 Case Study
 Study Basin: Boryeong Dam
 Dam Operations with Robust Optimization
 Stochastic Dynamic Programming
 Robust Stochastic Dynamic Programming
 Simulation
 Results
 Water Infrastructure Planning With Real Option Analysis
 Uncertainty in Climate
 Drought Damage Cost
 ROA Modeling Using the Decision Tree Approach
 Results
Case Study
Study Basin: Boryeong Dam
Boryeong Dam is located in the city of Boryeong in the South Chungcheong province, South Korea. The dam supplies fresh water to eight neighboring districts (Boryeong, Cheongyang, Dangjin, Hongseong, Seocheon, Seosan, Taean, and Yesan) through a regional water supply system (as shown in Figure 10.4). Since 2014, this region has suffered from a persistent drought for multiple years. For example, the total annual
Figure 10.3 Example of the “option matrix” evaluated by the real options on each node for an alternative (Ryu et al., 2017).
Figure 10.4 Boryeong Dam and the eight administrative districts to which Boryeong Dam supplies water (Ihm et al., 2019).
rainfall was approximately 800 mm in 2015, which is only 60% of the annual mean rainfall. Owing to the severe and persistent drought in this region, the reservoir stage remained below the severe warning level for 135 days. Consequently, the government imposed a water supply restriction from October 2015 to February 2016, which led to a massive shortage of drinking water for approximately 480,000 people.
The reservoir capacity of Boryeong Dam is not sufficiently large to store sufficient water to cope with an extreme drought, such as the multiyear drought event mentioned above. Therefore, the reservoir operation rule should be optimized to reduce the water shortage risk, which is discussed through the robust perspective in Section 10.3.2. Furthermore, the Korean government carried out a project for the construction of a water conduit (called the Conduit Project hereafter) that supplies additional water to Boryeong Dam from upstream (Ihm et al., 2019). The construction cost and operational expenses are estimated at 58 million USD and 1 million USD per year, respectively (KDI, 2016). Although the Conduit Project could reduce the risk of depletion of reservoir storage, it has been controversial in terms of economic feasibility due to its short period of construction. Therefore, Section 10.3.3 reevaluates the economic feasibility of this project through an adaptive perspective.
Dam Operations with Robust Optimization
The first case study linked the robust concept to the monthly operations of a single reservoir of the study basin (i.e., the Boryeong Dam). A conventional stochastic dynamic programming optimization model (CSDP hereafter) was first developed for the dam and then extended to the proposed robustSDP model (RSDP hereafter) to test a mathematical form of the robustness of the SDP objective function. The monthly releases according to the CSDP and the RSDP models were calculated using “the historical inflow series,” which is based on the historical mean (/to) and standard deviation (oo), and then evaluated with “the test inflow series” which has 400 sets of the mean (/zj) and standard deviation (o) values that are different from po and <7o The simulation results with test inflows were expressed in 2D climate response functions with different measures of evaluation metrics. The overall flowchart of this case study is as in Figure 10.5.
Stochastic Dynamic Programming
SDP has been widely applied for reservoir operations (Stedinger et al., 1984; Kim & Palmer, 1997; Kim et al., 2007) because it can include various types of objective functions in its recursive equation and explicitly consider uncertainty in probabilistic terms. In this study, SDP was used to determine the monthly optimal release rules R, with the discretized storage S,^{k} and the monthly inflow Q, as state variables. The probability
Figure 10.5 Overview of the first case study.
distribution of the monthly inflow was assumed to follow a lognormal distribution with historical statistics. The recursive equation and the constraints used in this study are shown in Equations (10.1310.16).
subject to
where Z,( ) = the objective function to be minimized; Eo,+q, [ ] = the expectation over the specified conditional distribution of period t + 1 inflow, given the value of <2,; S_{max} = the maximum reservoir storage (maximum water level); S_{m}j_{n} = the minimum reservoir storage (dead storage level).
A common objective for the water supply is to minimize the difference between the demand, Z),, and the release, R,, from the reservoir. This was selected as the main objective (ff_{v} in Equation (4)), with a quadratic form shown in Equation (10.17).
In Korea, the dam elevation at the end of drawdown season (i.e., June), S_{T}, is usually required to satisfy a specific standard elevation for the upcoming flood season, target This elevation is expressed in Equation (10.18), and it was included in the above objective.
However, note that the water supply objective is considerably more important than the flood preparation; hence, CSDP searches for the optimal solution with Equation (10.17) and then proceeds for the optimal solution with Equation (10.18). The objective function of the CSDP model was formulated as Equation (10.19). For further details of the SDP algorithm for reservoir operation, see Kim & Palmer (1997).
Robust Stochastic Dynamic Programming
To formulate the shape of the feasibility penalty function p(v _{lv}.., v_{s}) in Equation (10.4), this study conducted multiple discussions with the stakeholders of the study basin, including dam operators and local government officials who constitute a structured public participation council. To help the stakeholders with different backgrounds in understanding the concept easily, this study modified the term in the constraint so that the state variables can take values below S_{m}j_{n}, and penalized the possible violation.
The modified constraint in Equation (10.20) takes the value of instead of S_{m}j_{n},
making the model use the water storage between 5_{min} and .S_{min й} in case of emergency.
Then, the violation of the original constraint (v^ in Equation (10.6)) is included in the feasibility penalty function with a new term/3, where the magnitude of reservoir storage below the original constraint is incorporated into the penalty function.
Then, the objective function of the RSDP model was formulated as Equation (10.22) to derive the monthly optimal release of Boryeong Dam.
