Stage-wise optimizing operating rules for flood control in a multi-purpose reservoir

Accepted Manuscript Stage-wise optimizing operating rules for flood control in a multi-purpose reservoir Frederick N.-F. Chou, Chia-Wen Wu PII: DOI: Reference:

S0022-1694(14)00995-0 http://dx.doi.org/10.1016/j.jhydrol.2014.11.073 HYDROL 20088

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

23 May 2014 3 November 2014 25 November 2014

Please cite this article as: Chou, F.N., Wu, C-W., Stage-wise optimizing operating rules for flood control in a multipurpose reservoir, Journal of Hydrology (2014), doi: http://dx.doi.org/10.1016/j.jhydrol.2014.11.073

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Stage-wise optimizing operating rules for flood control in a multi-purpose reservoir Frederick N.-F. Chou 1 and Chia-Wen Wu2 1

Professor, Department of Hydraulic and Ocean Engineering, National Cheng-Kung

University, 1 University Rd., Tainan, Taiwan; email: [email protected] 2

Post doctoral fellow, Department of Hydraulic and Ocean Engineering, National Cheng-

Kung University, 1 University Rd., Tainan, Taiwan; email: [email protected]

Corresponding author: Chia-Wen Wu Postal address: Department of Hydraulic and Ocean Engineering, National Cheng-Kung University, 1 University Rd., Tainan, Taiwan. Email: [email protected] Phone: +886 (6) 275 7575 ext. 63215-18 Fax: +886 (6 )274 1463

(submit to Journal of Hydrology)

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Abstract This paper presents a generic framework of release rules for reservoir flood control operation during three stages. In the stage prior to flood arrival, the rules indicate the timing and release discharge of pre-releasing reservoir storage to the initial level of flood control operation. In the stage preceding the flood peak, the rules prescribe the portion of inflow to be detained to mitigate downstream flooding, without allowing the water surface level of reservoir to exceed the acceptable safety level of surcharge. After the flood peak, the rules suggest the timing for stepwise reduction of the release flows and closing the gates of spillways and other outlets to achieve the normal level of conservation use. A simulation model is developed and linked with BOBYQA, an efficient optimization algorithm, to determine the optimal rule parameters in a stage-wise manner. The release rules of Shihmen Reservoir of Taiwan are established using inflow records of 59 historical typhoons and the probable maximum flood. The deviations from target levels at the end of different stages of all calibration events are minimized by the proposed method to improve the reliability of flood control operation. The optimized rules satisfy operational objectives including dam safety, flood mitigation, achieving sufficient end-of-operation storage for conservation purposes and smooth operation. Keywords: multi-purpose reservoir, flood control operation, optimal release rules, BOBYQA

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1. Introduction Flood operations of gate-controlled reservoirs are generally carried out in three stages: (1) prior to flood arrival; (2) preceding flood peak, and (3) post peak. Each stage has its specific objective and operating characteristics. Prior to the onset of flooding, operators have the option of pre-releasing reservoir storage to the initial level of flood control operation (IL). The aim is to prepare sufficient detention capacity. During the stage preceding the peak, operators seek to alleviate downstream flooding by detaining the inflow in the available detention zone, without allowing the water surface level (WSL) of reservoir to exceed the acceptable safety level of surcharge (SL). After the peak has passed, spillway gates must be closed to reach the end-of-operation normal pool level of conservation use (NL); this level is capable of accommodating subsequent demands for water. There may be multiple sub-stages within each stage, repeated transitions between stages during a multi-peak flood, or different ways to define the span of each stage. Nonetheless, the operations of these three stages should remain the basic and major components due to the very nature of a multi-purpose reservoir in reaction to a flood: to prepare, mitigate and store. This generic concept of stage-wise flood moderating has been adopted by many reservoirs with flood control purpose around the world (Pitman and Basson, 1980; Kojiri et al., 1989; Faber, 2001; Government of India Central Water Commission, 2005; Chang, 2008; Wei and Hsu, 2009; Huang and Hsieh, 2010; Li et al., 2010; SEQwater, 2011; Chou and Wu, 2013).

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Among the above operating objectives, the prevention of dam overtopping takes precedence, followed by the objectives of flood mitigation and achieving the desired level of end-of-operation storage. In order to ensure the safety of dam, the IL is usually designed during the planning of reservoir to safely accommodate the probable maximum flood (PMF). For reservoirs with limited capacities and significant perennial water supply demands, the conservation zone usually overlaps with the flood control zone. This overlap yields an NL that is higher than the IL, thus prompting the necessary pre-release prior to the actual beginning of a flood. Pre-releasing the storage of reservoir to attain this requisite IL in the first operating stage not only satisfies the safety requirement, but also promotes flood attenuation as it generates more detention capacity. On the other hand, it may elevate the risk of water shortage if the subsequent floodwater fails to recover the pre-released storage. Decision makers may select a different pre-release target during real-time operation in response to the forecasted inflow volume and time of year, based on the tradeoff between flood control and water usage (Faber, 2001; Li et al., 2010; Chou and Wu, 2013). With the target pre-release level assigned, the balance between flood mitigation and water conservation prior to a flood’s arrival is determined, and the operating goal is simply to achieve the target pre-release level by the end of the first stage. During the stage preceding peak, the primary task is to detain flood and the WSL of reservoir will be raised consequently. A perfect mitigating strategy with comprehensive knowledge of inflow process would fully utilize the available detention volume to

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mitigate flooding and thus elevate the WSL of a reservoir to a maximum approaching the designed flood level (DL), which represents the maximum surcharge level of a reservoir. Nonetheless, the uncertainty about future inflows prevents the adoption of such a high risk strategy, since an underestimation of flood volume will directly lead to overtopping of a dam. In reality, a SL is usually assigned well below the DL in order to accommodate unexpected extreme inflow, while the zone below SL can still be used for flood detention. The value of SL is usually determined subjectively based on decision makers’ experience and risk tolerance with respect to flood mitigation and protecting dam safety. Regulating the WSL of a reservoir to reach the SL at end of the stage preceding the peak will maximize the prescribed detention function without compromising dam safety. The capacity above the SL can be regarded as backup space for storms that exceed forecasted levels, e.g. the occurrence of a PMF. Thus, a feasible mitigation strategy corresponding to a specific SL should also safely accommodate the PMF volume within the space between IL and DL. In the end of the flood, the reservoir should store sufficient water to support normal operation for all purposes over a long period. This target level NL specifies the upper limit of storage for conservation use. Its value may vary in accordance with monthly patterns of reservoir inflow and perennial water usage. Since the NL is usually lower than the SL, detention of flood waters preceding peak does not conflict with the goal of reaching desired storage at the end of the flood. Thus achieving the respective target

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values for IL, SL and NL at the end of each stage will inherently satisfy the assigned multiple objectives of the entire flood control operation. This characteristic allows stagewise evaluations of operating strategies, thereby simplifying real-time flood control operation in situ. Under the framework of stage-wise operation, the transition of stages is determined simply based on the real-time measured inflow of the reservoir, which is sufficient to judge whether the flood is arriving, rising or receding. The operation can then adapt to the associated requirements and objectives of the current stage. In order to achieve multiple objectives, many studies have applied optimization methods to identify real-time operating policies for reservoirs. The employed approaches include linear programming (Windsor, 1973), goal programming (Can and Houck, 1984), network flow programming (Brago and Barbosa, 2001), nonlinear programming (Unver and Mays, 1990), dynamic programming (Shim et al., 2002), mixed integer programming (Needham et al., 2000; Hsu and Wei, 2007; Chou and Wu, 2011), optimal control theory (Wasimi and Kitanidis, 1983; Karbowski et al., 2005; Kearney et al., 2011; Delgoda et al., 2012), genetic algorithm (Chang, 2008) and other heuristic algorithms (Li et al., 2010; Qin et al., 2010; Valeriano et al., 2010), or combinations of these (Niewiadomska-Szynkiewicz et al., 1996). These methods regard reservoir releases as decision variables to be optimized and require future inflow to be forecasted. However, forecasts of rainfall and runoff processes are always uncertain during real-time operations. These uncertainties limit the optimized

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policy to be only effective for the current period or a few following periods (Cheng and Chau, 2001). The processes of rainfall-runoff forecasting and dynamic optimization of future operations must be sequentially performed throughout the operating horizon to guide reservoir release in each period (Pitman and Basson, 1980; Shim et al., 2002). Forecasting methods that incorporate probability may serve to manage hydrological uncertainty in real-time. Additional works will be required to forecast the distribution of future inflows and evaluate operating responses to more generated realizations (Mediero, 2007; Kearney et al., 2011; Delgoda et al., 2012). Inevitably, when great uncertainty is introduced, the performances of all these approaches will degenerate due to their reliance on the reliability of real-time forecasts. The above assertion is supported by the operating experiences of Tsengwen Reservoir, the largest reservoir in Taiwan, during Typhoo Morakot in 2009, which is a worldwide well-known event and damaged the watershed of Tsengwen Reservoir severely. Prior to the invasion of this typhoon, the Central Weather Bureau (CWB) of Taiwan forecasted a total rainfall depth of 650 mm. However, the unexpected slow movement of Morakot brought record-breaking rainfall for southern Taiwan. The torrential storm caused malfunction of most of the telemetric rain gauges in the watershed of Tsengwen Reservoir, which originally recorded a 3-day rainfall depth of 1,711 mm and led to an unreasonable runoff coefficient of 1.47 according to the inflow measured at the dam site. It was modified to 2,485 mm in the post-flood review by Chou and Wu

