Trust-Based Multi-Agent Filtering for 

Increased Smart Grid Security

Ion Matei

John S. Baras

Vijay Srinivasan

NISTIR 7808
NISTIR 7808

Trust-Based Multi-Agent Filtering for 

Increased Smart Grid Security

Ion Matei

Vijay Srinivasan

Manufacturing Systems Integration Division

Engineering Laboratory

Gaithersburg, MD 20899

John S. Baras

University of Maryland

College Park, MD 20742

August 2011

U.S. Department of Commerce

Rebecca M. Blank, Acting Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Under Secretary for Standards and Technology and Director
Trust-Based Multi-Agent Filtering for Increased Smart Grid Security

Ion Matei John S. Baras Vijay Srinivasan

Abstract—We address the problem of state estimation of the

power system for the Smart Grid. We assume that the monitoring

of the electrical grid is done by a network of agents with

both computing and communication capabilities. We propose

a security mechanism aimed at protecting the state estimation

process against false data injections originating from faulty

equipments or cyber-attacks. Our approach is based on a multiagent filtering scheme, where in addition to taking measurements,

the agents are also computing local estimates based on their own

measurements and on the estimates of the neighboring agents.

We combine the multi-agent filtering scheme with a trust-based

mechanism under which each agent associates a trust metric to

each of its neighbors. These trust metrics are taken into account

in the filtering scheme such that information transmitted from

agents with low trust is disregarded. In addition, a mechanism

for the trust metric update is also introduced, which ensures that

agents that diverge considerably from their expected behavior

have their trust values lowered.

I. Introduction

Smart Grid refers to the modernization of the electric system

through the integration of new information-age technologies

and new strategic public policies. It is based on adding and

integrating new digital computing and communication technologies and services with the power-delivery infrastructure.

Bidirectional flows of energy and two-way communication

and control capabilities will enable new functionalities and

applications that go well beyond “smart” meters for homes

and businesses. Some of the characteristics of the Smart Grid

include an increased use of digital information and controls

technology to improve reliability, security, and efficiency of

the electric grid, dynamic optimization of grid operations and

resources, with full cyber security [2]. It is widely recognized

that one of the most challenging task in implementing the

Smart Grid is putting in place security policies that address

security threats to the infrastructure. The main goal of a cybersecurity strategy is the prevention of damages to, unauthorized

use of, exploitation of, and, if needed, the restoration of electronic information and communication systems and services

to ensure confidentiality, integrity and availability [3].

The part of the Smart Grid infrastructure used to control the

electricity generation and transmission is represented by the

Supervisory Control and Data Acquisition (SCADA) systems.

Ion Matei is with the Engineering Laboratory at the National Institute of

Standards and Technology, Gaithersburg, MD 20899 (ion.matei@nist.gov) and

with the University of Maryland, College Park, MD 20742 (imatei@umd.edu).

John S. Baras is with the Electrical and Computer Engineering Department

and with the Institute for Systems Research at the University of Maryland,

College Park, MD 20742 (baras@umd.edu).

Vijay Srinivasan is with the Engineering Laboratory at the National Institute

of Standards and Technology, Gaithersburg, MD 20899 (vsrini@nist.gov).

This material is based upon work supported by the NIST-ARRA Measurement Science and Engineering Fellowship Program award 70NANB10H026.

A SCADA system receives measurements of the state of the

power grid and computes an estimate on the state of the power

grid on which the control strategy is based. Power blackout

events due to the failure of the SCADA systems to recognize

load and stability restrictions, human errors, faults, etc. suggest

the need for improved system-wide monitoring, alarms and

power system state estimation programs [10]. As emphasized

in [1], the SCADA system state of the art architecture is

vulnerable to cyber and physical attacks such as sabotage by

disgruntled insiders, false data injection from faulty equipment

or from cyber attacks.

In this paper we focus on the security of the state estimation

of the power system. We propose a strategy aimed at improving the security of the SCADA systems, by implementing

algorithmic policies aimed at protecting the estimation process

against false data injection generated by faulty equipment or

cyber attacks. Our approach is based on a multi-agent filtering

scheme, where intelligent agents, geographically distributed in

the power grid, receive (local) measurements on the state of

the power grid and compute local estimates based on their own

measurements and on the estimates of the neighboring agents

as well. We combine the multi-agent filtering scheme with a

trust-based mechanism under which each agent associates a

trust metric to each of its neighbors. These trust metrics are

taken into account in the filtering scheme such that information

transmitted from agents with low trust is disregarded. In

addition, a mechanism for the trust metric update is introduced,

which is based on the long-term behavior characteristics of the

agents. Other approaches for dealing with false data injections

on SCADA systems can be found in [6], for example.

