INTRODUCTION
Operating reserve is used in electric power systems to respond to unforeseen
load changes and sudden generation outages and a wide range of techniques
have been used to determine operating reserve requirements (Fotuhi-Firuzabad
et al., 1999b). Usually, operating reserve requirements are determined
using deterministic criteria or rule-of-thumb methods. The most common
deterministic criterion dictates a reserve margin equal to the size of
the largest unit or to some percentage of the peak load (Billinton and
Fotuhi-Firuzabad, 1994). For example in Spanish and Ontario power system,
reserve is determined equal to some fraction of the peak load and to the
largest on line generator, respectively (Bouffard and Galiana, 2004).
Due to their simplicity of concept and ease of applying, the deterministic
criteria methods have widely used in practice. The basic weakness of the
deterministic criteria is that they do not consider the stochastic nature
of system behavior and component failures. In the probabilistic techniques,
the stochastic nature of system components is incorporated and a comprehensive
evaluation of system risk is provided. The first major probabilistic technique
for operating reserve assessment, known as the PJM method, was proposed
in 1963 (Billinton and Allan, 1996). This method evaluates the probability
of the committed generation just satisfy or failing to satisfy the expected
demand during a specified time into the future, known as the lead time.
A major task for a power system operator is to make rapid on-line decisions
based on the available information. This information should be easy to
understand and interpret, because more realistic and understandable information
will help the operator make an appropriate decision. On the basis of the
PJM method, the system operator can make a decision regarding the required
capacity based on the calculated risk, the forecast load and the specified
risk criterion. Although, PJM technique considers stochastic nature of
the power system, but it has not been employed widely in practice. Difficulty
in interpreting the risk index and the lack of system operating information
contained in the use of a single risk value are the two most important
reasons for this.
A well-being approach is introduced in (Billinton and Fotuhi-Firuzabad,
1994) to overcome these difficulties by incorporating system operating
states in operating reserve assessment. Deterministic criteria and probabilistic
indices for monitoring system well-being are combined in (Billinton and
Fotuhi-Firuzabad, 1994). In the previous treatment of well-being framework
probability indices are considered fix (Billinton and Fotuhi-Firuzabad,
1994; Billinton and Fotuhi-Firuzabad, 2000, Fotuhi-Firuzabad et al.,
1996, 1999; Fotuhi-Firuzabad and Billinton, 1999; Fotuhi-Firuzabad, 1999).
This may result in an overestimated solution and consequently higher operating
costs. Operating limits, such as system reserve requirement, is often
imposed to enhance security and does not represent a physical bound (Mantawy,
2004). In other words, reserve requirement and reliability indices constraints
are soft constraints. Fuzzy set theory provides a natural platform to
model fuzzy relationship such as Very Good, Not Bad and so on. Numbers
of application of fuzzy set in power system operation are presented in
(Attaviriyanupap et al., 2004; Mantawy, 2004).
In this study a hybrid method is proposed for operating reserve determining
in the well-being framework. In the proposed framework the probabilities
of being in the risk and healthy states are considered as soft limits.
A fuzzy well-being unit commitment is solved using Genetic Algorithm.
At first, probabilities of being in the risk and healthy states are fuzzified.
Then, fuzzy penalty function corresponding to each chromosome of generation
is calculated. Finally, fitness value is determined using total operating
cost of generating units plus penalty function. Algorithm is repeated
until the stopping criterion is met. The proposed method is applied on
IEEE-RTS and test results are also presented. This results show the out-performance
of the proposed method with respect to the crisp well-being unit commitment
(CWBUC) from the reliability and cost point of view.
MATERIALS AND METHODS
The concepts of PJM method for solving unit commitment problem are illustrated
by Billinton and Allan (1996). In this probabilistic method, the system performance
is identified as being in either the comfort or at the risk domains for a given
load and committed units. This is a pure probabilistic method and doesn`t give
any information about the degree of the system comfort. To solve this problem,
a well-being framework as shown in Fig. 1 is introduced by
Billinton and Fotuhi-Firuzabad (1994), which includes deterministic considerations
into the probabilistic indices for monitoring system well-being.
