1. Tree/Chain Approximation of Renewable Energy Generators Covariance Matrix
Navid Tafaghodi Khajavi and Anthony Kuh ({navidt,kuh}@hawaii.edu)
Electrical Engineering Department, University of Hawaii at Manoa
Objectives
A key to smart grids is getting information from
the grid in real time and especially at the distri-
bution level beyond the substation which we refer
to as the microgrid.
• Increasing the penetration of renewable sources
• Come up with a low complexity distributed
state estimation algorithm using greedy
approach and Cholesky factorization
• Approximate the first order Markov chain/tree
Structure for Microgrid Renewable Generators
Covariance Matrix
Background and Motivation
Due to large number of states in smart grid it is
maybe difficult to perform state estimation in real-
time fashion. In contrast, distributed state estima-
tors can give a reasonably good estimates for large
grid systems in real-time. This fact causes a trade
off between calculation time and accuracy of estima-
tion. Here, we seek ways to increase the accuracy of
distributed state estimators which are passing local
messages. We modified the distributed state estima-
tion method proposed in [1] such that it can perform
in real-time fashion. Since smart grid operators need
real-time state estimation, distributed state estima-
tors that achieve good accuracy are desirable but
present major challenges. Note that, with increas-
ing penetration of distributed correlated renewable
energy generators (REGs) this adds complexity to
the smart grid. A factor graph representation with
REG introduces many loops to the original graph.
This can create problems for the convergence of be-
lief propagation (BP) algorithm.
Micro Grid Model
Let X [V I]T
∈ Rn
be the state vector and G ∈
Rp
be the REGs’ current vector where p < n. Let
Y ∈ Rm
be the observation vector where m < n.
The microgrid is described as
X = AG and Y = BX + W
where A is microgrid characteristic matrix. G is
REGs’ random vector, G ∼ N (0, Σ). W is a zero-
mean iid Gaussian vector with covariance matrix
D = σ2
Ip and B is the sensor placement matrix.
Optimization Problem
Goal: To find a tree approximation, Σ∗
, that mini-
mizes Kullback Leibler divergence (maximizes aver-
age data log-likelihood) as follow:
Σ∗
= arg min
Σ∈T
D(fΣ(Y )||fΣ(Y ))
where T is the set of all tree structured matrices,
fΣ(Y ) is the observation distribution and fΣ(Y ) is
the observation approximated distribution.
To solve this problem we use Expectation-
Maximization (EM) algorithm.
The Factor Graph Representation of the Micro Grid
G G
G
G
G
G G
G
G
G
G G
G
G
G
Figure 1: Microgrid with correlated REGs (left), its Factor Graph (Middle) and reduced Factor Graph by Chain approximation (Right)
Tree/Chain Approximation Alg.
Initialization Step (l = 1):
• Σ1
= diag(diag(Σ)
While convergence occur, do (l-th Step):
• Compute Ωl
= E[EΣl−1(XXT
|Y )]
• For Tree approximation compute
Σl
= chow-liu(Ωl
)
• For Chain approximation compute
Σl
c = Burg-greedy(Ωl
)
The Tree approximated covariance is: Σ∗
= Σl
The Chain approximated covariance is: Σ∗
c = Σl
c
Note that, EM based Algorithms are con-
verging to local minimum.
Simulation Circuit
Figure 2: A typical radial simulation circuit (r = 1,R = 20)
Simulation Results
0 1 2 3 4 5
10
−3
10
−2
10
−1
10
0
σ2
KullbackLeiblerdivergence
Tree
Chain
Figure 3: Kullback Leibler divergence vs. the observation noise variance
Conclusion
The optimal MSE solution involves inverting large
matrices and so is infeasible due to complexity in
real time. This paper presents a distributed state
estimators by performing loopy GBP algorithm. To
assure the convergence of loopy GBP algorithm, we
approximate the covariance matrix of inputs using
EM based algorithm which gives us a simple dis-
tributed state estimator with good performance.
References
[1] Y. Hu, A. Kuh, T. Yang, and A. Kavcic, “A belief propagation based
power distribution system state estimator,” IEEE Computational
Intelligence Magazine, vol. 6, no. 3, pp. 36–46, 2011.
[2] N. Tafaghodi Khajavi, and A. Kuh, “First Order Markov Chain
Approximation of Microgrid Renewable Generators Covariance
Matrix,” in Proceedings IEEE International Symposium on
Information Theory (ISIT 2013), July 2013.
[3] J. P. Burg, “Maximum entropy spectral analysis,” in Proc.37th
Meet.Society of Exploration Geophysicists, 1967., pp. 34–41, 1978.
[4] C. K. Chow, C. N. Liu, “Approximating discrete probability
distributions with dependence trees,” IEEE Transactions on
Information Theory, pp. 462–467, 1968.
Acknowledgements
This work was supported in part by NSF grants ECCS-098344, 1029081,
D0E grant, DE-0E0000394, and the University of Hawaii REIS project.