INTRODUCTION
Availability of electricity at a cogeneration plant relies on the performance
of gas turbines. When the turbines are down, electricity must be purchased from
another producer. This is generally more expensive. Thus considering importance
of the electricity for plant operation and customer need, its availability should
be carefully evaluated in order to anticipate the performance of the plant.
Availability, as a performance measure, reflects the ability of a power system
to meet demand requirements. Barabady and Kumar (2007)
stated that the most important performance measures for repairable system are
system reliability and availability. Samrout et al.
(2005) described the availability as good evaluations of a system’s
performance.
Many availability analysis methods have been developed throughout the years,
which can be grouped into binary and multi-state methods. In binary availability
modeling, the system is assumed to be either in a working state or in a failed
one. However, in many real-life situations the performance of the system equipment
can degrade, which results in performance degradation of the system, so there
can be several states of degradation. This binary-state assumption may not be
adequate. Many papers have been devoted to estimate the availability of multistate
systems (Yu et al., 1994; Huang
et al., 2000; Levitin, 2004) and optimizing
the structure of the multistate systems (Levitin et al.,
1998; Ouzineb et al., 2008; Agarwal
and Gupta, 2007). In multi-state availability modeling, the system may have
more than two levels of performance varying from perfect functioning to complete
failure. A multi-State System (MSS) may perform at different intermediate states
between working perfectly and total failure (Lisnianski and
Levitin, 2003). The presence of setting and partial operation is a common
situation in which a system should be considered to be a MSS.
Practical methods of MSS availability evaluation are based on three different
approaches (Hoang, 2003): the structure function approach,
where Boolean models are extended for the multi-valued case; the stochastic
process (mainly Markov) approach and Monte Carlo simulation. Since the Markov
modeling approach can generate all possible states of a system, the number of
states can be extremely large even for a relatively small number of Markov elements.
In spite of these limitations the above-mentioned models are often used in practice,
for example in the field of power systems availability analysis (Billinton
and Allan, 1990). In the present work the Markov chain method has been selected
for the following reasons: it is appropriate for quantitative analysis of availability
and reliability of systems; it can be used with large, complex systems; it is
not only useful, but often irreplaceable, for assessing repairable systems.
Therefore, this study adopts random process (Markov model) that takes into consideration
of Multi- state model to analyze the availability of gas turbines operated at
a cogenerated District Cooling (DC) plant.
MODEL DESCRIPTION
System configuration: The gas turbines described in this paper are currently being used at a cogenerated gas district cooling plant of an academic institution. These gas turbines generate electricity with each turbine design capacity of 4.20 MW. The configuration of the gas turbines is as shown in Fig. 1.
The two gas turbines are connected in parallel and homogenous in features (Fig. 1). The output of the system electricity is the sum of the power output of each turbine. The plant system is down when both gas turbines completely failed. The electrical power produced by the plant is highly depending on these two gas turbines. Therefore, the system power availability is depending on the performance of these gas turbines.
States definition and probabilities: The electricity production highly
depends on the gas turbine states. Subtractive clustering (Romera
et al., 2007) analysis was used to cluster the performance for the
gas turbines to find the system states. The subtractive clustering method assumes
each performance data point is a potential cluster center and calculates a measure
of the likelihood that each performance data point would define the cluster
center, based on the density of surrounding data points.
|
| Fig. 1: |
System block diagram for the gas turbines at the cogenerated
DC plant |
| Table 1: |
Performance data cluster for gas turbines |
 |
|
Thus, the measure of potential for a data point is a function of its distances
to all other data points as per Eq. 1 and 2;
Where:
| pi |
= |
Potential value of data point i |
| xi |
= |
ith data points |
| n |
= |
Total number data points |
| ra |
= |
Radii or radius defining a neighborhood |
Three performances for gas turbines were developed using subtractive cluster
analysis as shown in Table 1. Therefore, the corresponding
performance, gs , associated with each state (s) are 0, 2.6 and 3.3
MW. Let Ps (t), s = {1, 2, ….., Kj} is the state
probabilities of the element’s performance process G(t) at time t as shown
in Eq. 3:
System of differential equations for finding the state probabilities Ps(t),
s = {1, 2, ….., Kj} for the homogeneous Markov process (Lisnianski
and Levitin 2003) defined as follows:
Where:
| Ps(t) |
= |
System-state probability vector at time t, whose entries are
the system state probabilities at t |
| V (t) |
= |
Transition-rate matrix, whose entries are the component failure, repair
and intensity rate |
State-space model for multi-state elements: A state-space method using
Markov Model (MM) was used for multi-state system availability analysis. This
method is flexible and gives realistic and dynamic model for availability. The
model enclosed intensity of reduced capacity, repair times and failure rates.
The state-space method is not limited to two states only, such as up and down.