II. Problem formulation

State estimation of power systems, using real-time measurements of active and reactive power flows in the network, is

used to build the model for the observable part of the power

grid. It was introduced to help the system operators having

a good image on the state of the power system in order to

increase the ability to tackle contingency conditions. Rather

than centralizing the measurements to the SCADA systems

for state estimate computations, we propose a strategy to

decentralize the estimation process, under which a network

of intelligent agents computes local estimates based on their

measurements and the estimates computed by other agents.

Thanks to their simplicity, the most common models used

to represent the dynamics of the power grid are linear [9],

[11], [22]. Similarly, throughout this paper we consider the

following stochastic linear equation as an approximate model

for the power grid dynamics

x(k+1) = A(k)x(k)+w(k), (1)
where k denotes the discrete time, x(k) ∈ Rn

is the state vector,

A(k) is the matrix connecting the current state with the one at

the next time instant, and w(k) ∈ Rnis the state noise, assumed

Gaussian with zero mean and covariance matrix Q. The state

variables usually include nodal voltages (voltage magnitudes,

voltage angles), transformer ratios, and complex power flows

(active and reactive power flows) [20]. The initial state x0 has

a Gaussian distribution, with mean μ0 and covariance matrix

P0. The parameters of the model, A(k) and Q, are assumed

to be determined a priori through a parameter identification

process [9]. Usually, the matrix A(k) is assumed slow timevarying; and, for simplicity in this paper we assume A(k) to

be constant, i.e., A(k) = A for all k.

In the context of this paper an agent is an intelligent

device with computation and communication capabilities; it

can receive measurements on the state of the power grid and it

is able to compute state estimates. A candidate for playing the

role of an agent is the Phasor Data Concentrator (PDC), which

collects the measurements from a set of GPS synchronized

Phasor Measurement Units (PMUs). Their ability to measure

positive sequence voltages at network busses and positive

sequence currents in transmission lines and transformers led to

an improvement of the state estimation capabilities [19]. The

estimation setup considered in this paper is somewhat similar

to the two-level estimator framework proposed in [25], where

the bus system is split into different areas where SCADAlike systems compute separately estimates. These estimates

are centralized and combined to obtain an overall estimate of

the entire power system. Unlike the aforementioned approach,

we do not centralize the estimates computed by different

areas, but use local collaboration to obtain an overall estimate

of the power grid. We assume that the agents (PDCs) can

communicate with each other and the SCADA system, thereby

forming a communication network. We denote by

Ni = {j such that i communicates with j}

the neighborhood of agent i, i.e., all agents it is capable to

communicate with. We assume linear sensing models for the

agents, given by

yi(k) = Cix(k)+vi(k), (2)

where yi(k) ∈ Rpiis the observation of the state x(k) made

by the agent i and vi(k) ∈ Rpiis the measurement noise,

assumed Gaussian with zero mean and covariance matrix Ri.

The measurements that are normally included in practical state

estimators are voltage magnitudes and angle differences, active

and reactive powers, current magnitude flows, magnitude of

turn ratios, phase shift angles of transformers, and active and

reactive power flows [20]. The above model is usually obtained

as a linearization of a nonlinear sensing model, where the

matrix Ciis the Hessian of a nonlinear function ci(x) computed

at some nominal point [11], relating the measurements to the

states.

Remark 2.1: We would like to point out that the entries of

matrix Ci reflect only components from the state vector x(k)

related to the bus(es) the agent i monitors. This implies that

the agent i has only a local view on the state of the network,

but through collaboration with other agents, this view can be

potentially enlarged.

We denote by ˆxi(k) and by ei(k) , x(k) − xˆi(k) the local

estimate and the estimation error computed by agent i, respectively. Each agent i associates to each of its neighbors a

trust metric denoted by Ti j, for j ∈ Ni. Intuitively, the trust

metrics designate the weight agent i gives to the information

received from its neighbors.

Problem: We assume that some agents can become faulty or

under the control of non-authorized entities that can cause the

respective agents to spread false data on the power grid to the

other agents. This false information can affect the computation

of control strategies for the power generation and transmission

needed to cope with changes in the state of the power grid.

