IEEE PES ISGT ASIA 2012 1569574515

Sensor Selection Schemes In Smart Grid

Xin Wang, Qilian Liang Department of Electrical Engineering

University of Texas at Arlington Email: xin.wang51@mavs.uta.edu

liang@uta.edu

Abstract-Wireless sensor and actuator networks (WSANs) have been widely implemented in smart grid to monitor and con­trol their environment. We consider the sensor selection schemes for parameter estimation in energy constraint WSANs. To pro­long the network lifetime and optimize energy consumption, only sensors experiencing favorable conditions will participate in the estimation process. In this paper, we study two different multiple access schemes between sensors and fusion center (FC): orthogonal and coherent. We compare these two schemes and derive a reasonable lower bound of distortion. Sensor selection scheme under equal power allocation is first discussed. We propose an opportunistic sensor selection scheme by solving an relaxed sensor selection problem. The asymptotic behavior is also investigated. Sensor selection scheme under optimal power allocation is then considered. We address this problem via convex optimization techniques and derive a reminiscent of the "water­filling" solution for this scenario. The validity of the proposed sensor selection schemes is finally demonstrated by numerical simulation results.

I. INTRODUCTION

Spread over the smart grid, wireless sensor and actuator networks (WSANs) monitor the functioning and the health of grid devices, monitor temperature, provide outage detection and detect power quality disturbances [1].

On the generation side, WSANs offer an ideal technology for monitoring and control of generation facilities in the smart grid. For example, WSANs enable clean energy to be generated more efficiently. Automated panels managed by sensors track sun rays to ensure that the sun's power is collected in a more efficient manner. On the transmission and distribution segment of smart grid, substation, overhead power lines and underground power lines are need to be monitored in near real-time. An equipment failure or breakdown may cause blackouts, or it may even be dangerous for public health. WSANs provide promising solutions for monitoring and securing the transmission and distribution segment. Mean­while, WSANs help realize more efficient energy routing, which could trace how energy is consumed across the grid in real-time. On the demand-side, WSANs provide the means to detect fluctuations, power outage at home, two-way flow electricity and real-time information. Above all, WSANs play a significant role in smart grid.

This work was supported by U.s. National Science Foundation under Grant CNS-1l16749.

Wei Wang, Baoju Zhang College of Physics and Electronic Information

Tianjin Normal University Email: wangweivip@tju.edu.cn

wdxyzbj@mail.tjnu.edu.cn

Typically, a WSAN consists of one Fusion Center (FC) and a potentially large number of energy-constraint sensor nodes deployed in order to observe a given phenomenon (temperature, humidity, etc). The FC then combines the ob­servations to generate an estimate of the observed quality. A proposing scheme for distributed estimation is analog amplify and forward [2], where observations from the sensors are transmitted directly to the FC without any coding. Analog amplify and forward scheme with optimal power allocation is studies in [3][4]. Asymptotic behavior is studies in [5]. Although these works deal with the power allocation problem, they fail to consider the sensor selection strategy, which is of paramount importance in WSANs.

One important property of many WSANs is their stringent power constraint and communication capability. In such net­works, sensors have only small-size batteries whose replace­ment can be costly if not impossible. If a sensor remains active continuously, its energy will be depleted quickly leading to its death [6]. To prolong the network lifetime, sensors need to alternate between being active and sleeping. The FC schedules only a subset of available sensors for transmission each time. [7] proposed a heuristic approach to the problem with convex optimization. However, the work of [7] only considered the classic AWGN interference scenario without the discussion of fading channel. The algorithm proposed did not take the power constraint into consideration either.

In this paper, we mainly focus on linear decentralized estimation problem in smart grid. we study two different multiple access schemes between sensors and FC: orthogonal and coherent. We compare these two schemes and derive a reasonable lower bound of distortion which will serve as the performance benchmark for later analysis. We first investigate the sensor selection scheme under equal power allocation. We propose a novel opportunistic sensor selection scheme. The asymptotic behavior is also investigated. Then we address sensor selection scheme under optimal power allocation via convex optimization techniques and derive a reminiscent of the "water-filling" solution for this scenario.

This paper is organized as follows. Section II will give the general system model. Section III analyzes the sensor selection scheme under equal power transmission strategy. Section IV addresses the case where power is allocated under optimal power allocation. Finally we conclude the paper in Section V.

Source

.,Ja; V2

· · · · · · · · : S \� , '

, Sm f------'

........ � '. ' .

Fig. 1. System Model

II. SYSTEM MODEL

Fusion Center

Source

Estimation.

