1

State estimation in power distribution networks withpoorly synchronized measurements

Saverio Bolognani, Ruggero Carli and Marco Todescato

Abstract—We consider the problem of designing a state esti-mation architecture for the power distribution grid, capable ofcollecting raw measurements from low-end phasor measurementunits, processing the data in order to minimize the effects ofmeasurement noise, and presenting the state estimate to thecontrol and monitoring applications that need it. The proposedapproach is leader-less, scalable, and consists in the distributedsolution of a least square problem which takes into explicit con-sideration the inexact time synchronization of inexpensive PMUs.Two algorithms are proposed: the first one is a specialization ofthe alternating directions method of multipliers (ADMM) forwhich a scalable implementation is proposed; the second one isan extremely lightweight Jacobi-like algorithm. Both algorithmscan be implemented locally by local data aggregators, or bythe individual PMUs, in a peer-to-peer fashion. The proposedapproach is validated via simulations on the IEEE 123 test feeder,considering the IEEE standard C37.118-2005 for PMUs. We alsoanalyze the performance of a distributed control algorithm thatmakes use of the grid state estimation and that we adopted asa prototype. We show that, via the proposed state estimationarchitecture, feedback control of the distribution grid becomesviable also with poorly synchronized sensors, and possibly alsowithout GPS-based synchronization modules.

I. INTRODUCTION

The electric grid is currently undergoing a deep renovationprocess towards the so-called smart grid, featuring largerhosting capacity, widespread penetration of renewable energysources, better quality of the service, and higher reliabil-ity. One of the major aspects of this modernization is thewidespread deployment of dispersed measurement, monitor-ing, and actuation devices.

In this paper, we consider one specific thrust, which isthe deployment of phasorial measurement units (PMUs) inthe medium and low voltage power distribution grid. Thepresence of PMUs is quite uncommon in today’s powerdistribution networks. However, this scenario has become thesubject of recent research efforts in the power systems commu-nity, including a 3-year research project involving Universityof California together with the Power Standards Lab andLawrence Berkeley National Lab [1]. Previous investigationon the subject includes the experimental deployment and theextensive measurement campaign in [2], and the analysisproposed in [3], [4], [5] for the possible applications ofsuch measurement devices, also called µPMUs, in the powerdistribution grid.

S. Bolognani is with the Laboratory on Information and Decision Sys-tems, Massachusetts Institute of Technology, Cambridge (MA), USA email:saverio@mit.edu.

R. Carli and M. Todescato are with Department of Information Engi-neering, University of Padova, Padova (PD), Italy email {carlirug |todescat}@dei.unipd.it

There is a substantial consensus on the many opportunitiesoffered by the deployment of a live network of µPMUs,including diagnostic routines (e.g. for unplanned islanding andfault detection), power flow estimation on the power lines (byknowing the phasorial nodal voltage differences across theentire grid), and feedback control strategies for coordination ofrenewable resources, islanding, and volt/VAR compensation.

There are a number of unresolved issues that need to beaddressed, in order to make these applications possible. Asthe cost of PMUs depends on their sensing accuracy, noisyand low-end device will necessarily be preferred in orderto keep the cost of large scale deployment acceptable. Inparticular, time synchronization between different PMUs is amajor technological issue, which is generally tackled via GPSmodules that can provide timestamping of the data.

Following the approach that is typically adopted in wirelesssensor networks [6], [7], in this paper we propose a solution tothese issues that exploits the large number of sensors and theircommunication capabilities to compensate for their modestperformance. We propose a leader-less and distributed stateestimation architecture, in which PMUs communicate withtheir neighbor peers in order to improve the quality of theirmeasurements. Such improved measurements are then madeavailable locally to the monitoring and control algorithmsthat may need them, in a transparent way and without anyaggregation of the data at a central server (see Figure 1).

We design two different state estimation algorithms that canbe adopted in order to implement the proposed architecture.Both have pros and cons, that we discuss. Compared to theexisting approaches to state estimation for power transmissionnetworks (which are reviewed later in this section), we focuson some specific features that are of crucial importance in thescenario of medium voltage power distribution grids:• we need to deal effectively with poorly synchronized

measurements;• we cannot assume large X/R impedance ratio of the power

lines;• we only allow measurements that can be performed by

devices connected to the grid buses, and no power flowand current measurements on the power lines;

• we do not assume redundancy of measurement locations(instead, we exploit the fact of being able to sense bothvoltage and current injection at each bus);

• the computational effort must remain limited as the gridgrows in size;

• we aim at leader-less algorithms (in which no gridsupervisor or central data aggregator is present);

• we allow a peer-to-peer architecture, in which each PMU

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PMU

PMU

PMU

PMU

PMU

power gridphysicallayer

measurementlayer

raw measurements

state estimates

controllayer

U (m), θ(m)u

U , θu

controller

I(m), θ(m)i

Fig. 1. The panel shows a functional representation of the proposed archi-tecture. Each bus is provided with a PMU, which measures bus voltage andcurrent. These device communicate with their neighbors, in order to processthe raw data and obtain a better estimate of the system state. They constitutethe middle measurement layer that we are considering in this paper. The stateestimates computed in this layer are then transmitted to the control layer,where different monitoring and optimization algorithm can coexist.

is provided with its own computational capabilities, andcan communicate with its neighbor PMUs (even if wealso consider the possibility of local data aggregators inthe algorithm, as it might be a convenient architecturewhen combined with some hierarchical communicationinfrastructures [8]).