Our goal is to propose a strategy aimed at limiting the effect

of false data injection on the state estimate computation, based

on the notion of trust.

III. Trust Model

Trust appears in various ways and meanings. We can refer

to the reduced trustworthiness of a sensor, meaning that the

sensor may have been compromised, or we can refer to the

trustworthiness of the data transmitted by a sensor. Similarly,

we can refer to a compromised link due to jamming, which

reduces the trustworthiness of the link. Thus trust in sensor

networks, and more generally in hybrid networks consisting

of collaborating humans and automated agents (sensors, actuators, computers) is a composite entity, represented by several

metrics and/or parameters.

There are various ways to represent trust weights numerically. In some trust schemes, continuous or discrete numerical

values are assigned to measure the level of trustworthiness. For

example, in [18], an entity’s opinion about the trustworthiness

of a certificate is described by a continuous value in [0, 1]. In

[24], a 2-tuple in [0,1]2 describes the trust opinion. In [12],

the metric is a triplet in [0,1]3, where the elements in the

triplet represent belief, disbelief, and uncertainty, respectively

(we denoted by [0,1]nthe n times Cartesian product of the

set [0,1]). Trust can also be interpreted as probability. In [13],

subjective probability is employed, while objective probability

is used in [14]. As a concept of uncertainty, entropy in

information theory is a natural measurement of trust as well

[23]. In the extreme case, trust can be binary: trust (trust

weight=1) or distrust (trust weight=0) because either there is

100% security in the network or the approach to evaluate trust

is very coarse. There is no absolutely right or wrong way to

represent these trust weights. All the aforementioned numerical representations are suitable for different environments and

management requirements.

In this paper, we assume each agent i assigns a trust metric

to each of its neighbors j, denoted by Ti j, which refers to

the reliability of data received from agent j. We represent

trust values as non-negative real numbers taking values in

the interval [0, Tmax], for some positive real Tmax. In the

following, these trust values will be used in conjunction with
a multi-agent estimation algorithm to limit the negative effect

on the state estimation process caused by false data injection.

IV. Trust-based Multi Agent State Estimation

In this section we present the trust-based multi-agent filtering scheme aimed at improving the security of the state

estimation process in the power grid. The section is divided

into three parts. First, we present a multi-agent filtering

scheme. Second, we describe the update mechanism for the

trust values, based on the behavior of the agents. Third, we

combine the multi-agent filtering scheme with the trust update

mechanism, which ensures that agents with low trust values

have limited influence on the estimation process.

A. Distributed Estimation

A fundamental problem in sensor networks is developing

multi-agent (distributed) algorithms for the state estimation

of a process of interest. Generically, a process is observed

by a group of sensors organized in a network. The goal

of each sensor is to compute accurate state estimates. The

distributed filtering (estimation) problem has received a lot of

attention during the past years, starting with the contributions

made by Borkar and Varaiya [7]. The main idea behind

distributed estimation, found in most of the papers addressing

this problem, consists of using a standard Kalman filter locally,

together with a consensus step in order to ensure that the local

estimates agree [8], [16], [21].

In what follows, we use a simplified version of the algorithm

proposed in [21], which is described next. For simplicity

Algorithm 1: Distributed Filtering

Input: μ0, P0

1 Initialization: ˆxi = μ0, Pi = P0

2 while new data exists

3 Compute the filtering gain Li

4 Compute the intermediate estimate of the state:

φi = xˆi + Li(yi −Cixˆi)

5 Estimate the state after a Consensus step:

ξi =

P

j∈Ni

wi jφj

6 Update the state of the local filter:

xˆi = Aξi

we omitted the time index in Algorithm 1. There are several

approaches for computing the filtering gains. In [21], the

authors propose the filtering gains to be computed using the

local Kalman filter equations. Other ideas include the off-line

computation of the (stationary) filtering gains using Linear

Matrix Inequalities techniques [16], which takes into account

the topology of the network. In line 5, the local information is

linearly combined with information received from neighbors.

Unlike the algorithm introduced in [21], we assume that only

local estimates are exchanged but not output measurements.

We will refer to line 5 as either the information fusion step or

the consensus step. Further on in the paper, we will focus our

analysis on the values of the weights wi j, which are positive

values summing up to one. Through these weights each agent

controls how the information received from neighbors is used.

