1. During operation according to this rule, the target on-peak energy is ... rules. The system is then simulated using ten long synthetic series of monthly data of.
Extreme Itydroloeical Events: Precipitation, Floods and Droughts (Proceedings of the Yokohama Symposium, July 1993). IAHS Pubf. no. 213, 1993.
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Simulation of a reservoir with standard operating rule R. HARBOE & U. RATNAYAKE Division of Water Resources Engineering, Asian Institute of Technology, GPO Box 2754, Bangkok 10501, Titailand
Abstract Standard operating rules for a reservoir have been derived with a stochastic dynamic programming model. The objective function was to maximize expected guaranteed on-peak energy while constraints on irrigation and flood control are satisfied. A special type of recursive equation was developed, which takes many iterations to converge. Several standard operating rules were designed based on results of the optimization model. For testing the reliability of these operating rules, simulation with long synthetic inflow series were performed. During the simulation the release required from the reservoir was calculated on the basis of the target energy, reservoir level and previous months inflow. The results showed that stochastic results were more conservative, (lower energy targets), but in turn had a higher reliability than higher targets obtained from deterministic models. The model was applied to Victoria Reservoir in Sri Lanka as an example.
INTRODUCTION The two most common purposes of the large number of multipurpose reservoirs are irrigation and hydropower production. These two purposes are generally in conflict as irrigation demands are usually higher during dry periods while the power is required throughout the year. The power requirement may need releases from the reservoir in addition to irrigation requirements, in such amounts which will not cause irrigation failures in future dry periods. Flood control, on the other hand may require some releases prior to the rainy season and also during it, in order to have an empty storage space in the reservoir as flood control reserve. In such situations the real time operation of the reservoir depends on how actually the system managers can determine reservoir releases which compromises all future and present needs. The energy generation planners are greatly assisted if a firm energy amount that can be supplied by the reservoir can be known beforehand. Reservoir operation in accordance to standard operating rule with an energy target is a way to operate such systems to satisfactorily meet above objectives. The standard operating rule used in this study can be shown as in Fig. 1. During operation according to this rule, the target on-peak energy is produced if the available water is enough to do so. When the available water is not enough all water is released and energy is produced according to a nonlinear relation. When the storage is more than the maximum allowable storage all the excess water is released and additional energy also produced. The rule described in this paper, defines the target in terms of annual on-peak firm energy. The annual distribution pattern is used to calculate the respective monthly targets. If a target can be found for the guaranteed on-peak energy production while meeting the irrigation and flood control
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R. Harboe & U. Ratnayake
c jo 'in
'o
03
Q Turbine Capacity
Target
Flood Control Reserve Dead Storage
Reservoir Capacity
- •
Available Water (Begining storage + Inflow)
Fig. 1 Standard operating rule.
requirements, then the reservoir system can be operated in real time without any forecasted knowledge of future inflows. The probabilities of occurrences of all inflows and the critical periods are considered in the analysis by formulating a stochastic optimization model. Improving the performance of an irrigation and hydropower system thus operated can be done by improving on the guaranteed on-peak hydropower production targets as well as the reliability of irrigation water supply. In this paper a method based on stochastic dynamic programming (SDP) is developed to arrive at this guaranteed onpeak energy targets necessary for system simulations. Boehle et al. (1981) and Harboe (1990) applied deterministic dynamic programming (DDP) in determining the targets for standard operating rules for low flow augmentations and firm power production respectively. The obtained targets were optimal if a critical period worse than the past critical period will not occur. In order to asses the stochastic nature of the inflows and thus to arrive at a target which considers the effects of all possible critical periods, the above methodology is extended to include the stochastic effects of inflows. The SDP model is based on an objective function which maximizes expected generated on-peak energy while deterministic constraints on irrigation and flood control are satisfied. The targets obtained through converged policy is used in designing the standard operation rules. The system is then simulated using ten long synthetic series of monthly data of hundred years each to asses the performance. These synthetic series are generated using the disaggregation model, 'LAST' software (Lane, 1990).