The paper is organized as follows. In the remainder of thissection, we review the main related technical literature. InSection II we introduce a model for the power grid and forthe measurement errors, where time synchronization errorsare explicitly considered. Based on this model, we detailthe least-square problem that has to be solved in order tofind the maximum-likelihood estimation of the grid state(Section III). In Section IV we illustrate how the problem canbe decomposed into sub-problems, each one correspondingto a subset of the PMUs (possibly to individual PMUs, inthe peer-to-peer case). In Section V we propose two differentalgorithmic approaches. The first approach is based on a dis-tributed implementation of the Alternating Direction Methodof Multipliers (ADMM, [9]). In particular we tailored thealgorithm introduced in [10] to the case of quadratic costfunctions showing that it reduces to a linear iterative algorithm.The second approach is a leader-less distributed Jacobi-likealgorithm, where each agent has to solve an extremely simpleoptimization problem, for which a closed form solution isprovided. Of note is the fact that both algorithms can beimplemented in a scalable way, in which every agent is onlyrequired to store a portion of the entire state of the system. InSection VI we show how both these algorithms are effectivein improving the quality of the measurements, providing aconsistent and accurate estimate of the node voltages, and weevaluate their performance both in the multi-area approachand in the special case of a peer-to-peer architecture. Wethen consider a specific application belonging to the classof problems that can be addressed by the deployment of alive network of µPMUs. Such application is the coordinatedcontrol of the power inverters of microgenerators to providevolt/VAR compensation, i.e. reactive power compensation forvoltage support and power loss minimization [11], [12], [13],[14]. We consider the strategy proposed in [15], and we show

that the measurement errors in standard low-end PMUs areunacceptable for such task. When the proposed state estimationalgorithms are employed, instead, the improved measurementsbecome accurate enough to be used effectively. The resultingstate estimation is particularly robust against synchronizationerrors, to a degree that it is possible to envision implemen-tations in which PMUs are not provided with GPS modules,relying instead on standard synchronization routines over thecommunication network.

A. Related work

Several solutions alternative to the centralized computationof the state of a electric grid have been proposed in theliterature. The most common approach is based in a two levelarchitecture requiring a global coordinator [16], [17], [18],[19], [20], [21]. In this multi-area approach the grid is split intomacro-areas, each of them equipped with a data processor withcommunication and computational capabilities. Each processorestimates the state of the sub-area which has been assignedto it, excluding boundary bus measurements, and sends thisestimate to the global coordinator. The global coordinatorcombines all these estimates making them compatible to theboundary bus measurements obtained at a interconnectionlevel. Of note is that this approach becomes uninteresting inthe case in which each macro area reduces to one single PMU,because the global coordinator would have to face the verysame problem of a centralized state estimator.

Leader-less solutions include approximate algorithms de-veloped from the optimality conditions involved [22], [23].Beyond requiring local observability, these algorithm are notalways guaranteed to converge. The auxiliary principle isinvoked in [24], with the drawback that several paramentershave to be tuned. Also the distributed algorithms proposedin [25], [26], adopt the multi-area approach. Remarkably,the algorithm in [25] is shown to be provably convergentin finite time to the optimal solution of a classical weightedleast problem. However each area is envisioned to maintaina copy of the entire high-dimensional state vector, drasticallyincreasing the communication and computation effort requiredat each iteration. In [26] the overall system is decomposed intosubsystems, each of them provided with a slack bus monitoredby a PMU. Each subsystem obtain its own local estimate,and these estimates are then coordinated based on the PMUmeasurements, re-estimating the boundary state variables atthe aggregation level. However, the global solution obtainedthrough this two-steps procedure is not guaranteed to coincidewith the optimal centralized solution, and no analysis ofpotential synchronization errors is included.

To address the conventional least-squares power systemstate estimation problem, the authors in [10] propose a novelalgorithm which is a local version of the classical ADMM.When dealing with the exact non-linear model of the electricgrid, the functional costs to be minimized are non-convexand, in general, the behavior of the ADMM is characterizedby slow transient performance. For this reason, typically, theexact models are iteratively linearized via the Gauss-Newtonmethod, or by resorting to the so-called DC approximation

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[27]. Either way, one arrives at classical linear models whichare just first-order approximations of the true model. Eventhough the ADMM-based algorithm introduced in [10] ex-hibits good performance in simulations, a complete proof of itsconvergence is not provided. We stress the fact that one of thetwo approaches that we propose in this paper is an extensionof the local ADMM algorithm of [10] to the minimizationof quadratic functions. These quadratic functions are obtainedby casting the state estimation problem as a classical least-squares problem and by adopting a specific linearized modelwhich also allows to deal with the synchronization errors in themeasurements provided by the PMUs. The ADMM algorithmwe propose in this paper reduces to linear iterations and it isshown to be provably convergent. Due to space limitations, wedo not report here the proof of its convergence and we referthe interested reader to the document [28], where we provideall the technical details.

II. POWER DISTRIBUTION GRID MODEL

In this section, we introduce some useful notation and weprovide a description and a mathematical model for the powerdistribution grid.

A. Power grid model

We model a power distribution grid as a directed graphG = (V, E), in which edges represent the power lines, andthe n nodes (n = |V|) represent buses and also the pointof common coupling (PCC), i.e., the point of connection ofthe power distribution grid to the trasmission grid. The gridtopology in represented via the incidence matrix A, defined as

[A]`v =

-1 if ` starts from node v1 if ` ends in node v0 otherwise.

We consider a balanced three-phase grid, in which all steadystate currents and voltages can therefore be represented as acomplex number y = Y exp(jθy) of absolute value Y andphase θy (measured with respect to an arbitrary common timereference). Exploiting this notation, the steady state of thenetwork is described by the following system variables:

• u ∈ Cn, where uv is the voltage at bus v;• i ∈ Cn, where iv is the current injected at bus v.

Let Z = diag(z`, ` ∈ E) be the diagonal matrix of lineimpedances, in which z` is the impedance of the power linecorresponding to edge `. From Kirchhoff’s laws we have

i = Lu, (1)

where L is the complex-valued weighted Laplacian matrix(also known as the grid admittance matrix)

L = ATZ−1A.

In the following, when we refer to the power grid state, werefer to the vector of bus voltages u, which describe the systemworking point uniquely.

B. PMU model

Every bus of the power distribution grid is provided witha PMU, which measures the bus voltage uv = Uv exp(jθuv

)and the injected current iv = Iv exp(jθiv ). Compared to othermeasurement setups presented in the power system literature,especially for the high voltage power transmission network,the setup presented here is therefore characterized by the factthat only nodal measurements are available, while power linesare unmonitored.

We assume that the different PMUs sense the grid atsynchronous, evenly distributed, times t ∈ {kT, k ∈ N}, whereT > 0 is a predetermined sample time, as dictated also by theIEEE standard C37.118-2005 [29].

At every measurement time t, the PMUs at every bus vobtain the following noisy data

U (m)v = Uv + eUv

θ(m)uv

= θuv+ eθuv

+ esync,v

I(m)v = Iv + eIv

θ(m)iv

= θiv + eθiv + esync,v

(2)

where the different measurement error terms are caused byindependent physical causes and can therefore be assumeduncorrelated:

• The magnitude error terms eUvand eIv are caused by

quantization of the acquisition devices and by harmonicdistortion of the signals.

