Microgrids offer several reliability benefits, such as the improvement of loadpoint reliability and the opportunity for reliabilitydifferentiated services. The primary goal of this work is to investigate the impacts of operating condition on the reliability index for microgrid system. It relies on a component failure rate model which quantifies the relationship between component failure rate and state variables. Some parameters involved are characterized by subjective uncertainty. Thus, fuzzy numbers are introduced to represent such parameters, and an optimization model based on Fuzzy Chance Constrained Programming (FCCP) is established for reliability index calculation. In addition, we present a hybrid algorithm which combines scenario enumeration and fuzzy simulation as a solution tool. The simulations in a microgrid test system show that reliability indices without considering operating condition can often prove to be optimistic. We also investigate two groups of situations, which include the different penetration levels of microsource and different confidence levels. The results support the necessity of considering operating condition for achieving accurate reliability evaluation.
1. Introduction
Microgrid reliability evaluation is a wellestablished tool to measure the ability of a microgrid of withstanding random failure events in a longterm period. Reliability indices achieved are important parameters in microgrid planning and operation.
The potential benefit of microgrids in grid reliability improvement is an important advantage as pointed out in
[1
,
2]
. However, it is not always true that the reliability of power supply to all customers can be improved. While microsource (
μ
S) may greatly increase power supply reliability for its owner, it may degrade the reliability index and the power quality of other customers on the same feeder
[3]
.
Reliability indices are defined to measure the ability of continuous power supply to customers. There are significant differences between microgrid and distribution system in structure
[4]
. Thus, four special indices were introduced in
[5]
and another two were proposed in
[6]
. However, there is no uniform standard defined for microgrid reliability indices so far. Many studies still follow reliability indices designed for the distribution system
[7
,
8]
. In addition, two critical characteristics of microgrid are integrated
μ
Ss and its autonomous control. Some publications were focusing on those two features
[9
,
10]
, A more recent study attempted to investigate the quantitative impact of cyber network failures on microgrid reliability
[11]
.
In above approaches, the failure rate of component is measured by a constant value, which is provided by the manufacturer or obtained from the statistical data. The impacts of operating condition are ignored in the indices calculation. However, the analysis in
[12]
indicates that the increase in transfer capacity at line will amplify its failure probability. Furthermore, the probabilities of generator tripping and load shedding increase with the continued deviation of voltage. Hybrid conditiondependent outage models for transformer and transmission line were further proposed to include the impacts of various operating conditions
[13]
.
Those previous results indicated that operating conditions have a significant impact on the reliability performance of transmission system only after multiple faults. The major reason for this phenomenon is that ‘N1’ security constraint is normally designed for transmission systems. However, no such preventive measure is normally designed to guarantee the security of microgrid. The purpose of this paper is to investigate the necessity of considering operating condition in the reliability evaluation of a microgrid.
This paper is organized as follows: in section 2, a component failure rate model is proposed to measure the relationship between component failure rate and state variables; in section 3, an optimization model based on FCCP is established for reliability index calculation; in section 4, scenario technology is used to draw distinctions among the operation modes of microgrid; furthermore, a fuzzy simulation is adopted to handle fuzzy factors involved; in section 5, proposed model and approach are tested in a small microgrid system; a summary is given in section 6.
2. Failure Rate Modeling for Feeders
Generally, the failure rate of a feeder is measured by its annual statistical result based on an assumption that this feeder is operating under its rated condition. However, the microgrid has versatile operating conditions due to its multiple operation modes and integrated smallscale
μ
Ss. Abnormal operating conditions occasionally happen, such as overcurrent, overvoltage and undervoltage. Furthermore, bidirectional power flows may occur in microgrids, which is different from the conventional distribution network. Thus, there is a significant necessary to integrate the operating condition in evaluating feeder failure rate.