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(2010). The reservoir inflow peaked at 11,729 m3/s, which is very close to 12,430 m3/s as the PMF of Tsengwen Reservoir. Without accurate data during this severe typhoon emergency, operators were forced to rely on documented procedures and their own experiences, rather than the forecast-optimization systems of Tsengwen Reservoir. After the flood, several projects were carried out in order to evaluate and adapt the operating strategies and rules of the reservoir for severe conditions. In contrast to the optimization approach, which relies upon accurate forecasts, a well-established operating rule can properly guide reservoir release based simply on realtime measurements available from the dam site. No matter how good the forecastoptimization models are, having another effective alternative available is always a gain, especially since rule-based operation is much simpler, more computationally efficient, more reliable when the forecast is highly uncertain, and also valuable as evidence to sufficiently support the decision maker in court when the operation invokes public controversy (Valdes and Marco, 1995). This study presents a method to develop the optimal release rules for reservoir flood control operation. In the following section, a comprehensive framework of rules covering all three stages of flood operation is proposed. These user-friendly rules are devised from operating insights and simple hydrologic concepts. A simulation model based on these rules is constructed to analyze flood control operations with respect to the PMF as well as historical flood events. The Bounded Optimization BY Quadratic

8

Approximation (BOBYQA) algorithm of Powell (2009) is used to calibrate the optimal rule parameters complying with the characteristics and objectives of different stages of flood operation. The target values of the IL, SL and NL levels are assumed to be given conditions when applying the proposed method. These levels are usually specified by the decision maker based on relative prioritization of dam safety, flood mitigation and water conservation objectives. With these conditions specified, the proposed method strives to achieve a target WSL at the end of each stage, thus achieving the assigned objectives. Case study results validate the effectiveness of the proposed method.

2. Methodology 2.1 Framework of the simulation-optimization procedure Most previous studies involving simulations of reservoir flood control treated reservoir release as a function of inflow and WSL of a reservoir (U.S. Army Corps of Engineers, 1987; Chang and Chen, 1998; Yang et al., 2004; Ahmad and Simonovic, 2006). Inflow may be replaced by the variation of a reservoir’s WSL to more precisely reflect the operating urgency (Bagis and Karaboga; 2004), or by inflow fluctuations to determine whether the flood is rising or receding (Huang and Hsieh, 2010). The rules have been expressed in the forms of figures, tables, equations and linguistic guidelines using “if and then” methods. Nonetheless, most of the previous proposed rules only focused on flood mitigation during the stage preceding the peak. The present study

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addresses this limitation by proposing a comprehensive rule framework that covers all three stages of a flood. Another distinction of the proposed rules is that they are devised based on the concept of stage-wise control of a reservoir’s WSL to the assigned target levels. As a result, different rule frameworks are employed according to real-time hydrologic conditions, operating objectives and constraints of different stages. After establishing the framework of rules, the optimal rule parameters are determined by the linkage of optimization-simulation approach. Following pre-defined release rules, this approach simulates flood control operations during historical flood events and the PMF. An optimizer is linked to the simulator to calibrate the optimal values of parameters in these rules. The inclusion of historical events and the PMF in this procedure ensures the robustness, effectiveness and representativeness of the optimized results. The presence of the PMF and extreme floods force the established rules to fully release the inflow in order to prevent overtopping of a dam under severe conditions. This leads to a conservative strategy that sacrifices certain flood detention functions during moderate storm events in exchange for strictly securing dam safety during extreme floods. This risk aversion ultimately serves reservoir managers well by providing the safest mitigation policy, especially in an age when severe hydrologic conditions are expected to occur more frequently due to the impact of climate change. In the above-mentioned procedure, the operation aims to satisfy the objectives of preparing detention capacity, mitigating flood and storing sufficient water for

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conservation use through achieving the target levels at the end of different stages. Therefore, the optimized releasing rules, or even a multi-objective real-time operation strategy, should independently achieve the operating objectives of each stage. This characteristic makes integrating the operation of all stages into a single complex, multiobjective optimization problem unnecessary. Instead, the optimal releasing rule of each stage is stage-wise obtained by minimizing the deviation of the WSL of a reservoir from the assigned target level at the end of each stage. The conditions simulated by the established rules of the previous stages are then included in calibrating the rules of later stages. Thus the operating consequences from previous stages are incorporated in the optimization of rules in later stages. Even the end-of-stage WSL may vary around the target level due to uncertain inflow during real-time operation, the adaptive mechanism of the established rules will be capable of regulating the WSL toward the target level of the next stage. Fig. 1 illustrates the flowchart of the simulation and optimization procedure.

Fig. 1 Flowchart of the linkage of optimization and simulation approach on calibrating rule parameters of the ith operating stage

2.2 Continuity equation for simulating reservoir flood control operation The simulation of flood control operation is based on the continuity of reservoir storage and reservoir release rules. The continuity equation is as follows:

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S (Ht +1) = S (Ht ) + I t +1 ⋅ ∆t − Ot +1 ⋅ ∆t

(1)

where, S( ) is the function to convert a reservoir’s WSL into storage volume, Ht is the WSL of a reservoir at hour t, ∆t is the unit time period for operating the gates of outlet works (set to 1 hour in this study), and I t+1 is the average inflow discharge from hour t to

t+1. In real-time operations, I t +1 is obtained by solving Eq. (1) after Ht +1 is measured at hour t+1. Ot +1 denotes the average outflow from the reservoir from hour t to t+1, and the timing of determining its value is between hour t-1 and t in real-time operation. Measurements available to support this decision include Ht −1 and I t −1 . Precipitation, evaporation, seepage, leakage, and other terms associated with reservoir pondage are ignored in Eq. (1) because their magnitude is insignificant with respect to flood inflow and reservoir release volumes. The following section outlines the framework of rules and the associated optimization formulations for different stages of flood control operation.

2.3 Pre-release rules for the stage prior to flood arrival The operation of the first stage commences when an impending flood is expected. It allows the outflow to exceed inflow to effectively pre-release the WSL of reservoir to the IL. In order to avoid inducing flooding damage, the pre-release discharge is required to be below the threshold that the downstream channel is capable of safely conveying. This requirement requires the first stage to end when the latest measured reservoir inflow exceeds the non-damaging discharge of the downstream channel. 12

In the beginning of this stage, the flood has not yet arrived; therefore, there may not be any precipitation in the watershed and the reservoir inflow is usually low. However, if the WSL does already significantly exceed the IL, the operator will nonetheless be required to commence pre-release in order to lower the detention volume in time. Small release can be carried to evacuate the storage gradually. When rainfall actually begins, the release rate could be raised beyond the rising inflow in order to lower the WSL back to the IL. Based on the above-mentioned concepts, the proposed rules for determining prerelease discharge are formulated as follows: 1. It is assumed that pre-release can only be performed with k+1 specific outflow discharges. These discharges can be denoted in order of magnitude as O1P , O2P ,…,

OkP and the non-damaging discharge OS . The value of OS is pre-determined by assessing the conveyance capacity of the downstream river channel. 2. The rules determining when to increase the pre-release discharge are conditional based on the latest measured WSL and cumulative rainfall in the watershed during the previous tc hours, denoted as Ht −1 and Rt −tc ~t −1 respectively. The tc is the concentration time of reservoir’s watershed. This pair of measurements thus portrays the current storage plus potential future inflow. Table 1 illustrates the reference table for determining the timing of increasing pre-release discharge, with the assumption of k = 2. The associated release rules are as follows:

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(1) A set of threshold cumulative rainfall for increasing reservoir release to O1P , O2P and OS is identified by interpolating the value of Ht −1 among the associated reference levels in row 1 and reference rainfall from rows 2 to 4 of Table 1. (2) If the current outflow is less than O1P , and Rt −tc ~t −1 exceeds the interpolated reference rainfall from rows 1 and 2 of Table 1, according to the value of Ht −1 , then Ot +1 is increased to O1P . (3) If the current outflow equals O1P , and Rt −tc ~t −1 exceeds the interpolated reference rainfall from rows 1 and 3 according to Ht −1 , then Ot +1 is increased to O2P . (4) If the current outflow equals O2P , and Rt −tc ~t −1 exceeds the interpolated reference rainfall from rows 1 and 4 according to Ht −1 , then Ot +1 is increased to OS . 3. The outflow is decreased if Ht −1 and Rt −tc ~t −1 no longer satisfy the conditions of increasing release discharge. This flexibility prevents excessive pre-release due to the delayed arrival of a flood. However, the release should not be reduced below O1P . In other words, the gates of major outlet works remain opened to ensure a smooth operating transition into the subsequent stage preceding peak.