Remark 4.1: It is reasonable to assume that the agents

cannot observe the entire state of the grid. However, through

collaboration (line 5 of Algorithm 1), provided that the power

grid is globally observable, the agents will potentially have a

global view on the state of the power grid.

B. Trust Update Algorithm

Due to the dynamic nature of agents (some of them may

become faulty, receive measurements from faulty equipment,

or may come under the control of unauthorized entities),

agents need to implement a mechanism for updating the trust

values Ti j. Examples of trust update mechanisms are presented

in [15], in the context of reputation systems where the update

is based on the notion of belief divergence and in [17] in the

context of distributed estimation.

In this paper we pursue a different avenue of investigation.

We assume that the agents “learn” the behavior patterns of

their neighbors, and when they determine significant changes

in these patterns, they adjust the trust values accordingly, i.e.,

by decreasing them. We assume that the learning period takes

place during the initial operation of the system, when it is

reasonable to assume that the agents function properly, i.e.,

the information provided by them is correct.

As mentioned earlier, agent i receives from its neighbors

their local estimates ˆxj(k), for j ∈ Ni. Let us denote by ei j(k)

the difference between the estimates of agents i and j, i.e.

ei j(k) = xˆi(k)− xˆj(k).

We note that ei j(k) can be equivalently written as

ei j(k) = ej(k)−ei(k),

where ei(k) = x(k) − xˆi(k) is the estimation error at agent i.

If we define the vector e(k) = (ei(k)), then it can be easily

shown that this vector has a multivariate normal distribution,

for all k, since e(k) is updated according to a linear dynamics

with initial state normally distributed. Consequently, it can

be shown that the vectors ei j(k) have multivariate normal

distributions as well. From the equations of the distributed

estimation algorithm we get that ei(k) have zero means, and

therefore ei j(k) have zero means as well, for all k ≥ 0. Let

us denote by Pi j(k) the covariance matrices of the vectors

ei j(k). Unfortunately, due to cross-correlations, computing

these matrices exactly is intractable for large values of k (in

fact the complexity increases exponentially with the time).

We consider that the statistics of the vectors ei j(k) determine

the behavior patterns of the agent i’s neighbors. We can

define confidence regions for ei j(k) based on the chi-square

distribution. It is well known that for a multivariate normally

distributed vector X ∼ N(μ,Σ) in Rn, the region

{x | (x−μ)

′Σ−1

(x−μ) ≤ χ

2

(α)}, (3)
contains (1 − α)100% of the probability in the distribution,

where χ

2

(α) is the chi-square distribution with n degrees of

freedom, computed at α. By varying α we can define different

confidence regions.

The update mechanism of the trust values is based on the

following idea. Every time ei j(k) is outside the confidence

region determined by parameter α, the trust value Ti j is

decreased up to a zero value, while every time ei j(k) is inside

the confidence region, the trust value Ti j is increased up to a

maximum value Tmax.

Learning - As mentioned earlier, the behavior patterns are

determined by the covariance matrices Pi j(k) whose exact

computation is unfortunately not tractable for large values of k

(computing Pi j(k) is in the same spirit as computing the gains

in a decentralized control problem; problem which is still open

for decades [5]). Therefore, the agents need to approximate

these matrices. Denoting by Pˆ

i j(k) the estimate of Pi j(k), we

propose the use of a time averaging filter given by

Pˆ

i j(k+1) =

1

k+1

ei j(k)ei j(k)

′ +

k

k+1

Pˆ

i j(k), (4)

to approximate the covariances matrices. The intuition behind

this approach is the following. Under the assumption that

the parameters of the stochastic process are time-invariant

(and under the assumption that the estimation errors are mean

square stable), the distributions of the estimation errors ei(k)

will converge to some stationary distributions. Consequently,

the same will happen for the random vectors ei j(k). The

covariance matrix of the stationary distribution of ei j(k) can

be approximated by taking the average over a sufficiently

large numbers of samples. Writing the averaging operation

iteratively, it results in a filter as in (4). Let P ̄

i j be the approximation of the covariance matrix of the stationary distribution

of the vector ei j(k), given by

P ̄

i j = Pˆi j(K),

where K is sufficiently large and represents the learning

horizon. Algorithm 2 summarizes the trust update mechanism

implemented by agent i (for brevity, the time index is ignored).