In this study, voltage
U
and current
I
are selected to represent the state variables of a feeder. Let
λ
_{0}
be the longterm statistical result of the failure rate and
λ
be the one considering operating condition.
λ
could be evaluated by two introduced coefficients
h
(
U
) and
h
(
I
),
 2.1 Overvoltage and undervoltage coefficients
The variation in voltage magnitude is a significant parameter in the measurement of power quality. According to the criterion of power supply in China, the allowable variation
ε
is set to 5% or 7% (the former has been adopted in this study). The maximum variation, which is represented by
δ
, is regarded to be related to customer demand on power quality or the setting value of protective relay. Thus, a piecewise function
h
(
U
) is designed to describe the voltage related coefficients, as shown in
Fig. 1
and explained as follows.
The impact of voltage on shortterm failure rate
1) (1−
ε
)
U_{N}
≤
U
≤ (1+
ε
)
U_{N}
. Here,
U_{N}
is the rated voltage. In the normal condition, conventional failure rate
λ
_{0}
is deployed to
λ
, or
h
(
U
)=1.
2)
U
≤(1−
δ
)
U_{N}
or
U
≥ (1+
δ
)
U_{N}
. Protective device should operate since the state variable exceeds its allowable limits. The feeder will be loss of power until fault recovery. If the shortterm period Δ
T
is comparable to the repair time of the feeder,
λ
can be approximated by 1/Δ
T
[14]
. The expression of
h
(
U
) is
3) (1−
δ
)
U_{N}
≤
U
≤ (1−
ε
)
U_{N}
. A linear function is deployed to fit the relationship between
h
(
U
) and
U
,
4) (1−
ε
)
U_{N}
≤
U
≤ (1+
δ
)
U_{N}
. A linear function is also chosen,
 2.2 Overcurrent coefficient
h
(
I
) is analyzed in a similar way. Assuming
I_{N}
is normal rating value and
I_{S}
is shorttime rating value, which are defined in
[15]
, the expression of
h
(
I
) is explained in (5) and
Fig. 2
.
The impact of current on shortterm failure rate
3. Shorttime Reliability Evaluation Model.
Scenario technology is introduced to handle the stochastic environment related to operating environment. In addition, parameters involved in failure rate models are simulated using fuzzy numbers and investigated based on credibility theory.
 3.1 Operating condition related reliability indices
Many
μ
Ss are fundamentally dependent on weather condition. For example, photovoltaic arrays rely on illumination and its ambient temperature must be in a special range for normal operation. Similarly, wind turbine generator delivers power only within a special speed limit. Although weather conditions could not be forecasted accurately with current weather forecasting technology, annual meteorological data follows certain regularity in the location of microgrid. Those meteorological data including ambient temperature
E
(
t
), illumination
S
(
t
) and wind speed
V
(
t
) can be obtained from local meteorological department.
Each combination of
E
(
t
),
S
(
t
) and
V
(
t
) is defined as a scenario represented by
σ
, as shown in (6). Scenario
σ
is distinguished by the values of three parameters.
Considering those three parameters are continuous variables, scenario number is a tremendous value. Therefore, meteorological data are discretized in the following way: the same value is deployed for all sampling parameters at the range within an individual step. Assuming
n
(
σ_{e}
) is the number of scenario
σ_{e}
,
N_{σ}
is scenario types and
f
is sampling frequency, their relationship is described in (7).
Subsequently, six reliability indices defined for distribution system
[16]
are modified to evaluate the reliability of microgrid considering operating conditions,
where,
N
is the number of load points;
ξ_{i}
(
σ_{e}
) is average failure rate of
i
th load point LP
_{i}
under scenario
σ_{e}
;
ω_{i}
(
σ_{e}
) is average outage time of LP
_{i}
;
m_{i}
is the number of customer at LP
_{i}
;
D_{i}
(
σ_{e}
) is load demand at LP
_{i}
.
 3.2 Optimization model
Although factors including
ε
,
δ
,
I_{N}
and
I_{S}
rely on the property of a feeder, human experience and decision always play an important role in determining those parameters. In addition, some operating constrains are mathematically soft constraints, such as
U
≤ (1+
ε
)
U_{N}
and
I
≤
I_{N}
. Thus, those parameters are subject to a subjective uncertainty and they are represented by a fuzzy variable
in this study. Assuming
is determined by a triangular fuzzy value (
ρ
_{1}
,
ρ
_{2}
,
ρ
_{3}
), its membership function
is shown in
Fig. 3
The membership of fuzzy number
According to the above analysis, each defined reliability index
F_{k}
(
k
=1, 2,…, 6) is also a fuzzy value and it is handled based on credibility theory
[17]
. In each scenario, shorttime reliability indices are calculated by an optimization model based on FCCP. The objective function is chosen as the
γ_{k}
pessimistic value of reliability index
F_{k}
. It is desired that constraint ‘
F_{k}
≤
F_{k,M}
’ must hold at a confidence level
γ_{k}
.
Subject to:
where,

