INTRODUCTION
Cogeneration power plant produces power and chilled water to meet the customer requirement. In order to meet the customer need, the plant has to perform at certain level and the equipment should be reliable. Therefore reliability assessment of the overall plant system is required to deliver the expected output and to keep the equipment in good conditions. This study focuses on study of system reliability of the plant. A Multistate system (MSS) reliability analysis is applied for this study.
The Multistate system was introduced in the middle of the 1970s (Murchland,
1975; Modarres et al., 1999; Barlow
and Wu, 1978). Griffith (1980) generalized the coherence
definition and studied three types of coherence. The reliability importance
was extended to MSSs by Griffith (1980) and
Butler (1979). An asymptotic approach to MSS reliability evaluation was
developed by Koloworcki (2000). An engineering method
for MSS unavailability boundary point estimation based on binary model extension
was formulated by Pourret et al. (1999). Practical
methods of MSS reliability evaluation are based on three different approaches
(Aven, 1993): the structure function approach, where
Boolean models are extended for the multivalued case; the stochastic process
(mainly Markov) approach; and Monte Carlo simulation. Since the Markov modeling
approach can generate all possible state of a system, the number of state can
be extremely large even for a relatively small number of Markov elements. Thus,
Markov modeling approach must become familiar with reduction techniques that
reduce the number of states. Simulation can be performed in order to assess
MSS reliability. The simulation technique is also very sensitive towards the
number of state in the model. It has the same problems during the model construction
stage and often requires enormous computational resources during the solution
stage. Universal Generating Function (UGF) which is based on algebraic procedure
can reduce the problem's dimension and extremely beneficial for reliability
analysis. Ushakov (Griffith, 1980) introduced the basic
ideas of the UGF method in the mid1980s. Since then, the method has been considerably
expanded (Lisnianski and Levitin, 2003). In the last ten
years, the UGF approach was further developed and completed by Lisniaski and
Levitin for evaluating and optimizing reliability indices of multistate systems
(Trivedi, 2002; Levitin et al.,
1998; Lisnianski and Levitin, 2003; Gnedenko
and Ushakov, 1995). Therefore, this study adopts combined random process
and the universal generating function (UGF) so as drastically reduces the number
of state in multi state model. The universal generating function procedure helps
to find the entire MSS performance distribution based on the performance distribution
of its elements by using algebraic procedures.
MODEL DESCRIPTIONS
Generic model for a multistate system (MSS): In order to determine
and analyze MSS behavior one has to know the characteristics of its components.
A functional and logical order of the blocks in Fig. 2 is
described by the system structure function and each block’s behavior is
defined by the corresponding performance stochastic process (Ushakov,
1986; Aven and Jensen, 1999; Gnedenko
and Ushakov, 1995). In a multistate analysis of cogeneration power plant
used in UTP, each block of the Reliability Block Diagram (RBD) as shown in Fig.
1 indicates one multistate element of the system. The GDC, UTP plant is
designed to provide 8.4 MW of electrical power and 5300 refrigeration tons (RT)
of cooling capacity to UTP. The plant consists of two gas turbine generators,
each rated at 4.20 MW. For chilled water production double effect steam absorption
system each rated to produce 1250 RT of cooling capacity. In addition there
are four electric chiller (EC) and one thermal energy storage (TES) each rated
to produce 250 RT and 1000RT/hr respectively. The reliability of block diagram
of GDC is as shown Fig. 1.
The electric power production of cogeneration power plant which is currently
working in universiti teknologi PPETRONAS highly depend on the performance of
the gas turbines which are connected in parallel. Basically these gas turbines
produced electric power directly to customer and exaust gas for chilled water
production. This study focuses on the reliability and availability analysis
of gas turbines for production of electricity. The functional relation and the
corrsponding associated performance is shown in Fig. 2.
States definition: The state of each gas turbine is highly depending
on the daily production performance. Subtractive clustering analysis is done
to cluster the performance for each gas turbine to find the system state for
1400 operation days. The subtractive clustering method assumes each production
performance data point is a potential cluster center and calculates a measure
of the likelihood that each data point would define the cluster center, based
on the density of surrounding data points. The algorithm of subtractive cluster
(Romera et al., 2007) does the following:
• 
Selects the data point with the highest potential to be the first cluster
center 
• 
Removes all data points in the vicinity of the first cluster center (as
determined by radii), in order to determine the next data cluster and its
center location 
• 
Iterates on this process until all of the data is within radii of a cluster
center. 
The daily production performance cluster points of gas turbines are show in
Fig. 3 and Table 1.

