PIECEWISE-LINEAR OPERATING RULE FOR SINGLE RESERVOIR USING NSGA-II Taesoon Kim (1), Jun-Haeng Heo (2) , Deg-Hyo Bae (1), Younghun Jung (2) (1)
Department of Civil and Environmental Engineering, Sejong University, Seoul, 143-747, Republic of Korea phone: +822 3408 3337; e-mail:
[email protected],
[email protected] (2) School of Civil and Environmental Engineering, Yonsei University, Seoul, 120-749, Republic of Korea phone: +822 393 1597; e-mail:
[email protected],
[email protected]
ABSTRACT Piecewise-linear operating rule for Soyanggang reservoir in Korea Peninsula is developed using multi-objective genetic algorithms (NSGA-II) and time series modelling. Implicit stochastic approach is applied and 100-year synthetic inflow is generated by autoregressive order one (AR-1) time series model. The simulation model is performed to evaluate the performance of the developed operating rule using historical average initial storage at the start of year and historical annual inflow. Regression analysis is not necessary to get the operating rule since each end point of piecewise-linear operating rule is calculated by optimization model directly, and multi-objectives which are to maximize sum of hydroelectric power production and to minimize water shortage index is considered to get a proper operating rule. Although the piecewise-linear operating rule is developed with only annual synthetic inflows which are larger than the sum of designated water supply, it can be used when annual inflow is not enough to supply water resources to downstream. Storage and release pattern also show good results comparing with historical operation. Keywords: piecewise-linear operating rule, single reservoir, NSGA-II 1.
INTRODUCTION The development of reservoir operating rule is classical problem in water resources engineering (Labadie, 2004; Yeh, 1985). Two approaches have generally been applied to this problem. Theoretically appealing approach is the application of explicit stochastic optimization methods (Loucks et al., 1981). However, these explicit stochastic methods suffer from great computational resources and time. Another approach is to conduct deterministic optimization methods to a probabilistic problem using synthetic inflow. This deterministic approach is usually called as implicit stochastic optimization. The large deterministic solution is intensively examined in order to obtain relationship between independent and dependent variables of reservoir system such as time, initial storage, inflow, and release (Karamouz et al., 1992; Young, 1967). Regression analysis is the most popular method to refine promising operating rule. However, piecewise-linear operating rule can be good alternative approach (Oliveira and Loucks, 1997). In this study, multi-objective genetic algorithms (MOGAs), NSGA-II is applied to develop piecewise-linear operating rule. NSGA-II is one of the most imitating MOGAs and it shows good performance in application to many engineering problems (Deb et al., 2002; ISI, 2004). 2.
APPLYING NSGA-II TO DEVELOP OPERATING RULE The key features of NSGA-II are elitism that passes the best chromosome in the current generation to the next and sharing function without any user-defined parameter like sharing parameter δ . Many applications of reservoir system optimization with NSGA-II show good performance (Kim et al., 2006; Liong et al., 2004). NSGA-II can effectively find Paretooptimal solutions from relatively complex decision variable space, and doesn’t deteriorate
performance even if fine-tuned initial trajectory that is essential to dynamic programming is not used. First, a 100-year synthetic inflow series is generated. Time series modelling is employed to formulate auto regressive model. Then, piecewise-linear operating rule is developed using the 100-year synthetic inflow series as input data of NSGA-II. A chromosome consists of five x and y coordinates pairs and time horizon of optimization model is a year (12 months) thus total length of a chromosome is 120 real variables. 3.
TIME SERIES MODELING Case study area is Soyanggang reservoir basin in Korea Peninsula (Figure 1). Soyanggang reservoir is located at the North Han River and has a key role in supplying municipal water demand of Seoul Metropolitan area. The storage capacity of Soyanggang reservoir is the largest one in South Korea that is 2,900MCM.