Consider a WSAN in smart grid with one FC and m sensors which have been deployed to estimate an unknown parameter e. The system model is shown in fig. 1. The observation Xi at sensor i is represented as a random signal e corrupted by observation noise Vi:

Xi = e + Vi· (I)

We assume that both e and Vi are i.i.d. Meanwhile e and Vi have zero mean and variance (J� and (J;, respectively. Then m sensors transmit their observations to the FC via different fading channels. Motivated by our analysis in the introduction section, we adopt an amplify-and-forward strategy at each sensor transmitter. Therefore, at sensor i, the transmitter can be simply modeled by a power amplifying factor ai and the average transmit power is given as

2 2 Pi = aiP.Ti = ai((Je + (Ji )' (2)

P.Ti represents the power of observation Xi. We investigate two different multiple access schemes between sensors and FC: orthogonal and coherent.

A. Orthogonal Access Scheme In this case, each sensor transmits its measurement to the

FC via orthogonal channels (e.g. using FDMA or CDMA), so the FC receives

Yi = hie+ni,i = 1,2, ... ,rn. (3)

Then the received signal vector

y= hB+n ( 4)

with h = [ylaI91,"" ylam9mjT and n standing for AWGN with diagonal covariance matrix R given as dia9[R] = [ 2 ,,2 2 ,,2 ]T .. (JI al91 + <"1,· · · , (Jmam9m + <"m . T denotes transposItIOn. Here we introduce the the power gain of the fading channel 9i and the variance of noise Wi from sensor i to FC denoted as a.

Based on the received signal, the FC generates an estimate fj to minimize the MSE. In this paper, we adopt the maximum

2

likelihood (ML) estimator to recover e. Accordingly, the estimate for e is given by [8]

fj [hTR-IhrlhTR-ly

(� ai9i )-1 � yiai!jiYi (5) � (J2ai9i + �2 � (J2ai9i + �2' 2=1 '[ 2 2=1 2 '[

The MSE of this estmator is given as m

DOTth = [hTR-1h]-1 = (L 2 ai9i

e)-I. i=1 (Ji ai9i + i

B. Coherent MAC

(6)

Another case is that all sensors transmit simultaneously. The transmitted signals from all sensors reach the FC as a coherent sum. In this case, The received signal at the fusion center can be expressed as :

m m m Y = L hie + n = L ylai9ie + L ylai9iVi + W. (7)

i=1 i=1 i=1 Note that in this case hi is scalar, while in the orthogonal scenario h is vector. We assume the variance of noise W from sensors to FC is e. Given y, FC generates an estimation of e. Similarly, the ML estimator is

m fj = (hT R-Ih)-lhT R-Iy = (L ylai9i)-ly. (8)

i=1 This estimator is linear and achieves a MSE given as

Dcmac m m

� -2 � 2 2 (� ylai9;) (� (Ji ai9i + � ). (9) i=1 i=1

Remark 2.1: When all sensor observations are directly available to the fusion center, we can get the estimator

(10)

which achieves a MSE

(11 )

This result could serve as a performance benchmark for later analysis.

Remark 2.2: Let's consider the scenario when the power approach infinity or the Gaussian noise variance e could be neglected. For the orthogonal multiple access case, D;;T�h = 2::7:1 �, which achieves the performance benchmark Do. As

for the �oherent multiple access case, D;:;'ac :::: 2::7:1 �. The detailed derivation is given in Appendix A. '

From the above analysis, we have the inequality

Dcmac � DOTth > Do, ( 12)

which means if the power approach infinity, the noise variance e introduced at the FC could be neglected, then the orthogonal access scheme could approach the low bound Do; however,

even if the power is large enough, the coherent multiple access scheme could not approach the low bound Do, unless the condition of Cauchy-Schwarz inequality is satisfied for all i. One reasonable explanation is that if there is no noise introduced at the FC, the individual observations from the sensors perform better than a combination of the observations.

Remark 2.3: Next we consider the case when the noise variance �2 is very large. If �2 dominates, Dorth - Dcmac 2 0 will stand. So under this assumption,

(13)

which means the coherent scheme will have smaller estimation variance than the orthogonal scheme.

Based on the above analysis, we can not say which one always performs better. Since the coherent MAC scheme requires perfect carrier-level synchronization among sensors, in the later sections of this paper we will adopt the orthogonal access scheme.

III. SENSOR SELECTION SCHEME UNDER EQUAL POWER

ALLOCATION

First suppose all sensors consume the same transmit power, and Ptat is the total power constraint. We propose a oppor­tunistic sensor selection scheme under equal power allocation, which could be characterized by the following algorithm:

Algorithm: 1. Initialization: Formulate the approximate relaxed sensor

selection problem without power constraint. 2. Sensor selection: Activate K favorable sensors from all

m available sensors. Only active sensors will participate in the estimation process.

3. Equal power allocation: The K active sensor will adjust their transmit power accordingly and send their observations to the Fe.