Table 1. Rules for determining release discharge during the stage prior to flood arrival

Assume that the pre-release can only be performed at k+1 specific discharges and the values of IL and OS have been specified. Then the parameters to be determined in the above rule include the first k incremental outflow discharges as well as the rainfall 14

references as shown in Table 1. The objective function to optimize these parameters is:

  Minimize ∑ ∑  H i , j , t I − H IL i =1 j =1   nf

nh

(

)

2

tI

+ wI ⋅ ∑ t =1

O i , j , t +1 − O i , j , t   S  O  

(2)

where, nf = number of historical flood events; nh = number of initial levels for different trial runs; t I = duration of the first stage; H i , j ,t = the WSL of reservoir at the end of the I

first stage by simulating the pre-release operation from the jth initial level using the hydrograph of the ith flood event; HIL = the IL; wI = weighting factor; O i, j ,t +1 = outflow discharge from hour t to t+1 operating against the ith flood event from the jth initial water level. Using Table 1 as an example, a feasible set of parameters should also satisfy the following constraints:

O1P < O2P < O S

(3)

Ri −1, j < Ri , j < Ri , j −1 for i = 1, 2, 3 and j = 1, 2,..., 5

(4)

where, R0,j = 0 for j = 1,2,...,5; Ri,0 = a pre-defined large value for i = 1,2,3. The first component in Eq. (2) aims to bring the WSL of a reservoir as close to the IL as possible at the end of this stage. The second term prevents frequent changes of release discharge. Weighting factor wI controls the trade-off between the precision in regulating levels and the smoothness of the pre-release process. Eqs. (3) and (4) ensure the pre-release discharge will not be increased unless more evidence from the measured accumulative rainfall shows that the flood has actually begun. By repeatedly solving Eqs. (2) to (4) with different values of wI, the decision makers can then identify the optimal pre-release 15

rule according to the acceptable deviation from IL and preferred number of altering prerelease discharge. 2.4 Release rules for the stage preceding flood peak The second stage begins when the latest measured inflow exceeds the nondamaging discharge, and it ends when the flood peak has occurred. The goal in this stage is to protect the dam, and to mitigate flood conditions as much as possible. The adopted operating principle is to reduce the reservoir outflow according to a flood reduction ratio (Chang and Chen, 1998). Accordingly, the desired outflow discharge can be expressed as:

O t + 1 = α ⋅ I t −1 (5) where, α indicates the flood reduction ratio with a feasible range between 0 and 1, I t −1 represents the average inflow during t-2 to t-1, which is the latest inflow measurements during t-1 to t when determining the value of Ot +1 in situ. If α = 0, all of the inflow is retained in the reservoir during the time step t to t+1; if α = 1, the latest measured inflow is released in order to protect the reservoir from further rapid increases of storage. α should not be greater than 1 in order to avoid inducing a man-made flood, which is the most undesirable situation for the operators when facing a liability investigation of flooding damage. The key factors for determining the value of α during real-time operation include the current available detention volume as well as the expected volume of future inflow. However, future inflow is always uncertain; a proactive strategy should thus allow real16

time adjustment of α to account for the varied detention volume throughout the stage preceding the peak. This idea prompts this study to relate α to both the real-time measured WSL value, H t − 1 , and the net inflow discharge dS t −1 , which equals I t −1 − O t . A set of reference values for these two measurements is pre-defined first. Each distinct pair of values for these two references is then assigned a specific value of α for reference. During real-time operations, the actual adopted flood reduction ratio is obtained by interpolating the pre-defined values of α attached to the references surrounding the measured H t − 1 and dS t −1 . As illustrated in Fig. 2, bi-linear interpolation is used to calculate the value of α according to ( α i , j , α i , j +1 , α i +1, j , α i +1, j +1 ) and the location of H t − 1 and dS t −1 . The adopted value of α varies continuously when H t − 1 and dS t − 1 change.

Fig. 2. Determination of α according to real-time measurements

Among the referenced measurements, H t −1 indicates the current available detention volume and dS t − 1 represents the rate of storage increase. This increase rate is expected to remain within a similar order for a few subsequent hours; potential future storage can thus be projected when coupling with H t −1 . A lower potential future storage, which means more detention space available, will enable the use of a smaller α to store water and mitigate flooding. On the other hand, a high potential future storage will

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require a large α to increase release for the safe accommodation of flood water. In order to model this mechanism, the value of α is designed to increase along with the WSL of a reservoir and its net inflow discharge. Consequently, it leads to a smaller α and reservoir release when the inflow has not yet risen and the detention zone is only partially filled. Along with the rise of inflow, the WSL will then be aggressively elevated due to the increasing difference between inflow and release rates. The reservoir release will then be increased to match the inflow and keep the WSL of reservoir from surpassing the SL. In this manner, the detention zone is mostly consumed in earlier periods during the stage preceding the peak, rather than maintained empty first (e.g. setting α = 1) to more effectively attenuate the flood peak later (e.g. releasing a constant discharge and retaining all the excess inflow). Although the later strategy reduces the peak release more effectively, reliable forecasts are necessary to accurately determine when to start retaining inflow in the reservoir. Otherwise, the WSL of reservoir will be elevated beyond SL if forecasted inflow is underestimated. This study, however, focuses on developing release rules solely based on real-time measurements. The adopted strategy may be less effective in alleviating the peak, but it is more conservative and robust when the forecast is highly uncertain. In addition to Eq. (5), other operating regulations can be easily incorporated in the framework of rule-based simulation. For instance, Taiwanese Water Law requires that (1) the reservoir release preceding the peak should not exceed the maximum reservoir inflow

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measured from the beginning of the flood till the current operating time, and (2) the incremental release between adjacent periods should not exceed the largest inflow increment over the previous adjacent periods. The operators also prefer maintaining the release to be monotonically non-decreasing prior to the peak. These regulations act as upper and lower bounds on the release suggested by Eq. (5) and are all included in the analysis in the section of the case study. For the above rules, the values of α at all combinations of the pre-defined reference levels and net inflow discharges are parameters to be determined. Optimizing these parameters should provide maximum flood mitigation without elevating the WSL of reservoir to surpass the SL for all historical floods. In addition, the maximum level when accommodating the PMF should be maintained below the DL. In practice, reservoir operators usually intend to control the WSL below the NL for ordinary flood conditions. In the case of an extreme flood, the storage zone above the NL can then be used to store and mitigate floodwater. To accommodate these characteristics, the formulation of optimizing the values of α can be listed as below:

)

(

 max O i , j , t +1  t I < t ≤ t II  Minimize ∑∑ + wII ⋅ Pi , j    nh ⋅ nf i =1 j =1   nf

nh

Pi , j =  max H i, j ,t − H NL  tI < t ≤tII 

(

)

2

(

)

if max Hi , j,t > H NL ; 0 otherwise tI < t ≤ t II

subject to

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(6)

(7)

(

)

max H i , j ,t ≤ H SL for i = 1,2,..., nf , j = 1,2,..., nh

tI < t ≤ tII

(

(8)

)

max H PMF , j ,t ≤ H DL for j = 1,2,..., nh

tI < t ≤ t II

(9)

0 ≤ α i−1, j < α i , j < α i , j +1 ≤ 1 for i = 1, 2,..., nr ds and j = 1,2,..., nr h (10)

(

)

(

)

where, t II = the period when the second stage ends; max O i, j ,t +1 and max H i , j ,t = tI < t ≤ tII

tI < t ≤ tII

maximum release and WSL during the stage preceding peak by operating against the ith event from the jth initial level; wII = the weighting factor; Pi , j = a penalty function which punishes situations when max (H i , j ,t ) exceeds NL; HNL = the NL; HSL = the SL; tI < t ≤ tII

(

)

max H PMF , j ,t = the maximum WSL of reservoir by accommodating the PMF from the

tI < t ≤ tII

jth initial level; HDL = the DL; α i, j = the value of flood reduction ratio at the pre-defined ith reference net inflow rate and jth reservoir level; nrds and nrh = numbers of reference net inflow and WSL of reservoir, respectively. The first component in Eq. (6) aims to reduce the maximum reservoir release and thus maximize flood mitigation. The second component aims to keep the WSL below the NL. By repeatedly solving Eqs. (6) to (10) with different values for wII, the optimal values for α can be established according to the preferences of decision makers with regard to balancing flood mitigation against strict protection of the dam.

2.5 Release rules for the stage post following flood peak

Following the stage preceding the peak, the post-peak stage ends after the gates of spillways and major outlet works for flood control are closed. Subsequently, a base 20

release, denoted as OB, might be maintained through the bottom outlets for hydropower generation or hydraulic sluicing. A portion of the remaining recession inflow should be stored to return the WSL to the end-of-operation desired target, which is set as the NL in this study. This study assumes that the release can only be monotonically reduced at l+1 specific discharges. Fig. 3 depicts an example of l = 2, where O2B and O1B represent the first and second reduced discharge, respectively. H 2T and H1T represent the target levels after the release is reduced to O2B and O1B , respectively. HNL denotes the level corresponding to NL. The release is reduced stepwise, from O2B , O1B to OB, to gradually elevate WSL of reservoir from H 2T , H1T to HNL.

