Application of Implicit Stochastic Optimization In the Han River Basin Taesoon Kim 1 , Jun-Haeng Heo 2
Abstract This paper describes the development of monthly operating rules for the multireservoir located at the upstream of the Han River Basin in Korea. These operating rules are estimated using implicit stochastic optimization and regression analysis. In this study, to generate the synthetic inflows, the periodic ARMA(1,1) model is taken and implicit stochastic optimization is applied to get optimal reservoir operating results. These results are regressed to derive the operating rules. The beginning storages, synthetic inflows and optimal releases for each site are used to estimate the operating rules. The graphical analysis of these operating rules illustrates the applicability and limitations of applying implicit stochastic optimization to development of operating rules.
1 Introduction The development of reservoir operating rules has been the focus of research for many years. Young (1967) first proposed methods to obtain general annual operating rules from the results of a deterministic optimization model. Yeh (1985) reviewed the state of the art reservoir management models. Karamouz and Houck (1992) developed operating rules for multireservoir systems by implicit stochastic optimization. Implicit stochastic optimization is also taken and tested using simplified simulation model (Lund and Ferreira, 1996). In the Han River Basin, three major reservoirs (Hwacheon, Soyanggang, and Chungju) are located at the upstream of the river and they play an important roles on water supply and flood control to downstream, especially Seoul Metropolitan Area. In the flood season, the multireservoir operating methods are applied, but in the normal season the reservoirs are operated separately, not considering the benefit gained from
1. Ph. D. Candidate, Department of Civil Engineering, Yonsei University, Seoul, 120749, Korea,
[email protected] 2 . Head and Associate Professor, Department of Civil Engineering, Yonsei University, Seoul, 120-749, Korea, jhheo@ yonsei.ac.kr
joint operation. Although there are several studies to develop operating rules for single reservoir in the Han River Basin (Ko et al., 1992; Lee et al., 1992; Ko et al., 1997), applications to multireservoir have fewer case studies. In this study, the monthly multireservoir operating rules were developed by implicit optimization and regression analysis. Because historical records are different from each other, univariate periodic ARMA (1,1) model is employed in order to generate the synthetic inflows. And, the incremental dynamic programming is used to get optimization results for multireservoir. The linear regression analysis is performed to develop operating rule using these optimization results.
2 Time series model The periodic ARMA (p,q) model can be expressed as (Salas et al., 1980) p
Zν ,τ = ∑ φ j ,τ Zν ,τ − j − j =1
q
∑θ i =1
i ,τ
εν ,τ − i + εν ,τ
(1)
where Zν ,τ = periodic time series; φ j ,τ and θ i ,τ = time varying autoregressive and moving average coefficients, respectively; and εν ,τ = an independent and identically distributed normal random variable. The lag-1 autoregressive and moving average time series model has been used. For the periodic ARMA (1,1) model, Eq. (1) can be written as: Zν ,τ = φ1,τ Zν ,τ −1 − θ 1,τ εν ,τ −1 + εν ,τ
(2)
In this model, the periodic inflow data Z at time τ depends on Z at time τ −1 . The synthetic generated data Zˆ are obtained by ν ,τ
Zˆν ,τ = φˆ1,τ Zˆν ,τ −1 − θˆ1,τ εν ,τ −1 + εν ,τ
(3)
where φˆ1,τ , θˆ1,τ = estimated parameters.
3 Dynamic programming model An incremental dynamic programming has been used to determine operating rules for multireservoir. The objective function of the multireservoir problem under consideration can be written as 2 TP TR F = ∑ ∑ F1 ( p ) − ∑ F2 (s ) p =1 i =1 s =1 T
maximize
(4)
where T = the time horizon; TP = the total number of hydropower plants; F1 ( p ) = the hydropower production at number p plant; TR = the total number of reservoirs; F2 (s ) = the ending storage at the number s reservoir. The continuity or a mass balance of the contents of the reservoir from the beginning of the season to the next must be included. St +1 − S t + Rt = I t
(5)
where Rt = release during season t; I t = inflow during season t; and St = storage volume at the beginning of season t. In addition to maintaining continuity within the reservoir operating model, it is necessary to ensure that all storage volumes and release observe physical restrictions. These constraints on maximum and minimum allowable release and storage during any season can be stated as Rtmax ≥ Rt ≥ Rtmin
(6)
Stmax ≥ S t ≥ S tmin
(7)
This mathematical program is solvable as a dynamic programming. The recursion relation can be written as 2 TP TR f t +1 ( St +1 ) = maximize ∑ F1 ( p ) − ∑ F2 (s ) + f t ( St ) p =1 s =1
(8)
where f t (St ) = the total maximum of objective function from the beginning of season 1 to the beginning of season t, when the storage volume at the beginning of season t is St .