To begin with, we attempt to select the sensors for parameter estimation. The sensor selection scheme is formulated as follows

m'in

s.t.

m ("""" gi ) -

1 �

Zi afgi + (f 1Tz = K Zi E {O, I}.

Here the vector 1 is a vector with all entries one and K rep­resents the target selected sensor number. The variable Zi can be chosen from 0 or 1, which decides whether the i th sensor observation will be selected or not. The original problem is an integer optimization problem which is nonconvex and hard to solve. By relaxing the nonconvex constraint Zi E {O, I} with the convex relaxation constraint 0 .-:; Zi .-:; 1, we reach the following relaxed sensor selection problem :

m gi max L Zi 2. e + , (lOg (Zi) + log (l - Zi)) i=l ai g, + i

S.t. 1Tz = K.

3

0.03

0.025-

0,02

0.015-

0.Q1

0.00 5

No Sensor Selection

I ....... Approximate Sensor Selection --Sensor Selection

�------�'0------2�O----�3�O ----�40------5�O----�60 Active Sensor Number K (Total sensor number is 100)

Fig. 2. Approximate Relaxed Sensor Selection (Note: Approximate sensor selection blue curve and sensor selection pink curve are almost overlapped )

where , is a positive parameter to control the quality of approximation.

Solving the above convex optimization problem, we could get the target sensors. Our strategy is to choose the K largest Zi from all available Zi choices, which means only sensors with favorable Channel Side Information (CSI) will participate in the estimation process. Next the power need to be allocated in a reasonable way due to the total power constraint.

As the total power budget for all sensors is Ptat, the equal power allocation scheme is given by:

1 '-:;i .-:; K. (14)

Implementing equal power allocation strategy, we can get an analytical distortion result:

m D ("""" Ptotgi )-1 5 arth = � Si 2 P . + Kt2( 2 + 2) . (1 )

i=l ai tatg, C,i a () ai where Si represents the approximate sensor selection decision. We need to clarify that relaxation parameter Zi is a fraction, from 0 to 1, however, the approximate relaxation parameter Si is an integer, 0 or 1. Let's discuss the following two asymptotic scenarios:

Remark 3.1: Ptot --+ 00 When total power approach infinity with equal power allo­

cation, we have Dorth --+ Do. This result suggests that even if the power is large enough, the distortion could only achieve the performance benchmark Do without approaching zero. This reason is that the observation noise Vi could not be eliminated even if power approach infinity. If the signal is amplified, the observation noise Vi is amplified as well.

Remark 3.2: K --+ 00 when K --+ 00, we can get a similar asymptotic behavior

as [ 4]:

Dorth

(16)

0.025

0.02

0.015

0.01

0.005

0.03

..... DistOliion with sensor selection

... Distortion without senosor selection

-O-Low Bound(Theory)

, , 10 20 30 40 50 60 70 80 Active Sensor Number K (Total sensor number is 100)

Fig. 3. Equal Power Allocation w/o Sensor Selection

-a-Active sensor number is 10 ,,*,Active sensor number is 20 ...... Active sensor number is 30 -Active sensor number is 40

I I I I I I I � � � � � � M 00 100 Power constraint in mw (Total sensor number is 100)

Fig. 4. Equal Power Allocation with Different Power Constraints

So with limited total power constraint for all sensors, the distortion could not decrease to zero even if K approach infinity. This is due to the fact that, under orthogonal access scheme, K different channel noise can not be eliminated even when K approach infinity.

We will provide several simulations to verify our proposed algorithm. The default simulation setup is Tn = 100 and (J� = 1. We assume the channel gain VJii follows i.i.d Rayleigh fading. Fig. 2 exhibits the proposed sensor selection performance. Apparently, our proposed approximate relaxed sensor selection scheme performs as well as the original relaxed sensor selection scheme, which means approximation from Zi to Si will not jeopardize the selection performance. Besides, we conclude that activating 20 sensors or more, the distortion performance will not make much difference. This conclusion verified the feasibility of the algorithm.

Then we verify our proposed sensor selection scheme under equal power allocation. Fig. 3 indicates that under equal power allocation, if we select more than 10 sensors from the whole 100 available sensors, our proposed sensor selection scheme will perform better than the original power allocation scheme without sensor selection; however, if we activate less than 10 sensors, our proposed sensor selection scheme will perform worse. One instinctive explanation is that if we select less than 10 sensors, the observation information collected is not enough for Fe to estimate the source information with

4

accuracy. Besides, if we select too many sensor at a time, the performance will also degrade. The reason is that more sensor selected means less power assigned to each sensor. Sensors with good channel condition will not be assigned enough power. So we need to select moderate sensors each time. This could be further illustrated in Fig. 4 . Also, Fig. 4 shows that with different sensor selection schemes, more power will always bring better estimation performance; however, the MSE performance could not approach the performance benchmark Do due to the power constraint and limited sensor number. Equal power allocation is easy to implement and integrates well with our proposed sensor selection scheme.