Fig. 3. Reducing reservoir release stepwise to gradually elevate reservoir level to a desired target

Fig. 3 provides an example to explain the rules associated with a reduction in release post peak. During the last stage, if the current outflow still exceeds O2B and the conditions shown in Eq. (11) are satisfied, release should be reduced to O2B in the following operating period to ensure that the target level H 2T can be achieved:

( )

τ O2B

S ( H t −1 ) +

∑

[Iˆ

t +i

]

− O2B ⋅ ∆t ≤ S ( H 2T )

(11)

i =0

where, Iˆt+i = inflow in the next i hours, which can be estimated by the recession flow

model according to the latest measured inflow I t −1 ; τ (O 2B ) = the expected number of 21

hours required for the inflow to recede from I t −1 to below O2B . A recession model based on the storage function method is used in this study to provide estimates of future recession inflow. It represents the relationship between storage in the watershed and recession flow as below:

( )

StW−1 = K ⋅ I t −1

p

(12)

W where St −1 = water storage in the watershed at hour t − 1 , K and p = parameters for the

recession analysis. Future recession flow can be determined by recursively solving Eq. (12) and the continuity equation of watershed storage. The hydrologic parameters K and p can be calibrated according to the records of recession processes from historical floods. The left hand side of Eq. (11) represents current storage plus the expected future storable recession inflow, while the right hand side represents the desired target storage when reducing release to O2B . Eq. (11) states the condition that the target storage cannot be achieved according to the latest measured inflow and WSL of reservoir. Thus the release should be reduced once this inequality holds. Similarly, the conditions for reducing release to O1B and OB to achieve the storage targets H1T and HNL are:

( )

τ O1B

S ( H t −1 ) +

∑

[Iˆ

t +i

]

− O1B ⋅ ∆t ≤ S ( H1T ) (13)

i =0

τ (O B )

S ( H t −1 ) +

∑ [Iˆ

t +i

]

− O B ⋅ ∆t ≤ S ( H NL )

(14)

i=0

In Eqs. (11), (13) and (14), the values of the intermediate reduced release discharges, O2B and O1B , and the associated target levels, H 2T and H1T , can be adjusted to 22

best achieve the ultimate goal, which is to regulate the end-of-operation WSL of reservoir to the HNL. Suppose that the release is reduced in steps of l+1 and a well-calibrated recession model is available, then O1B , O2B ,…, OlB and H1T , H 2T ,…, HlT are parameters that must be determined. Optimization analysis is conducted to calibrate these parameters according to the following formulation: nf

Minimize

nh

 max H − H NL  ∑∑ t
(

)

2

(15)

subject to

O1B > O B OiB > OiB−1

(16)

for i = 2,3,..., l

(17)

where, max (H i , j ,t ) = the maximum WSL after the gates of major outlet works are closed, t II
by operating against the ith historical event from the jth initial level, t III = the time when inflow discharge recedes to below OB. Following the rule framework and optimization formulation, if the release is reduced but the WSL is not regulated to meet the corresponding target level, the timing of reduction to the next step discharge will be adjusted to compensate for previous operating bias resulting from the variation of flood receding processes not captured by the recession model. Thus a higher value of l will provide more opportunities to adapt to the real-time recession discharge, although at the expense of more frequent gate operation.

23

2.6 Optimization algorithm

The BOBYQA algorithm of Powell (2009) is selected to solve the above problems due to its theoretical sophistication, derivative-free convenience, high efficiency in solving large-scale problems and the capability of directly incorporating simple bounds on decision variables. The algorithm is developed to solve the following nonlinear optimization problems: Minimize F ( x ) , x ∈ R n

s.t. a ≤ x ≤ b

(18)

where, F = the objective function, x = the vector of n decision variables, a and b = vectors of the upper and lower bounds of decision variables, respectively. The algorithm employs a quadratic function to approximate the objective function based on a set of interpolation points. The trust region approach is employed to iteratively update the interpolation points until a local optimal solution is reached. Since the algorithm cannot be guaranteed to converge to a global optimum, multiple randomly-generated initial solutions are tested during each implementation and the best solution is selected. More detail of the computational procedure of BOBYQA is provided in Appendix A. In the process of optimizing pre-release rules, the limit on decision variables is not in the form of simple bounds, as shown in Eq. (3) and (4). A procedure proposed by Chou and Wu (2010) is applied to convert the original variables into unconstrained surrogate variables. BOBYQA is then utilized to find the optimal values of the surrogates. Back

24

transformation of the unconstrained surrogate variables is then performed to determine the optimal release rule. In the process of optimizing rules for the stage preceding the peak, constraints such as Eqs. (8) and (9) cannot be expressed in terms of bounds of decision variables. The `

sequential unconstrained minimization technique (Fiacco and McCormick, 1967) is employed to solve this problem. Eqs. (8) and (9) are converted into a barrier function:

 nf nh  1  r ⋅ ∑∑  SL H i , j ,t  i =1 j =1  H − t max I < t ≤ t II

(

 nh  1  + ∑ DL  j =1  H − max H PMF , j ,t t I < t ≤ t II  

)

(

)

    

(19)

where, r = the weighting factor for the barrier function, which must be greater than 0. The barrier function is combined with the original objective function and BOBYQA is used to solve the compound objective function from an initial feasible solution. After an intermediate solution is found by BOBYQA, r is reduced and the above calculations are repeated. Solutions produced from consecutive iterations performed in this manner gradually approach the boundary of feasibility. The procedure terminates when the value of the barrier function falls below a specified threshold.

2.7 Performance indices

In the following section, the flood operating rules of the reservoir of case study are optimized according to their performances from simulating numerous calibration cases. These cases, represented by a set C, are generated by combing historical 25

inflow scenarios with different initial levels, which essentially renders the proposed method a stochastic optimization. Two nonlinear terms representing different objectives are included when optimizing rules of each stage. The primary objective is to regulate the end-of-stage WSL of a reservoir to the specific target level thus achieving the objectives of flood mitigation and water conservation, and the other considers the demands of operating smoothness or strict safety of the dam. These two terms are linearly combined with a weighting factor, the value of which controls the trade-off between different objectives. Since compromising with the secondary objective will inevitably deteriorate the performance associated with the primary objective, i.e. reduce the probability that the target level is reasonably achieved, the following index which measures the closeness of simulated end-of-stage WSL to the respective target level is adopted to facilitate proper selection of the values for weighting factors:

f kR ( Lk ,U k ) =

[

N C Lk ≤ H i, j ,tk ≤ U k , ∀ (i, j ) ∈ C nf ⋅ nh

], for k = I , II , III (20)

where, f kR ( Lk , U k ) = the empirical probability that the simulated end-of-stage WSL of reservoir is between Lk and Uk for the kth stage. Lk and Uk = the lower and upper bounds containing the target level of the kth stage; it serves as a confidence interval to evaluate the precision in achieving a target level. H i, j ,tk = the simulated WSL of a reservoir against the ith flood event from the jth initial level at the end of the kth stage. N( ) = a function to count the total number of elements in a set, e.g. N(C) will be nf ⋅ nh . Several Lk and Uk values can be used to summarize the distribution of simulated end-of-stage

26

WSL, of which maximum and minimum values are also used as auxiliary indices to evaluate the acceptability of optimized rules:

(

H tmax = max H i , j ,tk k

)

for i = 1,2.., nf , j = 1,2..., nh, k = I , II , III

(21)

(

H tmin = min H i , j ,tk k

)

for i = 1,2,..., nf , j = 1,2,..., nh, k = I , II , III

(22) 3. Case study 3.1 Overview of the Shihmen Reservoir Shihmen Reservoir in northern Taiwan is selected as a case study. Constructed in 1963, it was the first multi-purpose reservoir in Taiwan. The reservoir watershed covers an area of 763.4 km2. Its IL, NL and SL are El. 240.00, 245.00 and 248.00 m, with respective storage volumes of 171.94, 213.14 and 240.29 million m3 according to the survey in 2011. The existing outlet works of Shihmen Reservoir include a main spillway, tunnel spillways, outlet to a hydropower plant, PRO and the Shihmen Irrigation Canal, with respective maximum capacities of 11,400; 2,400; 137, 34 and 13.78 m3/s. A sluiceway has been constructed from one of the hydropower penstocks, with a maximum discharge of 300 m3/s. The simulation model in the case study assumes that the release priorities of these outlet works are ordered as follows: PRO and hydropower outlet, sluiceway, tunnel spillways, and finally the main spillway. The release from the Shihmen Irrigation Canal is neglected and PRO and hydropower outlet are assumed to constantly discharge water of 100 m3/s during the flood operation analysis, for their magnitudes are small compared to 27

the release from spillways. Fig. 4 depicts the location of Shihmen Reservoir. Records of reservoir inflow in 59 historical typhoons from 1963 to 2010 are used to develop the optimal release rules for flood control operation. Among these typhoons, 24 events had flood peaks between 1,000 and 2,000 m3/s, 22 events peaked between 2,000 and 4,000 m3/s, and 10 events were between 4,000 and 6,000 m3/s. The remaining three were Typhoon Herb in 1996 with a peak of 6,363 m3/s, Typhoon Aere in 2004 with a peak of 8,594 m3/s and Typhoon Gloria in 1963 with a peak of 10,120 m3/s. The PMF of Shihmen Reservoir has a total flow volume of 1.47 billion m3 with a peak of 14,500 m3/s. By operating against the PMF from the IL, the DL is El. 250.58 m according to the official safety assessment report of Shihmen Reservoir in 2009.

