Water Resources Management (2005) 19: 655–672 DOI: 10.1007/s11269-005-3275-3
C
Springer 2005
Reservoir Operation Using a Dynamic Programming Fuzzy Rule–Based Approach S. J. MOUSAVI1 , K. PONNAMBALAM2,∗ and F. KARRAY2 1 School of Civil and Environmental Engineering, Amirkabir University of Technology (Tehran Polytechniques), Tehran, Iran; 2 Department of Systems Design Engineering, University of Waterloo, Canada (∗author for correspondence, e-mail:
[email protected])
(Received: 27 November 2002; in final form: 8 September 2004) Abstract. A dynamic programming fuzzy rule–based (DPFRB) model for optimal operation of reservoirs system is presented in this paper. In the first step, a deterministic dynamic programming (DP) model is used to develop the optimal set of inflows, storage volumes, and reservoir releases. These optimal values are then used as inputs to a fuzzy rule–based (FRB) model to establish the general operating policies in the second step. Subsequently, the operating policies are evaluated in a simulation model. During the simulation step, the parameters of the FRB model are optimized after which the algorithm gets back to the second step in a feedback loop to establish the new set of operating rules using the optimized parameters. This iterative approach improves the value of the performance function of the simulation model and continues until the satisfaction of predetermined stopping criteria. This method results in deriving the operating policies, which are robust against the uncertainty of inflows. These policies are derived by using long-term synthetic inflows and an objective function that minimizes its variance. The DPFRB performance is tested and compared to a model, which uses the commonly used multiple regression–based operating rules. Results show that the DPFRB performs well in terms of satisfying the system target performances and computational requirements. Key words: dynamic programming, fuzzy inference system, reservoir operations
1. Introduction Among the main sources of complexities in optimization models for reservoir operation is the need to account for uncertainties of all sorts. Explicit stochastic methods that consider uncertainties usually do not represent the detailed characteristics of a system under consideration, especially in multiple reservoirs systems. Implicit stochastic methods on the other hand can deal with it. However, one of the important challenges is to combine the large set of results obtained from an implicit stochastic method to derive general operating policies. A summary of different methods used for surface water reservoir management can be found in studies by Yeh (1985) and Ponnambalam (2002). Implicit stochastic programming has been widely used to derive the general operating policies for reservoir systems. Young (1967) proposed the use of linear
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regression procedure to find the operational rules from the results of a deterministic optimization method. Bhaskar and Whitlach (1980) used a multiple linear regression to derive the operating rules as a function of inflows to reservoir and storage volumes. Karamouz and Houck (1982) extended this approach in a model called DPR. In the DPR model, an iterative method is used in which dynamic programming (DP) releases are progressively constrained to be close to the operating-rule values. Along with regression or using simple statistics and diagrams and tables to infer the operating policies (Lund and Ferreira, 1996), other methods included using the Artificial Neural Networks (ANN) (Raman and Chandramouli, 1996; Chandramouli and Raman, 2001; Cancelliere et al., 2002), and fuzzy rule–based (FRB) technique (Russel and Campbell, 1996; Shrestha et al., 1996; Panigrahi and Mujumumdar, 2000; Dubrovin et al., 2002) and the combination of FRB and ANN (Ponnambalam et al., 2001, 2003) have been used. Raman and Chandramouli (1996) noted that the multiple regression tends to smoothen the values of the release functions. They showed in a case study that ANN-based policies have better performance compared to multiple regression–based policies. There are other models dealing with the use of operation rules in multiple reservoir control models such as Johnson et al. (1991), Koutsoyiannis and Economou (2003), Lund and Guzman (1999), Oliveira and Loucks (1997), and Philbrick and Kitanidis (1999). In the dynamic programming fuzzy rule–based (DPFRB) model proposed in this paper, the optimal values obtained from a deterministic DP model are used in an FRB model to derive the operating policies. These fuzzy rules are then tested in a simulation model for evaluation of their performances. Also, an iterative method is proposed to find the best values of the FRB parameters, based on the results of the simulation model. The DPFRB and DPR models are applied to Dez and Karoon reservoirs system and the results are compared. This paper is organized as follows: In Section 2, the deterministic DP model used in the first step of DPFRB and DPR models is described. Section 3 outlines the principles of the FRB method. The complete description of proposed DPFRB model is presented in Section 4. Analysis of the results after applying the earlier models to the case study is discussed in Section 5. Finally, we conclude in Section 6 by summarizing the findings and providing suggestions for improvement. 2. Dynamic Programming–Based Modeling Dynamic programming is a widely used numerical tool for optimization of reservoirs operation. Yakowitz (1982) has described the various uses of DP in water resources. We use a deterministic DP model for optimization of a multiple site reservoir system. The recursive function and constraints of such a system could be mathematically presented as follows: f t+1 (St+1 ) = min[Loss(Rt , Dt ) + f t (St )]