According to the aforementioned algorithm, every time the

error ei j(k) lies in the confidence region {x | x

′P ̄−1

i j x ≤ χ

2

(α)},

the trust value Ti j is increased by δ1 (a positive real value)

up to Tmax, while every time the trust values lie outside the

aforementioned confidence region, they are decreased by δ2

(assumed positive). We note that even a correctly functioning

agent j can have its trust value decreased. In the long run,

the trust value Ti j is decreased approximately αχ2(α)100% of

the total time. A good idea is to choose the parameters δ1

and δ2 such that they reflect how close or far the errors are

from the regions of the type (3). For example, if e

′

i jP ̄−1

i j ei j

is large compared to χ

2

(α), then δ2 should be chosen large

as well, such that the corresponding trust value is decreased

rapidly. We chose an incremental procedure for updating the

trust values, through the parameters δ1 and δ2. We can also

envisage a model where the increase/decrese rate of the trust

values is exponential. In such a case, we would have

Ti j =







min{δ1Ti j,Tmax} e

′

i jP ̄−1i j ei j ≤ χ

2

(α)

δ2Ti j e

′

i jP ̄−1i j ei j > χ2

(α)

where δ1 > 1 and 0 < δ2 < 1.

Algorithm 2: Trust update

Input: P ̄

i j, Tmax, δ1, δ2, α

1 while new data exists

2 Compute the errors between estimates:

ei j = xˆi − xˆj

3 Update the trust values:

Ti j =







min{Ti j +δ1,Tmax} e

′

i jP ̄−1

i j ei j ≤ χ

2

(α)

max{Ti j −δ2,0} e

′

i jP ̄−1i j ei j > χ2

(α)

4 end while

C. Trust-based Distributed Estimation

In this subsection we introduce an estimation algorithm

where the distributed estimation scheme presented in Algorithm 1 is combined with the trust update mechanism presented

in Algorithm 2.

We note that the weights wi j control how much the neighbors influence the update of the estimates ˆxi(k). A small value

of the weight wi j means that agent j will have little influence

on the agent i. Therefore, it makes sense to choose the weights

wi j to be proportional to the trust values Ti j. We propose to

choose the weights wi j as weighted trust values, such that they

sum up to one, i.e.,

wi j =

Ti j

P

j∈Ni

Ti j

.

This way the weights decrease with the trust values, such that

agents with low reliability will have little influence in the

computation of the local estimates. Algorithm 3 presents the

trust-based distributed estimation algorithm. We would like to

emphasize that Algorithm 3 acts on two levels. On one level

it extends the scope of the agents’ view with respect to the

grid by making the estimation process distributed. This way,

through collaboration, the agents can potentially have a more

accurate image on the global state of the grid, which otherwise

would be more difficult. On another level, the agents limit the

effect of false data injection by updating the agents’ trust in

their neighbors, according to their recorded behavior. Thus,

the estimation process becomes more robust.

V. Numerical example

We consider an example of a power grid with three generators and nine buses [4], shown in Figure 1. We assume

that a PMU is placed at each bus that measures the complex

voltages and currents (in the case of adjacent buses). We use an

estimation model similar to the one presented in [26]. Under

this model, the state vector is formed by the voltages measured

at buses, i.e., X = (Ui), where Uiis the complex voltage at bus
Algorithm 3: Trust-based Distributed Estimation Algorithm

Input: μ0, P0,Tmax, α, δ1, δ2, K

1 Initialization: ˆxi = μ0, Pi = P0, Ti j = Tmax

2 For a learning horizon K, apply Algorithm 1 together with (4)

to approximate the covariance matrices Pi j(k).

Set P ̄

i j = Pˆ 3 i j(k)

4 while new data exists

5 Compute the filtering gain Li

6 Compute the intermediate estimate of the state:

φi = xˆi + Li(yi −Cixˆi)

7 Compute the errors between estimates:

ei j = xˆi − xˆj

8 Update the trust values:

Ti j =







min{Ti j +δ1,Tmax} e

′

i jP ̄−1i j ei j ≤ χ

2

(α)

max{Ti j −δ2,0} e

′

i jP ̄−1i j ei j > χ2

(α)

9 Update the consensus weights

wi j =

Ti j

P

j∈Ni

Ti j

.

10 Compute the state after a Consensus step:

ξi =

P

j∈Ni wi jφj

11 Update the state of the local filter:

xˆi = Aξi

12 end while

Fig. 1. The 3-generators, 9-bus system

i. The measurement models are as follows. In the case where

the buses (i, j) are adjacent (e.g., (2,7), (3,9) and (1,4) in

Figure 1), the complex measurement model for each PMU is

given by

Zi(k) =

 

1 0

Yi j −Yi j ! Xi(k)

Xj(k)

!