• The phase error terms eθuvand eθiv depend on the

sampling time of the acquisition devices and on thealgorithm employed for the estimation of the voltage-current phase difference, traditionally denoted by φ.

• The synchronization error esync,v represents the phase er-ror due to inexact time synchronization between differentPMUs, and is the same for both the voltage measurementand the current measurement at the same PMU.

We assume that PMUs are homogeneous (even if theproposed formulation could easily account for heterogeneityof the measurement devices) and thus their errors exhibit thesame probability distribution. We adopt a Gaussian distributionfor all these error terms, and therefore we have for all v ∈ V

eUv∼ N (0, σ2

U )

eIv ∼ N (0, σ2I )

eθuv, eθiv ∼ N (0, σ2

θ)

esync,v ∼ N (0, σ2sync).

(3)

A discussion about the variance of these error terms is post-poned to Section VI, where typical values given by industrialstandards, are reported. Notice that, because we are modeling atime synchronization error as a phase angle, a wrapped normaldistribution [30] (or the approximated Fisher-von Mises distri-bution) should be adopted for that specific term. The differenceis however absolutely negligible, because of the extremelysmall size of typical phase measurement errors.

Let us stack all the measurement errors in the vectors

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eU , eθu , eI , eθi , esync, and let

e =

eU

eθu + esynceI

eθi + esync

be the cumulative vector of all noise terms, in the orderintroduced in (2). Then the correlation matrix R for the noiseis

Re = E[eeT ]

=

σ2UIn

(σ2θ + σ2

sync)In σ2syncIn

σ2IIn

σ2syncIn (σ2

θ + σ2sync)In

.where In denotes the n-dimensional identity matrix.

It is worth remarking that the variance σ2sync of the synchro-

nization error is typically larger than the variance of the anglemeasurement error σ2

θ . Therefore, the correlation matrix Re isnot diagonal (i.e., the different cumulative error terms are notindependent). Proper modeling of this fact (as we did here) isthe crucial point that allows to tackle the problem of poorlysynchronized PMUs in an effective way.

III. WEIGHTED LEAST SQUARE STATE ESTIMATION

We adopt the classical approach in which the power state es-timation problem is posed as a weighted least square problem[16], [27]. We therefore consider the optimization problem(

U , θu

)= arg min

U,θuJ(U, θu) (4)

where the cost function J(U, θu) is defined according to themeasurement model of Section II-B, as

J(U, θu) = zTR−1e z, (5)

with z defined as

z =

U (m) − Uθ(m)u − θu

I(m) − I(U, θu)

θ(m)i − θi(U, θu)

. (6)

The functions I(U, θu) and θi(U, θu) in (6) derive fromthe grid model presented in Section II-A and are nonlinearfunctions of the decision variables U and θu. Therefore theoptimization problem (4) is in general non-convex, and canbe difficult to tackle via standard iterative algorithms.

As proposed in other state estimation approaches based onPMUs, we introduce a rectangular representation of the state,so that we will be able to introduce a convenient linearizedmeasurement model and, ultimately, a convenient closed formsolution to (4). Let us define the complex valued measurements

u(m)v = U (m)

v exp[jθ(m)uv

]i(m)v = I(m)

v exp[jθ

(m)iv

].

(7)

The measurement model can be rewritten as a linear measure-ment model in the form

m = H x + η (8)

in which1

m =

<(u(m))=(u(m))<(i(m))=(i(m))

, x =

[<(u)=(u)

], η =

<(eu)=(eu)<(ei)=(ei)

,where the vectors eu and ei are defined element-wise as

euv= uv − u(m)

v , eiv = iv − i(m)v ,

and

H =

In 00 In<(L) −=(L)=(L) <(L)

. (9)

Problem (4) can now be equivalently reformulated as

minx

(m−Hx)T [Rη(x)]−1(m−Hx) (10)

where Rη(x) = E[ηηT ].The nonlinearity of the original model is now entirely

contained in the error covariance matrix Rη(x), which dependson x and makes the problem (10) hard to solve in general.This matrix is not diagonal, because both magnitude andphase errors will be reprojected into both real and imaginarycomponents in the error vector η of the linear model (8).

A suitable approximation for Rη (that maintains the correctmodeling of the time synchronization errors between PMUs)has been obtained in the Appendix, and takes the form

Rη ≈

Σ<(u) Σ<(u)=(u) Σ<(u)<(i) Σ<(u)=(i)

Σ=(u)<(u) Σ=(u) Σ=(u)<(i) Σ=(u)=(i)Σ<(i)<(u) Σ<(i)=(u) Σ<(i) Σ<(i)=(i)Σ=(i)<(u) Σ=(i)=(u) Σ=(i)<(i) Σ=(i)

(11)

where the matrices Σ<(u), Σ<(u)=(u), Σ<(u)<(i), Σ<(u)=(i),Σ=(u), σ=(u)<(i), Σ=(u)=(i), Σ<(i), Σ<(i)=(i), Σ=(i), ∈ Rn×nare all diagonal matrices whose v-th diagonal elements dependon the measurements and are given by[

Σ<(u)]vv

= σ2U cos2(θ(m)

uv) + σ2

θ(U (m)v )2 sin2(θ(m)

uv)

+ σ2sync(U

(m)v )2 sin2(θ(m)

uv)[

Σ=(u)]vv

= σ2U sin2(θ(m)

uv) + σ2

θ(U (m)v )2 cos2(θ(m)

uv)

+ σ2sync(U

(m)v )2 cos2(θ(m)

uv)[

Σ<(i)]vv

= σ2I cos2(θ

(m)iv

) + σ2θ(I(m)

v )2 sin2(θ(m)iv

)

+ σ2sync(I

(m)v )2 sin2(θ

(m)iv

)[Σ=(i)

]vv

= σ2I sin2(θ

(m)iv

) + σ2θ(I(m)

v )2 cos2(θ(m)iv

)

+ σ2sync(I

(m)v )2 cos2(θ

(m)iv

)[Σ<(u)=(u)

]vv

=(σ2U + (σ2

θ + σ2sync)(U

(m)v )2

)·

· sin(θ(m)uv

) cos(θ(m)uv

)[Σ<(i)=(i)

]vv

=(σ2I + (σ2

θ + σ2sync)(I

(m)v )2

)·

· sin(θ(m)iv

) cos(θ(m)iv

)

1The notations <(·) and =(·) represent the real and imaginary part of avector, respectively.