Furthermore, components contained different states such as operational, partial
and down. Therefore, the Markov model of two parallel gas turbines was developed
as shown in Fig. 2. The following assumptions and conditions
are adopted for the model.
| • |
The system is repairable |
| • |
The system is subject to repair and maintenance |
| • |
The outcome of each individual repair and maintenance measure is random |
| • |
The system can be in a working state but not operating at full capacity |
Where:
| μ |
= |
Repair rate |
| λ |
= |
Failure rate |
| Tc |
= |
Cycle time |
| PO |
= |
System operating at reduced capacity (61% of the design capacity) |
| tp |
= |
The mean duration of the peak |
| ψ |
= |
Transition intensity rate from PO to up state |
| ε |
= |
Transition intensity rate from up state to PO |
| Up state |
= |
The turbine working at nominal capacity 3.3 MW as shown Table
1 which means 78.5% of the design capacity |
In this model, all transitions are caused by the element’s failures and
repairs corresponding to the transition intensities are expressed by the element’s
failure and repair rates. Every element state there is associated performance
of the element as in Table 1. Failure and repairs cause element
transition from one state to only adjacent state. As can be seen in Fig.
2, with assumption state 1 is the best state of the system, there is transition
from state 1 to state 2 and 3, if failure (λ1) and ε occurs
in the state 1, if the repair (μ1) will be completed, the system
will be back to the previous highest state 1. Similarly, if state 3 fail or
subjected to demand variation, it goes to state with failure and intensity rate
λ1 and ψ.
|
| Fig. 2: |
State-space diagram for the system |
Based on the developed state space diagram, the mathematical equations using
Markov chain were developed. The corresponding system of equations are written
as Eq. 5-7:
Assume that the initial state is the state 1 with the best performance. Therefore, by solving system (5) using Laplace transformation under the initial condition Pk(0)=1, Pk-1(0)=……= P2(0) = P1 (0) = 0, the state probabilities were determined.
Equations for multi-state system availability: Based on state probabilities
which are determined in Markov model for all elements, availability defined
as a measure which depicts the probability of maintaining normal working of
systems/components under determinate time and task conditions. Therefore, the
availability of the system is defined as Eq. 8 and 9,
| • |
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 state probabilities: In order to evaluate the availability of the entire system, it is necessary to determine the probability of each system states with corresponding system performance. Using Eq. 5 with initial conditions pi(0) is zero, for all i ≠ 1 and P1(0) = 1, the evaluated states probabilities as function of time are shown in Fig. 3.
In Fig. 3 each value of performance corresponds to the probability that the element provides a performance rate. As can be observed from Fig. 3, state 1, which is the best sate of the system was greater than 90.23% during this operation days. Analogously, State 4 (worst state of the system) almost did not occur in the system. The probability that the plant runs under state 4 conditions is almost negligible or zero.
|
| Fig. 3: |
Probability different performance level |
|
| Fig. 4: |
Availability of generated electricity |
Availability of generated electricity: Depending 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 Eq.
8 is defined by the sum of probability of only acceptable state. The availability
of generated electricity is shown in Fig. 4.
Figure 4 shows that the availability of the system with respect to time between 0 to 1400 operation days. As an over all trend it is clear that the availability of electricity decreased through time due to either the performance degradation or high demand need. If the required demand or generation of electricity between 0 and 3.3 MW, all states delivered the required amount of electricity except state 4. In this case state 4 would be an unacceptable state and the Eq. 8 of the system was the sum of probabilities state one, state 2 and state 3. So the system availability is greater than 99%.
Expected output performance of the system: The expected output performance as a function of time is shown in Fig. 5 using Eq. 9. The expected output decreases at the beginning for 400 days of operation due to setting and degradation and then constant for the remaining operation days. The expected output performance was compared with the actual output. The results were statistically evaluated using T-test for comparison of actual and expected output for evidence of significant difference. Table 2 displays summary of statistics and the variances of the two samples. According to the data presented in Table 2, the P value for actual and expected output (p = 0.0865) is greater than the conventional p-value (0.05). Therefore, the actual output is significantly more precise with expected output. Deviation of mean and the standard error of the outputs were minimal which means the model can predict the output of the plant without significant difference.
| Table 2: |
T-test: Two-sample assuming unequal variances |
 |
|
|
| Fig. 5: |
Instantaneous mean performance (kW) |
CONCLUSION
This study shows that random process method takes into account multistate models for all system components to predict the availability of the electricity in a gas district cooling plant. Results indicate that the availability of the system is more than 99.5% and the expected performance is greater than 6.6 MW for 1400 operation days. As an overall trend, availability of the system decreased with time due to performance degradation and setting.
Comparison of the outputs from the model and actual plant was done using statistical analysis. Result shows that expected output is more precise with the actual mean output. Therefore, the model can predict the availability and expected output performance of the plant.
Even though this model is very essential to analyze availability of the system, it has its own limitation particularly when the numbers of states or components increase. To address these difficulties this model can be integrated with universal generating function to reduce the number of states and iteration.
ACKNOWLEDGMENTS
The authors would like to thank University Technology PETRONAS for providing grant and facilities for the research.
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