The definitions of these three states are as follows (Billinton and Fotuhi-Firuzabad,
1994):
| • |
Healthy: A system operates in the healthy state when it has
enough reserve to withstand the deterministic criterion, i.e., any
single unit outage |
| • |
Marginal: A system operates in the marginal state when it
does not have sufficient margin for withstanding specified deterministic
criterion |
| • |
At risk: A system operates in the risk state when the system
load is greater than or equal to operating capacity |
According to the above definitions, the total system state probabilities
can be expressed by Eq. 1 as follows:
|
| Fig. 1: |
System well-being states and their interactions in this framework |
where, P
H+P
M and P
R are probabilities of
the system being in the healthy, marginal and at risk states, respectively.
The operating criterion which can be used in the unit commitment, are satisfying
an acceptable risk level, satisfying an acceptable healthy level or both.
Selecting an operating criterion depends on the required reliability level.
If a single criterion is adopted, the goal is to satisfy the following constraint:
The above relationship means that, the generating units are committed
so that the probability of system risk is not greater than the Specified
Risk Level (SRL), which is determined by the system operator. If multiple
criteria are adopted, the following constraints should be satisfied:
The above conditions mean that, generating units should be committed
such that, not only the probability of the risk is smaller than SRL, but
also, the probability of the healthy state is greater than Specified Healthy
Level (SHL). These criteria are determined by the system operator.
Fuzzy well-being framework (proposed framework): In the crisp
well-being framework, the minimum value of the probability of being in
the healthy state and the maximum value of the probability of being in
the risk state are crisp values. In the proposed framework, these constraints
and also objective function are considered as fuzzy values. In order for
considering the amount of fuzzy constraints satisfaction in the unit commitment
problem, a penalty factor function is also introduced. This penalty factor
is added to the operating cost, which is an objective function in the
unit commitment problem. The value of penalty factor depends on the degree
of the satisfaction of fuzzy constraints. The genetic algorithm is used
to solve the combinatorial optimization problem of the proposed fuzzy
well-being unit commitment problem. The genetic algorithm test allows
the acceptance of any solution at the beginning of the search, while only
good solutions will have higher probability of acceptance as the generation
number increases.
A general fuzzy system has basically five components, fuzzification,
application of the fuzzy operator (AND or OR), implication, aggregation
and defuzzification. The first step is to take the inputs and determine
the degree to which they belong to each of the appropriate fuzzy sets
via membership functions. The probability of risk and the probability
of healthy are input variables for proposed framework that proper membership
functions should be defined for them. In the proposed framework AND operator
has been chosen as the fuzzy operator. Zero-order Sugeno inference is
used for implication. In the Sugeno inference method aggregation and defuzzification
is done simultaneously and the final output of the system is the weighted
average of all rule output. In our framework, each rule is weighted by
its firing strength which has been calculated using product method.
Membership function for probability of risk: The fuzzy set of
input for probability of risk is divided into five fuzzy values. These
values are: risk is very low (RVL), risk is low (RL), risk is medium (RM),
risk is high (RH) and risk is very high (RVH). In Fig. 2
the parameters R1, R2, R3, R4
and R5 are determined from the following relationships:
where, α is determined by the system operator.
Membership function for probability of healthy: The fuzzy set
of input for the probability of healthy is divided into five fuzzy values.
These values are: healthy is very low (HVL), healthy is low (HL), healthy
is medium (HM), healthy is high (HH) and healthy is very high (HVH). In
Fig. 2 the parameters H1, H2,
H3, H4 and H5 are determined from the
following relationships:
where, β is determined by the system operator.
|
| Fig. 2: |
(A) Membership function of the probability of risk, (B) Membership
function of probability of healthy |
Penalty factor: Penalty factor is used to guide the solving of
the fuzzy well-being optimization problem. Membership function of the
probability of healthy and membership function of the probability of risk
are used to calculate the fuzzy penalty factor. For this purpose zero-order
Sugeno implication has been implemented based on the decision matrix of
Table 1 and according to Table 2.
The final output of system which is the so-called penalty factor is then
calculated as the weighted average of the fired rules outputs. In this
stage, each rule is weighted in accordance with the production of the
membership value of its inputs (product method). Finally, using the following
equation, total cost of the committed units is computed.