5

[Σ<(u)<(i)

]vv

= σ2syncU

(m)v I(m)

v sin(θ(m)uv

) sin(θ(m)iv

)[Σ<(u)=(i)

]vv

= −σ2syncU

(m)v I(m)

v sin(θ(m)uv

) cos(θ(m)iv

)[Σ=(u)<(i)

]vv

= −σ2syncU

(m)v I(m)

v cos(θ(m)uv

) sin(θ(m)iv

)[Σ=(u)=(i)

]vv

= σ2syncU

(m)v I(m)

v cos(θ(m)uv

) cos(θ(m)iv

).

Notice that the value of these elements in the proposedapproximation depends on the value of the measurements, anddoes not depend on the state x. Therefore, the optimal solutionx∗ of problem (10) can be written in closed form as

x∗ =(HTR−1η H

)−1HTR−1η m. (12)

IV. PROBLEM DECOMPOSITION

To compute the optimal solution x∗ directly as in (12),one needs to know the entire measurement vector and theentire matrices H and Rη . Moreover, the matrix HTR−1η Hneed to be inverted. For a large-scale power distribution grid,such computation imposes a limit to the dimension of thematrix H , and hence on the number of measurements that canbe efficiently processed to obtain a real-time state estimate,given the available computation time between subsequent mea-surement instants. To reduce the computational complexity ofthe problem we adopt the multi-area decomposition approachproposed in [23], and also advocated in the IEEE standardC37.118-2005 [29]. Following this approach, we assume thatthe grid is partitioned into non-overlapping sub-areas, each ofthem provided with a data processor with computational andcommunication capabilities. Each processor• can collect raw data measurement from the PMUs be-

longing to its sub-area;• has a local knowledge of the grid topology, namely, it

knows the grid topology pertaining its sub-area and italso knows which power lines connect the sub-area withother adjacent sub-areas (called neighboring areas);

• can communicate with the monitors of the neighboringsub-areas.

. We aim to design a distribute state estimation algorithmso that, based on the available information, each processorultimately obtains the optimal estimate of the portion of staterelated to its sub-area.

In order to derive a formal notation for the problem de-composition, let the set of buses V be partitioned into the ssub-areas A1, . . . ,As, where |Aj | = nj , with

∑sj=1 nj = n.

Without loss of generality, assume that the bus indices areconveniently ordered, so that the vectors i and u are partitionedas

i =

iA1

...iAs

, u =

uA1

...uAs

.Two sub-areas Ah and Ak are said to be neighboring sub-areas if there is a power line connecting them, i.e. if there existv ∈ Ah and v′ ∈ Ak such that the edge (v, v′) or the edge(v′, v) belong to E . In such case, we say that sub-area Ah isa neighbor of sub-area Ak and vice-versa. For h ∈ {1, . . . , s}we denote by Nh ⊆ {1, . . . , s} the set of the indices of theneighbors of Ah.

The above partition also induces a corresponding blockstructure for the Laplacian L, i.e., L = [Lhk], h, k = 1, . . . , s,such that

iAh=

s∑k=1

LhkuAk=

∑k∈Nh∪{h}

LhkuAk.

The area Ah is assigned to monitor h which1) knows the matrices Lhh, {Lhk}k∈Nh

;2) collects the measurements U (m)

Ah, θ(m)

uAh, I(m)Ah

and θ(m)iAh

;3) knows the variances σ2

U , σ2I , σ

2θ , and σ2

sync.The linear model (8) can therefore be expressed in the

corresponding partitioned form. Specifically

<(u(m)Ah

) = <(uAh) + <(euAh

) (13)

=(u(m)Ah

) = =(uAh) + =(euAh

)

<(i(m)Ah

) =∑k∈Nh∪{h}

(<(Lhk)<(uAk)−=(Lhk)=(uAk

)) + <(eiAh)

=(i(m)Ah

) =∑k∈Nh∪{h}

(=(Lhk)<(uAk) + <(Lhk)=(uAk

)) + =(eiAh)

Now, let

mAh=

<(u

(m)Ah

)

=(u(m)Ah

)

<(i(m)Ah

)

=(i(m)Ah

)

, xAh=

[<(uAh

)=(uAh

)

], ηAh

=

<(euAh

)

=(euAh)

<(eiAh)

=(eiAh)

.Then we can write that

mAh= HhhxAh

+∑k∈Nh

HhkxAk+ ηAh

where the matrices Hhh ∈ R4nh×2nh , Hhk ∈ R4nh×2nk areobtained by reordering the elements of the matrix H definedin (9), according to the Equations in (13).

Finally, observe that the matrix Rη can be rearranged intoa block diagonal matrix diag

(RηA1

, . . . , RηAn

)where the

matrix RηAh∈ R4nh×nh is the covariance matrix associated

to the noise ηAh. By introducing the function

βh(xAh

, {xAk}k∈Nh

)= mAh

− HhhxAh−∑k∈Nh

HhkxAk,

we have that the problem (10) is equivalent to the problem

minxA1

,...,xAs

s∑h=1

Jh(xAh

, {xAk}k∈Nh

)(14)

whereJh(xAh

, {xAk}k∈Nh

)= βThRηAh

βh.

In next section we propose two distributed algorithms tosolve optimization problems in the separable quadratic form(14), and that can therefore be used in our application.

Remark. The case of a peer-to-peer architecture is onespecial case of the architecture that we just described. Suchspecial case can be easily obtained just by considering

6

“atomic” areas, each one encompassing one single PMU. Wetherefore have nj = 1 for all areas, and the set of neighborsof each area simply corresponds to the neighbor buses in thepower distribution grid.

V. DISTRIBUTED OPTIMIZATION ALGORITHMS

The minimization problem formulated in (14) belongs to acommon setup in distributed optimization that we review herein a general form, in order to reduce the complexity of thenotation to a minimum.