Λiis the set of nodes which are connected to nodei;

θijis the phase angle difference of feederij;

gijis the resistance of feederijandbijis reactance;

GP,i/GQ,iis the power output ofμS;

DP,i/DQ,iis active/reactive load demand at nodei;
Constraints (16) and (17) represent power flow equations; constraints (18) and (19) set limitations for state variables. In fuzzy chance constraint (15), credibility measure ‘Cr{
F_{k}
≤
F_{k,M}
}’ indicates the satisfaction that
F_{k}
is not larger than
F_{k,M}
· Cr{
F_{k}
≤
F_{k,M}
} can be calculated by possibility measure ‘Pos{
F_{k}
≤
F_{k,M}
}’ and necessity measure ‘Nec{
F_{k}
≤
F_{k,M}
}’,
4. Solution Process
A failure mode and effects analysis based hybrid approach is proposed to calculate reliability indices.
 4.1 Load point classification
All feeder failures contributed to the reliability indices of load points are enumerated. Protective devices including relays and fuses then operate to limit failure impacts in a certain area. Subsequently, the whole system is divided into several subsystems according to the switch statuses of breaker, disconnector and tie switch. For a good explanation, load points LP
_{i}
(
i
=1, 2, …,
N
) under
l
th fault (
l
=1, 2, …,
M
,
M
is the feeder number) are classified into four classes, as given in
Table 1
.
Classification of load points
Classification of load points
Class A
: this class is not connected to any backup power sources and they will be loss of power until fault recovery. Its outage time
r_{il}
is depended on mean repair time
t_{r}
.
Class B
: no influence.
Class C
: the load recovery of this class is after disconnector operation.
r_{il}
is fault isolation time
t_{d}
.
Class D
:
r_{il}
relies on available power from
μ
S or tie switch with surplus capacity.
r_{il}
is the sum of
t_{d}
and load transfer time
t_{c}
. In some cases, only
y
% of load demand can be restored for a load point; and then
r_{il}
is corrected by
t
’,
This classification is based on a premise that a fault has happened. LP
_{i}
may be categorized into different classes under different faults.
Four classes of load points have different priorities in load recovery. Breadthfirst search strategy is adopted to determine load restore sequence. Load point with the closest electrical distance from
μ
Ss is recovered first. If more than one load points have the same electrical distance, load importance is used as a priority rule.
 4.2 Calculation of customer reliability indices
Three hypotheses are first introduced to evaluate customer reliability indices:

1) Circuit breakers are installed only in the power source end of main feeder or access points ofμSs;

2) Disconnectors are allocated in all subsections of main feeder and fuses are installed in all lateral distributors;

3) Fuse operation is instantaneous and accurate.
For LP
_{i}
, its average failure rate
ξ_{i}
, average outage time
ω_{i}
and average annual outage time
T_{i}
are calculated according to
λ_{l}
and
r_{il}
analyzed above, as shown in (22), (23) and (24), respectively. Here, sgn(·) is the signum function in mathematics. Subsequently, system reliability indices are evaluated according to their definitions.
Taking LP
_{1}
in
Fig. 4
as an example, the evaluation process of its reliability indices is summarized in
Table 2
and analyzed as follows,

1) If a fault occurs at main feederLij, LP1is classified as class C or D andr1lis equal totdor ‘td+tc’.

2) If a fault occurs at main feederLjk, LP1belongs to class C andr1lequals ‘td’.