t = 1, . . . , T
(1)
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Such that: St+1 = St + It − Rt
(2)
Rmin ≤ Rt ≤ Rmax
(3)
Smin ≤ St ≤ Smax
(4)
where f t (St ) is the the total minimum losses of operation from stage 1 to stage t when the state of storage volumes of reservoirs at the beginning of season t is St , T the time horizon, St = [S1t , S2t , . . . , Snt ] the vector of storage volumes of a system of n reservoirs at the beginning of season t, It = [I1t , I2t , . . . , Int ] the vector of inflow volumes into corresponding reservoirs during season t, Rt = [R1t , R2t , . . . , Rnt ] the vector of release volumes from corresponding reservoirs during season t, Dt the vector of the amount of water demands in season t, Loss (Rr , Dt ) the immediate loss of operation during the season t, Rmin = [R1 min , R2 min , . . . , Rn min ] the vector of minimum releases from reservoirs, Rmax = [R1 max , R2 max , . . . , Rn max ] the vector of maximum releases from reservoirs, Smin = [S1 min , S2 min , . . . , Sn min ] the vector of minimum storage volumes of reservoirs, and Smax = [S1 max , S2 max , . . . , Sn max ] the vector of maximum storage volumes of reservoirs. The function of Loss(Rt , Dt ) is defined as follows: Loss(RR) = A × 10(RMN−RR) − 1 if RR ≤ RMN (5) (RR−RMX) − 1 if RR ≤ RMX (6) Loss(RR) = C × 10 Rt is the proportion of release over demand, RMN is a parameter Dt that defines what percentage of demand target must be satisfied. Here we selected RMN = 0.8, RMX is a parameter defining safe range of release over the demand. Here we selected RMX = 1.2, A = 1.58 × 104 and C = 3.88 × 103 are the constants of the function which are the same as those values used in the paper in which DPR model is presented (Karamouz and Houck, 1982). This function is a discontinuous loss function where the shortage is more steeply penalized than surpluses with zero penalties for ±20% deviation from demand.
where RR =
3. Fuzzy Rule–Based (FRB) Modeling Reservoir operating policies might be presented to operators in the form of rule curves, tables, or regression equations. Because of the difficulties in including the operators’ expert knowledge and the assumptions involved in the mathematical models along with possible other approximations, applying the policies in realworld situations is a difficult and challenging problem. The FRB modeling could be regarded as a substitute for representing the operational policies. In this method, the human reasoning is easily incorporated in the decision-making process and therefore human operators are better able to apply
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the results of the model. They should also be able to incorporate their experimental findings through membership functions, the basic concept of fuzzy set theory. The FRB method is a mathematical model in which the rule-based system is defined by fuzzy rules. A fuzzy rule is an “if–then” proposition where “if’ is associated with the premise variables and “then” is associated with the fuzzy or crisp consequences such as: If St is A1i and It is A2i then Rt is Bi where St , It , and Rt are inflow to reservoir, storage volume, and release from reservoir, respectively, and Aki is the kth linguistic explanatory variable and Bi is the consequences of rule “i”. A linguistic variable is a variable represented by a label such as small, medium, or large. Each value of a linguistic variable is represented by a fuzzy set. Therefore, Aki is a fuzzy set, which is defined through its membership function. These fuzzy rules are the linguistic representation of an inference engine of a Fuzzy Inference System (FIS) and are determined by welltrained experts or through a learning mechanism using available input–output data. We have used the optimal solutions derived from the DP as input–output data to calibrate the fuzzy rules. In our FIS, the inflow and storage are taken as the fuzzy premises while the releases from reservoirs are considered as the crisp consequent variables. This is a simple form of the well-known Sugeno-type (Sugeno, 1985) FIS. To establish a FIS, the lower and upper limits of each premise variable are determined from the training data set. Then, the range between upper and lower limits of each premise variable is divided into a set of overlapped classes in which each class is a fuzzy number. Here we used triangular fuzzy numbers (TFN). Figure 1 shows the partitioning of storage volume of and inflow to reservoir, as our premise variables. It was found that dividing each premise variable into only three to five fuzzy numbers is sufficient in the FRB model. This is due to the fact that the
Figure 1. Discretization of storage volume of or inflows to reservoirs and their membership functions.