Figure 1. Soyanggang reservoir basin in Korea Peninsula Historical monthly inflows from 1974 to 2005 (Figure 2) are examined in order to generate a synthetic inflow of Soyanggang reservoir. Generally, over 70% of annual precipitation is occurred during summer season and thus very high inflow can be seen also. Time series model with periodic coefficients is one of the alternatives, but because the number of parameters is too large to satisfy parsimony of time series model, our study is confined to modelling with constant coefficients (Salas et al., 1980). 800
Soyanggang Inflow 700 600
Inflow (cms)
500 400 300 200 100 0 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Year
Figure 2. Historical inflow of Soyanggang reservoir
First, autocorrelation function (ACF) and partial autocorrelation function (PACF) are used to identify time series model. In Figure 3, ACF and PACF shows typical properties for AR(1) that are infinite in extent, consisting of damped waves (ACF) and finite in extent, having peak at lags 1 then cutting off (PACF). Therefore, AR(1) with constant coefficients is used to generate synthetic inflow. The AR(1) time series model is generally written as Eq. (1).
zt = φ1 zt −1 + ε t
(1)
where zt is time dependent variable, φ1 is the 1st autoregressive coefficient, and ε t is independent variable with mean zero and variance σ ε2 . 1.0
1.0
Lag 1 Sample ACF=0.370
Lag 1 Sample PACF=0.370
0.5
Sample PACF
Sample ACF
Lag 2 Sample ACF=0.184 Lag 3 Sample ACF=0.155 Lag 4 Sample ACF=0.121 95% confidence limits (0.100) 0.0
Lag 2 Sample PACF=0.0547
0.5
Lag 3 Sample PACF=0.0384 95% confidence limits (0.100) 0.0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
0
2
4
6
8
10
12
Lag
14
16
18
20
22
24
26
28
30
32
Lag
Figure 3. Autocorrelation function (ACF) and partial autocorrelation function (PACF) of historical inflow Various statistical properties of sample and synthetic inflows are compared to verify the goodness of fit time series model. All basic statistics of synthetic inflow resemble those of sample inflow of Soyanggang reservoir including minimum and maximum (Table 1). Table 1. Basic statistics of sample and synthetic inflow Month
AVG
STD
Sample inflow
1 2 3 4 5 6 7 8 9 10 11 12
6.5 9.2 29.0 66.2 60.3 56.3 191.7 213.0 133.5 30.2 21.5 11.3
3.6 9.9 21.7 34.7 44.7 65.0 112.0 159.5 146.2 29.9 18.2 8.8
Synthetic inflow
1 2 3 4 5 6 7 8 9 10 11 12
7.3 9.9 26.0 61.2 54.0 50.5 191.2 203.9 119.6 28.2 20.0 12.4
4.0 10.2 19.6 33.0 42.1 52.1 119.0 145.8 111.8 25.5 14.9 9.9
SKEW
KUR
MIN
MAX
0.8 2.3 1.4 0.7 1.5 2.2 0.6 1.2 2.2 2.6 2.6 2.1
0.3 5.4 1.8 -0.3 2.5 4.2 -0.3 1.5 4.6 8.0 9.5 5.2
1.0 1.1 5.1 17.7 4.8 9.3 30.0 24.4 14.1 7.0 4.1 2.8
14.9 41.2 88.3 150.4 204.3 269.3 442.1 703.1 607.3 151.2 98.1 43.1
1.7 2.3 2.0 1.3 2.1 2.6 0.9 1.9 2.7 3.4 2.4 3.2
5.4 6.9 6.6 1.8 5.5 7.9 0.6 4.6 9.5 15.4 9.1 16.6
-0.5 0.4 1.2 8.6 -1.0 4.0 -12.8 21.3 7.7 5.8 3.1 1.9
26.2 55.9 127.0 174.4 229.5 307.0 544.8 812.8 696.3 184.6 101.6 75.3
AVG is average, STD is standard deviation, SKEW is skewness coefficient, KUR is kurtosis coefficient, and MIN/MAX is minimum and maximum values of sample and synthetic inflow.
