978-1-4673-8562-61631.002016 European UnionFrom Technical Design Structures to Bayesian [615202]

978-1-4673-8562-6/16/$31.00©2016 European UnionFrom Technical Design Structures to Bayesian
Networks in Power Engineering
F. Munteanu, Alexandra Ciobanu, C. Nemes
Department of Power Engineering
“Gheorghe Asachi” Technical University
Iasi, Romania
[anonimizat]

Abstract —Bayesian networks, known as belief or causal networks
also, exploded in last decade as a convenient method used in
different domains: medical diagnosis, availability and reliability
studies, risk analysis, decision procedures, map learning, etc.
This paper is mainly dedicated to the first step in using belief
networks consisting in real technical systems formalization as
Bayesian networks. This step ca n be a simple one but, sometimes
risky and exposed to great errors, related to the equivalent
structure as well as variabl es involved and information
transferring. After a short mathematical background
introduction, the authors focu s on the equivalent Bayesian
networks of some of the main components and structures of
power systems like high voltage circuit-breakers, distribution
networks and usual nodal architectures.
Keywords-Bayesian networks, mo deling, design structures,
power engineering
I. INTRODUCTION
The reverend, statistician, philosopher and mathematician
Thomas Bayes (1701-1761) is the author of well-known Bayes’
theorem or conditional probability theorem. The next axioms
and rules of probability calculus help to understand it [1]:
1. 0 ≤ p(a) ≤ 1 is the probability of event a to occur, while
p(a) = 1 in the case when event is certain;
2. If the events a and b are mutually exclusive,
)( )( ) ( )or ( bp ap bap bap  (1)
For n events, pairwise exclusive,

 


n
ii nn
ii ap ap ap a p
11
1)( )( ………)( (2)
3. The fundamental rule of prob ability or the probability of
the events a and b to simultaneously appear is
)()|( )()|(),( ) ( ) and (
bpbap apabpbap bap b ap
 (3)
The Bayes’ theorem is derived from Eq. (3):
)()()|()|(bpapabpbap (4)
which can be seen as ] evidence)[(] prior)[(] likelihood)[(] posterior)[(bpap abpbap (5)
where a and b are probabilistic events having p(a) and p(b)
their probabilities to appear, p(a│b) is conditional probability
of event a to appear given b is observed and similar to p(b│a).
Bayes' rule tells us how to update our prior knowledge with the data generating mechanism. The prior distribution p(a)
describes the information we ha ve about the variable before
seeing any data. After data b arrives, we update the prior
distribution to the posterior p(a│b) / p(b│a)p(a).
Furthermore, given n mutually exclusive and exhaustive
events a
1, a2, …, a n with p(a i ) ≠ 0 for all i and for 1 ≤ i ≤ n,
)]()|( ..)()|()()|(/[)]()|([)|(
2 21 1
n ni i i
apabp apabpapabp apabp bap
   (6)
Calculating a conditional proba bility using Eq. (4) or Eq. (6)
means the Bayesian inference.
II. WHAT IS A BAYESIAN NETWORK , HOW IT CAN BE
CONSTRUCTED AND WHAT WE CAN USE IT FOR?
Belief networks have the capab ility to include, for analysis
and synthesis the probabilistic i nformation about an event in a
general method based on specific algorithms allowing to reach
important conclusions, making d ecisions, time evolution study
and corresponding effects, utility functions evaluation.
Usually, a probability distribution of random variables can
be extremely difficult to handle using the input data given by
statistics, specific equations or probabilistic systematic
studies. Belief networks, with their graphical representation are considered as a suitable t ool for modeling and reasoning
under uncertainty. “Many appli cations can be reduced to
Bayesian network inference a llowing one to capitalize on
Bayesian network algorithms instead of having to invent
specialized algorithms for each new application” [2].
A.
What is a Bayesian network?
In principle, a Bayesian (belief) network consists in a set of
nodes representing ra ndom variables, each of them having a
finite set of states [3]. Between nodes there are, or not, a set of directed edges showing the causal dependence between random variables. A direct acyclic gr aph (DAG = no loops) is the
formalization of a Bayesian network as shown in Fig.1 where

“A and B are so called ‘parents” or marginal variables and both
are parents of the ‘child’ C, whereas C is a ‘parent’ of both D and E. in this example, C is diverging into D and E. The marginal probabilities to be specified are P(A) and P(B) by their corresponding CPT = conditional probabilities tables in the case of discrete variables or by the CPD = conditional probability distribution in the cas e of continuous variables or
combining CPT and CPF in the same DAG. The Bayes’
theorem based on conditional pro babilities is expressed by p(C
| A,B), p(E | C), p(D | C), p( F | D) and p(G | D,E,F)”.
For example, when A receiv es evidence, then it will
directly influence all the subse quent probabilities. Considering
the DAG in Fig.1, evidence on A can change belief concerning
B because of their connection thro ugh C. It will not affect P(C |
A, B), which is constant (and is part of the variable domain
specification), but it may lead to a different posterior distribution.