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rule-based system becomes incomplete if finer discretization is used, i.e. some of the rules are not fired as there would be no data matched with them. In fact, the number of partitions and hence the number of fuzzy rules are determined by a trial and error procedure so that the fuzzy rule base is complete (Sheresta et al., 1996). In some cases, as it is essential to keep the number of fuzzy rules as minimum as possible, clustering methods such as fuzzy c-mean clustering is used to optimize the number of rules (Ponnambalam et al., 2003). Two different methods were used to determine the supports (the range between minimum and maximum) of each fuzzy number. In the first method, the premise variables are divided into equally distant classes where different number of training data exists for each class. In the second method, the supports of the fuzzy numbers are determined in such a way as to have equal frequencies of the training data located in the different classes. The two methods were also used to determine the mean value of fuzzy numbers. In the first method, the mean value of each fuzzy number would be the expected value of all training data located on the support of this number and is independent of other premise variables. In the second method, the mean value of each fuzzy number in rule “i” depends on the other premise variables of this rule. Suppose each fuzzy rule “i” has m premise variables. Then, the set Pi is defined as Pi = {a1i (s), a2i (s), . . . , aki (s), . . . , ami (s); bi (s), s = 1, . . . , n i | aki (s) ∈ li− (k), li+ (k) } and the mean value of fuzzy numbers are calculated as: n i aki (s) m i (k) = s=1 i = 1, . . . , NR ni
(7)
where n i is the number of elements belonging to Pi and aki (s) the sth (s = 1, . . . , n i ) element of Pi . Each of these elements is located on the support of the kth (k = 1, . . . , m i ) premise variable of rule “i”. This support is the range between li− (k) and li+ (k). Therefore, aki (s) is the input part and bi (s) the output part of the training data of the set Pi . If the kth premise variable of the fuzzy rules divides into nn k fuzzy numbers, the total number of rules NR is then given by: NR = nn 1 × nn 2 × · · · × nn m
(8)
The consequence of each fuzzy rule “i” is calculated by combining the outputs of all the data which belong to Pi and then satisfy at least partially this rule. The degree to which the sth element of Pi belongs to the rule “i” is measured by the so-called degree of fulfillment (dof ). In fact, the dof represents the weight or degree with which each input data satisfies each fuzzy rule. An input data (aki (s)) consists of m elements each of which belongs to a fuzzy set by the degree of µ(aki (s)). Therefore, the final weight (dof ) which is actually an aggregated membership function of this
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input data, can be selected as the minimum degree of satisfaction among these elements (“min” operator) or product of different weights (“product” operator). In general, any T-norm operator could be used. Here we apply a parametric “product” operator as follows: dofi (s) =
m
g µ(aki (s))
if
dofi (s) ≥ e
(9)
k=1
where µ(aki (s)) is the membership function of the sth element of Pi in the kth premise variable of the rule “i” and “g” and “e” are the parameters of the dof values. Parameter “g” is called a “hedge” in literature and parameter “e” a “α-cut”. The consequence of the rule “i” (Bi ) is then calculated through aggregation of all the output data belonging to Pi by using a weighed average method given by: n i dof (s) × bi (s) ni i Bi = s=1 (10) s=1 dofi (s) For this type of FIS the consequence of each rule after aggregation is a crisp value, so there is no need to use a defuzzification method to determine the output. It should be noted that “g” and “e” are two important parameters which are described in the next section. It was noted that the fuzzy rules take the simple form of “If St is A1i and It is A2i then Rt is Bi for a single reservoir case. A1i and A2i are fuzzy numbers whose membership functions are determined by partitioning storage and inflow variables between their range of variation, i.e. [Smin , Smax ] and [Imin , Imax ], as shown in Figure 1. According to the example in Figure 1, the first fuzzy rule (Rule 1) takes the following form: If St is between [800, 900] and It is between [10, 100], then R1 equals B1 where B1 is computed by using equation 10 with all (St , It ) pairs of input–output data resulting from the DP model whose range is St ∈ [800, 900] and It ∈ [10, 100]. Note that Stc (1) = 800 and Itc (1) = 100 and other values of Stc and Itc in Figure 1 are determined by Equation (7). 4. Overview of the DPFRB Model As we mentioned earlier, the DPFRB model is a combination of dynamic programming, fuzzy rule–based, and simulation models. In DPFRB, the monthly historical inflows to reservoirs are divided to two different sets. One set is used for the DP and FRB models as training inflows while the other is used for the simulation model as test inflows. At first, the DP optimization model is executed and an optimal set of reservoir releases and storage volumes are determined. These optimal storage
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Figure 2. The flow chart of DPFRB model.
