Fuzzy reliability-based optimization of a hydropower reservoir

Reservoir operation modeling and optimization are inevitable components of water resources planning and management. Determination of reservoir operating policy is a multi-stage decisionmaking problem characterized by uncertainty. Uncertainty in inflows and power demands lead to varying degrees of the working of a reservoir from one period to another. This transition, being ambiguous in nature, can be addressed in a fuzzy framework. The different working states of the reservoir are described as fuzzy states. Based on the degree of success in meeting the power demand and randomness associated with inflows, hydropower production is considered as a random fuzzy event. This paper examines the scope of profust reliability theory, a theory used in the reliability analysis of manufactured systems, in the performance optimization of a hydropower reservoir system. The operating policy derived from a profust reliability-based optimization model is compared with a simulation model. The model is then used to derive the optimal operation policy for a hypothetical reservoir fed by normally distributed inflow, for a period of five years. The results show that the model is useful in deriving optimal operating policies with improved reliabilities in hydropower production. doi: 10.2166/hydro.2019.078 om https://iwaponline.com/jh/article-pdf/21/2/308/534013/jh0210308.pdf 2019 Sukanya J. Nair (corresponding author) K. Sasikumar Department of Civil Engineering, National Institute of Technology Calicut, Calicut, Kerala, India E-mail: sukanya.syamanthakam@gmail.com

ures. According to Koutsoyiannis (), the failure of a system may be attributed to structural failure or inability to perform the intended function. The failure of a system can be defined using various criteria or factors. Structure, performance, cost, and even subjective intention may be used for defining failure. Operational failures, in other words, performance degradation, are more significant among these failures. As far as reservoir operation is concerned, operational failure may be the inability to meet the specified target in terms of drinking water supply, agricultural water, hydropower, or flood control. Whatever a failure is, if the effect of it tends to be critical, research on it becomes essential (Cai ).
The failure of a system such as a reservoir can be ascertained in terms of performance measures. As such, reliability is an important performance measure to study system failure. Reliability has been defined in different ways.
Time-based reliability, volume-based reliability, and occurrence-based reliability are different forms of reliability (Kundzewicz & Kindler ). Time-based reliability is defined as the probability of no failure performance in a particular interval. Occurrence-based reliability is the ratio of the number of periods the system has entered into the satisfactory state to the total number of periods of operation.
Volume-based reliability is the ratio of the volume of water supplied to the volume of water demanded, and is relevant in the case of water supply reservoirs.
Reservoir reliability studies can be broadly classified into two typesanalytical studies and simulation-based studies. Some of the studies based on analytical methods include reliability analyses of water supply reservoirs by

METHODOLOGY Profust reliability
Conventional reliability is based on binary state assumption.
It examines whether the system is working or failed. There are only two possible states. By contrast, profust reliability is Let X ¼ {X 1 , X 2 , X 3 , X 4 , … , X n } be the set of possible states of a system. Let W be the event that the system is working and F be the event that the system has failed.
Each one of the states X 1 , X 2 , X 3 , . . . X n is a fuzzy state with a certain degree of membership μ i in the set F and consequently a membership value of 1-μ i in set W.
Initially the system is in a particular state X i . Depending on the input to the system and the target output for the subsequent time period the system can enter into another state X j , which may be either a more working state or a less working state compared to X i . The transition is considered as failure if the system enters into a less working state from a more working state. Since W and F are fuzzy sets, the transitions between them are fuzzy in nature.
If we denote the failure transitions as T SF , it may be defined as a fuzzy event.
where μ T ij is the membership value of failure transition.
In the present study, four partially working states are considered (i.e., n ¼ 4) and these states are designated as X 1 , X 2 , X 3 and X 4 : X 4 is the most working state and X 1 is the most failed state. The states are discretized based on the degree of failure. The continuation of the system in the fully failed state X 1 is the worst failure, with a membership value of 1 in the set T SF . Continuation of the system in the fully working state X 4 is not considered as failure, and hence, the membership value is assumed as zero for this transition. Similarly, transition from X 1 to X 4 is given a membership value of zero and transition from X 4 to X 1 is assigned a membership value of 0.9, a value close to 1.
In order to derive the membership values of other state transitions, each transition is assigned two parameters -(x, y). x represents the parameter value for the discrete state at period t and y represents the parameter value for the discrete state at period t þ 1. The parameter values for each transition are given in Table 1 The membership value for the transition from current state i with parameter value x to the next state j with parameter value y is assumed to be of the form On applying the boundary conditions, The values of μ T ij obtained by substitution of parameter values mentioned in Table 1 in Equation (4) are given in Table 2.
Conventional reliability of the system during the interval (t 1 , t 2 ) ¼ P (failure transitions do not occur in the interval (t 1 , t 2 )) ¼ 1 À P (failure transitions occur in the interval (t 1 , t 2 )) In profust reliability, failure transition is the fuzzy event T SF . Therefore, the profust reliability for the time interval (t 1 , t 2 ) ¼ 1À P (T SF occurs in the interval (t 1 , t 2 )) 1 À X n i¼1 X n j¼1 (μ T ij )P(T ij occurs during (t 1 , t 2 )) where μ T ij is the membership value of transition from state X i to state X j without passing via any intermediate state (Cai ).     The optimization model is formulated to derive the operating policy for a small hydropower reservoir, with the objective of maximizing the fuzzy reliability of power production. It is assumed that the head-storage relationship is linear and evaporation losses are negligible. Since the probability of a fuzzy event is the product of the membership function of the fuzzy event and the probability of the event, the product of μ T ij and P ij is to be minimized for all time periods. The values of μ T ij are taken from Table 2.
The objective function is The different constraints associated with the problem are discussed next.
1. The degree of failure associated with each state: The degree of failure during any time period t is defined such that it is unity when the power produced P t is equal to demand, D t and zero when the power produced is zero ( Figure 2). However, since the working states are defined as fuzzy, the cases of complete working and complete failure are not discussed, or in other words, extreme values of the degree of failure are not significant.
2. Expected value of degree of failure: The exact value of the degree of failure for a particular period is unknown. Thus, the expected value is considered.
The expected value of μ t F is given by the above constraint, where P t i is the probability that the system is in state X i during period t and μ i is the representative value of degree of failure corresponding to state X i , as given in Table 3. 3. Sum of state probabilities: This constraint indicates the fact that the system is in any one of the possible states in a particular time period or, in other words, the sum of probabilities is equal to 1.