Consider a network with set of nodes V = {1, . . . , s} andfixed undirected communication graph G = (V, E). Let Nidenote the set of neighbors of node i, that is, Ni = {j ∈V | (i, j) ∈ E}. The graph G is assumed to be connected.Consider the minimization of a cost function in the form

minx

s∑i=1

Ji(x) (15)

where each Ji is a convex function of x and it is known onlyto node i. Assume problem in (15) admits a unique solutionx∗. In this section we consider problems as in (15) with aspecific structure, that is a partition-based structure. Let thevector x (and, similarly, the vectore x∗) be partitioned as

x =[xT1 , . . . , x

Ts

]Twhere, for i ∈ {1, . . . , s}, xi ∈ Rmi for some mi ∈ N suchthat

∑si=1mi = N . The sub-vector xi represents the relevant

information at node i, referred to, hereafter, as the state ofnode i. Additionally, let us assume that the each objectivefunctions Ji depends only on the state of node i and on itsneighbors, that is, on xi and {xj}j∈Ni

, where xi and xj arethe portions of the state vector x that correspond to node iand to each neighbor node j, respectively. Then the problemwe aim at solving distributively is in the form

minx

s∑i=1

Ji(xi, {xj}j∈Ni). (16)

Because of the specific least square estimation problem thatwe are considering, the functions Ji has a specific quadraticform, and therefore we further restrict to the case where

Ji(xi, {xj}j∈Ni) = (17)qi −Aiixi −∑j∈Ni

Aijxj

T

Qi

qi −Aiixi −∑j∈Ni

Aijxj

.

To solve (16), and therefore to ultimately solve our stateestimation problem, we now propose two iterative algorithms.

A. ADMM-based algorithmThe method we propose in this subsection is a partition-

based version of the classical ADMM method which exploitsthe equivalence between problem in (16) and the followingproblem

minx

s∑i=1

Ji(x(i)i , {x(i)j }j∈Ni

)

subject to x(i)i = z

(i,j)i ; x

(i)j = z

(i,j)j

x(i)i = z

(j,i)i ; x

(i)j = z

(j,i)j , ∀ j ∈ Ni,

(18)

where with the notation x(i)j we denote a copy of state xj

stored in memory by node i, and where the z’s are auxiliaryvariables that are introduced by the ADMM algorithm. Ob-serve that the connectedness of the graph G and the presenceof the bridge variables z′s ensures that the optimal solutionof (18) is given by x(i)i = x∗i and x(i)j = x∗j .

The redundant constraints added in problem (18) with therespect to problem (16), allow to find the optimal solutionthrough a distributed, iterative, partition-based implementationwhich optimizes the standard augmented Lagrangian defined,for ρ > 0, as

L =

s∑i=1

Ji(x(i)i , {x(i)j }j∈Ni) +

∑j∈Ni

[λ(i,j)i

(x(i)i − z

(i,j)i

)+λ

(i,j)j

(x(i)j − z

(i,j)j

)]+∑j∈Ni

[µ(i,j)i

(x(i)i − z

(j,i)i

)+µ

(i,j)j

(x(i)j − z

(j,i)j

)]+ρ

2

∑j∈Ni

[‖x(i)i − z

(i,j)i ‖2

+‖x(i)j − z(i,j)j ‖2 + ‖x(i)i − z

(j,i)i ‖2 + ‖x(i)j − z

(j,i)j ‖2

]}.

At each iteration of the algorithm, node i, i ∈ {1, . . . , s},alternates dual ascent step on the Lagrange multipliers λ(i,j)i ,{λ(i,j)j }j∈Ni

and µ(i,j)i , {µ(i,j)

j }j∈Ni, with minimization steps

on the variables x(i)i , {x(i)j }j∈Ni and z(i,j)i , {z(i,j)j }j∈Ni .However, for the case where the functions J ′is have the

particular quadratic structure illustrated in (17), these opti-mization steps can be greatly simplified. Indeed in this case thepartition-based ADMM algorithm reduces to a linear algorithmrequiring, during each iteration of its implementation, onlyone communication round involving the x

(i)i , {x(i)j }j∈Ni ,

i ∈ {1, . . . , s}, variables. To show that, it is convenient tointroduce the following compact notation. Consider node iand, without loss of generality, assume Ni =

{j1, . . . , j|Ni|

}.

Then let

X(i) =

x(i)i{

x(i)j

}j∈Ni

,Ai =

[Aii Aij1 . . . Aij|Ni|

],

Mi = diag{|Ni| Imi , Imj1

, . . . , Imj|Ni|

}Additionally we introduce the following auxiliary variables,

G(i) =

G

(i)i

G(i)j1...

G(i)j|Ni|

, F (i) =

F

(i)i

F(i)j1...

F(i)j|Ni|

, B(i) =

B

(i)i

B(i)j1...

B(i)j|Ni|

where G(i)

i , F(i)i , B

(i)i ∈ Rmi and G

(i)jh, F

(i)jh, B

(i)jh∈ Rmjh .

It turns out that Ai ∈ Rri×γi , Mi ∈ Rγi×γi andG(i), F (i), B(i) ∈ Rγi , where γi = mi +

∑|Ni|h=1mjh .

The partition-based ADMM algorithm for quadratic func-tions is formally described as follows. The standing assump-tion is that all the matrices ATi QiAi+Mi, i ∈ {1, . . . , n} areinvertible.

7

Processor states: For i ∈ {1, . . . , s}, node i stores a copy ofthe variables X(i), G(i), F (i), B(i).

Initialization: Every node initializes the variables it stores inmemory to 0.

Transmission iteration: For t ∈ N, at the start of the t-thiteration of the algorithm, node i transmits to node j,j ∈ Ni, its estimates x(i)i (t), x(i)j (t). It also gathers thet-th estimates of its neighbors, x(j)j (t), x(j)i (t), j ∈ Ni.

Update iteration: For t ∈ N, node i, i ∈ {1, . . . , s}, basedon the information received from its neighbors, performthe following actions in order:1) it computes G(i)(t+ 1) by setting

G(i)i (t) =

ρ

2

∑j∈Ni

(x(i)i (t)− x(j)i (t)

)G

(i)jh

(t) =ρ

2

(x(i)jh− x(jh)jh

), 1 ≤ h ≤ |Ni|

2) it computes F (i)(t+ 1) by

F (i)(t+ 1) = F (i)(t) +G(i)(t)

3) it computes B(i)(t+ 1) by

B(i)(t+1) = 2ρMiX(i)(t)−G(i)(t+1)−2F (i)(t+1)

4) it updates X(i) as follows

X(i)(t+ 1) =[ATi QiAi +Mi

]−1 [ATi Qizi +

1

2B(i)(t+ 1)

]The following proposition characterizes the performance of

the above algorithm.