and reservoir volumes as well as training inflows are then taken as the input values of the FRB model to derive the operating policies in the second step. The inflows and optimum storage volumes are taken as the fuzzy premise variables while the reservoir releases are taken as the consequences of fuzzy rules. Figure 2 shows the flow diagram of the DPFRB model. The rule system was calibrated for each month of a year, separately. The parameters of rule-based in
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Equation (4), “g” and “e” are set equal to 1.0 and 0.0, respectively, in the first iteration of DPFRB model. Following this, the operating policies or fuzzy rules, determined from the FRB model, are tested in a simulation model. Initial reservoir volumes are supposed known and inflows to reservoirs are presumed available from the set of test data. The reservoir releases are then calculated using the weighted average method similar to the way the consequences of fuzzy rules are determined: dof (vt ) · Bi (FRB) Rt = i i (11) ∀i | dofi (vt ) ≥ e i dofi (vt ) In Equation (11), Rt is the vector of the reservoir releases during the season t for our case study of two reservoirs system; νt = [I1t , I2t , S1t , S2t ] is the vector of the state fuzzy variables and Bi (FRB) = [R1t , R2t ] is the vector of responses of rule “i” obtained from the FRB model. After calculating the reservoir releases in each time period, the reservoir storage volumes are determined from the continuity equation for the next period and the process is repeated over the simulation horizon. To find the best values of the “g” and “e” parameters, the simulation model was imbedded in a direct search optimization procedure to minimize the value of the objective function in the simulation. This means that, the parameters “g” and “e” were discretized within their range and the simulation was performed for each grid point of the two-dimensional search space of those variables to find out the best values of “g” and “e” resulting in a minimum value of the simulated objective function. Subsequently, the best values of these parameters as well as the test inflows are fed back to the FRB model, and the fuzzy rules are established again by the new values of “g” and “e” and the inflows. This iterative procedure continues until the best value of the objective function in the simulation model is obtained. The DPFRB algorithm can be summarized as follows: (i) Derive optimal releases and storages using DP. (ii) Derive operating rules using FRB. The input data and the parameters “g” and “e” of the FRB model are obtained from (i) initially and from (iii) during iteration. (iii) Determine the optimal parameters “g ∗ ” and “e∗ ” using a direct search of “g” and “e” in simulation. (iv) Check for satisfaction of stopping criterion. (v) If yes then stop else go to step (ii). In this model, the operating policies are parameterized in a way as to be able to find the best values of parameters even for different objective functions in simulation. For example, the reliability of meeting the different types of demands under any other risk or reliability consideration could be taken into account as the objective function in the search algorithm. Nalbantis and Koutsoyiannis (1997) have discussed the advantages of parameterized operating rules. In addition to the flexibility of parameterized operating rules in satisfying the different objectives of the
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system, we use these parameterized policies as an interface between the DP and the simulation model. Usually, after a few iterations, the best value of the simulated objective function is obtained and further iteration does not result in further improvement. The stopping criterion was selected in such a way that the value of objective function has no improvement in successive iterations. 5. Implementation and Discussion The DPFRB model was applied to the reservoirs system of Dez and Karoon in Iran. Dez and Karoon drainage basins are located in the southwest region of Iran, carrying more than one-fifth of the surface water available in the country. Total area of these basins is about 45,000 km2 . The reservoirs are constructed on Karoon and Dez rivers and they are named after them. The two rivers are joined together at a location called Band-e-Ghir, north of the city of Ahwaz and form the “Great Karoon River”. This river passes Ahwaz and reaches the Persian Gulf in the south of the city of Ahwaz. Average annual streamflow to Dez and Karoon reservoirs are 8.5 and 13.1 billion m3 (BCM), respectively. Figure 3 shows the schematic representation of this system. These two rivers downstream of Karoon and Dez dams supply water for the domestic, industrial, agricultural, and agro-industrial sectors. Total irrigated land, downstream of Dez and Karoon dams are estimated to be about 250,000 ha. Besides the irrigation benefit from the lands, which are the main consumer of water in these basins, these two rivers supply domestic water demand for cities and towns as well as to main industrial units such as steel industries and thermal power plants. Total water demand from Dez and Karoon rivers for all purposes is estimated to be 1.91 BCM. Thirty-five percent of this amount
Figure 3. The schematic of the Karoon and Dez systems.
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is allocated to the downstream of Karoon dam between Karoon reservoir and Band-e-Ghir (first reach), 42% is allocated to downstream of Dez dam (second reach), and the rest to downstream of Band-e-Ghir to Persian Gulf (third reach). To evaluate the DPFRB model, the DPR model of Karamouz and Houck (1982) is also applied to this system. As noted in Section 1, the DPR is an extended form of implicit stochastic DP, which uses multiple linear regression analysis to derive the general operating rules. In this model, the set of optimal solution of DP is regressed and operating policy is presented by a linear equation. These regression equations are then tested in a simulation model. Following the first iteration, the DP is repeated by imposing an additional constraint in which the DP releases are constrained in a bound of linear releases of the previous iteration. Using this iterative procedure, the value of the objective function is improved. Forty years of historical monthly inflows into each reservoir are available. Of that, 30 years of data is used in the DP and the FRB models and the remaining 10 years’ data are used in the simulation model. Statistical characteristics of these inflows show the first 10 years of the data belong to a dry period and the last 10 years to a wet period. Table I compares the mean and the standard deviation values of the first and last 10 years of inflows with the value of these statistics for the total 40 years of available inflows. Because of significant differences between the statistics of these two parts of inflow time series, the DPFRB and DPR models are evaluated for two scenarios. In one of them, the first 30 years of inflows are used in the DP model and the last 10 years in the simulation model. In the other scenario, the first 10 years of inflows are used in the simulation model while the last 30 years of inflows are used in the DP model. In another test and under each of the mentioned scenarios, the amount of water demands have been increased by multiplying them with a demand factor of two (demfact = 2) so that meeting them becomes more challenging for the DPFRB and the DPR models. Table II shows the results obtained from the DPFRB and the DPR models in each of the scenarios. It should be noted that from the different types of water demands, the domestic, industrial, and agro-industrial usages were satisfied over all the times of the simulation period. Therefore, only the reliability of meeting the agricultural water demand has been reported. In Table II, NS and NI are, respectively, the numbers of fuzzy intervals of storage and inflow premise variables in the FRB model. Optimization and simulation costs are the average monthly loss of reservoir operation in DP and simulation models, respectively, and the demand reliability is the percentage of time the target demand Table I. Statistics of different parts of historical inflows (million m3 )
Statistics
First 10 years of inflows
Last 10 years of inflows
Total 40 years of inflows
Mean
Mean
Mean
S.D.
