i j} (t)

for the sake of simplicity) is smaller than the distance threshold constituted by the maximum between their transmission ranges, i.e.,

d_{i j} (t) \leq max (R_{TX}^{i}, R_{TX}^{j})

. Hence, it results that the transmissions of a certain HAN gateway belonging to the i-th NAN, say

H_{ℓ}^{i}

, collide with the transmissions of the j-th NAN if it exists al least one active HAN gateway belonging to the j-th NAN within the distance threshold. We denote the probability of this event as

P_{C}^{H_{ℓ}^{i}, j}

. As a consequence, there is no collision between the transmissions of NANs i and j if none of the active HANs belonging to NAN i is within the distance threshold of any of the active HANs belonging to NAN j. We denote the probability of this event as

{\bar{P}}_{C}^{i j}

. In the following, by accounting for these definitions, with some algebraic manipulations, we derive the expressions of

P_{C}^{H_{ℓ}^{i}, j}

and

P_{C}^{i j} \overset{▵}{=} 1 - {\bar{P}}_{C}^{i j}

. Specifically:

P_{C}^{H_{ℓ}^{i}, j} \overset{▵}{=} 1 - {\bar{P}}_{C}^{H_{ℓ}^{i}, j} = 1 - [P_{in}^{i} + P_{a}^{i} {(1 - P_{a}^{j} P_{d_{i j}})}^{G_{j}}]

(1)

where

P_{d_{i j}}

denotes the probability of the event

\{d_{i j} (t) \leq max (R_{TX}^{i}, R_{TX}^{j})\}

. By accounting for Equation (1) and the definition of

P_{C}^{i j}

, in the reasonable hypothesis of independence of the HAN transmissions, it results

\begin{matrix} P_{C}^{i j} \overset{▵}{=} 1 - {\bar{P}}_{C}^{i j} = 1 - \prod_{l = 1}^{G_{i}} {\bar{P}}_{C}^{H_{ℓ}^{i}, j} = 1 - {[P_{in}^{i} + P_{a}^{i} {(1 - P_{a}^{j} P_{d_{i j}})}^{G_{j}}]}^{G_{i}} \end{matrix}

(2)

Since in a typical SG scenario it is reasonable to assume statistically the NANs having similar traffic and communication characteristics, the collision probabilities between two NANs are symmetric, i.e.,

P_{C}^{i j} = P_{C}^{j i} = P_{C}

. Hence the overall collision probability

P_{C}^{O}

, i.e., the probability that the transmissions of at least one NAN collide with the transmissions of at least one of the remaining

(K - 1)

NANs, can be calculated as follows:

\begin{matrix} P_{C}^{O} \overset{▵}{=} 1 - {({\bar{P}}_{C})}^{(\binom{K}{2})} = 1 - {({\bar{P}}_{C})}^{\frac{K (K - 1)}{2}} \end{matrix}

(3)

where

{\bar{P}}_{C} = {\bar{P}}_{C}^{i j}

is given in Equation (2) and the binomial coefficient

(\binom{K}{2})

is due to the symmetry of the collision probabilities

{P_{C}^{i j}}_{i, j \in {1, \dots K}}

. By substituting

{\bar{P}}_{C}

in Equation (3), one has:

\begin{matrix} P_{C}^{O} = 1 - {[P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}]}^{\frac{G K (K - 1)}{2}} \end{matrix}

(4)

As mentioned in the introduction,

P_{C}^{O}

has to be under a threshold, depending on the considered application, to assure a certain level of protection on the NANs transmissions, i.e.,

P_{C}^{O} \leq T_{C}

(5)

With

T_{C}

denoting the largest value of the overall collision probability tolerated by the SG scenario. In the following we refer to

T_{C}

as collision constraint. Satisfying Equation (5) is equivalent to satisfy:

1 - {[P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}]}^{\frac{G K (K - 1)}{2}} \leq T_{C}

(6)

The largest value of K satisfying Equation (6), say the optimal value

K_{opt}

, is the maximum number of NANs that can be deployed in the region of area

A

satisfying the collision constraint

T_{C}

. Specifically, by applying the logarithm function with basis

P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}

to Equation (6) and by recognizing that this logarithm is a decreasing function of its argument since

P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G} \leq 1

, one has:

P_{C}^{O} \leq T_{C} \Leftrightarrow K (K - 1) \leq \frac{2}{G} {log}_{P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}} (1 - T_{C})

(7)

Hence,

K_{o p t}

is obtained as the largest integer less than or equal to the maximum value satisfying Equation (7):

K_{opt} = ⌊\frac{1}{2} (- 1 + \sqrt{1 + \frac{8}{G} {log}_{P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}} (1 - T_{C})})⌋

(8)

With

⌊ \cdot ⌋

denoting the floor operator. It’s easy to verify that

1 + \frac{8}{G} {log}_{P_{in} + P_{a} {(1 - P_{a} P_{d})}^{G}} (1 - T_{C}) \geq 1

, i.e.,

K_{opt}

is positive. From Equation (8) it results that, for a given value of

T_{C}

K_{opt}

decreases as G increases. The same consideration holds if

P_{a}

increases.