3) If a fault occurs at lateral distributorl1, LP1is class A. It will be recovered only afterl1is repaired.
A part of microgrid system
Reliability indices of LP1
Reliability indices of LP_{1}
 4.3 Fuzzy simulation forγkpessimistic value Fk,M
Fuzzy simulation is used to approximate the fuzzy chance constraint ‘Cr{
F_{k}
≤
F_{k,M}
}≥
γ_{k}
’ in the optimization model. A monotonous function
L
(
f_{k}
) is introduced for the calculation of
F_{k,M}
, as in (25). A bisection search is employed to find the maximal value
f_{k}
which is used to estimate
F_{k,M}
.
The procedure of fuzzy simulation is shown in
Fig. 5
and main steps are given as follows.
The flowchart of approach
Step 1)
Generate a fuzzy vector
ρ_{m}
= (
ρ
_{1}
,
ρ
_{2}
, ⋯
ρ_{NF}
) from credibility space randomly,
m
=1,2,…
N_{M}
(
N_{F}
is the number of fuzzy variables and
N_{M}
is fuzzy sampling number);
Step 2)
Calculate a membership vector
v_{m}
=
μ
(
ρ_{m}
);
Step 3)
Find
f_{k}
satisfied the inequality constraint ‘
L
(
f_{k}
) ≥
γ_{k}
’;
Step 4)
Return
f_{k}
as the estimation value of
F_{k,M}
.
Step 5)
Repeat steps 1 ~ 4
N_{σ}
times.
Step 6)
Calculate the reliability indices according to (8)(13).
5. Case Study
The microgrid system proposed in
[18]
is chosen as a test system and some modifications were made by the authors in
[14]
. This system comprises 11 nodes, 12 main feeders and 10 load points (LP).
U_{N}
is 400V and
δ
is set to 10%. This study generates four kinds of fuzzy parameters by multiplying a triangular fuzzy factor (0.9, 1.0, 1.1).
 5.1 Comparisons of reliability indices
A simulation was implemented to compare the proposed approach with the conventional approach, in which
λ
_{0}
is used to assess the failure rate of each feeder. The reliability performances of feeders, customers and the system are presented as follows.
 5.1.1 Feeder reliability
The failure rates of feeders except for two tie lines are compared in
Fig. 6
. It can be noticed that feeder failure rates increase remarkably in the proposed approach. For example, maximalratio of two cases is close to 2.86 times at No.8 feeder. In addition, the failure rates of No.1 and No.2 feeders have two biggest values due to the fact that they are two longest feeders. Some effective measures are needed on them to improve their reliability indices.
The comparisons of feeder failure rate
 5.1.2 Customer reliability
Average annual outage times of load points are compared in
Fig. 7
. Load points including LP3, LP5, LP6, LP8, LP9 and LP10 have relatively low values. It is because there are
μ
Ss in or near those load points. The proportions of the results achieved by two approaches are also indicated in
Fig. 7
. Load points far away from
μ
Ss are influenced much less.
Average annual outage time of customers
On the other hand,
μ
S development will not help customers to decrease their failure rate of power supply. However, it can shorten outage time and recover power as soon as possible.
 5.1.3 System Reliability
Six indices defined in (8) ~ (13) are compared in
Table 3
. In the conventional approach, it is assumed that all feeders are operating under normal conditions. Therefore, only statistical results of feeder failure rates are used in the evaluation process. The achieved reliability indices turn out to be optimistic results. It can be found in
Table 3
that the most significance difference between two cases is SAIFI. Its deviation reaches more than 50%.
Reliability indices of microgrid system
Reliability indices of microgrid system
 5.2 Impacts of operating condition
Reliability indices related to system and customers are calculated based on the failure rates of feeders; therefore, this section only studies the relationship between operating condition and feeder failure rate. In
Fig. 8
, the horizontal axis shows main feeders; the vertical axis shows load rate (
I
/
I_{N}
) and voltage deviation Δ
U
(Δ
U
=
U