CV
S.D.
CV
S.D.
CV
Karoon
632.0
336.5
0.53
1211.8
842.3
0.70
972.6
729.1
0.75
Dez
493.1
355.8
0.72
885.50
784.4
0.89
724.8
652.0
0.90
665
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Table II. Comparison between DPFRB and DPR models Model
Demand factor
NS∗
NI∗∗
DP cost
Simulation cost
Demand reliability
For first scenario, 10 last years’ inflows as test data DPFRB 1 (2) 3 (3) 4 (4) 0.00 (2070)
0.00 (2180)
100.0 (87.5)
DPR
459 (3840)
97.9 (84.2)
0.00 (22,500) 7020 (58,400)
100.0 (54.2) 93.3 (49.8)
–
–
For second scenario, 10 first years’ inflows as test data DPFRB 1 (2) 3 (3) 4 (4) 0.00 (324) DPR – –
demfact = 1 (or demfact = 2). ∗ Number of fuzzy clusters of the storage variable in the FRB model. ∗∗ Number of fuzzy clusters of the inflow variable in the FRB model.
has been satisfied during the simulation horizon. It should be noted that the NF, NS, and NI values were set the same for each reservoir. To avoid the discretization error in the case of demfact = 2 for which the optimization cost is not equal to zero, 30 discrete storage states for each reservoir was used in the DP model. However, it was found that dividing each premise variable into only three to five fuzzy numbers is sufficient in the FRB model. As mentioned earlier, this is due to the fact that some of the rules will not be fired if finer discretization is used. Both methods described in Section 3 for discretizing the premise variables had approximately similar results. Also, the first method of determining the mean value of each fuzzy number showed better results. Therefore, the mean values of the kth fuzzy number of each premise variable are the same for all the rules. From Table II and in the case of usual water demands (demfact = 1) it is seen that the value of objective function is zero in both optimization and simulation in some scenarios for the DPFRB model. For the DPR model, there is significant difference between the values of the objective function in the DP and the simulation models where the simulation costs for two wet and dry scenarios are equal to 459 and 7020, respectively. Also, the reliability of meeting the agricultural water demand in the DPFRB model is higher than that in the DPR model. We can observe in Table II that the DPFRB performs better than the DPR with respect to the value of the objective function in simulation as well as in terms of the reliability in meeting the demands. The results of both the DPFRB and the DPR models in the second scenario are not as good as the first scenario, especially for demand factor = 2.0, as can be expected but still DPFRB does perform much better than DPR. It should be noted that the difference between the values of the objective functions in DP and simulation sub-models of DPFRB and DPR comes from two sources. The first is due to difficulties of not fitting the operating rules to optimal values. In fact, the actual relation between release, storage and inflow may be highly nonlinear and this nonlinearity should be adequately modeled through the operating rules.
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When the inflows in the DP and the simulation models are the same, the operating rules are used for interpolation. However, when these inflows are different, we may have some inflows in simulation beyond the range of the inflows used in the training of the operating rules. Therefore, in addition to the capability of policies to fit to a complex and nonlinear pattern of data, the capability of operating rules for extrapolation should also be explored. This extrapolation can be converted into interpolation if the policies could be retrained when subjected to new data. Therefore, there are two different characteristics of an interface model (FRB or regression) that are important for developing the operating rules: its capability to fitting optimal data obtained from DP and its flexibility in retraining due to uncertainty of inflows. To distinguish between the aforementioned aspects, Table III compares the DPFRB and the DPR models in two different cases: when the inflows into reservoirs are supposed to be the same in the DP and the simulation sub-models and when they are not. This table shows the results of the first and the best iterations of the models in the two earlier cases. The A11 and A12 portions of Table III show the performance of the DPFRB and the DPR models in fitting the optimal solutions of the DP model because the inflows into reservoirs are the same for both the DP and the simulation models. Comparison between the results of A11 with A12 shows that the DPFRB outperforms the DPR with respect to fitting capability. For example, the best values of simulation cost are 4.36 (2640) in A11 (DPFRB) and 1080 (10,200) in A12 (DPR) cases. We see that the minimum value of the simulation cost in the DPFRB is less than the DPR cost for all the cases. This implies that the fuzzy rules and the tuning of their parameters in successive iterations result in a good fitness to the optimal solution of the DP model and hence the fuzzy rules can fit to a nonlinear structure of optimal data better than the regression-based rules. Comparing B11 to B12 is similar to the results of Table II, but comparing the A11 to B11 or A12 to B12 shows the effect of adding uncertainty on the performance of Table III. Effect of the uncertainty of inflows on the value of simulated cost in DPFRB and DPR models for demfact = 1 (demfact = 2) Same inflows in DP and simulation
Different inflows in DP and simulation
K∗
A11∗∗ DPFRB
A12∗∗ DPR
B11∗∗∗ DPFRB
B12∗∗∗ DPR
1
15.6 (2930)
1080 (10,200)
309 (30,900)
7110 (66,700)
2
4.36 (2640)
1080 (10,200)
0.00 (22,500)
7020 (58,400)
Optimum cost in DP = 0.00 (324). ∗ K = 1 is corresponding to the first iteration and K = 2 is corresponding to the best. ∗∗ A11/A12 are the values of simulation cost when the last 30 years’ inflows are used both in the DP and the simulation in the DPFRB/DPR models. ∗∗∗ B11/B12 are the values of simulation cost when the last 30 years’ inflows are used in the DP and the first 10 years’ inflows in the simulation in the DPFRB/DPR models.