4 Case-study application The Han River multireservoir system is a large multireservoir system. There are three large reservoirs such as Hwacheon, Soyanggang, and Chungju located at the upstream of the Han River Basin, as shown in Fig. 1. Because the historical inflow records of three reservoirs are different and the high flows are formed mainly by large rainfall, the univariate periodic ARMA (1,1) model is applied to generate synthetic inflow data. One hundred and fifty years of monthly data were generated to dynamic programming. Tables 1-2 show the statistics for historical and generated data. Incremental dynamic programming is used to obtain optimization results. It is very efficient methods to solve dynamic programming and it needs an initial trajectory to
begin computation. In this study, initial trajectory is computed by considering the system constraints (maximum and minimum outflow, storage capacity) and the amounts of synthetic inflow.
Hwacheon
Chuncheon
Soyanggang
Uiam
Chungpyong
Chungju
Storage Reservoir
Paldang
Flow -Through Reservoir
FIG 1. Han River Multireservoir System Table 1. Statistics of Historical Monthly Inflows Years
Hwacheon 1971 – 1998 Standard Deviation
Soyanggang 1974 – 1998
Month
Mean (MCM)
Mean (MCM)
1
14.02
5.32
6.21
2
15.90
9.53
3 4
35.14 66.30
5
Chungju 1956 – 1998
Standard Deviation
Mean (MCM)
Standard Deviation
3.25
30.52
13.57
9.78
10.71
35.03
30.68
22.38 37.91
31.00 68.90
23.03 33.99
88.63 198.92
71.20 136.63
68.19
43.18
63.82
46.98
126.03
75.21
6 7
72.35 249.07
82.29 176.76
63.81 194.34
69.83 106.29
135.97 508.47
136.76 267.78
8
308.21
236.93
190.60
154.69
366.83
243.99
9
169.84
143.33
119.20
150.44
286.02
247.25
10 11
38.99 35.34
23.39 37.43
29.53 22.30
32.24 18.78
92.90 63.72
70.35 35.50
12
21.01
9.91
12.26
9.20
44.88
22.26
The objective function was taken to be Eq. (4), which is 2 TP TR maximize F = ∑ ∑ F1 ( p ) − ∑ F2 (s ) (4) p =1 i =1 s =1 The ending storage of the year is the mean value of each reservoir in December, which T
is 576MCM(million cubic meters, 106 ×m3 ) at Hwacheon, 1607.5MCM at Soyanggang, 2134.1MCM at Chungju. And, in the flood season, the allowable maximum storages of these reservoirs are lowered to secure the flood control storage. Table 2. Statistics of Generated Data
Month
Hwacheon Mean Standard (MCM) Deviation
Soyanggang Mean Standard (MCM) Deviation
Chungju Mean Standard (MCM) Deviation
1 2
15.12 15.97
8.88 10.32
6.89 10.60
5.36 19.74
31.93 34.98
16.25 32.28
3
36.36
23.27
33.85
34.85
92.94
67.72
4 5
68.93 74.62
46.74 56.76
68.52 66.69
36.10 53.92
199.12 137.46
131.94 108.46
6
74.16
79.37
64.81
83.47
143.08
153.88
7
229.13
253.46
185.20
158.91
472.50
351.94
8 9
330.56 164.86
282.45 146.32
205.87 104.81
170.02 99.43
383.89 288.08
242.15 231.37
10
38.19
21.8
27.42
22.70
89.37
58.58
11 12
34.10 20.86
21.33 9.07
22.82 12.02
15.41 7.60
65.70 45.38
32.61 21.76
From these optimal releases, storages, and inflows for each reservoir, operating rules were obtained for future operation of the system. In this study, the two linear relationships are assumed as follows. Rt = a1 I t + b1 St + c1 (9) where Rt = release during month t at each reservoir; I t = inflow to the each reservoir during month t; St = storage in reservoir at the beginning of month t; and a1 , b1 , c1 = regression coefficients determined through multiple linear regression. Another one is St +1 = a2 St + b2 ST (t ) + c 2 IT (t ) (10) where ST (t ) = total storage in the reservoirs within the system at the beginning of month t; IT (t ) = total inflow to the system during month t; and a 2 , b2 , c 2 = regression coefficients. Bhaskar and Whitlach (1980) tested more complex nonlinear forms of the operating rules and found that the simple linear operating rules are as good as or better than the more complex rules in many cases.