IV. SENSOR SELECTION SCHEME UNDER OPTIMAL

POWER ALLOCATION

In section III, we proposed an opportunistic sensor selection scheme under equal power allocation. Now we will address sensor selection scheme under optimal power allocation. Un­like equal power allocation scenario, if we want to derive the optimal sensor selection scheme, the active sensor number should not be fixed. The number will depend on the obser­vation quality, channel quality and so forth. Sensor selection scheme under optimal power allocation is formulated as

'max m

S.t. i=l ai � O.

Obviously, this problem is convex and ai is the variable we need to optimize. The Lagrangian G is given as

C m m

(17) i=l i=l

From the Lagrangian function we can derive the following Karush-Kuhn-Tucker (KKT) conditions [9]:

8C 8ai

gia 2 2 ( 2 e) 2 + A( (J Ii + (Ji ) - J.Li (Ji aigi + i

m

L 2 2 A ( ai((J1i + (Ji ) - Ptat) i=l

ILiai A

J.Li ai

0

0

0 > 0 > 0 > O.

Solving the KKT conditions, we obtain the reminiscent of the "water-filling" solutions in wireless communications,

a ai = -2 - ( (Ji gi (18)

0.01

0.025

::: 0.02 " "2 � 0.015 is

0.01

0.005

10

... Equal power without sensor selection

..... Equal power with sensor selection

...... OpLimal power with sensor selection

-0- Lmv Bound(Theory)

Optimal Point

, 20 30 40 50 Active Sensor Number K

, 60 70 80

Fig. 5. Optimal Power Allocation VS Equal Power Allocation

where x+ equals to 0 when x is less than zero, and otherwise equals to x. The solution is derived in Appendix B. We can express the T/i = (7 (ort-un' So for sensor i, if T/i > A, then the corresponding sensor will be active; otherwise the corre­sponding sensor will be switched off for energy efficiency.

From the above analysis, the proposed sensor selection scheme under optimal power allocation should outperform the sensor selection scheme under equal power allocation. The system parameters are set the same as in section III, only the power constraint is different for analysis purpose. We note from Fig. 5 that the proposed sensor selection scheme under optimal power allocation will always perform better than the sensor selection scheme under equal power allocation, and of course better than the scenario without sensor selection. Besides, our proposed optimal scheme approach the perfor­mance benchmark Do. However, the improvement in MSE performance of sensor selection scheme under optimal power allocation comes with the price of more complex computation.

V. CONCLUSION

This paper has investigated the sensor selection schemes in smart grid. First we propose an opportunistic sensor se­lection scheme under equal power allocation by solving an approximate relaxed sensor selection problem. The asymptotic behavior is also studied. We have further addressed sensor selection scheme under optimal power allocation via convex optimization techniques and derived a reminiscent of the "water-filling" solution for this scenario. Numerical simula­tions verify our proposed algorithms. Further study of related problems will form the main topic of succeeding research.

ApPENDIX

A. Derivation of (12) When the power approach infinity or the Gaussian noise

variance � could be neglected, D;;r�h = 2:�1 � is easy to be derived. As for the coherent multiple access case,

(2:�1 yfai9i) 2 (2:�1 yfai9i) 2

"m 2 c2 "m 2 L.i=l Ui aigi + <" L.i=l Ui aigi D-l cmac

<

5

The last inequality could be get from the Cauchy-Schwarz in­equality. So the Dcmac could approach Do only if the Cauchy­Schwarz equality condition u;aigi = Ljaigi is satisfied for all i, here l is a constant.

B. Derivation of (18) We notice that, if A = 0, then from the first KKT condition

would imply that fLi < ° for all sensor i. This will contradict with the fifth KKT condition. Thus, we must have A > 0, which means 2:�1 ai(u� + un - Ptat = O.

From the first KKT condition, we obtain the solution

( 19)

We know that for those sensor which will be active, ai should satisfy ai > O. Then the third KKT condition tell us that if ai > 0, then fLi = 0 holds. Thus the proof of equation (18) is completed.

To determine A, let us assume that the sensors are ordered such that T/l 2 T/2 . . . 2 T/m· Clearly, this ranking favours the sensors with better channel conditions and higher observation quality.

Combining the first and second KKT conditions, we can get

(20)

The number of active sensor number K' (which has been shown to be unique [10]) can be solved if we substitute A back to (18). Now the optimal strategy is that we will active the corresponding sensor if sensor index 1 :::; i :::; K'; and we will switch off the sensor for all sensor i > K'.

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