A B
C
D E
F G

Fig. 1. DAG as formaliza tion of a Bayesian network
To better understand the essence of such kind of networks it
is necessary to introduce [4]:
– Conditional probability distributions (model
parameters). The relationships between variables are encoded by conditional probability distri butions (CPDs) of the form
p(b│a) = the probability of b given a. For discrete variables,
probability distributions are expressed as conditional probability tables (CPTs) contai ning probabilities that are the
model parameters. For each node, the probability that the variable will be in each possible state given its parent’s states can be calculated based on the frequency observed in a set of training data. It is often necessa ry to use a prior distribution for
the model parameters, as, without a prior, a possible
configuration that was not seen in the training examples would be incorrectly assigned a zero probability of ever being observed.
–
Joint probability distribution (JPD): the JPD over all
variables in a DAG is of a great interest and importance even the number of model parameters for JPD increase dramatically with the number of variables. To diminish this we can use the conditional independence between variables For a Bayesian network (BN) defined over U={A
1, A2, ….., An), JPD = p(U) is
the product of all conditional probabilities specified in BN:

ii ApaAp Up ))( ( )( (7)
For the DAG in Fig.1, Eq. (7) becomes ),, ()(() () (), ().()(),,,,,,(
FEDGpDFpCEpCDpBACPBpApGFEDCBAp
 (8)
– Inference in Bayesian networks : generally, the
inference related to the values of a set of variables can be made given evidence for another set of variables, by marginalizing
over unknown variables. Marginalization means considering
all possible values the unknown variables may take, and
averaging over them. A simple but detailed marginalization
example is given here: let’s consider a two state reliability model of a circuit-breaker failing due to its mechanism or breaker itself (contacts, arcing chamber, or isolation). The breaker and its causal BN corresponding structure is that presented in Fig. 2. A more complex BN for a circuit-breaker is given further in this paper.

Fig. 2. A circuit-breaker (a) and the corresponding maximum simplified BN
(b) to demonstrate de marginalizatio n process; the random variables are P –
pole failures, M mechanism failures and CB – circuit-breaker failures.
The probability distributions of marginal variables
recommended in [6] are
)002.0,998.0()( ) 0002.0, 9998.0()(   Mp Pp (9)
The corresponding CPT is:
P(CB│P,M): P = operating P = failed
M = operating M =failed M =operating M =failed
CB=operating 0.9978 0.002 0.0002 0.0000004
CB = failed 0.0022 0.998 0.9998 0.9999996
The joint probabilit y distribution is
)()(), ( ),,( MpPp MPCBp MPCBp   (10)
Consequently, we can write:
P(CB,P,M): P = operating P = failed
M = operating M =failed M =operating M =failed
CB=operating 0.9956 3.9992·
10-6 3.992·10-8 1.6·10-13
CB = failed 0.0219516088 0.19956 0.00019956 3.999·10-7

Marginalization of M from p(CB,P,M) means calculating

MMPCBpPCBp
),,(),( P = operating P = failed
CB=operating 0.995603992 4.0008·10-7
CB = failed 0.2215116088 0.0001999599
The final step means generally, the normalization through a
probability potential ΓX defined on domain dom(X). ΓX is Mechanism M P
CB
a) b)Pole
Circuit-breaker

transformed in a probability distribution p(X). We note υ(ΓX) to
define normalization of ΓX:

XX X X / )(defined
 (11)
From here, p(X)=υ(ΓX) and for the above presented
example, the CPD p(CB,P) is obtained from JPD p(CB,P)
using the conditional normalization with respect to CB:
)|()(),(
),(),()],([defined
CBPpCBpPCBp
PCBpPCBpPCBp
PX     (12)
The result of normalization is:
P(P|CB)=
ΓCB[p(CB,P)] CB=fail CB=operating
P=fail 0.9999996 0.999098
P=operating 0.0000004 0.000902