5 Results and analysis The coefficients of determination (R2 ) are shown in Table 3 and the graphical relationships among optimal releases, storages, and inflows in Figs. 2-5. In Table 3, the means of coefficient of determination, which are 0.749, 0.553 and 0.724 respectively, have larger values at Eq. (9) in all reservoirs. And, the R2 of Eq. (9) is usually larger than that of Eq. (10). This shows that Eq. (9) is more proper regression relationship than Eq. (10). The lower R2 values in January resulted from the fixed initial storages (the means in January) and the little difference in sum of initial storage and inflow. But this boundary effect is diminished after only one month passed. In July, August and September, the mean values of R2 of Eq. (9) are much higher than those of Eq. (10). This higher value in this flood season is because the release is simply proportional to x coordinates and the ending storage is fixed at maximum capacity. In Korea, July to September belong to the flood season in which we have a heavy rainfall. More than 70% of annual rainfall comes down in this season. Thus, the annual water storage usually makes in this season and it is beneficial to make maximum storage. Table 3. Coefficient of determination (R2 ) Month
Hwacheon Eq. (9) Eq. (10)
Soyanggang Eq. (9) Eq. (10)
Chungju Eq. (9) Eq. (10)
1 2
0.043 0.510
0.136 0.441
0.037 0.896
0.275 0.865
0.029 0.879
0.034 0.864
3
0.615
0.481
0.763
0.586
0.478
0.521
4 5
0.709 0.743
0.376 0.234
0.466 0.346
0.316 0.341
0.778 0.792
0.698 0.697
6
0.938
0.477
0.374
0.176
0.627
0.540
7
0.902
0.402
0.547
0.384
0.750
0.561
8 9
0.981 0.977
0.663 0.367
0.723 0.819
0.351 0.412
0.946 0.969
0.685 0.819
10
0.829
0.588
0.587
0.585
0.561
0.513
11
0.896
0.747
0.640
0.646
0.869
0.846
12 Mean (7-9)
0.834 0.953
0.829 0.477
0.437 0.696
0.547 0.382
0.979 0.888
0.985 0.688
Mean (1-12)
0.749
0.479
0.553
0.457
0.724
0.647
In Fig. 2(b), suppose that the sum of storage, total storage and total inflow exceed 7960 (MCM), the ending storage of Chungju is almost fixed to its maximum capacity (2385 MCM). The similar thing occurred in Fig. 2(a). If the sum of the beginning storage and inflow is over 2670 (MCM), the release is simply proportional to the value of x
coordinate and the ending storage is almost fixed at the maximum capacity too. In Fig. 3, the optimization results of Hwacheon are similar to those of Fig. 2. 2400
2700
2300
2400
2200
Ending Stora ge (MCM)
3000
Release ( MCM)
2100 1800 1500 1200 900
2100 2000 1900 1800 1700
S t + ST( t) + IT( t) = 7950 (M C M )
1600
600
S t + I t = 2 670 (M CM )
300
1500
0
1400 1500
2000
2500
3000
3500
4000
4500
5000
5500
5000
B eginning Stor age + Inflow (M C M)
6000
7000
8000
9000
10000
11000
Storage + Total Storag e + To ta l Inflow ( MC M )
(a)
(b)
FIG. 2 Optimization Results (Chungju, September) 850
1600
800
1400
750
Ending Stora ge (MCM)
Release (MCM)
1200
1000
800
600
700 650 600 550 500
400 450 200
400
0
350 400
600
800 1000 1200 1400 1600 1800 2000 2200 2400
Beginning Storage + Inflow (M CM )
(a)
5000 5500 6000 6500 7000 7500 8000 8500 9000 9500
Stor age + To tal Storage + Total Inflow (M CM )
(b)
FIG. 3 Optimization Results (Hwacheon, August) Figures 4-5 are the optimization results of non-flood season (April and May). In the non-flood season, the releases are classfied into three categories. The differences of inflows rather than storages seem to cause these classifications, because the means of inflows of gathered points increased from lower to higher in almost all cases but there is no definite trend of the storages. In the non-flood season, these classified-releases are good criteria for making decision in the watershed management.
350
350
R eleas e = 305 ( MC M )
R eleas e = 300 (M C M )
300
Relea se = 220 (M C M )
Release (MCM)
Release (MCM)
250
300
200
150
100
Relea se = 273 (M C M )
250
200
150
Relea se = 120 (M C M ) 100
50
0
Re le ase = 123 (M C M )
50 600
800
1000
1200
1400
1600
1800
Beginning Storag e + Inflow (M C M )
FIG. 4 Optimization Results (Soyanggang, April)
2000
400
600
800
1000 1200 1400 1600 1800 2000 2200
Beginning Stor age + Inflow ( M CM )
FIG. 5 Optimization Results (Soyanggang, May)
6 Conclusion In this study, the monthly operating rules are developed by implicit stochastic method. The synthetic inflows are generated by univariate periodic ARMA (1,1) model and the optimization method is applied to obtain optimal releases and optimal storages. The regression analysis is performed using these optimal releases, storages and inflows. Multiple linear regression analysis can be accomplished in two ways. The first is to show the relationship among the beginning storage, the synthetic inflow and the release. The other can relate the storage in any reservoir at the end of the month to its beginning storage, the total beginning storage in the system, and the total inflows during that month. Between these two methods, the first one has more proper ability to make an operating rule. The operating rules are considered for the cases of the flood season and non-flood season. In the flood season, because the release is simply proportional to x coordinates and the ending storage is fixed at maximum capacity, the values of R2 are relatively high. In the non-flood season, the developed monthly operating rules are divided into two or three categories in releases. This classification is resulted from the difference of inflows rather than storages. And, these classified releases are useful tools for making decisions.
Reference 1.
Bhaskar, N. R., and Whitlatch, E. E. (1980). “Derivation of monthly reservoir release policies.” Water Resources Research, 16(6), 987-993.
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