U_{N}
). Bar graph represents outage rate per unit length, which is different from the curves in
Fig. 6
.
Operating conditions and failure rates of main feeders
Two conclusions are deduced from
Fig. 8
: (1) The outage rates of feeders which are close to main power source or the slack bus are more sensitive to load rate than voltage deviation, exemplified by four feeders including No. 1, No. 2, No.3 and No.8. (2) Feeders far away from power sources are more likely to be influenced by voltage deviation, such as, No. 5, No. 7, No. 9 and No. 10.
 5.3 Reliability indices of microgrid with different penetrations
μ
Ss are normally installed near to load points; therefore, they can potentially improve customer and system reliability indices. This section investigates system reliability indices and the penetration level of
μ
S. A penetration factor
η
is defined as,
where
G
(
σ_{e}
) is total power output of
μ
S in scenario
σ_{e}
and
D
(
σ_{e}
) is total load demand.
The simulation of SAIFI is shown in
Fig. 9
. SAIFI could not be improved when
η
is larger than 0.4528. In other words, there is a limitation for SAIFI improvement through
μ
S installment. This conclusion holds a significant value in microgrid planning. Furthermore,
η
is not the only reason for SAIFI variation. Small margin fluctuations in the curve are jointly caused by the types or locations of
μ
S.
SAIFI under different penetrations
 5.3.1. Pessimistic values under different confidence levels
The confidence level is one of the most important parameters in the FCCP based model.
Fig. 10
shows that
γ_{k}
pessimistic value
F_{k,M}
is monotonic increasing along with confidence level
γ_{k}
. The larger
γ_{k}
is, the larger value
F_{k,M}
is needed to satisfy the inequality ‘
F_{k}
≤
F_{k,M}
’. Thus, only a small reliability index can be expected when a large value of confidence level is demanded.
γ_{k}pessimistic values of ENS
6. Conclusion
This paper investigates the impacts of operating condition on microgrid reliability. The simulation on a microgrid test system demonstrates the necessity for analyzing component failure rate in the microgrid reliability evaluation. It also shows that customers in a microgrid could not have the same reliability benefit of power supply at different locations and periods.
Acknowledgements
This work is supported by National Natural Science Foundation of China (51407128).
BIO
Xufeng Xu He received the B.E. degree and the Ph.D. degree in electrical engineering from Zhejiang University, Hangzhou, China. His research interests include microgrid reliability, power system planning, and optimization algorithms.
Joydeep Mitra He received the B. Tech. (Hons.) degrees in electrical engineering from Indian Institute of Technology, Kharagpur, India, and the Ph.D. degree in electrical engineering from Texas A&M University, College Station, TX, USA. His research interests include power system reliability, distributed energy resources, and power system planning.
Tingting Wang She received the Ph.D. degree in electrical engineering from Clemson University, Clemson, USA. Her research interests are renewable energy integration, power system transient and electricity market.
Longhua Mu He received the B.E., M.E. and Ph.D. degrees in electrical engineering from China University of Mining and Technology, Xuzhou, China. His current research interests include protective relaying of power system, power quality and microgrid control.
Costa P. M.
,
Matos M. A.
2009
“Assessing the contribution of microgrids to the reliability of distribution networks”
Elect. Power Syst. Res.
79
(2)
382 
389
DOI : 10.1016/j.epsr.2008.07.009
Kennedy S.
“Reliability evaluation of islanded microgrids with stochastic distributed generation,”
Power Eng. Soc. Gen. Meeting
Calgary, AB, Canada
July 2630, 2009
Dugan R. C.
,
McDermott T. E.
“Operating conflicts for distributed generation on distribution system”
Rural Electric Power Conferenc
Little Rock, Arkansas, USA
April 29May 1, 2001
Luo Y.
,
Wang L. J.
,
Zhu G.
,
Wang G.
“Network analysis and algorithm of microgrid reliability assessment”
AsiaPacific Power and Energy Engineering Conference (APPEEC)
Chengdu, China
March 2831, 2010
Yokoyama R.
,
Niimura T.
,
Saito N.
“Modeling and evaluation of supply reliability of microgrids including PV and wind power”
Power and Energy Society General Meeting – Conversion and Delivery of Electrical Energy in the 21st Century
Pittsburgh, USA
July 2024, 2008
Li Z.
,
Yuan Y.
,
Li F.
“Evaluating the reliability of islanded microgrid in an emergency mode”
45th International Universities Power Engineering Conference (UPEC)
Aug. 31 Sep. 3, 2010
Bae I. S.
,
Kim J. O.
2007
“Reliability evaluation of distributed generation based on operation mode”
IEEE Trans. Power Syst.
22
(2)
785 
790
DOI : 10.1109/TPWRS.2007.894842
Bae I. S.
,
Kim J. O.
2008
“Reliability evaluation of customers in a microgrid,”
IEEE Trans. Power Syst.
23
(3)
1416 
1422
Kennedy S.
,
Marden M. M.
“Reliability of islanded microgrids with stochastic generation and prioritized load”
IEEE Bucharest Power Tech Conference
Bucharest, Romania
June 28July 2, 2009
Bie Z.
,
Zhang P.
,
Li G.
,
Hua B.
,
Meehan M.
,
Wang X.
2012
“Reliability evaluation of active distribution systems including microgrids”
IEEE Trans. Power Syst.
27
(4)
1 
9
DOI : 10.1109/TPEL.2011.2178735
Falahati B.
,
Fu Y.
,
Wu L.
2012
“Reliability assessment of smart grid considering direct cyberpower interdependdencies”
IEEE Trans. Smart Grid
3
(3)
1 
10
DOI : 10.1109/TSG.2012.2235092
Sun Y. Z.
,
Cheng L.
,
Liu H. T.
,
He S.
“Power system operational reliability evaluation based on realtime operating state”
the 7th International Power Engineering Conference
Nov. 29Dec. 2, 2005
722 
727
He J.
,
Sun Y. Z.
,
Wang P.
,
Cheng L.
2009
“A hybrid conditions dependent outage model of a transformer in reliability evaluation”
IEEE Trans. Power Deli.
24
(4)
2025 
2033
DOI : 10.1109/TPWRD.2009.2028771
Xu X.
,
Mitra J.
,
Wang T.
,
Mu L.
2014
“Evaluation of operational reliability of a microgrid using a shortterm outage model”
IEEE Trans. Power Syst.
29
(5)
2238 
2247
Billinton R.
,
Allan R. N.
1996
Reliability evaluation of power systems
(Edition 2)
Plenum Press
New York, NY
Liu B.
2009
Theory and Practice of Uncertain Programming
(Edition 2)
SpringerVerlag
Berlin, Germany
Rudion K.
,
Orths A.
,
Styczynski Z. A.
,
Strunz K.
2006
“Design of benchmark of medium voltage distribution network for investigation of DG integration”
proc. Power Engineering Society General Meeting
Montreal, QC, Canada