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models in addition to the fitting capability. As we expect in this case, by adding the effect of uncertainty, the value of simulation cost is increased in the first iteration. For example, in the DPFRB model, the simulation cost has increased from 15.6 (2930) to 309 (30,900). After a few iterations, this effect has been compensated in the DPFRB model and the simulation cost has reached 0.0 (22, 500). This shows that for demfact = 1, the iterative procedure used in the DPFRB model result in a simulation cost of 0.0, which is equal to the value of optimization cost. On the other hand, we observe that even in the case of not including the uncertainty effect (A11), the best value of the simulation cost of the DPR model is equal to 4.36, which is greater than 0.0. For demfact = 2, the iterative method has improved the simulation cost from 30,900 to 22,500, but this value is still larger than 2640, which is the best value of the simulation cost when the uncertainty is not included. This shows the significant effect of the uncertainty in the case of demfact = 2. Similarly, if we compare the results of A12 with B12, it could be observed that the simulation cost of the DPR model is also increased due to the effect of uncertainty. For example, the simulation cost has been changed from 1080 (10,200) to 7110 (66,700) in the first iteration. These values have been decreased to 7020 (58,400) in the best iteration that is not as good as the improvement due to iterative procedure of the DPFRB model. This advantage of the DPFRB over the DPR is important, because in the DPFRB model, the time consuming DP sub-model has to be executed only in the first iteration, while in the DPR model it is executed in all the iterations. Compared with DPR, this feature of the DPFRB is attractive in terms of minimal computational resources required, especially when it is applied to a large-scale optimization problem. However, running a multiple reservoir DP model even once can be expensive if DP is used in the first step of the model. Although the DPFRB model is limited in terms of resolving the dimensionality problem associated with multiple reservoir DP models, the three-step methodology presented is independent of the type of optimization model used in the first step. In other words, although DP is used in the paper, it can be replaced by any other optimization method suitable for multiple reservoir systems. 6. DPFRB and Robust Optimization The previous section showed that the DPFRB model has advantages with respect to flexibility against uncertainty of inflows. This important advantage can be used to achieve robust operating rules, which are to hedge against uncertainties. For such a purpose, we generated long-term monthly synthetic inflows into the Karoon and Dez reservoirs by fitting the best-fitted Auto-Regressive Moving Average (ARMA) models to their historical inflows. These long-term time series represent the probable variations of inflows and include the extreme inflows, which are important for evaluation of future expected performance of the system. A time series of 250 years (6000 months) of synthetic inflows was generated. Table IV compares the statistics
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Table IV. Statistics of historical and synthetic inflows (million m3 ) Historical inflows Statistics
Mean
S.D.
Synthetic inflows CV
Mean
S.D.