Fig. 1: 
System block diagram for GDC 
Table 1: 
Performance data cluster for GT 1 and 2 


Fig. 2: 
RBD of gas turbine 

Fig. 3: 
Clustering production performance of Gts 
Therefore based on the data as shown in Table 1, using subtractive
cluster method the performance the state of the two gas turbines are determined.
Gas turbine GT1 has contained three states which are complete failure, partial
failure and zero failure at maximum performance level and GT2 contains only
two states which are complete failure and zero failure. A state of total failure
for both turbines corresponds to a capacity of 0 and the maximum operational
states 3.6 MW and 3.0 Mw. GT1 has partial failure, which is the capacity of
2.6MW. The state space diagram and system state are described in the next section.
State space diagram and determination of state probabilities: Multistate
system was considered to have constant demand. In practice, it is often not
so. A multistate element can fall into a set of unacceptable states in two
ways: either through performance degradation because of failures or through
an increase in demand. If all failures and repair times are distributed exponentially
then the performance stochastic process will have a Markov property and can
be described by a Markov model (Trivedi, 2002). The state
space diagram of the system developed as follows;
Every element state there is associated performance of the element. Minor failure and repairs cause element transition from one state to only adjacent state. As can be seen in the Fig. 4. with assumption state 1 is the best state of the system, there is transition to the state 2 from the state 1, if failure (λ_{1}) occurs in the state 2, and there is in transition to the state 1 (μ_{1}), if the repair will be completed. Similarly, there will be transition from state 3 and state 5 to state 4 and state 6 respectively with failure rate of λ_{1} if there is performance degradation. If state 1 and 2 fail and goes to state 3 and 4 respectively, there will be failure rate give by λ_{2}. Analogously if state 3 and 4 fail and go to state 5 and 6 respectively, there will be failure rate give by λ_{3}. If state 5 and 6 getting minor repair μ_{3}, the states will be up in to state 3 and 4. The state of the system and state space diagram is defined. The corresponding performance g_{s} is associated with each state s. Let P_{S}(t), s={1,2........k_{j}} is the state probabilities of the element’s performance process G(t) at time t:
Then the probability of each state has to be defined using Eq.
1. For a Markov process, each transition from the states to any state m
(s, m=1; . . . ; k) has its own associated transition intensity designated as
a_{sm}. In this study, any transition is caused by the element’s
failure or repair. If m<s, then a_{sm}=λ_{sm}, where
λ_{sm} is a failure rate for the failures that cause the element
transition from state s to state m. If m<s, then a_{sm} = μ_{sm},
where μ_{sm} is a corresponding repair rate. System of differential
equations for finding the state probabilities P_{S}(t), s={1,2........k_{j}}
for the homogeneous Markov process is defined (3) as follows:
In this case, all transitions are caused by the element’s failures and
repairs corresponding to the transition intensities a_{is} and are expressed
by the element’s failure and repair rates.

Fig. 4: 
Sate space diagram of the system 
Therefore, the corresponding system of differential equations for the power
system as shown in Fig. 4 are written as:
Assume that the initial state is the state k with the best performance. Therefore
, by solving system (27) of differential equations under the initial condition
P_{k}(0)=1, P_{k1}(0)=……= P_{2}(0)= P_{1}(0)=0,
the state probabilities P_{s}(t), s= 1,……….k is obtained.
Moreover, the power output of each state is the sum of the output of each turbine which is connected in parallel as shown Fig. 2. The system state performances of the six states are indicated in Table 2.
Model for Multistate system reliability and its demand: Based on state
probabilities which are determined in Markov model for all elements, reliability
can, in the general sense, be defined as a measure which depicts the probability
of maintaining normal working of systems/components under determinate time and
task conditions.
Table 2: 
System state and performance 