State transition equation:
The transition of the reservoir from one state to another can be represented as a non-homogenous Markov chain.

P tþ1
j is the probability that the system is in state j during period t þ 1, P t i is the initial probability and P t ij is the probability that the transition from i to j occurs during period t.

Storage continuity equation:
The fifth constraint is the storage continuity equation.-S tþ1 , S t are reservoir storages at time periods t and t þ 1, respectively, in Mm 3 , R t is the release during period t in Mm 3 , I t is the inflow during period t in Mm 3 and O t is the spill from the reservoir during period t in Mm 3 .
6. Hydropower equation: This constraint gives the monthly power production in megawatts, ρ is the unit weight of water, g is the acceleration due to gravity, H t is the average head during (t, t þ 1) in m and ε is the efficiency of the turbine.
7. Equation for spill from reservoir: This constraint limits the value of spill to zero when net storage is less than capacity, to the amount exceeding capacity when net storage is more than capacity.

Storage limits:
S min S t S max The storage during any period must lie within the limits of S min and S max , where S min is the dead storage and S max is the reservoir capacity.
9. Limits on release: The lower limit of release is fixed as zero and the upper limit is fixed as the reservoir capacity as no other downstream flow requirements are considered.
10. Head-storage relation: H t is the average head during the interval (t, t þ 1) and a and b are the slope and intercept of the head-storage curve.
11. Limits on degree of failure: The degree of failure, being defined as a fuzzy quantity, should lie between 0 and 1. But, in order to ensure that 75% of the demand is satisfied in all periods, the degree of failure is limited to a maximum value of 0.25 as already mentioned.
12. Limits on probability: The probability of the system in state i during period t and the probability that the transition from i to j occurs during period t must lie between 0 and 1. In the simulation model, the release policy is formulated such that power produced is equal to the constant demand of 2 MW, if possible. If available water is less than S min , no release is made. During any period, if S tÀ1 þ I t > S max , then extra water after meeting power demand is spilled. If there is not enough water to generate the required power, power is generated to the extent possible.

Hypothetical case study
The optimization model is now applied to the same hypothetical reservoir with monthly inflows that are generated using the Thomas-Fiering model. The flows in all the months are assumed to follow normal distribution with statistical parameters listed in Table 4. The reservoir details are given in

RESULTS AND DISCUSSION
The results of the validation are presented in Figures 3 and   4. The power produced as per the simulation and optimization models are compared with the specified demand in     manner that it is available for release during periods of low flows. However, simulation and optimization models perform alike during periods receiving high inflow.

SUMMARY AND CONCLUSIONS
A profust reliability-based optimization model is developed to determine the optimal operating policy for a hydropower reservoir with the objective of minimizing the transition from more working state to less working state. The results from the model are compared with those from a simulation model for a hypothetical reservoir. The optimal operating policy is obtained for a period of five years, on a monthly time scale basis, with the assumed normally distributed inflows and specified power demand. The following conclusions may be drawn from the study: • The optimization model performs better than the simulation model. • The results of the case study demonstrate the applicability of profust reliability in reservoir operation.