Total cost = (1+ Penalty factor)x(Operating
cost of committed units) |
(6) |
Applying genetic algorithm: Genetic algorithms are widely used
in science, business and engineering (Belkadi et al., 2006; Borji,
2008; Farshadnia, 2001; Hashemi et al., 2008; Kangrang and Chleeraktrakoon,
2008; Rabi, 2006; Reyes-Garcia et al., 2008; Tlelo-Cuautle and
Duarta, 2008; Ustun, 2007). Coding of chromosomes, fitness function, selection,
crossover, mutation (Haupt and Haupt, 2004) and stopping criteria are
the main steps of genetic algorithm. These steps have been applied for
present optimization problem via following steps:
| Table 1: |
Decision matrix of fuzzy rules |
 |
| V: Very; H: High; L: Low; M: Medium; G: Good |
| • |
Coding of chromosomes: A chromosome should contain information
about the solution that it represents. The most common way for coding
is a binary string, which is used in this study. The solution in well-being
framework is represented by a vector of N dimension. Each column of
the vector determines the ON or OFF status of each generating unit |
| • |
Fitness function: The fitness function, which is used in
this study, is the summation of total operating cost plus production
of the total operating cost and penalty factor which is determined
by fuzzy risk membership function and fuzzy healthy membership function |
| • |
Selection: In this study a roulette wheel selection method
is used for selecting parents to produce the next generation. The
size of the section in the roulette wheel is proportional to the value
of the fitness function of each chromosome |
| • |
Crossover: Two points crossover is used for creating new
generation from the selected parents. Two crossover sites are produced
randomly and the data of the two parents between these sites are swapped |
| • |
Mutation: In this study a uniform method is used for mutation.
Uniform mutation is a two-step process. First, the algorithm selects
a fraction of the vector entries of an individual for mutation, where
each entry has a probability rate of being mutated. In the second
step, if the produced probability corresponding to each gene is smaller
than the mutation rate then this gene is mutated |
| • |
Stopping criteria: The algorithm stops when the number of
generation reaches the value of generations or if there is no improvement
in the objective function for a sequence of consecutive generations
of length stall generations |
Summary of the proposed algorithm: In summary, the proposed fuzzy
genetic well-being unit commitment algorithm can be implemented through
the following steps:
| Step 1: |
Determine the required risk level, required healthy level, α
and β |
| Step 2: |
Construct membership functions of the probability of being at the
risk state and the probability of being at the healthy state using
Fig. 2 |
| Step 3: |
Create an initial population of genetic algorithm randomly |
| Step 4: |
Calculate the operating cost corresponding to each member in the
population |
| Step 5: |
Calculate the probability of being at the risk state and the probability
of being at the healthy state for each member in the population |
| Step 6: |
Calculate fuzzy penalty factor for each member; |
| Step 7: |
Determine fitness function for each member in the population using
operating cost and penalty factor |
| Step 8: |
Sort the population members according to their fitness functions |
| Step 9: |
Check stopping criteria. If satisfied stop, otherwise go to the
step 10 |
| Step 10: |
Generate next generation using genetic algorithm operators (elitism,
crossover and mutation) |
| Step 11: |
Go to step 4 |
RESULTS AND DISCUSSION
The IEEE-RTS (Billinton and Allan, 1996) has been used as our benchmark
to examine the applicability of the proposed method. The total system
generation is 3405 MW and the system annual peak load is 2850 MW. It is
assumed that the system lead time is 4 h.