CV
Karoon
972.6
729.1
0.75
976.0
723.3
0.74
Dez
724.8
652.0
0.90
725.7
656.4
0.90
of synthetic inflows and historical inflows and as can be noticed, we see that these statistics are very close to each other. The DPFRB model was first executed during the first 3000 months of synthetic inflows to find the values of “g” and “e” parameters. In this case, the objective function in the simulation would be the minimization of standard deviation of the monthly simulation cost, which may be considered as a risk aversion objective function. In other words, the parameters of the FRB were obtained by using 3000 months of synthetic inflows in which the variation of monthly operation cost should be minimal. After calculating the best values of the parameters, we predicted that the fuzzy rules of the DPFRB model are robust against uncertainty of inflows if these parameters are used. To validate this prediction, both of the DPFRB and the DPR models were utilized by using the other 3000 months of synthetic inflows in the simulation sub-models for both of them. It should be noted that in the stages of fixing the parameters and testing the fuzzy rules based on these fixed parameters, the 40 years of historical monthly inflows were used in the DP sub-models of the DPFRB and the DPR models. For further evaluation of the proposed approach, two versions of the DPFRB model were examined. In the first version (DPFRB1), we did not consider any risk consideration, and the parameters of “g” and “e” are selected in such a way that the expected value of the monthly loss is minimized during simulation. In the second version, these parameters are determined by considering the risk aversion characteristic as described earlier and by also considering the trade-off between the expected value and the standard deviation of the simulated monthly loss. To evaluate the sensitivity of the operating rules against the uncertainty of the inflows, each model was implemented in the following two situations: (a) when the inflows in the DP and the simulation are the same, and (b) when the historical inflows are used in the DP and the long-term synthetic inflows are used in the simulation. The two situations represent the performance of the models without and with the consideration of the uncertainty of inflows, respectively. Table V shows the best result among the different iterations of the models for these situations. Similar to the other tables, the values inside the parentheses are related to demfact = 2.0. Each of the earlier three models are sensitive to the uncertainty of inflows because both, the mean value and the standard deviation of the simulated loss, increase
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Table V. Robustness of DPFRB and DPR rules against the uncertainty of inflows for demfact = 1 (demfact = 2)
Model
Mean of simulation cost
Standard deviation of simulation cost
Demand reliability
g∗
e∗
DPR a) b)
949 (16,137) 8757 (35,491)
11,717 (51,283) 53,432 (130,607)
99.4 (70.2) 96.4 (78.0)
– –
– –
DPFRB1 a) 56.5 (6053) b) 4365 (22,050)
705 (21,192) 31,734 (98,475)
99.4 (78.1) 96.5 (81.4)
1.25 (0.90) 0.60 (1.0)
0.02 (0.09) 0.21 (0.00)
DPFRB2 a) 65.0 (6718) b) 1873 (23,480)
651 (23,076) 14,180 (88,735)
99.2 (78.5) 96.6 (65.2)
1.35 (0.55) 0.60 (0.80)
0.10 (0.15) 0.225 (0.12)
significantly when we use a long-term synthetic inflows instead of the historical inflows in simulation. For example, for demfact = 1.0, the mean and the standard deviation values of (949, 11,717) have been changed to (8757, 53,432) in the DPR model. In the DPFRB1, the values of (56.5, 705) have been increased to (4365, 31,734). Due to uncertainty of inflows, both the expected value and the standard deviation of the simulated loss are increased. These values are however less in the DPFRB1 than in the DPR model. In other words, even without considering the risk consideration for optimizing the parameters, the performance of DPFRB1 is better than the DPR. For the DPFRB2 model, these values have been changed from (65, 651) to (1837, 14,180). We observe that in scenario (a), these values are close to the values obtained by DPFRB1 model, but in scenario (b) the standard deviation of the simulated loss of DPFRB2 is 0.45 of this value in the DPFRB1 model. This shows the capability of the DPFRP2 model in deriving robust operating rules. It should be noted that it does not matter much which of the model to use, the DPFRB2 or the DPFRB1 model, when the uncertainty of inflows is not considered. This is the case, especially if we compare their results in demfact = 2. For example, in scenario (a), the results of the DPFRB2 (6713, 23,076) are no better than the DPFRB1 (6053, 21,192). In scenario (b) and where uncertainty exists, the standard deviation of the simulated loss of the DPFRB2 (88,735) is smaller than this value in DPFRB1 (98,475), when their expected values are approximately the same (23,480, 22,050). However, both of the DPFRB1 and the DPFRB2 models perform better than the DPR model in the two scenarios (a) and (b). 6.1.