+Vi(k),

where the measurement vector Zi(k)

′ = (Ui(k),Ii j(k)) encompasses the complex voltage at bus i and the complex current

on the line (i, j), Yi j is the admittance of line (i, j) and Vi(k)

is the complex measurement noise. In the case bus i has no

other adjacent bus, the measurement model gets simplified to

Zi(k) = Xi(k)+Vi(k).

Thus, we can generically represent the complex measurement

model as

Zi(k) = HiX(k)+Vi(k).

The real valued measurement model is given by

yi(k) = Cix(k)+vi(k),

where

yi =

 

Re(Zi)

Im(Zi)

!

,Ci =

 

Re(Hi) −Im(Hi)

Im(Hi) Re(Hi)

!

, vi =

 

Re(Vi)

Im(Vi)

!

,

and x

′ = (Re(X′

),Im(X

′

)).

We consider the power system to be reasonably stable, and

where the oscillations in the state variables are assumed to be

small and induced by a white Gaussian noise. Thus, we model

the dynamics of the power system by

x(k+1) = x(k)+w(k), x(0) = x0,

where a solution to the power grid in Figure 1 is used as initial

state for the dynamics of the state variables.

We assume that each PMU plays the role of a PDC and that

they form a communication network, as shown in Figure 2,

where each node represents a PMU.

2 7 8 9 3

5

6

4

1

Fig. 2. PMU network

In the numerical simulations that follow, we assume the

estimation is performed over 1500 time units and that during

the interval [500,1000] agent (PDC) 8 (color red in Figure 2)

shares false data with the neighbors.

Figure 3(a) shows the voltage at bus 1 (u

1

(k)) and the

estimates of the voltage at bus 1 made by agents 4, 7 and

9 (ˆu

1

4

(k),uˆ

1

7

(k),uˆ

1

9

(k)), together with the false data injected by

agent 8 (ˆu

1

8

(k)), when Algorithm 1 is used (and assigning

equal consensus weights), i.e., the trust update mechanism is

not applied. We note that the estimates of the aforementioned

agents are significantly affected. We have repeated the numerical simulation using Algorithm 3. The results are presented

in Figure 3(b). We note that although agent 8 shares false

information, its neighbors are not affected this time. This is

because the neighbors of agent 8 adjust their trust values such

that the data coming from agent 8 is rejected. Figure 3(c)

shows the time evolution of the consensus weights of agent

4. We note that between the interval [500,1000] the weight

w4,8(k) is lowered to zero, as a result of decreased trust in

agent 8.
0 500 1000 1500

16

16.5

17

17.5

18

18.5

19

19.5

20

20.5

21

k

u

1

(k)

uˆ

1

4 (k)

uˆ

1

7 (k)

uˆ

1

9 (k)

uˆ

1

8 (k)

(a)

0 200 400 600 800 1000 1200 1400 1600

16

17

18

19

20

21

22

k

u

1

(k)

uˆ

1

4(k)

uˆ

1

7(k)

uˆ

1

9(k)

uˆ

1

8(k)

(b)

0 500 1000 1500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

k

w45(k)

w46(k)

w48(k)

(c)

Fig. 3. (a) Estimates of the voltage at bus 1 using Algorithm 1, with agent 8 injecting false data; (b) Estimates of the voltage at bus 1 using Algorithm 3,

with agent 8 injecting false data; (c) The evolution of agent 4’s weights.

Remark 5.1: The local filtering gains were computed using

only the observable part of the pairs (A,Ci). However, as it can

be seen from Figure 3(b), through collaboration (consensus

step) even agents that do not measure directly the voltage

at bus 1 are still able to compute good estimates for the

aforementioned voltage.

VI. Conclusions

In this paper we proposed an algorithm for the state estimation of the power grid, aimed at making the estimation

process robust to false data injection. Our approach consisted

of combining a multi-agent filtering algorithm with a trust

metric, where agents with low trust values have little influence

on the computation of the estimates. In addition, we proposed

a trust update mechanism such that the trust values of the

agents are updated according to their recorded behavior.

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140, Title XIII, Sec. 1301, 2007.

[3] Report to NIST on the Smart Grid Interoperability Standards Roadmap.

Electric Power Research Institute, August 2009.

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