In the hypothesis of NANs uniformly distributed in the region of area

A

and

R_{TX} < < \sqrt{A}

(We assumed a bi-dimensional squared network region. This hypothesis is not restrictive. In fact, if the network region has dimensions a and b, respectively, one has to compare the transmission range with the smallest dimension, i.e.:

R_{T X} < < min {a, b}

P_{d} = P (d (t) < R_{TX})

can be evaluated as:

P_{d} ≃ \frac{π R_{TX}^{2}}{A}

. By substituting this in Equation (6), the optimal NAN coverage

R_{TX}^{opt}

, i.e., the maximum value of the transmission range that satisfies the collision constraint

T_{C}

, is obtained by solving Equation (6) with respect to

R_{TX}

and by taking the maximum value:

R_{TX}^{opt} = \sqrt{\frac{A}{π P_{a}}} \sqrt{1 - \sqrt[G]{\frac{\sqrt[\frac{G K (K - 1)}{2}]{(1 - T_{C})} - P_{in}}{P_{a}}}}

(9)

From Equation (9) it results that the optimal coverage increases as the SG traffic demand

P_{a}

decreases, as expected.

In Table 1 we summarized the notation used for the parameters involved in the analysis along with their meaning, to improve the clarity of the manuscript.

4. Numerical Results

Here, we validate the theoretical results through Monte Carlo simulations. We consider

K = 3

NANs placed randomly in a squared region of area

A

. Figure 2 shows the overall collision probability

P_{C}^{O}

as a function of the SG traffic demand

P_{a}

, for three different values of the normalized transmission range, i.e.,

R_{TX} / \sqrt{A} = {0.01, 0.02, 0.03}

. First, we note that the theoretical results match very well the simulation results. Then, when

P_{a}

decreases,

P_{C}^{O}

decreases as well, since the probability of an HAN gateway being active decreases. Moreover, when

R_{TX} / \sqrt{A}

increases

P_{C}^{O}

increases, since it is more likely that two HAN gateways are within a distance smaller than their transmission range. In the same figure, we also plot the optimal transmission range

R_{TX}^{opt} / \sqrt{A}

given in Equation (9) as function of

P_{a}

for a value of the collision threshold equal to

T_{C} = 0.05

. By comparison with the previous plot, we note that the theoretical results are once again confirmed. In fact, only if

R_{TX} \leq R_{TX}^{opt}

, the collision constraint is satisfied. Figure 3 shows

P_{C}^{O}

as a function of

P_{a}

, for three different values of the number G of HAN gateways, i.e.,

G = {10, 20, 30}

. Again, the theoretical results match very well the simulation results. Moreover, when G increases,

P_{C}^{O}

increases as well. This result is reasonable, since it is more likely to find active HAN gateways.

5. Conclusions

We developed a theoretical framework for the optimal deployment of multiple closely-located NANs over TVWS spectrum. Specifically, we derived through closed-form expressions the optimal values for both the number of NANs and their coverage, i.e., the maximum values assuring the given collision constraint. Such closed-form expressions reveled the highly non-linear relationship among the involved parameters as the number of NANs, their traffic demand, the number of HAN gateways and their transmission ranges. The developed analysis is crucial and preliminary for designing any effective coexistence protocol. Such an issue will be addressed in a future work along with optimization strategies for the DAU deployment.

Acknowledgments

This work has been partially supported by the National PON grants Databanc Chis and Fersat and the regional POR grant myOpenGov.

Author Contributions

The authors equally contributed to this work.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. Smart Grid Scenario: Multiple Closely-Located Neighborhood Area Networks (NANs).

Figure 2.

P_{C}^{O}

versus

P_{a}

, for different values of

R_{TX} / \sqrt{A}

Figure 2.

P_{C}^{O}

versus

P_{a}

, for different values of

R_{TX} / \sqrt{A}

Figure 3.

P_{C}^{O}

versus

P_{a}

, for different values of G.

Figure 3.

P_{C}^{O}

versus

P_{a}

, for different values of G.

Table 1. Adopted notation.

**Table 1.** Adopted notation.
Symbol	Definition
K	number of NANs deployed in the network region
$A$	Area of the network region
$H_{ℓ}^{i}$	The ℓ-th HAN gateway belonging to the i-th NAN
$H_{m}^{j}$	The m-th HAN gateway belonging to the j-th NAN
$P_{C}^{i j}$	Collision probability of the transmissions of NANs i and j
$P_{C}^{O}$	Overall collision probability
$P_{a}$	Probability of an arbitrary HAN gateway belonging to the i-th NAN being active
$P_{in}$	Probability of an arbitrary HAN gateway belonging to the i-th NAN being inactive
G	Number of HAN gateways
$R_{TX}$	Transmission range of an HAN gateway
$P_{d}$	Probability of the distance $d (t)$ between two arbitrary HAN gateways belonging
	to two different NANs being smaller than $R_{TX}$
$T_{C}$	Collision constraint, i.e., the largest value of the overall collision probability tolerated by the SG scenario
$R_{TX}^{opt}$	Optimal NAN coverage, i.e., the maximum value of $R_{TX}$ satisfying the collision constraint $T_{C}$
$K_{opt}$	The largest value of K satisfying the collision constraint $T_{C}$

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Cacciapuoti, A.S.; Caleffi, M.; Paura, L. On the Probabilistic Deployment of Smart Grid Networks in TV White Space. Sensors 2016, 16, 671. https://doi.org/10.3390/s16050671

AMA Style

Cacciapuoti AS, Caleffi M, Paura L. On the Probabilistic Deployment of Smart Grid Networks in TV White Space. Sensors. 2016; 16(5):671. https://doi.org/10.3390/s16050671

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Cacciapuoti, Angela Sara, Marcello Caleffi, and Luigi Paura. 2016. "On the Probabilistic Deployment of Smart Grid Networks in TV White Space" Sensors 16, no. 5: 671. https://doi.org/10.3390/s16050671

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On the Probabilistic Deployment of Smart Grid Networks in TV White Space

Abstract

1. Introduction

2. System Model

3. Optimal Deployment Analysis

4. Numerical Results

5. Conclusions

Acknowledgments

Author Contributions

Conflicts of Interest

References

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