1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.
-
Upload
leo-harper -
Category
Documents
-
view
219 -
download
0
Transcript of 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.
![Page 1: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/1.jpg)
1
© A. Kwasinski, 2015
Cyber Physical Power Systems
Fall 2015
Power in Communications
![Page 2: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/2.jpg)
2
© A. Kwasinski, 2015
Information and Communications Tech. Power Supply
• ICT systems represent a noticeable (about 5 % of total demand in U.S.) fast increasing load.• Increasing power-related costs, likely to equal and exceed information and communications technology equipment cost in the near to mid-term future.
Example of a server in a data center normalized to 100 W:• 860 W of equivalent coal power is needed to power a 100 W load
![Page 3: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/3.jpg)
3
© A. Kwasinski, 2015
Information and Communications Tech. Power Supply
• In addition to been efficient, ICT power plants need to be highly reliable/available.
![Page 4: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/4.jpg)
4
© A. Kwasinski, 2015
Land-line Telecommunications Network
![Page 5: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/5.jpg)
5
© A. Kwasinski, 2015
Land-line Telecommunications Network • Power infrastructure is for telecommunication networks as cardiovascular
system is for humans.
• Power needs to be provided to the switch (nowadays it is a “big computer” routing packets of information) and sometimes to remote terminals.
• CATV systems are similar
![Page 6: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/6.jpg)
6
© A. Kwasinski, 2015
Wireless Telecommunications Network
![Page 7: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/7.jpg)
7
© A. Kwasinski, 2015
Wireless Telecommunications Network
• Power needs to be provided to the switch (called Mobile Telecommunications Switching Office or MTSO) and to the remote terminals (the based stations).
![Page 8: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/8.jpg)
8
© A. Kwasinski, 2015
Power plants architectures
• DC: For telephony and wireless communication networks
• AC: For data centers RECTIFIER + DC-DC CONVERTER
![Page 9: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/9.jpg)
9
© A. Kwasinski, 2015
• Typical configuration in data centers:
• Total power consumption: > 5 MW (distribution at 208V ac)
Power plants architectures
![Page 10: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/10.jpg)
10
© A. Kwasinski, 2015
Power plants architectures
• Typical power plant for telephony networks:
![Page 11: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/11.jpg)
11
© A. Kwasinski, 2015
• Typical centralized architecture for telephony networks:
Centralized architecture
Only (centralized) bus bars
Power plants architectures
![Page 12: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/12.jpg)
12
© A. Kwasinski, 2015
Distributed architecture
Each cabinet with its own bus bars connected to its own battery string and loads. Then all cabinets’ bus bars are connected
• Typical distributed architecture for telephony networks:
Power plants architectures
![Page 13: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/13.jpg)
13
© A. Kwasinski, 2015
• 13 x 200 Amps. Rectifiers• 11 x 1400 Ah Batteries
TelecomPower Plant
Telecom central office power plant
![Page 14: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/14.jpg)
14
© A. Kwasinski, 2015
Telecom central office power plant
![Page 15: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/15.jpg)
15
© A. Kwasinski, 2015
Batteries
![Page 16: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/16.jpg)
16
© A. Kwasinski, 2015
Distribution frames
![Page 17: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/17.jpg)
17
© A. Kwasinski, 2015
Distribution frames
![Page 18: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/18.jpg)
18
© A. Kwasinski, 2015
Inverters
![Page 19: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/19.jpg)
19
© A. Kwasinski, 2015
Base station power plant
![Page 20: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/20.jpg)
20
© A. Kwasinski, 2015
Base station power plant
![Page 21: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/21.jpg)
21
© A. Kwasinski, 2015
Base station power plant
![Page 22: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/22.jpg)
22
© A. Kwasinski, 2015
22
Telephony outside plant
• Digital Loop Carrier and other outside plant broadband remote terminals may provide service up to 500 subscribers in average.
• Local backup is usually provided by batteries with 8 hrs of autonomy• Significant variations in power consumptions:
![Page 23: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/23.jpg)
23
© A. Kwasinski, 2015
RECTIFIERS
Telephony outside plant
![Page 24: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/24.jpg)
24
© A. Kwasinski, 2015
Telephony outside plant
![Page 25: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/25.jpg)
25
© A. Kwasinski, 2015
Outside plant power supply
• Traditional emergency power solutions during long grid outages
![Page 26: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/26.jpg)
26
© A. Kwasinski, 2015
• Reliability applies to components. Once they fail, they cannot be repaired.