SYSTEM DESCRIPTION The applicability of the suggested methodology is emphasized by a real world example.
423
Simulation of a reservoir with standard operating rule
The Victoria Reservoir in the Mahaweli River system of Sri Lanka is selected for this purpose. The geographic map of the river system and the reservoir are shown in Fig. 2. The reservoir and hydrological dimensions of the Victoria reservoir are as follows: Catchment area Average annual discharge Maximum flood water level Normal maximum water level Minimum operating level Capacity of the reservoir Storage of the reservoir at flood control level Dead storage Spillway capacity Low level outlet capacity Number of turbine units Rated power Rated head Rated discharge
= = = = = = = = = = = = = =
1891 km2 1984 x l 0 6 m 3 441.2 m a.m.s.l. 438.0 m a.m.s.l. 370.0 m a.m.s.l. 720.0 x 106 m3 697.0 x 1 0 s m3 34.0 x l 0 6 m 3 7900 m3/sec 760 m3/sec 3 3x70 MW 190 m 3x46.7 m3/sec
Certain assumptions regarding the regulation of the inflows and the constraints are necessary as this reservoir is a part of a complex system with one small reservoir and a diversion located upstream and several downstream reservoirs. Simulated inflows of the reservoir, where the upstream control points are managed under a known policy, are therefore used in this research. The releases of Victoria reservoir goes into Randenigala reservoir which is located immediately downstream. This downstream reservoir acts as a regulating reservoir of 874 xlO 6 m3 capacity with an average annual incremental inflow of 544 X106 m3. As the on-peak firm energy requirement does not have seasonal variations, the reservoir operation for generation of this energy requirement will results
- I ] TO
IRRIGATION
SYSTEMS IN THE NORTHERN REGION 0RY ZONE____
INTERMEDIATE
.
IRRIGATION AREA EXISTINQ OR UNDER CONSTRUCTION PROPOSED IRRIGATION AREA EXISTING RESERVOIR ^ £ g r l PROPOSE0 RESERVOIR IRRIGATION SYSTEM BOUNDARY A , B . . . . IRRIGATION SYSTEMS RAINGAUGE STATION
ZONE
* - ' ' Polqollo
PROJECT BOUNDARY RIVER/CANAL TUNNEL EXBT1N0 HYDROPOWER STATION PROPOSED HYOROPOWtR STATION
BarTaqn? KANDY/*
VICTORIA RESERVOIR
Fig. 2 Map of Mahaweli Ganga River Basin.
R. Harboe & U. Ratnayake
424
in monthly discharges of nearly equal amounts unless other constraints, such as flood control, are violated. Considering this discharge pattern and relying on the degree of regulation provided by the Randenigala reservoir, the Victoria reservoir is operated to provide a minimum discharge of 107 X106 m3/monfh as the supply for downstream demands. The lumped annual downstream demand for irrigation is about 1554 X106 m3 with a peak demand of 304 x 106 m3/month. The other constraint imposed on reservoir operation is the flood control. This is to be achieved by not violating the flood control reserve on monthly basis.