– Conditional independence . The independence
between random variables is important due to simplification of calculus. Two variables a and b are conditionally independent
given the third variable c if:
)|()|( )|,( cbpcap cbap  (13)
Graphically, the independence (d-separation) or
dependence (d-connection) can be illustrated using the principle of information flow between the nodes of a DAG. For the elementary connections this principle is presented as it
follows:
–
serial connection, Fig. 3:
Fig. 3. Serial connection
The corresponding JPD of BN from Fig.3 is given by:
)|()|()( ),,( bcpabpap cbap  (14)
The information may be transmitted from a to c unless the
state of b is known.
– diverging connection is given in Fig.4, where
)|()|()( ),,( bcpbapbp cbap  (15)
Fig. 4. Diverging connection
The information can be transmitted through the diverging
connection unless the state of b is known.
The converging connection is de picted in Fig. 5 for which
we can write:
),()()( ),,( cabpcpap cbap  (16) Fig. 5. Diverging connection
Information can pass through the converging connection if
evidence on b or one if its descendants is known.
A better method to detect the (in) dependence of two
variables is based on Markov global oriented method as clearly described in [7]. It can be used for any kind of BN.
B.
How a Bayesian network can be constructed?
A Bayesian network can be constructed starting by
knowing very well the technical or natural system we want to
model. This is a step usually neglected when many people are focusing on probabilities, Bayesi an theorem, and statistical
inference. After this, three major steps are involved [5].
–
– select the sets of variables as well as their discrete (CPT) or
continuous (CPD) values for marginal type; the values are not of an extremely importance at this stage even it is
recommended they be accurate as much as possible;
–
– build the Bayesian network struc ture and attach the variables
to a DAG based on their causal (in)dependence;
– – construct the CPT’s for each ra ndom variable of the network;
this is not an easy task du e to possible large volume of
information to be processed.
Keeping in mind the above mentioned steps, a more
complex and difficult process mean s an intelligent analysis of
the problem (system) states and variables as well as their causality relationship. Synthetically, the process includes:
– Very well-defined variables. As an example, for a circuit-
breaker (CB), the relevant elem ents are: the mechanism, the
protection subsystem, the asso ciated current and voltage
transformers, the foundation, the automation subsystem as random variables consider ing their reliability primary
parameters. Referring to CB states, there are: fail to open or to close when necessary, closing or opening without order as relevant probabilistic real situations.
– Detailed structured system so the cause-effect relationship
between variables can be easy identified. This means good knowledge not only about struct ure (nodes and directed links)
but about CPT or CPD and, if it the case, about utilities
(generally these are outgoing variables like energy flow,
temperature, power, pollution, etc.).
– Time-changing systems can be studied based on “time-
slice” technique [8].
– Optimization of expected utility is usually, combined with
a decision variable. Referring to th e CB in Fig. 2, the decision
of maintenance due to a higher failure rate can be an example of optimization. a b c
a b
c a c
b

C. What we can use Bayesian networks for?
A justified question is what kind of results we can obtain
using a well-defined Bayesian netw ork. Indirectly the answer is
deductible from a “collection” of queries the Bayesian networks can be related to:
– the probability of evidence (see Eq. 5); – prior marginal probability; given a joint probability
distribution p(a1, a2, ….., an), the marginal distribution p(a1,
a2, ….., am),m ≤ n, defined as

n
mn m a aap a aap
12 1 2 1 ) ,…..,,( ) ,…..,,( (7)
– posterior marginal probability given some evidence e:

n
mn m ea aap ea aap
12 1 2 1 )| ,…..,,( )| ,…..,,( (8)
– most probable explanation and maximum a posteriori
hypothesis are also queries well detailed in [4].
Behind these queries, more tec hnical, we can use Bayesian
networks for quantitative purposes:
– calculate the conditional probabilities of different
variables;
– calculate the JPD of all involved variables; – calculate and analyse the causal relationship between
network variables using direct and back propagation of values
for considered sets of variables;
– availability and reliability studies; – optimization of utilities and decisions; – risk analysis.
III.
FROM TECHNICAL STRUCTURES TO BAYESIAN
NETWORKS FOR AVAILABILITY STUDIES
A. Modeling a simple usual nodal architecture
Fig. 6 shows a simple nodal architecture of an electricity
distribution network having tw o sections connected by a
recloser, three power supply s ources. To study the availability
of supply with respect to load point L considering human
restoration errors and upstrea m failures, the corresponding
Bayesian network was established and its structure is presented in Fig.7.
Fig. 6. The simple busbar architecture Fig. 7. Equivalent Bayesian networ k of the simple busbar architecture
To analyse the availability of supply of load L, the random
variables considered, in terms of the corresponding probabilities, are: s1 – source S1, s2 – source S2, si – supply
interruption conditionally depending on S1 and S2, us –
upstream short-circuits, pi – power supply interruption of load
L, rc – recloser failures, hr – human errors to restore the
supply, cd – customer (L) damages.
The links between nodes are reflecting the causal
relationship. The related JPD is given by:
), (),, ()2,1()()()()2()1(),,,,,,2,1(
hrpicdprcussipipsssiphrprcpuspspspcdpisihrrcusssp
 