• Reliability, R, is defined as the probability that an entity will operate without a failure for a stated period of time under specified conditions.
• Unreliability is the complement to 1 of reliability (F = 1 – R)
F(t) = Pr{a given item fails in [0,t]}
• F(t) is a cumulative distribution function of a random variable t with a probability density function f(t).
• Both F(t) and f(t) can be calculated based on a hazards function h(t) defined considering that h(t)dt indicates the probability that an item fails between t and t + dt (“event A”) given that it has not failed until t (“event B”). From Bayes theorem
Reliability and Availability
• Reliability
Pr{ | }Pr{ } Pr{ }( ) Pr{ | }
Pr{ } Pr{ }
B A A Ah t dt A B
B B
![Page 27: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/27.jpg)
27
© A. Kwasinski, 2015
• Since • Pr{B|A} = 1• Pr{A} = f(t), • Pr{B} = 1 - F(t).
• Then
• and
Reliability and Availability
• ReliabilityPr{ | }Pr{ } Pr{ }
( ) Pr{ | }Pr{ } Pr{ }
B A A Ah t dt A B
B B
( )( )
1 ( )
f th t dt
F t
0( )
( ) 1th d
F t e
![Page 28: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/28.jpg)
28
© A. Kwasinski, 2015
• The hazards function may take various forms and is a combination of various factors. Typical forms for electronic components (solid lines) and mechanical components (doted lines) with the three most characteristics components (early mortality, random and wear out) are
Reliability and Availability
• Reliability
![Page 29: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/29.jpg)
29
© A. Kwasinski, 2015
• Considering electronic components during the useful life period, the hazards function is constant and equals the so called constant failure rate λ. So,
F(t) = 1 – e- λt
f(t) = λe- λt
R(t) = e- λt
• And,
• The inverse of λ is called the Mean Time to Failure. I.e.,it is the expected operating time to (first) failure
Reliability and Availability
• Reliability
R(t)
t
0
1[ ( )] ( )E f t tf t dt
![Page 30: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/30.jpg)
30
© A. Kwasinski, 2015
• The failure rate of a circuit is in most cases the sum of the failure rate of its components.
• General form for calculating failure rate (from MIL-Handbook 217):
• Aluminum electrolytic capacitors tend to be a source of reliability concern for PV inverters. Although their base failure rate is low (about 0.50 FIT), the adjusted failure rate is among the highest (about 50 FIT). Compare it with a MOSFET adjusted failure rate of about 20 FIT.
• NOTE: FIT is failures per 109 hours.
adj base Q T E O
Production quality
Thermal stress
Electrical stress
Other factors (power and operational
environment factors)
Reliability and Availability
• Reliability
![Page 31: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/31.jpg)
31
© A. Kwasinski, 2015
• Availability applies to systems (which can operate with failed components) or repairable entities.
• Definitions depending application:• Availability, A, is the probability that an entity works on demand. This
definition is adequate for standby systems.
• Availability, A(t) is the probability that an entity is working at a specific time t. This definition is adequate for continuously operating systems.
• Availability, A, is the expected portion of the time that an entity performs its required function. This definition is adequate for repairable systems.
• Consider the following Markov process representing a repairable entity:
Reliability and Availability
• Availability
![Page 32: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/32.jpg)
32
© A. Kwasinski, 2015
λ is the failure rate and μ is the repair rate. The probability for a repairable item to transition from the working state to the failed state is given by λdt and the probability of staying at the working state is (1-λ)dt. An analogous description applies to the failed state with respect to the repair rate.