Reliability is a result of the interaction between the task (demand) and the
performance (capacity) in the timevarying probability space. It can be describes
task and performance random variables by the UGFs firstly, enumerates state
combinations by composition operators step by step, and then obtains the reliability
of systems/ components finally.
By applying composition operators over UGF of individual elements and their
combinations in the entire MSS structure, the resulting UGF for the entire MSS
is obtained by using simple algebraic operations. UGF characterizes the output
performance distribution for the entire MSS at each time instant t. MSS reliability
indices easily derived from this output performance distribution. The following
steps are executed:
• 
Having performances g_{ji} and corresponding probabilities
P_{ji} (t)for each element jε{1,2.......n}; iε{1,2............K_{j}}
UGF for this element is defined in the following form: 
• 
The composition operators Ω_{φs} (for elements
connected in a series), Ω_{φp} (for elements connected
in parallel) and Ω_{φB} (for elements connected in a bridge
structure) should be applied over the UGF of individual elements and their
combinations. These operators were described in (Lisnianski
and Levitin 2003), where corresponding recursive procedures for their
computation were introduced for different types of systems. Based on the
above procedures, the resulting UGF for the entire MSS is obtained: 
where, K is the number of the entire system states and gji is the entire system
performance in the corresponding state I, I ε{1,2..........K_{j}}
• 
Applying the operator’s δ_{A}, δ_{E},
δ_{D} introduced in (Aven and Jensen 1999)
over the resulting UGF of the entire MSS, the reliability indices of MSS
is obtained 
• 
MSS availability A(t, w) at instant t>0 for random constant demand
w: 
• 
MSS expected output performance at instant t > 0 for arbitrary
constant demand w: 
RESULTS AND DISCUSSION
Estimation of probabilities: In order to evaluate the performance distribution
of the entire system, it is necessary to determine the probability of each system
states with corresponding system performance. Using Eq. 27
with initial conditions p_{i}=0, for all i≠1 and P_{1}(0)=1
and based on the failure and repair data for 1400 operation days given in the
Table 3, the sate probability defined as show in Fig.
5.
Table 3: 
Transition intensity rate for failure (λ) and repair
(μ) 


Fig. 5: 
Probability different performance level 

Fig. 6: 
Availability of the system 
In this Fig. 5, each value of performance corresponds to
the probability that the element provides a performance rate. As can be observed
from Fig. 5, state 4 and 6 do not occur in the system. Whenever
these states occurred, the plant uses electric power from Tenaga Nasional Berhad
(TNB) to meet the required demand. The probability that the plant runs under
state 4 and 6 conditions are almost negligible or zero. In the other way the
plant run under state 1 over 75% to satisfy the requirement of high demand.
Availability of the system: Depend on the demand required, each state
constitute the set of acceptable states. The states which have the output performance
lower than the demand required will be combined in one state called absorbing
state (unacceptable states). Therefore the instantaneous availability (10) is
defined by the sum of probability of only acceptable state.

Fig. 7: 
Reliability of the system for different level demands 

Fig. 8: 
Instantaneous mean performance 
The availability of the system for different demand level is shown in Fig.
6.
Figure 7 shows that the availability of the system with respect to time between 0 to 1400 days. As an overall trend it is clear that the availability of the system went down through time due to either the performance degradation or high demand need. If the required demand between 0 and 3 MW, the system delivers almost 99% of availability. When the demand between 3.6 and 5.6MW, the system availability become 98% and above. If the demand goes to 6.6 MW, the availability went down to around 94%. Therefore increasing of demand has an impact on the availability of the system.
Reliability of the system: The reliability function R(t, W) is defined
by combing all unacceptable state into absorbing state, forbid repairs that
return the MSS from this state to the acceptable states and replace the failure
rate from each acceptable state to the absorbing state 0 by the sum of the failure
rates from acceptable states to all unacceptable states. Therefore the reliability
of the GTG system is the sum of the probability of absorbing state:
As can be seen from the graph, the reliability of the system went down when
the load increased. In reliability 1, state 4 and 6 are absorbing state because
the plant demand requires is not less than 3 MW and the reliability is greater
than 93%. Analogously, the demand requirement greater than 5.6 MW, all state
except state one will be absorbing state and the reliability is going to be
65%. Therefore the system reliability will be highly affected by high demand
requirement. This will bring the performance degradation.
Expected output performance: The expected output performance is defined in Fig. 8 using Eq. 11. The expected output performance is decreasing through time. The efficiency of the GTG is also reduced due to frequent failure or over load.
CONCLUSION
This study predicts the availability and reliability of the power generated from gas turbines which are connected in parallel using universal generating function and the random process method and takes into account multistate models. The result indicates that the availability and reliability of the system at each different state and performance level. As can be seen in the above graphs the reliability and availability of the system went down through time due to performance degradation and overload.
ACKNOWLEDGMENT
I would like to thank University Technology PETRONAS for providing grant and facilities for the research.