MATHEMATICAL MODEL The stochastic mathematical model for a single reservoir optimization is formulated to maximize the expected annual on-peak firm energy production. In general the monthly on-peak firm energy requirement may vary from one period to another but the distribution within an year may not change. The generation of on-peak energy also should follow this distribution for optimality. The monthly energy distribution, which can be known beforehand, is used to convert the monthly generation into the annual energy generation. The ratio EGt/at in (1) thus calculates the corresponding potential annual firm on-peak energy generation for a given monthly generation (EGJ using a known distribution (aj. Formulation of the model is as given in Ratnayake & Harboe (1992). This mathematical model is solved using dynamic programming with the following recursive equation (backward recursion):
ft{St,Qt_x)
= max R,
mm
.Q,
(i)
^T'£
where: EG, = on-peak energy generation in month t in GW h; Q, = inflow during month t in 106 m3; R, = release during month t (decision variable) in 106 m3; S, = storage at the beginning of month t (state variable) in 106 m3; a( = monthly on-peak energy distribution coefficient; 12 Y, a, = 1.0;
t =1 P(QJQ,.\) = conditional probability of inflow Q, given that inflow Q,A occurred in the previous month; and f,(S,Q,A) = optimal expected annual firm energy if initial state is S, and inflow in the previous month was QtA, from stage t through the last stage. The above equation is solved subjected to the following constraints: (i) Mass balance: S, +1 = S, + Q, - R, - LOSS, - SPILL t
(2) 6
(ii)
3
where: LOSS, = evaporation loss during month t in 10 m ; and SPILL, = spill during month t in 106 m3; Storage constraint:
Simulation of a reservoir with standard operating rule
D < S,
(iii)
425
< C - FC,
(3)
where: C = reservoir capacity in 106 m3; D = dead storage in 106 m3; and FC, = flood control reservation of month t in 106 m3. Release constraint: min(/D, ,AW,) < R, < TC
(4) 6
3
where: AW, = available water in the reservoir in month t in 10 m ; IDt = irrigation demand of month t in 106 m3; and TC = turbine capacity. Application of SDP requires discretization of state and decision variables into a finite set of values. The performance of the simulation depends on the targets of the standard operating rule. A good representative value is therefore necessary. This in turn requires smaller discretization of the state and decision variables. The state variable, which is the reservoir storage, is discretized into 40 equal sized classes while the decision variable, which is the release from the reservoir through turbines, is discretized into a class size of 25% of the class size of the state variable. Smaller increments of these variables caused significant increases in computer time without a significant improvement to the objective function value. The inflows of a month are divided into a maximum number of ten classes of class width equal to half the standard deviation (2a) of inflows of the month. This inflow class arrangement is illustrated in Fig. 3. Mean of the fifth inflow class (JM) is positioned to coincide with the mean of the inflow distribution. The interval of the lowest class is taken to be from zero to the upper limit of the nearest full width class. The highest inflow class is taken to be an open class (with no defined upper limit). Representative inflow for the first inflow class to the one before last inflow class is taken as the mean value of the class interval. The representative inflow value (Q) for
Case 1:^J - 7 a > 2 a Mean of class no. 5 = \i
Lowest class: (1) i 2a - Minimum limit « 0.0 ta & •Class width > 2a
Class 10 No upper limit
a I a
T 0 Class number
-j
2
3
4 s\ 6 Mean of the overall distribution
7
8
9
10
Inflow
Case 2 : / J - 7 a < 2a Lowest class: (2) - Minimum limit = 0.0 • Clas.s width > 2a
2a
Mean of class no. 5 = /j i
Class 10 No upper limit
i
a^ a
0 Class number
2
3
4 5| 6 Mean of the overall d stribution
7
8
9
Inflow 10
a = 0.25 * Standard Deviation of the distribution Fig. 3 Discretization of inflows.
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R- Harboe & U. Ratnayake
the last inflow class (10) is taken as the value corresponding to half the exceedance probability of the lower limit of the class (ju+9a). The corresponding probability is equal to the above exceedence probability divided by the class width [2(Q—fx— 9a)].
COMPUTATIONAL RESULTS The recursive equation is iteratively solved considering yearly periodicity of hydrological data (i.e. one iteration is the solution of the recursive equation for one year period using monthly data). The objective function value given by the considered maxmin type objective function should converged to a constant value corresponding to maximized minimum expected value for a given set of inflows. Another factor that affects the obtained energy targets is the number of iterations of the SDP algorithm. Curtailment of the number of iterations depending on two criterion is tested. One criteria is to terminate the iterations when the number of changes in the policy is less than a reasonable amount. The other criteria is to terminate the iterations when the change in objective function is considerably smaller. With different levels of curtailment the energy targets are obtained and the system is simulated. Figure 4 shows the pattern of variation of the number of changes in the policy and the objective function value. Judging from the system performance it is recommended to curtail the iterations when the number of policy changes are less than 10%. This corresponds, in this case, to objective function value change of about 0.4%.