Fig. 4 Location of Shihmen Reservoir

3.2 Optimal pre-release rules prior to flood arrival In evaluating the optimal pre-release rules for the Shihmen Reservoir, the nondamaging discharge is 1,000 m3/s and the concentration time of a watershed is assumed to be 10 hours. Pre-release is restricted to be performed with three specific discharges. Simulations are conducted for 413 cases, representing a combination of 59 typhoons with 7 different initial levels ranging from El. 239 to 245 m. BOBYQA is used to analyze the optimal pre-release rules for different values of wI. To perform optimization with a 28

specific wI, the distribution of end-of-stage WSL can be established using the optimized samples from 413 cases. Table 2 shows the statistics of end-of-stage WSL corresponding to wI of different values.

Table 2. Statistics of end-of-stage level corresponding to the optimization results using various values for wI

In Table 2, f IR (239.5,240.5) , which represents the probability that the end-of-stage level is between El. 239.5 and 240.5 m, will fall below 0.9 by increasing wI from 0.2 to 0.3, while the reduction of changes of pre-release discharge is insignificant. Hence the optimal rule corresponding to wI = 0.2 is selected with its related parameters shown in Table 3. Figs. 5 illustrates the simulated pre-release processes during Typhoons Clara based on the optimal rules with wI = 0.0 and 0.2, respectively. As shown in the figure, the rule associated with wI = 0.2 significantly reduces the number of changes of reservoir release, representing much smoother operations, while maintaining the end-of-stage WSL of the reservoir very close to El. 240 m.

Table 3. Optimal release rules in the stage prior to flood arrival, corresponding to wI = 0.2

29

Fig. 5 Simulated pre-release process during Typhoon Clara in 1981

3.3 Optimal release rules preceding flood peak According to the operating direction of Shihmen Reservoir, the stage preceding the peak begins when the latest measured inflow exceeds 1,000 m3/s, and ends before the inflow drops 90% below the flood peak. This percentage is established from the inflow records of reservoir and site experiences. It prevents repeated transitions between flood stages preceding and post peak when encountering a minor local peak of inflow. To analyze the optimal rules preceding the peak, eight specific reference levels and four reference net inflow discharges are predefined. These levels include the minimum operational levels of main spillway and tunnel spillways, IL, NL, SL, and DL. The specific net inflow discharges, which are defined as the differences between inflow and outflow discharges, are 250; 750; 1,500 and 2,500 m3/s respectively. An initial value of

α is assigned to each combination of values for these two references. Simulations are conducted for 840 cases, representing a combination of 60 flood events (one of which is the PMF) with 14 initial levels ranging from El. 195 to 245 m. The optimal values of α are calibrated by using the pre-release rule corresponding to wI = 0.2. After the first run, the calibrated results are examined and five additional intermediate reference levels and net inflow discharges are tested to ensure that the difference of α ’s values between adjacent predefined references is within a reasonable range. Table 4 shows the

30

distributions of end-of-stage WSL levels from 840 cases in the optimization results using various wII values. As shown in Eq.(6) and Table 4, increasing the value of wII will penalize and reduce the value of f IIR (245,248) , which represents the probability that the end-of-stage WSL is between El. 245 and 248 m, to prevent unnecessary surcharge above the NL. Nonetheless, f IIR (245, 248) is only partially decreased after wII is increased to 2. Further increase of wII cannot prevent H tmax from exceeding NL during Typhoons Gloria, II Herb and Aere, which brought the highest flood peaks since the construction of Shihmen Reservoir. Thus the result from wII = 2 is selected as a final rule for its better flood mitigation performance. Fig. 6 shows the optimal values of α pertaining to wII = 2. The last row of Table 4 also lists the result of simulations using pre-release rules corresponding to wI = 0 and wII = 2. It shows only minor differences compared to the optimized results corresponding to wI = 0.2 and wII = 2. In other words, the selection of wI which sacrifices certain pre-release precision in exchange for higher operating smoothness does not impose significant impact on the performance of flood mitigation in the second stage. This is because the deviation from the IL is restricted within the acceptable range when selecting the value of wI.

Table 4. Statistics of end-of-stage WSL and release of reservoir preceding peak from the optimization results

Fig. 6 Optimal values for α under wII = 2

31

3.4 Optimal release rules following flood peak After the stage preceding the peak, three sub-stages are considered before the closure of spillway gates. The first sub-stage applies when the WSL of a reservoir exceeds the NL due to the detention of flood water during the previous stage. Reservoir release is allowed to exceed the latest measured inflow by an excess of 600 m3/s to accelerate drawdown of WSL in preparation for the possibility of subsequent peaks (Sinotech Consultants Inc, 1979). If the WSL of reservoir is below the NL but not low enough to close the gates of spillways, then the second sub-stage will release the expected inflow of the next hour to maintain a near-constant WSL. The operation enters into the final sub-stage of elevating the WSL once the conditions of reducing the release rate, as shown in Eqs. (11), (13) and (14), are satisfied. In the analysis of optimal rules for reducing release rate, the target end-of-operation level is set to El. 245 m. After the spillway gates are closed, a release rate of OB= 400 m3/s from the bottom outlets is maintained for hydraulic sluicing and hydropower generation. Flood control operations are simulated for 826 cases, representing a combination of 59 flood events with 14 different initial levels, ranging from El. 195 to 245 m. BOBYQA is employed to determine the optimal rules to decrease the release discharges and close the gates of spillways, assuming that operations in the previous stages used rules corresponding to wI = 0.2 and wII = 2. The optimal rules established for reducing release in various numbers of steps are tabulated in Table 5.

32

As shown in Table 5, the WSL is targeted at El. 245.00 m before the release is reduced to OB. This moves the post-peak operation into the last sub-stage of reducing release and elevating the WSL of reservoir earlier. The probability that the end-ofoperation level falls below the target level thus can be reduced. By maintaining a higher WSL before the release is reduced to OB, the timing for closing gates of spillways can be postponed. Hence even if rain unexpectedly falls again as the flooding recedes, the reservoir can still discharge a more adequate amount of water to counteract the second rise in inflow. Therefore the probability for a final WSL to exceed NL can be reduced. Table 6 shows the distribution of end-of-operation WSL for different values of l. It can be shown that f IIIR (0,244) , the probability that the end-of-stage WSL is below El. 244 m, is greater than 0.40 for all values of l. For this situation, a similar strategy can be adopted to close the gates of the sluiceway. After it is closed, only 100 m3/s will be released through the PRO and hydropower plant. A portion of the inflow between 400 and 100 m3/s can be stored to achieve the target storage. Using l = 2 as an example, f IIIR (0,244) can be reduced from 0.426 to 0.196 when the inflow recedes to below 100

m3/s. This 19.6 percentage is from cases with flood peaks below 3,000 m3/s and initial levels below El. 225.00 m. In other words, for an adequate flood magnitude and initial operating level, the optimized rule is capable of achieving sufficient end-of-operation storage. The last row of Table 6 also lists the result simulated from the release reduction

33

rule of l =2 but based on the flood mitigating rule of wII = 0. Again no significant difference is measured compared with the rules based on wII= 2 and l=2. The increase of wII from 0 to 2 reduces the value of f IIR (245,248) as shown in Table 4. This conservative

strategy of wII = 2 leads to a lower WSL at the end of the second stage, which may be counterproductive to the water conservation objective of the third stage. However its impact is minor and can be compensated by advancing the timing of reducing release to store more recession flow to achieve adequate end-of-flood storage. To facilitate easy implementation in real-time operation, the optimal rule of l = 2 is transformed into the lookup Table 7, which serves to determine the timing of reducing release to different discharges. Using the rule of reducing release to 1,600 m3/s as an example, it is derived by first defining a set of reference reservoir WSL, as shown in the first row of Table 7. Each of these levels is substituted as H t −1 into the equality form of Eq. (11). This equation is then solved to determine the corresponding values of reservoir inflow I t −1 as shown in the third row of Table 7, each of which can elevate the reservoir WSL from the respective reference level to the target level of El. 245 m after reducing the release to 1,600 m3/s. Therefore, as the reservoir WSL is measured in real-time, the minimum required inflow to ensure achieving the target level can be identified by interpolating the measured WSL according to the first and third rows of Table 7. Comparing this minimum required inflow with the latest measured inflow will then yield the decision of whether reducing the reservoir release to 1,600 m3/s or not. The

34

establishment and real-time implementation of the fourth and fifth rows of Table 7 follow the same procedure.