Proposition V.1. Consider the partition-based ADMM algo-rithm described above. Let ρ be any real number. Assume thatthe matrices ATi QiAi + Mi, i ∈ {1, . . . , s}, are invertible.Then the trajectory t →

{X(i)(t)

}converge exponentially to

the optimal solution, namely, for i ∈ {1, . . . , n}, x(i)j (t)→ x∗jfor all j ∈ Ni and, in particular,

x(i)i (t)→ x∗i .

Proof: The proof can be found in [28].

Remark. We point out that it is be possible to provide asimplified a version of the partition-based ADMM algorithm,where two monitors communicate with each other only theinformation related to those nodes which are on the boundaryof their sub-areas of interest. To be more precise, monitor ireceives from monitor j only those components of x(j)j (resp.of x(j)i ) such that the corresponding terms in Aij (resp. in Aji)are different from zero. This leads to a vector X(i) which issmaller in size, since monitor i keeps in memory only a copyof the states of the nodes of the other sub-regions which areon the boundary with sub-region Ai. In turn, also the vectorsG(i), F (i), B(i) and the matrices Mi, Ai result to be smallerin size.

It is worth mentioning that this approach has been taken in[10]. In this paper, for the sake of the notational simplicity, andbecause we are mostly interested in cases in which the size ofeach area is small (possibly just one agent), we have preferrednot to provide the details of this implementation aspect.

B. Jacobi-like algorithm

The method we propose in this subsection is inspired bythe Jabobi technique used to iteratively solve systems of linearequations [31]. In the sequel we refer to this method as theJacobi-like algorithm. The algorithm is formally describedas follows. The standing assumption is that the matricesATiiQiAii, i ∈ {1, . . . , s} are all invertible.Processor states: For i ∈ {1, . . . , s}, node i stores an esti-

mate xi(0) ∈ Rmi of its own state.Initialization: Every node initializes its estimate to an arbi-

trary value.Transmission iteration: For t ∈ N, at the start of the t-th

iteration of the algorithm, node i transmits its estimatexi(t) to all its neighbors. It also gathers the t-th estimatesof its neighbors, xj(t), j ∈ Ni.

Update iteration: For t ∈ N, node i, i ∈ {1, . . . , s}, basedon the information received from its neighbors, updatesits estimate as follows

xi(t+ 1) = argminxi

Ji

(xi; {xj(t)}j∈Ni

)=(ATiiQiAii

)−1ATiiQi

zi − ∑j∈Ni

Aijxj(t)

To establish the convergence properties of the Jacobi-like

algorithm it is convenient to introduce the following blockmatrix K = [Kij ], i, j = 1, . . . , s, where Kij ∈ Rmi×mj isdefined as

Kij =

ATiiQiAii if j = iATiiQiAij if j ∈ Ni, j 6= i

0 if j /∈ NiIn the following Proposition, by diag {K} we denote theblock diagonal matrix having in the diagonal the blocksK11, . . . ,Kss. We have the following result.

Proposition V.2. Assume the spectral radius of the matrix(diag {K})−1 (K − diag {K}) is strictly less than one, i.e., allits eigenvalues are strictly inside the unitary circle. Then, thereexists x ∈ RN , such that the trajectory t→ x(t) generated bythe Jacobi-like algorithm converges exponentially to x. If thematrix K is invertible then

x = K−1z,

where z =[(AT11Q1z1)T , . . . , (ATssQszs)

T]T

.

Proof: Observe that the updating step can be written as

x(t+ 1) = (diag {K})−1 (z − (K − diag {K})x(t)) (19)

where (19) represents the standard iteration of the Jacobimethod. The result established in the Proposition follows fromthe classical results on the Jacobi method [31].

Some remark are now in order.

Remark. In general the state x is different from the optimalstate x∗, i.e., the Jacobi-like algorithm does not converge to theoptimal solution. However we will see in Section VI that, when,applied to Problem (14), the Jacobi-like algorithm convergesto an estimate x which is very close to x∗.

8

Areamonitor

4

Areamonitor

1

Areamonitor

1

Areamonitor

2

Areamonitor

2

Areamonitor

3

Areamonitor

3

communication

Fig. 2. IEEE test feeder 123 divided into four non-overlapping areas.

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

mag

nitu

de [p

.u.]

Distance from the real state

measurementsadmm

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

phas

e [p

.u.]

nodes

Fig. 3. Residual error at the steady state of the ADMM algorithm, comparedto the raw measurements.

Remark. It is worth stressing that the computation of theupdate iteration of the Jacobi-like algorithm is simpler, and,in turn, less time-consuming, than the update iteration of thepartition-based ADMM algorithm. This fact is emphasized anddiscussed, also numerically, in Section VI.

VI. SIMULATION RESULTS

In this section we validate and compare the two proposedalgorithms via simulations on standard medium voltage IEEEtestbeds [32]. Specifically, in Subsection VI-A we providea comparison between the performance of the ADMM-basedalgorithm presented in Section V-A and the Jacobi-like algo-rithm presented in V-B on the IEEE 123 test feeder. In SectionVI-B, instead, we evaluate the effect of measurement errorson a prototypical feedback control algorithm on the IEEE 37test feeder, and we quantify the beneficial effects obtained viathe proposed state estimation approach.

A. State estimation accuracy

In order to evaluate and compare the performance of thetwo proposed algorithms, we adopted the IEEE 123 test feeder,

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

mag

nitu

de [p

.u.]

Distance from the real state

measurementsjacobi

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

phas

e [p

.u.]

nodes

Fig. 4. Residual error at the steady state of the Jacobi-like algorithm, in thecase of a 4-area architecture, compared to the raw measurements.

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

mag

nitu

de [p

.u.]

Distance from the real state

measurementsjacobi p2p

0 20 40 60 80 100 120−4

−2

0

2

4x 10

−3

phas

e [p

.u.]

nodes

Fig. 5. Residual error at the steady state of the Jacobi-like algorithm, in thecase of a peer-to-peer architecture, compared to the raw measurements.

whose parameters can be found in [32]. For both algorithms,we consider a 4-area architecture, in which four monitorscollect and process the measurements of the PMUs of theirrespective area (see Figure 2), and a peer-to-peer architecture,in which all PMUs independently process their data and thedata from their neighbors (i.e. areas correspond to individualbuses).