Especially, minimum and maximum of historical inflow should be preserved. If these values are much different from sample data, operating rule developed can be inappropriate and thus it cannot be applied in real situation. Minimum and maximum of the synthetic inflow show a similar pattern to those of sample (historical) inflow (Figure 4). Negative values in minimum are replaced by zero. 1000
min : sample max : sample min : synthetic max : synthetic
800
Inflow (cms)
600
400
200
0
1
2
3
4
5
6
7
8
9
10
11
12
Month
Figure 4. Comparison of maximum and minimum between historical and synthetic inflow 4.
PIECEWISE-LINEAR OPERATING RULE Piecewise-linear operating rule consists of several linear segments that are connected with each other (Oliveira and Loucks, 1997). In order to apply NSGA-II, a chromosome is formulated using x and y coordinates of search space. X-coordinate means a sum of monthly inflow and initial storage at the start of month, and y-coordinate is monthly release. Figure 5 shows an example of how chromosome is formulated in this study. A six-segmented piecewise-linear operating rule is developed thus seven end points should be determined. The x and y coordinates of each point are larger than those of previous end points.
5
800
4
700
Release (MCM)
600
X = probable maximum + maximum inflow Y = maximum release
3 2
500
5 points x 2 pair x 12 months = 120 genes
400
1
300 200
X = probable maximum Y = minimum release
100 0
200
400
600
800
1000
Initial Storage + Inflow (MCM)
Figure 5. Chromosome formulation
1200
Among these seven end points the coordinates of the first and last points are known. Generally, dead storage or active storage capacities are directly used for calculating the coordinates of the first and last points, and frequency analysis techniques are applied for getting a better result in this study. Table 2 shows maximum and minimum storage capacities based on the assumption that historical storages at the start of month are distributed according to GEV distribution. The 99% nonexceedance and exceedance quantiles are used as maximum and minimum storages to calculate the limits of the first and last points, respectively (Table 2). Table 2. Maximum and minimum limits based on probability (MCM) Jan M a x M i n
95 % 99 % 95 % 99 %
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2153.8 1969.4 1778.2 1682.9 1726.9 1750.9 1847.0 2198.7 2501.8 2452.7 2390.5 2304.8 2328.8 2147.9 1920.5 1816.6 1890.4 1960.0 2423.8 2448.4 2680.2 2555.6 2516.8 2456.1 1076.9 973.5
874.9
843.7
916.6
907.4
915.8 1038.3 1305.6 1330.9 1278.7 1198.4
866.9
695.6
678.3
778.0
780.9
847.9
790.2
847.6 1060.5 1034.4 1014.8 960.5
The objective functions of NSGA-II are a shortage index and a sum of hydroelectric power production. The shortage index is equal to the sum of the squares of the annual shortages over a 100-year period when each annual shortage is expressed as a ratio to the annual requirements (USACE, 1997), and defined as N ⎡S ⎤ 100∑ ⎢ A ⎥ i =1 ⎣ DA ⎦ SI = N
(2)
where, SI is shortage index, N is number of years, S A is annual shortage (annual demand volume minus annual volume supplied), and DA is annual demand volume. Hydroelectric power production is defined as KWH t = 9.81QtT H t e / 3600
(3)
where, QtT is total flow through the turbines in period t , H t is falling height, and e is a plant efficiency (Loucks et al., 2005). Five constraints such as the limits about x and y coordinates, storage capacity limit of Soyanggang reservoir, minimum water supply demand, and target terminal storage at the end of year are used. Minimum water supply demand is set to 95% and 90% of the designated water supply demand of Soyanggang reservoir. The population size is 1,000 and the number of generation is 500. Crossover and mutation probability is 0.7 and 1/120, respectively. The distribution indices for SBX and polynomial mutation operator are 5.0. 5.