Table 5. Optimal rules for reducing release and closing gates of spillways

Table 6. Distribution of end-of-operation levels when reservoir inflow recedes below 400 m3/s

Table 7. The lookup table for reducing reservoir release in the stage post peak

3.5 Validation analysis Four typhoons, which were not used in the calibration, are selected to test the effectiveness of the optimized rules. Among these events, Typhoons Saola, Soulik and Trami from 2012 to 2013 serve to evaluate the performance of rules established from all recorded major floods through 2011, except Typhoon Nari, against future challenges. Typhoon Nari in 2001 is excluded from the set of calibration events due to its distinct pattern of multiple peaks. Typhoon Nari serves the purpose of demonstrating the effectiveness of established rules against multi-peak floods. The following presentation of validation analysis follows the chronological order of these events.

35

The Typhoon Nari invaded Taiwan during the days between September 8 to 19 in 2001, brought a total rainfall of 922 mm on the reservoir watershed and led to two separate inflow peaks, 3,801 and 4,123 m3/s, respectively. Fig. 7 depicts the simulated flood control operation during Typhoon Nari, using the optimized rules corresponding to wI = 0.2, wII = 2 and l = 2. The result shows that the simulated maximum release at the

end of the first peak is 3,408 m3/s. Afterwards, the operation proceeds into the post-peak stage and the WSL of the reservoir is steadily kept around El. 242.5 m for 7 hours. The gates of tunnel spillways are held open during the first recession, thus allowing a quick response to the rise in inflow preceding the second peak. The simulated maximum release and WSL at the end of the second peak are 3,930 m3/s and El. 245.07 m, which are both less than the records of 4,111 m3/s and El. 245.21 m that actually occurred during this typhoon. In addition to the above simulation, a pseudo-optimization analysis, which assumes the forecast is perfect and thus the entire inflow process is known in advance, is also performed. It serves to evaluate the difference between the proposed method, which determines release solely based on real-time measurements, and the potential optimal solution with perfect knowledge of future inflow. The pseudo-optimization is carried out by setting α = 1 and applying a threshold discharge to limit the maximum reservoir release. The setting of α = 1 ensures that the reservoir release closely follows the inflow hydrograph and thus preserves the detention zone until inflow exceeds the threshold

36

discharge. Afterwards the release is steadily kept at the threshold discharge and all the excess inflow is retained in the preserved detention zone. Different threshold discharges are tested to find the one which elevates the WSL to reach the NL at the end of the second stage, thus maximizing peak reduction by filling the prescribed detention zone of reservoir. The result is also shown in Fig. 7, where the maximum reservoir release can be further reduced to 3,215 m3/s if the inflow can be perfectly forecasted. Further reduction of maximum release may be pursued by lowering the WSL prior to the flood’s arrival. The identification of the most appropriate target pre-release level can be found in Chou and Wu (2013) and is beyond the scope of the present study.

Fig. 7 Simulated operations during Typhoon Nari in 2001, using optimal rules under wI =0.2, wII =2 and l = 2

Typhoon Saola invaded Taiwan during July 31 to August 2 in 2012, bringing a total rainfall of 808 mm on the watershed of Shihmen Reservoir and an inflow peak of 5,083 m3/s. Fig. 8 shows the simulated flood control operation during Typhoon Saola. As depicted in Fig. 8, the simulated maximum reservoir release, maximum WSL preceding the peak and end-of-flood WSL are 4812 m3/s, El. 244.73 and 244.97 m, while the corresponding historical operating records are 5,079 m3/s, El. 244.53 and 241.56 m respectively. The pseudo-optimization analysis shows that the maximum release can be

37

reduced to 3,800 m3/s if the inflow preceding the peak is perfectly forecasted. Nonetheless, the optimized rules still provide a lower pre-release WSL, smaller maximum release, slightly higher maximum WSL of reservoir preceding peak and a significantly higher end-of-operation WSL compared to their actual historic counterparts.

Fig. 8 Simulated operations during Typhoon Saola in 2012, using optimal rules under wI =0.2, wII =2 and l = 2

Typhoon Soulik invaded Taiwan from July 11 to 13 in 2013, bringing a total rainfall of 399 mm on the watershed of Shihmen Reservoir and an inflow peak of 5,457 m3/s. Fig. 9 shows the simulated flood control operation during Typhoon Soulik. The simulated maximum reservoir release, maximum WSL of the reservoir preceding the peak and end-of-flood WSL are 4,505 m3/s, El. 244.02 and 245.02 m, while the corresponding historical operating records are 4,949 m3/s, El. 244.00 and 243.04 m respectively. The pseudo-optimization analysis shows that the maximum release can be reduced to 3,700 m3/s if the inflow preceding the peak is perfectly forecasted.

38

Fig. 9 Simulated operations during Typhoon Soulik in 2013, using optimal rules under wI =0.2, wII =2 and l = 2

Typhoon Trami invaded Taiwan from August 20 to 22 in 2013 with a total rainfall of 541 mm on the watershed of Shihmen Reservoir and inflow peak of 2,412 m3/s. As shown in Fig. 10, the simulated maximum reservoir release, maximum WSL of reservoir preceding the peak and end-of-flood WSL are 1,700 m3/s, El. 242.78 and 244.31 m, while the corresponding historical operating records are 1,955 m3/s, El. 242.59 and 242.67 m respectively. The pseudo-optimization analysis shows that the maximum release can be reduced to 1,000 m3/s if the inflow preceding the peak is perfectly forecasted.

Fig. 10 Simulated operations during Typhoon Trami in 2013, using optimal rules under wI =0.2, wII =2 and l = 2

From the simulations of four validation events, it can be shown that the proposed rules consistently achieve WSL levels closer to the target levels at the end of each stage than were actually achieved during the respective typhoon events. The ideal pseudooptimal solution reduces the maximum reservoir releases to 41 to 78 percent of the inflow peaks, and this performance of peak reduction degenerates to 70 to 95 percent with rulebased operation. This result is still satisfactory given that only real-time measurements

39

are applied to perform flood control with rule-based operation.

4. Conclusions This study scrutinized the flood control operations of reservoirs during three stages: prior to flood arrival, preceding the peak and following the peak. A method is proposed to derive the optimal release rules for each stage. Shihmen Reservoir in northern Taiwan is selected as a case study to verify the effectiveness of the proposed method. From the operations of 59 historical typhoons and the PMF, the deviations from target level at the end of each stage are minimized by the BOBYQA algorithm to produce optimal rule parameters. The results from the calibration and validation analysis show that the optimized rules can adapt to the encountered real-time contingencies, with the ability to compensate for previous operating bias due to uncertain inflows. The rule is robust since dam safety is guaranteed even if the PMF occurs. It can reasonably reduce the maximum reservoir release during floods and is effective in achieving desired end-of-flood storage. Because moderate operating bias without either endangering the safety of a dam or inducing man-made flooding can be compensated in later operations, there exist multiple acceptable solutions to the reservoir flood control operation problem. In this practical sense, the proposed method aims to provide a satisfactory instead of an exact optimal solution for mitigating peak discharge. It is a simpler yet feasible and cost-effective alternative in real-time operation.

40

This study treats determining and achieving target levels as two separate problems. This problem delineation conforms well to the actual hierarchical framework for making flood control decisions that decision makers employ. While the identification of the IL has been discussed in Chou and Wu (2013), how to objectively determine the most appropriate SL still remains an area of potential future research. In real-time operation, there may be multiple thresholds of sequential SLs in a decision maker’s mind against different degrees of unexpected flood severity. Although the proposed method only adopts a constant value of each target level, several sets of rules can be pre-established for different values of target levels to allow changes of IL, SL or NL during real-time operation.

Acknowledgements This work was supported by the National Science Council (Grant No. NSC 1002221-E-006-201), Taiwan, R.O.C.