• The probability of finding the entity at the failed state at t = t +dt is identified by Prf(t + dt) then this probability equals the probability that the item was working at time t and experiences a failure during the interval dt or that the item was already in the failed state at time t and it is not repaired during the immediately following interval dt. In mathematical terms,
Prf(t + dt) = Prw(t)λdt + Prf(t)(1-µ)dt
Reliability and Availability
• Availability
![Page 33: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/33.jpg)
33
© A. Kwasinski, 2015
• Hence,
• Which leads to the differential equation
• With solution (considering that at t = 0 it was at the working state)
Reliability and Availability
• Availability
Pr ( ) Pr ( )Pr ( ) Pr ( )
f fw f
t dt tt t
dt
Pr ( )( ) Pr ( )f
f
d tt
dt
( )Pr ( ) 1 tf t e
( )1Pr ( ) t
w t e
![Page 34: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/34.jpg)
34
© A. Kwasinski, 2015
• When plotted:
• If we denote the inverse of λ as the Mean Up Time (MUT), TU, when the system is operating “normally” and the inverse of μ as the Mean Down Time (MDT or off-line time), TD, then as t tends to infinity
• That is,
Reliability and Availability
• Availability
Pr ( ) U Uw
U D
T TA t
MTBF T T
Availability = Expected time operating “normally”
Total time (“normal” operation + off-line time)
![Page 35: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/35.jpg)
35
© A. Kwasinski, 2015
• Notes:
• Unavailability is defined as
• Mean time between failures (MTBF) is the sum of TD and TU
• Ways of improving availability• Modularity• Redundancy (parallel operation of same components)• Diversity (use of different components for the same function• Distributed functions
Reliability and Availability
• Availability
aMDT
UMTBF
UP
DOWN
![Page 36: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/36.jpg)
36
© A. Kwasinski, 2015
• About the common claim of data center operators of having “diverse power feeds.” Two power paths imply redundancy, not diversity because the grid is one.
Reliability and Availability
![Page 37: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/37.jpg)
37
© A. Kwasinski, 2015
• Now consider a two-components system (A and B). The Markov process is now
• So,
• Where,
Reliability and Availability
• Availability
TTd
dt P
P A
( ) 0
( ) 0
0 ( )
0 ( )
A B A B
A A B B
B B A A
B A A B
A
1 2 3 4
Pr ( ) Pr ( ) Pr ( ) Pr ( )TS S S St t t tP
![Page 38: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/38.jpg)
38
© A. Kwasinski, 2015
• The expected time that the system remains in each of the states is given by
• The probability density function of being at state Si is
• the frequency of finding the system in state Si is
Reliability and Availability
• Availability
1
1 1S
i Nii
ijjj i
Ta
a
( ) ii
i
aT i iif T a e
Pr ( )ii ii Sa t
![Page 39: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/39.jpg)
39
© A. Kwasinski, 2015
• Hence, for the two-components system (A and B).
Reliability and Availability
• Availability
![Page 40: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/40.jpg)
40
© A. Kwasinski, 2015
• If in a system all components need to be operating in order to have the system operating normally, then they are said to be connected in series. This “series” connection is from a reliability perspective. Electrically they could be connected in parallel or series or any other way. The availability of a system with series connected components is the product of the components availability.
• If in a system with several components, only one of them need to be operating for the system to operate, then they are said to be connected in parallel from a reliability perspective. The system unavailability equals the product of components unavailability, where the unavailability, U, is the complement to 1 of the availability (U = 1 – A).
S iA a
Reliability and Availability
• Availability
P iU u
![Page 41: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/41.jpg)
41
© A. Kwasinski, 2015
• For a series two-components system
(both A and B need to operate
for the system to operate).
Reliability and Availability
• Availability
Working stateFailed states
System availability
![Page 42: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/42.jpg)
42
© A. Kwasinski, 2015
• For a parallel two-components system
(either A or B need to operate
for the system to operate).
Reliability and Availability
• Availability
Working states
Failed state
System unavailability
![Page 43: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/43.jpg)
43
© A. Kwasinski, 2015
• n +1 redundant configuration. But more modules is not always better:
• Availability decreases when n increases to a point where A < a
1( 1) n nA n a u a
a = 0.97
• The most common redundant configuration is called n + 1 redundancy in which n elements of a system are needed for the system to operate, so one additional component is provided in case one of those n necessary elements fails.
A
Reliability and Availability
• Availability
![Page 44: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/44.jpg)
44
© A. Kwasinski, 2015
• For more complex systems, availability can be calculated using minimal cut sets• A minimal cut set is a group of components such that if all fail the system
also fails but if any one of them is repaired then the system is no longer in a failed state. The states associated with the minimal cut sets are called minimal cut states.• Much simpler than Markov approaches.