2.000
Obj. fn. value
A
No. of policy changes 1.600
\
c
' * * • •
-
1.2O0
O
T5 c Oi =
600
03
«0
~
--
'A-
"A. ''•A..
i •*
0
5
10
15
20
A
i.
, "*•
A
25
Number of iterations Fig. 4 Rate of convergence of the SDP iterations.
The reservoir was optimized again under the same objective function, but using deterministic dynamic programming (DDP) formulation as given by Harboe (1990), to obtain the base for comparison of performance. The target arrived using DDP model is 644.4 GWh while that for stochastic dynamic programming (SDP) model is 583.4 GWh.
427
Simulation of a reservoir with standard operating rule
Five standard operating rules are designed based on the targets obtained via the optimization models as described in the introduction and Fig. 1. One standard operating rule is designed with its target as the maximum objective function value from DDP optimization, corresponding to the state evaluation function for the first month [^(Sj)]. Another has the maximum objective function value from SDP optimization [max fj(S|,Q0)]. The other three operating rules have intermediate values as shown in Table 1. The simulation of the reservoir system is carried out on ten long synthetic inflow series of 100 years each. The generation of these synthetic series are done using a disaggregation method based inflow data generation software, LAST (Lane, 1990). The reservoir is operated to meet the energy target read from the standard operating rule or the flow constraints (irrigation, flood control release requirements) whichever is larger. Months in which the system fails to meet any one of the demands are counted and averaged over the ten synthetic series to obtain average number of failures. The reliability of the system is defined as the average number of months where system performed satisfactorily. Table 1 gives the results of these simulations.
Table 1 System reliability under each operating rule. Energy target (GWh/yr)
Irrigation target (Xl0 3 m 3 /month)
Number of months with failures with each synthetic series (100 yr) 1 2 3 4 5 6 7 8 9
Average number of system failures
System reliability
10
644.4
107.0
14 6 9 0 0 0 17 0 1 0
4.7
99.61
628.1
107.0
14 6 9 0 0 0 17 0 1 0
4.7
99.61
613.9
107.0
14 6 9 0 0 0 17 0 0 0
4.6
99.62
598.6
107.0
11 3 9 0 0 0 17 0 0 0
4.0
99.67
583.4
107.0
5 0 6 0 0 0 13 0 0 0
2.4
99.80
CONCLUSION The maximum yearly energy target from SDP optimization is about 9.5% smaller than the energy target obtained from DDP optimization while the reliability of operation increased for SDP based target. The tradeoff relation between a given energy target and the system reliability is established for the targets between the two of above mentioned energy targets.
Acknowledgments Research leading to this paper was made possible by the provision of financial support by the German Agency for Technical Cooperation (GTZ) through the Division of Water Resources Engineering, Asian Institute of Technology. The authors also thank Prof. Kuniyoshi Takeuchi for his interesting comments on this paper.
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REFERENCES Boehle, W., Harboe, R. & Schultz, G. A. (1981) Sequential optimization for the operation of multipurpose reservoir systems, Proc. International Symposium on Real-Time Operation of Hydrosystems, Waterloo, Ontario, Canada, Vol II, 583-598. Harboe, R. (1990) Optimal power output for operation of a reservoir system, Proc. VII Congress of the Asian and Pacific Division (IAHR), Beijing, China, Vol II, 171-174. Lane, W. L. et al. (1990) Applied Stochastic Techniques - User's Manual, Bureau of Reclamation, US Department of the Interior, Denver, Colorado, USA. Ratnayake, U. & Harboe, R. (1992) Optimal operating rule for a reservoir with stochastic inflows, Proc. VIII Congress of the Asian and Pacific Division (IAHR), Pune, India.