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26. Niewiadomska-Szynkiewicz, E., Malinowski, K., Karbowski, A., 1996. Predictive methods for real-time control of flood operation of a multireservoir system: Methodology and comparative study, Water Resour. Res., 32(9), 2885–2895, doi:10.1029/96WR01443. 27. Pitman, M.V., Basson, M.S., 1980. Operation of reservoirs for flood control in areas with limited water resources, Hydrological forecasting proceedings of the Oxford symposium, London, United Kingdom. 28. Powell, M., 2009. The BOBYQA algorithm for bound constrained optimization without derivatives. Tech. Rep., DAMTP 2009/NA06., University of Cambridge. 29. Qin, H., Zhou, J., Lu, Y., Li, Y., Zhang, Y., 2010. Multi-objective cultured differential evolution for generating optimal trade-offs in reservoir flood control operation, Water Resour. Manage. 24(11), 2611-2632. 30. SEQwater, 2011. Manual of operational procedures for flood mitigation at Wivenhoe Dam and Somerset Dam. 31. Shim, K.C., Fontane, D.G., Labadie, J.W., 2002. Spatial decision support system for integrated river basin flood control, J. Water Resour. Plann. Manage., 128(3), 190201. 32. Sinotech Engineering Consultants, Inc., 1979. A final report on the research of reconstructing tunnel spillways of Shihmen Reservoir. Tech. Rep., Northern Region Water Resources Office, Water Resources Agency, Taiwan. (in Chinese)

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33. Unver, O.I., Mays, L.W., 1990. Model for real-time optimal flood control operation of a reservoir system, Water Resour. Manage., 4(1), 21-46. 34. U.S. Army Corps of Engineers, 1987. Management of Water Control Systems. Engineer Manual, EM 1110-2-3600, Washington D.C. 35. Valdes, J.B., Marco, J.B., 1995. Managing reservoirs for flood control. U.S.-Italy Research Workshop on the Hydrometeorology, Impacts, and Management of Extreme Floods, Perugia, Italy. 36. Valeriano, O., Koike, T., Yang, K., Yang, D., 2010. Optimal dam operation during flood season using a distributed hydrological model and a heuristic algorithm, J. Hydrol. Eng., 15(7), 580-586. 37. Wasimi, S.A., Kitanidis, P.K., 1983. Real-time forecasting and daily operation of a multireservoir system during floods by linear quadratic Gaussian control, Water Resour. Res., 19(6), 1511–1522, doi:10.1029/WR019i006p01511. 38. Wei, C.C., Hsu, N.S., 2009. Optimal tree-based release rules for real-time flood control operations on a multipurpose multireservoir system, J. Hydrol. 365(3-4), 213224. 39. Windsor, J.S., 1973. Optimization model for the operation of flood control systems. Water Resour. Res. 9(5), 1219–1226, doi:10.1029/WR009i005p01219.

46

40. Yang, D., Koike, T., Tanizawa, H., 2004. Application of a distributed hydrological model and weather radar observations for flood management in the upper Tone River of Japan, Hydrol. Process. 18(16), 3119–3132.

Appendix A. BOBYQA algorithm In order to solve Eq.(18), a quadratic function Q is constructed during each iteration of BOBYQA to approximate F, based on a set of interpolation points. This quadratic function must satisfy

Q k ( y ki ) = F ( y ki ) , i = 1 ... m

(23)

where Qk = the quadratic function approximating F during the kth iteration, y ik = the ith interpolation point used in the kth iteration, m = the number of interpolation points, which

1   should be within the range of n + 2, (n + 1) ⋅ (n + 2) as suggested by Powell. 2  , Assuming that point x k has the lowest objective function value among the interpolation points, then

[

F (x k ) = min F (y ik ) : i = 1,2,..., m

]

(24)

The quadratic function can be expressed as follows:

1 Qk (x) = ck + (x − x 0k )T ⋅ g k + (x − x 0k )T ⋅ G k ⋅ (x − x 0k ) 2

(25)

where, x 0k = a base point which is varied throughout the course of computation to

47

minimize round-off error, ck, gk and Gk = coefficients of the quadratic function, which can be obtained by solving the equation below, based on the interpolation conditions shown in Eq. (23).

Minimize where,

Gk − Gk −1

s.t. equation ( 23)

(26)

is the function to calculate the Frobenius norm. Once Qk has been determined,

a new reference point can be found by solving

Minimize Qk (x k + d) s.t. a ≤ x k + d ≤ b and d < ∆ (≥ ρ )

(27)

where, d = a vector used to find new reference points around x k ; ∆ = the trust region radius; ρ = the lower limit of ∆ to keep interpolation points well separated. The following steps are iterated: 1. Solve Eq. (27) to obtain d. 2. If F(xk+d) ≥ F(x k), a point y kj which y kj − x k > 2 ρ is moved to satisfy y kj − x k < ρ to improve quadratic function Q. 3. If F(xk+d) < F(xk), an interpolation point (other than x k) in [ y ik , i = 1,..., m ] is replaced by xk+d. 4. Solve Eq. (26) to determine the quadratic function Qk+1 for the following iteration, based on the updated set of interpolation points. 5. Reduce ∆ and ρ . 6. Repeat these iterations until the terminal ρ is reached, whereupon a local optimal solution is obtained. 48

Appendix B. A list of variables Symbol

Description

∆

The trust region radius used in the BOBYQA algorithm

∆t

The unit time period for operating the gates of outlet works, which is set to 1 hour in this paper

α

The flood reduction ratio

ρ

The lower limit of ∆ to keep interpolation points well separated in the BOBYQA algorithm

τ (O B )

The expected number of hours required for the current reservoir inflow to recede below O B

ck , g k , G k

Coefficients of the quadratic function used in the BOBYQA algorithm

C

The set which contains calibration cases generated by combing historical inflow scenarios with different initial levels

d

A vector used to find new reference points in the BOBYQA algorithm

dS t −1

The net inflow discharge which equals I t −1 − O t

f kR ( Lk ,U k )

the empirical probability that the simulated end-of-stage WSL of reservoir is between Lk and Uk for the kth stage

H IL

The initial level of flood control operation

H NL

The normal level

H SL

The acceptable safety level of surcharge

H DL

The designed flood level

49

H i, j ,t

The reservoir level at time t by simulating the operation from the jth initial level using the hydrograph of the ith flood event

H iT

The target level after the release is reduced to OiB in the stage post peak

H tmax k

The maximum simulated water surface level of reservoir among all calibration cases by the end of the kth stage

H tmin k

The minimum simulated water surface level of reservoir among all calibration cases by the end of the kth stage

I t+1

The average inflow discharge from time t to t + 1

Iˆt +i

The estimated recession inflow in the next i hours

k

Number of increase of pre-release discharge in the first stage

K, p

The parameters for the storage-function-based recession analysis

l

Number of release reduction in the post peak stage

m

Number of interpolation points used to approximate the objective function in the BOBYQA algorithm

n

Number of decision variables to be determined by the BOBYQA algorithm

nf

Number of historical flood events

nh

Number of initial reservoir levels for different trial runs

nrds

Number of reference net inflow

nrh

Number of reference reservoir level

OB

The base release maintained through bottom outlets for hydropower generation or hydraulic sluicing after the closure of spillway gates

OiB

The ith specific reduced discharge, i = 1~2 and O1B < O2B

OiP

The ith specific pre-release discharge, i = 1~2 and O1P < O2P 50

OS

The non-damage inducing discharge which downstream channel is capable of safely conveying

Ot+1 i, j

Ot +1

the average outflow from the reservoir from time t to t + 1 The outflow discharge from time t to t+1 operating against the ith flood event from the jth initial water level.

Qk

The quadratic function approximating the original objective function during the kth iteration in the BOBYQA algorithm

r

The weighting factor for the barrier function in the SUMT algorithm

Ri, j

The rainfall reference corresponding to the ith specific pre-release discharge and jth reference level

Rt−tc ~t −1

The cumulative rainfall in the watershed from t − tc to t − 1

S()

The function to convert reservoir level into storage volume

tc

The concentration time of reservoir watershed

tI

Duration of the stage prior to flood arrival

t II

The time when the stage preceding peak ends

t III

The time when the flood inflow recedes below OB

wI

The weighting factor in the formulation of optimizing pre-release rule

wII

The weighting factor in the formulation of optimizing flood mitigation rule

x 0k

A base point varied throughout the course of BOBYQA computation to minimize round-off error

y ik

The ith interpolation point used in the kth iteration in the BOBYQA algorithm

51

Table1

Table 1. Rules for determining release discharge during the stage prior to flood arrival Reference water surface level of reservoir

H1R

H 2R

H 3R

H 4R

H 5R

Reference cumulative rainfall for increasing

R11

R12

R13

R14

R15

R21

R22

R23

R24

R25

R31

R32

R33

R34

R35

release to O1P Reference cumulative rainfall for changing release between O1P and O2P Reference cumulative rainfall for changing release between O2P and O S Note: (1) H1R , H 2R , …., H 5R are pre-defined levels for reference. The levels are ordered from lowest to highest as H iR  H iR1 . (2) The reference rainfall for Ht-1 is determined by interpolating the values corresponding to levels H iR and H iR1 such that

H iR  H t 1  H iR1 .

1

Table2

Table 2. Statistics of end-of-stage level corresponding to the optimization results using various values for wI wI

f IR (239.5,240.5)

f IR (239,241)

H tmax I

H tmin I

(El. m)

(El. m)

Average number of times discharge is changed during pre-release

0.0

0.98

0.99

241.87

239.52

5.17

0.1

0.91

0.96

242.35

239.46

3.32

0.2

0.91

0.96

242.38

239.47

3.28

0.3

0.87

0.96

242.53

239.26

3.26

0.4

0.81

0.94

242.48

239.47

3.14

0.5

0.77

0.94

242.64

239.38

3.02

0.6

0.77

0.94

242.64

238.38

3.01

Note: (1) f IR ( L, U ) is the empirical probability that the water surface level of reservoir at the end of the first stage is between L and U. = the maximum and minimum simulated level at the end of the first (2) H tmax and H tmin I I stage.