RESULT Figure 6 shows Pareto-optimal solutions which can supply 90% and 95% designated water downstream. Shortage index of 90% water demand is generally larger than that of 95% water demand except for one solution around 0.0125. Both cases show well-distributed Pareto-optimal solutions in Pareto-front. Supplying more water resources to downstream would produce more hydroelectric power because water resources are generally supplied through turbine discharge. Therefore, hydroelectric power production of 95% water demand is
usually larger than that of 90% water demand. -40700
Hydropower Generation (GWh)
-40800 -40900
90% water demand 95% water demand
-41000 -41100 -41200 -41300 -41400 -41500 -41600 -41700 0.002
0.004
0.006
0.008
0.010
0.012
0.014
Shortage Index
Figure 6. Pareto-optimal solutions if only sufficient annual synthetic inflow used with 90% and 95% water demand constraints in the optimization The designated annual water demand of Soyanggang reservoir is about 1,457MCM. In order to secure enough water resources during the following year, terminal storage is set to 95% ~ 110% of initial storage in this study. However, if annual synthetic inflow is much smaller than 1,457MCM it is impossible to satisfy a constraint about terminal storage. Hence, a piecewise-linear operating rule is developed using only sufficient annual synthetic inflow that is larger than 1,457MCM and the performance of the developed operating rule is assessed through examination of the results using both sufficient and insufficient historical inflows. Table 3 shows the summary statistics of the four different optimization results. Case A uses 95% water demand constraint and case B uses 90%, thus the shortage indices of case A are expected to have smaller value than those of case B. However, it is noted that the shortage indices of cases B-I and B-II have about a half smaller than those of cases A-I and A-II. And, the average number of shortages a year is much smaller also. These results show that even though the smaller water demand is used, more water resources can be supplied through tradeoff relationship among Pareto-optimal solutions. In cases C and D annual historical inflow is insufficient to satisfy a constraint about terminal storage. Therefore, the water shortage in fourth row is changed to negative value, different from cases A and B. The shortage indices of case C have smaller value than those of case D. It means that average annual water shortage of case D is severer than case C. Also, water shortage of case C is smaller than that of case D and the amount of water supply is larger than that of case D. As a result, it can be said that although the water supply constraint is set to smaller value like 90% than 95%, more water resources can be supplied downstream according to trade-off relationship among Pareto-optimal solutions. This is because inflow is sufficient to satisfy the constraint about terminal storage. On the contrary, if inflow is insufficient as like in cases C and D the amount of water supply demand has a key role to determine release. The piecewise-linear operating rule is developed using only sufficient annual synthetic inflow. However, all the four cases of Table 3 show proper trade-off relationship even though insufficient annual synthetic inflow that is smaller than the designated water supply demand (1457MCM) is used. It is noted that the developed piecewise-linear operating rule can be used in the case of relatively insufficient annual inflow.
Table 3. Summary statistics from four different cases sufficient inflow rule + sufficient inflow simulation A-I
A-II
A-III
B-I
B-II
B-III
Shortage Index
0.015073
0.015311
0.017090
0.007180
0.008019
0.018715
Hydropower Sum
14482.25
14626.72
14668.22
14647.39
14812.19
14845.80
Water Supply
2266.17
2280.70
2285.75
2286.08
2300.93
2305.27
Shortage
809.22
823.75
828.80
829.13
843.98
848.32
Hydropower
536.38
541.73
543.27
542.50
548.60
549.84
NO. shortage
3.19
3.19
3.56
1.74
2.26
2.11
sufficient inflow rule + insufficient inflow simulation C-I
C-II
C-III
D-I
D-II
D-III
Shortage Index
0.381013
0.374277
0.345559
0.578095
0.528182
0.622525
Hydropower Sum
1587.70
1587.20
1590.73
1561.70
1566.10
1563.58
Water Supply
1418.96
1418.33
1421.56
1397.07
1400.29
1395.85
Shortage
-37.99
-38.62
-35.39
-59.88
-56.66
-61.10
Hydropower
317.54
317.44
318.15
312.34
313.22
312.72
NO. shortage
9.20
9.40
9.40
8.40
9.60
8.60
1) Cases A and C use 95% water supply demand as constraint and cases B and D use 90% water supply demand. 2) Each initial I, III, and II note that the both extreme Pareto-solutions in terms of water shortage and hydroelectric power production, and compromising solution, respectively.