Performance analysis: Table 3 shows system operating
state probabilities with the Crisp Well-Being Unit Commitment (CWBUC)
framework and when the system is required to satisfy both a specified
risk and a specified healthy probability. Experience of the operator and
conditions under which the system is being operated are used for determining
risk and healthy probabilities. Table 4 shows system
operating state probabilities with the fuzzy genetic well-being unit commitment
(FGWBUC) framework and with H4 = 0.9, R2 = 0.01,
α = 0.005 and β = 0.05. From the Table 3 and
4 the out-performance of the FGWBUC method with respect
to the CWBUC is obvious from the reliability point of view. Using the
proposed method, not only the risk value is smaller than which is obtained
via the CWBUC method but also the obtained healthy probability is greater
than which is obtained by the CWBUC method. This is one of the most important
features of the proposed method.
| Table 3: |
Unit commitment with the CWBUC method and a specified risk of 0.01
and a desired healthy state probability of 0.9 |
 |
| Table 4: |
Unit commitment using the FGWBUC method and with H4 =
0.9, R2 = 0.01, α = 00.005 and β = 0.05 |
 |
| Table 5: |
Comparison of the operating cost for CWBUC and FGWBUC methods |
 |
Table 5 shows the total operating cost with the Crisp
Well-Being Unit Commitment (CWBUC) and the Fuzzy Genetic Well-Being Unit
Commitment (FGWBUC) methods. It is obvious that the FGWBUC is superior
to the CWBUC from the operating cost point of view. The saving of the
total operating cost is varied from 6.1 to 25.5%. The average of saving
in the total operating cost for these ranges of load is 15.8%. This is
the other good feature of the proposed method.
Sensitivity analysis: The probabilities associated with the system
health and margin depends on many factors such as system lead time, system
load and generating unit failure rates. The effects of the variations
in lead time and generating unit failure rates on the system operating
state probabilities and total operating cost are considered and also the
performance of the CWBUC and the FGWBUC under such situations are compared.
The effect of lead time variation: Here, it is assumed
that each generation can be represented by a two-state model which includes
up and down states.
|
|
| Fig. 3: |
(A)The effect of lead time on the operating cost using CWBUC and
FGWBUC, (B) The effect of failure rate ratio factor on the operating
cost using CWBUC and FGWBUC |
In this case, the probability of the unit failing
during a short interval of time T can be modeled as:
where, λ is failure rate. If repairing process is neglected and
λT<<1, then the above equation can be simplified as (Billinton
and Allan, 1996):
P(down)≈λT = ORR (Outage
and Replacement rate) |
(8) |
For showing the effect of the lead time on the operating cost, the lead
time is varied from 1 to 12 h. The operating cost associated with the
CWBUC and the FGWBUC methods for load level of 2280 are shown in Fig.
3A. It can be seen that the operating cost increases as the lead time
increases. It is obvious that in this case operating cost using FGWBUC
is so much smaller than CWBUC.
The effect of generating unit failure rates variation: The effect
of generating unit failure rates on the operating cost, generating failure
rates of each generation is varied from 100 to 300% of their nominal values
and for the load level of 2280 MW. The effect of the generating failure
rate variation on the total operating cost is given in Fig.
3B. As seen, as the failure rate ratio factor (failure rate/nominal
failure rate) is increased, the operating cost associated with both CWBUC
and FGWBUC are increased, but the amount of the increment for the FGWBUC
method is very smaller than the CWBUC.
In this research for showing the effect of generating failure rate variation
and lead time on the total operating cost, sensitivity analysis was done.
The results showed that the FGWBUC gives smaller operating cost versus
CWBUC in all cases.
CONCLUSION
In this study a hybrid method for determining operating reserve in the
well-being framework is presented. The proposed method is based on the
fuzzy genetic well-being framework. In the proposed method, the probabilities
of being in the healthy and risk states are considered as soft limits.
Probability of being in the healthy and probability of being in the risk
are represented by fuzzy set. A penalty factor for computing the degree
of the satisfaction of reliability constraints is introduced in this study.
For solving the fuzzy well-being unit commitment problem, a genetic algorithm
is used. Fuzzy genetic well-being unit commitment (FGWBUC) method is compared
with the crisp well-being unit commitment (CWBUC) method with two different
aspects. These aspects are reliability and operating cost. Using the FGWBUC
not only the obtained risk is smaller than which is obtained from the
CWBUC, but also the obtained healthy probability is greater than which
is obtained form the CWBUC. It is shown that the FGWBUC is superior to
the CWBUC from the operating cost of view point. Finally, for doing sensitivity
analysis the effect of lead time and also the effect of failure rate of
generation on the operating cost are considered. Out-performance of the
proposed method for wide ranges of variations of these parameters is noticeable
again.
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