DISCUSSION ON USING A PARAMETRIC FUZZY RULE BASE
An important advantage of the DPFRB lies in the flexibility created by the FRB parameters, i.e. “g” and “e”. In this way, the operating policies are parameterized
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in such a way as we are able to find the best values based on a preferred objective function in simulation. We can optimize these parameters to take into account any other reliability or system performance indices such as resiliency or vulnerability. The feedback loop, used between the FRB and the simulation steps of the DPFRB model, helps the model minimize the effect of using different inflows in the FRB and the simulation steps. As such, the model attempts to find the parameters and adjust the operating policies in which the use of different set of inflows results in minimum difference between the values of objective functions of the DP model and the simulation model. In this technique, we applied first a deterministic optimization model and then investigated the effect of uncertainty of inflows, which has been summarized in the set of long-term synthetic inflows. In other words, the DPFRB model divides a complex stochastic optimization problem to two simpler deterministic problems, i.e. the DP model and the optimization of parameters in the simulation. The FRB and the iterative feedback loop perform as an interface between these two simple models to make the value of their objective functions as close as possible. This difference between the value of the objective functions of the DP and the simulation models, which comes from the fitting problem and also from the uncertainty of inflows, is controlled in the DPFRB model through this interface model and the adaptive procedure. Finally, and regarding the direct search optimization method used for getting the optimal values of “g” and “e”, it should be noted that the objective function shows further improvement if a finer grid search is used around the best obtained values of “g” and “e”. However, it was observed that the rate of these later improvements is very slow and does not warrant further computations. 7. Summary and Conclusions A three-step DPFRB model was presented and tested in this paper. This model integrates a DP model, a FRB model, and a simulation model. The optimal solutions of the DP model are used to establish the general operating policies in the FRB model and these policies are then evaluated in the simulation model. A direct search optimization method is used to find the best values of the parameters of the FRB model. These optimal parameters are used by the FRB model to create a new set of operating rules and the procedure continues until the satisfaction of the stopping criteria. This model is compared to the DPR model, which uses multiple linear regression to derive the operating rules. The better values of the simulated objective function and the higher reliability of meeting the demand shows the capability of the DPFRB model in satisfying the objectives of the system when compared to the DPR model. General operating rules in form of fuzzy “if–then” rules are easy to introduce to human operators. This form of policies is consistent with the human thinking and can include explicit judgments in decision-making process through adequate definition of the membership
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functions (MFs). The parameterized form of the operating rules in the DPFRB model allows us to include risk and reliability considerations in deriving the policies. By optimizing these parameters, we can obtain such robust policies that could hedge against the uncertainty of inflows. Potential improvements could be achieved by complementing this work by using other tools of soft computing including ANNs for tuning the shape of the MFs, or genetic algorithms for optimizing the parameters of the FRB. References Bhaskar, N. R. and Whitlach Jr., E. E., 1980, ‘Deriving of monthly reservoir release policies’, Water Resources Res. 16(6), 987–993. Cancelliere, A., Giuliano, G., Ancarani, A., and Rossi, G., 2002, ‘A neural networks approach for deriving irrigation reservoir operating rules’, Water Resources Manage. 16, 71–88. Chandramouli, V. and Raman, H., 2001, ‘Multireservoir modeling with dynamic programming and neural networks’, J. Water Resources Plan. Manage. 127(2), 89–98. Dubovin, T., Jolma, A., and Turunen, E., 2002, ‘Fuzzy model for real-time reservoir operation’, J. Water Resources Plan. Manage. 128(1), 66–73. Johnson, S. A., Stedinger, J. R., and Staschus, K., 1991, ‘Heuristic operating policies for reservoir system simulation’, Water Resources Res. 27(5), 673–685. Karamouz, M. and Houck, M. H., 1982, ‘Annual and monthly reservoir operating rules’, Water Resources Res. 18(5), 1337–1344. Koutsoyiannis, D. and Economou, A., 2003, ‘Evaluation of the parameterization-simulationoptimization approach for the control of reservoir systems’, Water Resources Res. 39(6), 1170, 1–17. Lund, J. R. and Guzman, J., 1999, ‘Derived operating rules for reservoirs in series or in parallel’, J. Water Resources Plan. Manage. 125(3), 143–153. Lund, J. R., and Ferreira, I., 1996, ‘Operation rule optimization for Missouri River reservoir system’, J. Water Resources Plan. Manage. 122(4), 287–295. Nalbantis, I., and Koutsoyiannis, D. 1997, ‘A parametric rule for planning and management of multiple-reservoir systems’, Water Resources Res. 33(9), 2165–2177. Oliveira, R. and Loucks, D. P. 1997, ‘Operating rules for multi-reservoir systems’, Water Resources Res. 33(4), 839–852. Panigrahi, D. P. and Mujumdar, P. P., 2000, ‘Reservoir operation modeling with fuzzy logic’, Water Resources Manage. 14, 89–109. Philbrick, C. R. and Kitanidis, P., 1999, ‘Limitations of deterministic optimization applied to reservoir operations’, J. Water Resources Plan. Manage. 125(3), 135–142. Ponnambalam, K., 2002, ‘Optimization in water reservoir systems’, in P. M. Padalos and G. C. R. Mauricio (eds), Handbook of Applied Optimization, Oxford University Press; ISBN: 0195125940, pp. 933–943. Ponnambalam, K., Karray, F., and Mousavi, S. J., 2003, ‘Minimizing variance of reservoir systems operations benefits using soft computing tools’, Fuzzy Sets Syst. 139(2), 451–461. Ponnambalam, K., Mousavi, S. J., and Karray, F., 2001, ‘Regulation of Great lakes reservoir systems by a neuro-fuzzy optimization model’, Int. J. Comput. Anticipat. Syst. 9, 273– 285. Raman, H. and Chandramouli, V., 1996, ‘Deriving a general operating policy for reservoirs using neural networks’, J. Water Resources Plan. Manag. 122(5), 342–347. Russel, S. O. and Camplell, P. E., 1996, ‘Reservoir operating rules with fuzzy programming’, J. Water Resources Plan. Manage. 122(3), 165–170.
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