1

Table3

Table 3. Optimal release rules in the stage prior to flood arrival, corresponding to wI = 0.2 Reference

reservoir

level (EL. m) corresponding

storage

(million m3) Reference rainfall increasing

238

239 239.5

240 240.5

241

242

243

157.0 164.3 168.1 171.9 175.9 179.8 187.8 196.0

cumulative (mm) the

for total 277.4

98.6

29.2

7.2

5.6

3.9

1.5

0.0

total 480.1 290.3

50.9

11.5

7.5

4.3

3.1

1.8

total 547.0 450.6 253.8 113.2

66.9

29.0

28.6

28.1

release of reservoir to 225 m3/s Reference rainfall increasing

cumulative (mm) the

for

release of reservoir from 225 to 600 m3/s Reference rainfall increasing

cumulative (mm) the

for

release of reservoir from 600 to 1,000 m3/s

1

Table4

Table 4. Statistics of end-of-stage WSL and release of reservoir preceding peak from the optimization results wI

wII

f IIR (240,244) f IIR (244,245) f IIR (245,248) H tmax II

(EL. m)

Number

of Average

floods

maximum

inducing

release

H tmax II

3 to (m /s)

exceed 245 m 0.2

0

0.526

0.066

0.068

247.68

4

1982

0.2

2

0.536

0.078

0.043

247.39

3

2002

0.2

4

0.545

0.051

0.032

246.99

3

2044

0.2

6

0.545

0.038

0.032

246.66

3

2083

0.0

2

0.539

0.071

0.049

247.54

3

2003

Note: (1) The average inflow peak of historical typhoons is 2,912 m3/s, (2) f IIR ( L, U ) is the empirical probability that the WSL of reservoir at the end of the second stage is between L and U, (3) H tmax represents the maximum simulated WSL at the end of the second stage.. II

1

Table5

Table 5. Optimal rules for reducing release and closing gates of spillways Number

of Reduced

steps

in discharge in in each step steps

reducing

each

discharge

(m3/s)

Target level Number

step (El. m)

l=0

400

245.0

l=1

800

l=2

l=3

of Reduced

Target level

in discharge in in each step

reducing

each

discharge

(m3/s)

l=4

step (El. m)

1,950

243.6

245.0

1,750

243.9

400

245.0

900

244.4

1,600

245.0

600

245.0

700

245.0

400

245.0

400

245.0

2,200

243.7

1,500

243.7

1,800

243.9

900

244.3

1,200

244.0

600

245.0

800

244.2

400

245.0

600

245.0

---

---

400

245.0

l=5

Note: OB = 400 m3/s, target end-of-operation level = El. 245 m

1

Table6

Table 6. Statistics of end-of-operation levels when reservoir inflow recedes below 400 m3/s wII

Number

of

steps

in

f IIIR (0,244)

f IIIR (244,246)

f IIIR (246, )

reducing discharge 2.0

l=0

0.530

0.410

0.060

2.0

l=1

0.440

0.530

0.03

2.0

l=2

0.426

0.558

0.016

2.0

l=3

0.423

0.563

0.015

2.0

l=4

0.424

0.562

0.015

2.0

l=5

0.432

0.552

0.016

0.0

l=2

0.418

0.567

0.016

Note: f IIIR ( L, U ) = the empirical probability that the water surface level of reservoir at the end of the third stage is between L and U.

1

Table7

Table 7. The lookup table for reducing reservoir release in the stage post peak Reference reservoir

level

238

239

240

241

242

243

244

245

157.0

164.3

171.9

179.8

187.8

196.0

204.5

212.7

4510

4230

3935

3620

3280

2980

2610

1600

3040

2855

2690

2535

2360

1875

1390

700

2540

2400

2188

1830

1480

1160

850

400

(EL. m) corresponding storage (million m3 ) Reference inflow

(m3/s)

for reducing the release to 1600 m3/s Reference inflow

(m3/s)

for reducing the release to 700 m3/s Reference inflow

(m3/s)

for reducing the release to 400 m3/s

1

Figure1

1. Set initial rule parameters of the i-th stage 2. Use previously optimized parameters as simulating conditions for the stages prior to the i-th stage

j=1 No Yes

Is the terminating criterion of optimization met?

Input the hydrograph of the j-th event

Adjust rule parameters of the i-th flood stage

k=1

Output optimal parameters

Simulate operation in the k-th stage

j = j+ 1 k = k+ 1

Nonlinear optimization algorithm

Is k less than i ? Yes No

Is j less than the No number of calibration Yes flood events

1. Output simulated objective function 2. Evaluate the satisfactions of constraints

Fig. 1 Flowchart of the linkage of optimization and simulation approach on calibrating rule parameters of the ith operating stage

1

Figure2

Hi+1, dSj, αi+1,j

Hi+1, dSj+1, αi+1,j+1

(Ht-1, dSt-1, α) Hi, dSj+1, αi,j+1

Hi, dSj, αi,j

Fig. 2. Determination of  according to real-time measurements

1

Figure3

Discharge

Inflow Release

O2B

B 1

O

O

B

T B achieving target level H 2 after the release is reduced to O2

T achieving target level H1 after the release is reduced to O1B

NL B achieving the final target H after the release is reduced to O0

Time

Fig. 3. Reducing reservoir release stepwise to gradually elevate reservoir level to a desired target

1

Figure4

Shihmen Reservoir Reservoir Watershed 0

5

10 Kilom eter s

N

Tahan River W

Taiwan

E

S

Fig. 4 Location of Shihmen Reservoir

1

Figure5

Initial level = 244.00 El. m

Pre-released level: 239.95 El. m

Pre-released level: 240.26 El. m

Inflow Release

(a) wI= 0.0

(b) wI= 0.2

Fig. 5 Simulated pre-release process during Typhoon Clara in 1981

1

Figure6

Fig. 6 Optimal values for  under wII = 2

1

Reservoir Level (El. m)

Figure7

246 245 244 243 242 241 240 239 238 237 236

1st stage

2nd 3rd

2nd

3rd stage

Simulated level Historical level

Discharge (m3/s)

7,000 6,000 5,000

Inflow Historical release

4,000 3,000 2,000 1,000 0

Simulated release of spillway

Discharge (m3/s)

7,000

Simulated release of tunnel spillways

6,000

Simulated release of sluiceway

5,000

Simulated release of PRO and hydro power plant

4,000 3,000

Pseudo-optimal release process

2,000 1,000 0 14-Sep-01

15-Sep-01

16-Sep-01

17-Sep-01

18-Sep-01

19-Sep-01

Fig. 7 Simulated operations during Typhoon Nari in 2001, using optimal rules under wI =0.2, wII =2 and l = 2

1

Reservoir Level (El. m)

Figure8

246 245

1st stage

2nd

3rd stage

244 243 242

Simulated level Historical level

241 240

Discharge (m3/s)

7,000 6,000

Inflow Historical release

5,000 4,000 3,000 2,000 1,000 0

Simulated release of spillway

Discharge (m3/s)

7,000

Simulated release of tunnel spillways

6,000 5,000

Simulated release of sluiceway

4,000

Simulated release of PRO and hydro power plant

3,000

Pseudo-optimal release process

2,000 1,000 0 31-Jul-12

1-Aug-12

2-Aug-12

3-Aug-12

4-Aug-12

5-Aug-12

6-Aug-12

7-Aug-12

Fig. 8 Simulated operations during Typhoon Saola in 2012, using optimal rules under wI =0.2, wII =2 and l = 2

1

Reservoir Level (El. m)

Figure9

246 245 244 243 242 241 240 239 238

1st stage

2nd

3rd stage

Simulated level Historical level

Discharge (m3/s)

7,000 6,000

Inflow Historical release

5,000 4,000 3,000 2,000 1,000 0

Simulated release of spillway

Discharge (m3/s)

7,000

Simulated release of tunnel spillways

6,000

Simulated release of sluiceway

5,000

Simulated release of PRO and hydro power plant

4,000 3,000

Pseudo-optimal release process

2,000 1,000 0 11-Jul-13

12-Jul-13

13-Jul-13

14-Jul-13

Fig. 9 Simulated operations during Typhoon Soulik in 2013, using optimal rules under wI =0.2, wII =2 and l = 2

1

Reservoir Level (El. m)

Figure10

246

1st stage

2nd

3rd stage

245 244 243 242 241

Simulated level Historical level

240 239

Discharge (m3/s)

7,000 6,000

Inflow Historical release

5,000 4,000 3,000 2,000 1,000 0

Simulated release of tunnel spillways

Discharge (m3/s)

7,000 6,000

Simulated release of sluiceway

5,000 4,000

Simulated release of PRO and hydro power plant

3,000

Pseudo-optimal release process

2,000 1,000 0 20-Aug-13

21-Aug-13

22-Aug-13

23-Aug-13

Fig. 10 Simulated operations during Typhoon Trami in 2013, using optimal rules under wI =0.2, wII =2 and l = 2

1

1. We present release rules for a single reservoir during three flood stages 2. The rules aim to regulate reservoir to the target levels in the end of stages 3. An efficient algorithm is used to stage-wise optimize parameters in the rules 4. The deviation to the target level in the end of each stage is minimized 5. The results satisfy dam safety, flood mitigation, and sufficient final storage

52