We considered the following standard deviations of themeasurement errors:

voltage amplitude: σU = 10−3UN [Volt]

current amplitude: σI = 10−3Imax [A]

angle: σθ = 10−3 [rad]

sync: σsync = 3 · 10−3 [rad]

where UN is the nominal voltage of the grid and whereImax is the largest current magnitude in the network, i.e.,maxv∈V Iv . These parameters are in accordance with themaximum measurement errors allowed by the IEEE stan-dard C37.118-2005 [29], which specifies only an aggregateconstraint on the measurement errors, without differentiating

9

between magnitude and phase constraints:

|u(m)v − uv||uv|

≤ 1%,|i(m)v − iv||iv|

≤ 1%.

In the upper panel of Figures 3, 4, 5, we plotted, for eachnode v of the distribution test feeder, a comparison between themagnitude errors Uv−Uv and U (m)

v −Uv , while, in the lowerpanel of the same figures, we plotted a comparison betweenthe phase errors θuv−θuv and θ(m)

uv −θuv , where by Uv and θuv

we denote the steady state estimates computed by estimationalgorithms. The state estimates in Figures 3 have been obtainedvia the ADMM-based algorithm, and do not depend on theadopted decomposition of the grid (4 macro areas or peer-to-peer), as the steady state is provably the unique optimalestimate. On the other hand, the estimates in Figures 4 and 5have been obtained via the Jacobi-like algorithm, in the caseof a 4-area architecture and of a peer-to-peer architecture,respectively. In all these cases, the measurement errors aredrastically reduced by both the algorithms, which provide aconsistent and accurate estimate of the node voltages. Noticethat the Jacobi-like algorithm does not converge to the trueoptimal estimate, as a small steady state error is noticeablein the peer-to-peer case. It is however stable, as predictedby Proposition V.2, whose assumptions are all verified in thisscenario.

We also consider the case in which the synchronization erroris quite larger, up to

sync: σsync = 10−2 [rad],

which is comparable to the synchronization accuracy that onecould achieve even without GPS, if the PMUs are synchro-nized via data network synchronization protocols (see forexample [33], [34] and references therein).

In Figures 6 and 7 we plotted the norm of the magnitudeand phase error as a function of the standard deviation σsynchof the time synchronization error between PMUs. It can beseen that the ADMM algorithm outperforms the Jacobi-likealgorithm, even if they both reduce the measurement errorspresent in the raw data. As expected, the curves generated bythe ADMM algorithm correspond to the optimal centralizedestimate.

In order to analyze the computational complexity of thetwo proposed algorithm, we studied the evolution of the errornorm ‖u(t) − u∗‖, where u(t) = U(t)ejθu(t) denotes theestimate computed after t iterations of the algorithms, andu∗ is the optimal state estimate. We first plotted the errornorm as a function of the number of iteration, in Figure 8(for the 4-area architecture) and in Figure 9 (for the peer-to-peer architecture). As expected, the estimate u(t) generatedby Jacobi like algorithm does not converge to u∗; instead,when adopting the ADMM-based algorithm, u(t) convergesexponentially to u∗. Observe however that the updating stepof the Jacobi-like algorithm is computationally much sim-pler than the corresponding iteration in the ADMM-basedalgorithm; for this reason, we then compared the evolutionof the error norm as a function of computational time inFigure 10 (for the 4-area architecture) and in Figure 11 (for

0 0.002 0.004 0.006 0.008 0.01−5

−4.5

−4

−3.5

−3

−2.5

synchronization noise std

Distance from the real state (magnitude)

measurements

jacobi−like

admm−based

centralized

Fig. 6. Comparison of the norm of the magnitude estimation error for differentvalues of the standard deviation σsync. Both algorithms have been reported,together with the raw measurements and the optimal centralized estimate(which corresponds to the estimate generated by the ADMM algorithm). Theplot represents the average of 1000 trials, and the norm of the error has beenplotted in logarithmic scale.

0 0.002 0.004 0.006 0.008 0.01−5

−4.5

−4

−3.5

−3

−2.5

−2

synchronization noise std

Distance from the real state (phase)

measurements

jacobi−like

admm−based

centralized

Fig. 7. Comparison of the norm of the phase estimation error for differentvalues of the standard deviation σsync. Both algorithms have been reported,together with the raw measurements and the optimal centralized estimate(which corresponds to the estimate generated by the ADMM algorithm). Theplot represents the average of 1000 trials, and the norm of the error has beenplotted in logarithmic scale.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

Iterations

log(

J)

Cost function

admmjacobi

Fig. 8. Behavior of the error norm ‖u(t) − u∗‖ for both the ADMM andthe Jacob-like estimation algorithm, in the case of a 4-area architecture, asfunction of the number of iterations. The norm of the error has been plottedin logarithmic scale.

the peer-to-peer architecture). In this comparison, the Jacobi-like algorithm exhibit a faster transient, and may be preferredif the system sampling time is short, compared to the availablecomputational capabilities at the agents.

10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

Iterations

log(

J)Cost function

admm p2pjacobi p2p

Fig. 9. Behavior of the error norm ‖u(t)−u∗‖ for both the ADMM and theJacob-like estimation algorithm, in the case of a peer-to-peer architecture, asfunction of the number of iterations. The norm of the error has been plottedin logarithmic scale.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

Time (s)

log(

J)

Cost function

admmjacobi

Fig. 10. Behavior of the error norm ‖u(t)− u∗‖ for both the ADMM andthe Jacob-like estimation algorithm, in the case of a 4-area architecture, asfunction of time. The norm of the error has been plotted in logarithmic scale.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3.8

−3.7

−3.6

−3.5

−3.4

−3.3

−3.2

−3.1

Time (s)

log(

J)

Cost function

admm p2pjacobi p2p

Fig. 11. Behavior of the error norm ‖u(t)− u∗‖ for both the ADMM andthe Jacob-like estimation algorithm, in the case of a peer-to-peer architecture,as function of time. The norm of the error has been plotted in logarithmicscale.

B. Application example: Volt/VAR compensationIn order to evaluate the expected benefits of the proposed

measurement layer and state estimation protocols to the smart

grid operation, we investigated the effects of noisy measure-ments on the application presented in [15], which is a feedbackcontrol algorithm for voltage support and minimization ofpower distribution losses via optimal reactive power compen-sation.