(9)
B. Modeling a power supply subsystem
The supply network is depicted in Fig.8. The primary
switching equipment is neglected here due to irrelevance.

Fig. 8. The part of a power supply network
The reliability equivalent structural model of this network
based on power capacity of the components to supply the load L inclusively, is shown in Fig.9
Fig. 9. The reliability equivalent stru ctural model of the network in Fig.8 S
T2 T1
L1 L2
T3
L220kV
110k V
20kVBUS2BUS1
BUS3
BUS4
S1 S2 S3
R
Ls1s 2
hrus sirc
pi
cd
ST1
T2 L2L1
T3

For the bridge structure in Fig.9 and using, for example, the
cut-set or tie-set technique it is quite easy to calculate the probability to supply the load L. The Markov chain method is
also an adequate one to co mpute the reliability indices
following the well known steps: es tablish the states table and
states space diagram, calculate the absolute probabilities of
states considered and, finally , calculate the usual system
reliability indices [9].
To take the advantages of the Bayesian network approach,
the supply system presented in Fig.8 can be modelled, for reliability studies, by the corre sponding network in Fig.10.

Fig. 10. The equivalent Bayesian networ k of the supply subsystem in Fig.8 for
availability analysis
The variables s, t1, t2, l1, l2 and t3 are related to the
reliability of the corresponding supply subsystem components.
The intermediary variables bus21, bus22, bus2, bus31, bus32
and bus3 are necessary to model and calculate the voltage
presence on the corresponding subsystem nodes while variable subsystem allows evaluating the entire network reliability. A
energized busbar here means implicitly the rated power can be transferred through and from the upstream components. That means also a perfect reliable bus bar which is much closed to
reality.
C.
Modeling a bridge type structure using the tie-set method
for reliability studies
Fig. 11. A typical bridge nodal structur e; A,B,C,D and E are the equivalent
serial reliability components
The equivalent reliability bl ock and the minimal tie-sets
diagram are presented in Fig. 12. The successful state involves
AC or BD or AED or BEC components to be in good state and
connected. Supposing all com ponents are two states (up and down) from reliability point of vi ew with given probabilities,
the corresponding Bayesian network is shown in Fig.13.
Fig. 12. The minimal tie-sets for the nodal bridge structure

Fig. 13. Bayesian network for relia bility evaluation of the nodal bridge
structure
The results concerni ng the cut-sets a nd system probability
starting from the marginal two st ates (up and down) variables
probabilities, 0.85 and 0.15 respec tively, are shown in Fig. 14.

Fig. 14. The reliability of the bridge type nodal structure calculated using the
Bayesian network based on tie-sets method.
IV. CONCLUSIONS
The Bayesian networks are an extreme versatile tool for
automated reasoning for syst ems driven by probabilistic
variables. They can be constructed according to the axioms and rules of probabilistic calculus and based on the generalised
Bayes’ theorem. Correlated with the aim of study: decision making, risk analysis, system m onitoring or fault detection, the
Bayesian networks can be structured so the results can be useful for engineers. A C BD E
T1 T3 T2 T4
SP Components
reliability

Tie-sets
conditional
reliability

System
conditional
reliability s
t2 t1
bus
21 bus
22
bus
2
bus
3 bus
31 bus
32 l1 l2
t3
sub
system
BS1 S2
L2 L1 A
C DE BA
C E D E D B C A

The authors presented how to m odel simple real systems in
the field of power engineeri ng using conditional probability
theorem trying to cover the initial step indicated by the title of this paper.
Future work is to be dedicated to more complex systems
modelling: multi state circuit-br eakers and power transformers
monitoring and fault diagnosis and availability electricity
distribution subsystems.
R
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