Proper storage and release pattern is important of developing reservoir operating rule. Figure 7 shows storage and turbine discharge when the developed piecewise-linear operating rule is only used. Initial storage of Soyanggang reservoir is set to 1606.6MCM and historical inflow is used. All of storage and release is a typical one as can be seen in real reservoir operation result of Soyanggang reservoir. There is no abrupt change and inappropriate values.
Inflow (MCM)
1500 1200 900 600 300 0 1
2
3
4
5
6
7
8
9 10 11 12 13
1
2
3
4
5
6
7
8
9 10 11 12 13
1
2
3
4
5
6
7
8
9 10 11 12 13
Turbine Release (MCM)
Storage (MCM)
2400 2200 2000 1800 1600 1400 1200 700 600 500 400 300 200 100 0
Month
Figure 7. Historical inflow, simulated storage and turbine release (MCM). Water supply demand 95% is used and piecewise-linear operating rule (A-I) in Table 3 is used.
6.
Conclusion Conventional implicit stochastic approach for developing reservoir operating rule usually introduce regressing analysis in order to convert optimization results into proper operating rule. However, piecewise-linear operating rule doesn’t need any regression procedure because each end point is calculated by optimization model itself. In addition, multi-objective genetic algorithm (NSGA-II) permits decision maker to consider multi-objectives, not considering just single objective for reservoir operation. Although the piecewise-linear operating rule is developed only using annual synthetic inflow that is greater than the designated annual water supply demand, it can be used when annual inflow is not enough to supply sufficient water resources to downstream. And, storage and release computed by the developed piecewise-linear operating rule show good results that are not much different from historical operation results. AKNOWLEDGMENTS This research was supported by a grant (1-9-2) from Sustainable Water Resources Research Center of 21st Century Frontier Research Program. References Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II." IEEE Transactions on Evolutionary Computation, 6(2), 182-197. ISI. (2004). "Fast breaking papers." http://www.esi-topics.com/fbp/fbp-february2004.html. Karamouz, M., Houck, M. H., and Delleur, J. W. (1992). "Optimization and simulation of multiple reservoir systems." Journal of Water Resources Planning and Management, 118(1), 71-81. Kim, T., Heo, J.-H., and Jeong, C.-S. (2006). "Multireservoir system optimization in the Han River basin using multi-objective genetic algorithms." Hydrological Processes, 20(9), 2057-2075. Labadie, J. W. (2004). "Optimal operation of multireservoir systems: state-of-the-art review." Journal of Water Resources Planning and Management, 130(2), 93-111. Liong, S.-Y., Tariq, A. A.-F., and Lee, K. S. (2004). "Application of evolutionary algorithm in reservoir operations." Journal of The Institution of Engineers, Singapore, 44(1), 3954. Loucks, D. P., Beek, E. v., Stedinger, J. R., Dijkman, J. P. M., and Villars, M. T. (2005). Water resources systems planning and management : An introduction to methods, models and applications, UNESCO, Paris. Loucks, D. P., Stedinger, J. R., and Haith, D. A. (1981). Water resource systems planning and analysis, Prentice-Hall International, Englewood Cliffs, New Jersey 07632. Oliveira, R., and Loucks, D. P. (1997). "Operating rules for multireservoir systems." Water Resources Research, 33(4), 839-852. Salas, J. D., Delleur, J. W., Yevjevich, V., and Lane, W. L. (1980). Applied modeling of hydrologic time series, Water Resources Publications, Littleton, Colorado. USACE. (1997). Hydrologic engineering requirements for reservoirs, U.S. Army Corps of Engineers, Davis, CA. Yeh, W. W.-G. (1985). "Reservoir management and operations models: a state-of-the-art review." Water Resources Research, 21(12), 1797-1818. Young, G. K. (1967). "Finding reservoir operating rules." Journal of the Hydraulics Division, 93(HY6), 297-321.