We consider the IEEE 37 test feeder, on which 6 mi-crogenerators have been deployed and connected to the dis-tribution grid via power inverters (see [15] for the detailsof this scenario). In the mathematical analysis and in thesimulations proposed in [15], noiseless voltage measurementshave been considered. Here we considered the case in whichvoltage measurements are affected by the typical noise thatcharacterize standard medium voltage PMUs, together with asynchronization between PMUs that has standard deviation

σsync = 10−2[rad]

(therefore larger than the accuracy that is typically provided byGPS, and more similar to the accuracy provided by networksynchronization protocols). We show in Figures 12 and 13that, if driven by these raw data, the feedback control law isnot able to reduce power distribution losses on the grid. Weinstead considered the case in which the raw measurementsare processed according to the algorithms proposed in thispaper, and the resulting state estimates are then presented tothe control application. Because the control application is aniterative protocol, we need to enforce a stopping time in theexecution of the state estimation algorithms. We explore twoquite different scenarios:• in the first case, represented in Figure 12, we implement

the Jacobi-like algorithm, and we perform only 5 itera-tions of the algorithm before presenting the state estimateto the control application; because the Jacobi-like algo-rithm is extremely lightweight, this choice allows veryshort sampling time, and fast algorithm execution, alsoin the presence of narrow-band channels like powerlinecommunication;

• in the second case, represented in Figure 13, we insteadimplement the ADMM-based algorithm, and we allowa quite larger time for computation, 50 iterations; thisscenario corresponds to the case in which the control ap-plication is possibly executed at a slower pace, and a fastand reliable communication infrastructure is available.

The simulative results show how both the proposed ap-proach are effective, enable the control algorithm to operatecorrectly, and ultimately allow the reduction of power distri-bution losses on the grid.

VII. CONCLUSIONS

We considered the motivating scenario of a smart powerdistribution grid, in which we assumed that a number ofcontrol applications require precise measurements of the gridstate in order to optimize the grid operation, monitor it againstfault and attacks, and improve grid efficiency. We modeled apractical case in which these measurements are provided byPMUs located at the grid buses, and are affected by noise.In particular, we model the measurement error that is causedby inexact synchronization of the PMUs, which is a major

11

0 5 10 15 20

7.5

8

8.5

9x 10

4Losses

Iterations

True state

Estimates (5 iterates of Jacobi)

Noisy Measurements

Fig. 12. Behavior of the losses minimization feedback control algorithmwhen driven by noiseless voltage measurments (true state), by the raw noisymeasurements, and by the state estimate obtained via the Jacobi-like algorithmwith a stopping time of 5 iterations at each step of the control algorithm.

0 5 10 15 20

7.5

8

8.5

9

9.5x 10

4

Lo

sse

s

Iterations

True state

Etimates (50 iterations of ADMM)

Noisy Measurements

Fig. 13. Behavior of the losses minimization feedback control algorithmwhen driven by noiseless voltage measurments (true state), by the raw noisymeasurements, and by the state estimate obtained via ADMM with a stoppingtime of 50 iterations at each step of the control algorithm.

issue in the deployment of low-end measurement solutions.We designed an intermediate measurement layer that collectsthe raw data from the PMUs, processes it, and presents animproved state estimate to the applications in the control layer.We derived two different distributed and scalable algorithmsin order to process these data and to compute the optimal stateestimate, and we proved their convergence. Via simulations,we showed how these algorithms are extremely effective incanceling the measurement noise, and we showed that the stateestimate computed by the proposed intermediate measurementlayer is accurate enough to be used by the applications inthe control layer. Even if we considered only one specificcontrol application, this result suggests that, via the proposedapproach, time synchronization between PMUs in the powerdistribution network becomes a tractable issue. It also sug-gests that GPS-less PMUs could be used, if proper timesynchronization algorithms are implemented via the availablecommunication channel. This scenario is extremely motivatingfor further investigation in the field of distributed networkedcontrol in the smart power distribution grid.

APPENDIX

In this appendix we derive a suitable expression for thematrix in (11). To do so, we start by expressing the error mea-surements in a convenient form, starting from the knowledgeof σ2

U , σ2I , σ2

θ and σ2sync. Observe that the errors in the vector

η of the linear model (8) are correlated as, in general, bothmagnitude and phase errors (which are uncorrelated) will beprojected into both real and imaginary components. To betterunderstand this fact consider two generic vectors

x = ρ exp(jψ) = ρ(cosψ + j sinψ),

and

x = ρ exp(jψ) = (ρ+ δρ) exp(j(ψ + δψ))

= (ρ+ δρ)(cos(ψ + δψ) + j sin(ψ + δψ))

where δρ and δψ represent the errors affecting the exactvalues ρ and ψ. The quantity x can be rewritten by exploitingclassical trigonometric relations and by taking a first orderapproximation of the sine and cosine functions assuming δψsmall enough, as

x ' ρ(cosψ + j sinψ) + δρ(cosψ + j sinψ)

− ρ(sinψ − j cosψ)δψ − δρ(sinψ − j cosψ)δψ.

Discarding the second order term δρ(sinψ − j cosψ)δψ, weobtain

x ' x+ (δρ cosψ − ρ sinψ δψ) + j(δρ sinψ + ρ cosψ δψ).

Let us go back to our specific context, and to the expressionsin (7). According to the above approximation we have

<(euv) = eUv

cos θ(m)uv − (eθuv

+ esync,v)U(m)v sin θ

(m)uv

=(euv) = eUv

sin θ(m)uv + (eθuv

+ esync,v)U(m)v cos θ

(m)uv

and

<(eiv ) = eIv cos θ(m)iv− (eθiv + esync,v)I

(m)v sin θ

(m)iv

=(eiv ) = eIv sin θ(m)iv

+ (eθiv + esync,v)I(m)v cos θ

(m)iv

.

Consider now[Σ<(u)

]vv

. This term can be computed by

calculating the expectation E[(<(euv ))

2], where θ(m)

uv , U(m)v

are fixed and the noises eUv, eθuv

, esync,v are uncorrelatedGaussian random variables, therefore obtaining[

Σ<(u)]vv

= σ2U cos2(θ(m)

uv) + σ2

θ(U (m)v )2 sin2(θ(m)

uv)

+ σ2sync(U

(m)v )2 sin2(θ(m)

uv).

All the other terms of the matrix Rη are computed similarly.

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