Reliability and Availability
• Availability
![Page 45: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/45.jpg)
45
© A. Kwasinski, 2015
• Unavailability with minimal cut sets:
• Calculation:
• Approximation with highly available components:
1
PCM
S jj
U K
1
1 2 1 1
P( ) P( ) 1 [1 P( )]c c cM M Mi
i i j S ii i j i
K K K U K
1
P( )cM
ii
K
,1 1 1
P( )jC C
cM M
S j l jj j l
U K u
u
Reliability and Availability
• Availability
![Page 46: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/46.jpg)
46
© A. Kwasinski, 2015
Ac mains: 99.9 %
Genset: 99.4 % (includes TS) (failure to start = 2.41 %)
- 48 V
Power plant: 99.99 % (without batteries)
Each rectifier: 99.96 %n+1 redundant configuration is used for improved availability
Standby Power Plants
• Typical availabilities
![Page 47: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/47.jpg)
47
© A. Kwasinski, 2015
• Binary representation of Markov states: • 1st digit: rectifiers (RS) with n+1
redundancy• 2nd digit: ac mains (MP)• 3rd digit: genset (GS) (failure to start
probability given by ρGS
• Availability of power plant without batteries:
where
Standby Power Plants
• Availability Calculation
1
( )GS GS MP MP
PP TS RSMP MP GS
A A A
2 ( 1)
( 1)R
RSR R
n n
n
2 11
11
10
2 n nr r n
RS ni i n i
n r ri
C
C
!
C( )! !
kn
n k
k n k n
![Page 48: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/48.jpg)
48
© A. Kwasinski, 2015
• System availability equation:
• Failure probability (in time):
• The probability density function fPPf(t) associated with the probability of leaving the set of failed states after being in this set from t = 0 and entering the set of working states at time t + dt is
where
( ) ( ) 1 ( )i i
i i
PPf S SS F S W
P t P t P t
( ) 0 (1 ) 0 0 0( ) 0 0 0 0
0 ( ) 0 0 00 ( ) 0 0 0
0 0 0 ( ) 0 (1 )0 0 0 ( ) 00 0 0 0 ( )
MP RS GS MP GS MP RS
GS GS MP RS MP RS
MP GS MP RS GS RS
MP GS GS MP RS RS
RS MP RS GS MP GS MP
RS GS GS MP RS MP
RS MP GS MP RS
A
0 0 0 0 ( )GS
RS MP GS GS MP RS
( ) ( )Tt tP A P
Standby Power Plants
• Availability Calculation
aF = 3μRS + μMP + μGS
( ) a tPPff t a e F
F
![Page 49: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/49.jpg)
49
© A. Kwasinski, 2015
• Notice that is the sum of the transition rates from failed states (called minimal cut states) to immediately adjacent working states.
Standby Power Plants
• Availability CalculationaF = 3μRS + μMP + μGS
![Page 50: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/50.jpg)
50
© A. Kwasinski, 2015
• The probability of discharging the batteries is, then
• System unavailability or outage probability:
• Two cases are exemplified:• Case A: With a permanent
genset.• Case B: Without genset
lim ( )F BAT F BATa T a TO PPf at
P e P t e U
Standby Power Plants
• Availability Calculation
0( ) 1 ( )
BATBAT
T a TBD BAT PPfP t T f d e
F
![Page 51: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/51.jpg)
51
© A. Kwasinski, 2015
,
/ / /1MCS i BAT
i mcs
T
w B w oB w BU U e A
Total unavailabilityBase unavailability (without batteries) Batteries (local
energy storage) autonomy
Repair rate from a minimal cut state to an
operational state(Depends on logistics,
maintenance processes, etc.)
• In general, when batteries are considered the unavailability is
Total availability
Heavily depends on unavailability of the electric grid tie
Optimal sizing of energy storage depends on expected grid tie performance and local power
plant availability
Local energy storage contributes to reduce unavailability
Standby Power Plants
• Availability Calculation
![Page 52: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/52.jpg)
52
© A. Kwasinski, 2015
,
/ / /1MCS i BAT
i mcs
T
w B w oB w BU U e A
Total unavailabilityBase unavailability (without batteries) Batteries (local
energy storage) autonomy
Repair rate from a minimal cut state to an
operational state(Depends on logistics,
maintenance processes, etc.)
• In general, when batteries are considered the unavailability is
Total availability
Heavily depends on unavailability of the electric grid tie
Optimal sizing of energy storage depends on expected grid tie performance and local power
plant availability
Local energy storage contributes to reduce unavailability
Standby Power Plants
• Availability Calculation
![Page 53: 1 © A. Kwasinski, 2015 Cyber Physical Power Systems Fall 2015 Power in Communications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ee05503460f94bf108d/html5/thumbnails/53.jpg)
53
© A. Kwasinski, 2015
,
/ / /1MCS i BAT
i mcs
T
w B w oB w BU U e A
Related with minimal cut states
• In general, when batteries are considered the unavailability is
Standby Power Plants
• Availability Calculation