Simulation
As mentioned earlier, each tested inflow series has a different set of average and standard deviation values estimated from historical data (19982018), (ju_{0} =4.08 m^{3}/s and Co = 6.39 m^{3}/s). The difference between the historical and tested series considers the nonstationary characteristic of climate change. In other words, the future may not be just a repetition of the past.
This study tested different values of a ranging from 0.1 to 2.0, where Ц = ах/л_{0} (or Cj = a x Co). The range was selected based on the streamflow projections made by Seo and Kim (2018) which inputted 27 general circulation model (GCM) precipitation projections to a tank model (Sugawara, 1974) for the study basin. Figure 10.6 shows that the selected range (a = 0.12.0) sufficiently covers the possible changes of average and standard deviation of the future streamflow for the study basin. This study tested 400 values of average and standard deviation (i.e., 20 Ц x 20
The evaluation metrics selected in this study aim to account for the different measures of system performance and ensure that the measures considered important by the stakeholders are included in the metrics. It has been proven that, in theory, decisionmakers with responsibility over dam operations are interested in a wide range of risks, including (i) risk of cost overruns (solution robustness), (ii) risk of system failure (reliability), (iii) magnitude of those failures (vulnerability), and (iv) risk of performance deterioration (sustainability) (Ray et al., 2014). Furthermore, Kim et al. (2019) studied this basin and observed that the duration of failure is more vulnerable to climate change and is also harder to predict than the other metrics designed to measure failure.
Figure 10.6 Average and standard deviation ratio based on 27 GCM outputs compared with the historical values.
By considering the opinions of multiple stakeholders and past research results, this study selected three metrics to express various aspects of potential risk effectively. The selected metrics represent the average magnitude, average frequency, and maximum duration based on the initial concepts suggested by Hashimoto et al. (1982). In Table 10.2, ADM_{r} was chosen to express the magnitude of the annual water deficit; ADF,, was chosen to express the average frequency of the annual water deficit; ReT_{max}
Table 10.2 Selected evaluation metrics
Metric 
Classification 
Equation 
Unit 
ADM_{y} 
Magnitude of annual water deficit 
Ratio 

ADF_{y} 
Frequency of annual water deficit 
Ratio 

^max 
Duration of the longest failure 
Months 
ADM_{y}, average deficit magnitude; ADF_{y}, average deficit frequency; ReT_{max}, recovery time (maximum).
was selected as the longest time for the system to return from failure to success instead of average recovery time since the average recovery time showed no variation among different scenarios. All the metrics with higher values mean greater, more recurrent, and longerlasting failure.
Results
Comparing the optimal release rules derived from CSDP and RSDP, it revealed notably significant features. Figure 10.7 compares the derived optimal October release rules averaged with respect to the inflow for both CSDP and RSDP. Due to the modified constraint, RSDP starts its release rule from a lower level (2.8 MCM) when compared with the starting point of CSDP (8.2 MCM). According to the optimal release rule for RSDP, it releases a slightly smaller amount than that released by CSDP when the storage is low and compensates for the low release when the storage is relatively higher.
It was also observed that, among the paths that were first neglected due to the harsh constraint in CSDP, several of them were made possible with RSDP where the constraint violation became possible. It was observed that, among all the possible state variables, 3.33% were below S_{min} (8.2 MCM) in the CSDP model, whereas 6.78% were below 5_{m}i_{n} in RSDP. The difference between these ratios indicates that RSDP transforms 3.45% of the possible state variables those were infeasible under CSDP into feasible state variables that could potentially be selected as an optimal solution.
In terms of the selected evaluation metrics, both CSDP and RSDP showed tradeoffs under different conditions and different metrics. In terms of magnitude, the overall
Figure 10.7 Release rule comparison of CSDP and RSDP (October).
results show no significant difference between models, but in extreme cases (decreased averages and increased standard deviations), RSDP performed slightly better than CSDP. How'ever, in terms of frequency, RSDP performed better than CSDP overall, whereas CSDP performed better under cases with increased averages. In terms of the maximum duration of failure, CSDP performed better both in extreme cases and under increased averages of inflow'. The contour plots of all the metrics are shown in Figure 10.8 and the results are summarized in Table 10.3.
Figure I0.8 2D contour plots of CSDP and RSDP models of each evaluation metric.
Table 10.3 Comparison of results
Metric 
Overall 
Extreme cases 
Increased averages 
ADM, 
Similar 
RSDP > CSDP 
Similar 
AFM, 
RSDP> CSDP 
Similar 
CSDP> RSDP 
R^{e} Tjnax 
CSDP> RSDP 
CSDP> RSDP 
Similar 
Note: “>” indicates a better performance of the left model than that of the right model.
Water Infrastructure Planning With Real Option Analysis
In this section, a case study of ROA that assesses water resources infrastructure planning for drought mitigation is introduced. This section is based on the study by Ihm et al. (2019), who developed a ROA model with the adaptive perspective to reevaluate the economic feasibility of the Conduit Project described in Section 10.3.1.
Uncertainty in Climate
Climate uncertainties are estimated as the occurrence probabilities of different drought conditions. Drought probabilities are calculated as follows: (i) Monthly streamflow series are synthetically generated by a periodic autoregressive moving average model, (ii) Monthly reservoir storage series is simulated by a reservoir model based on a drought contingency plan that was established for the target dam basin, (iii) The number of days in a year during which the reservoir stage is under a certain threshold level is calculated over the entire planning horizon, (iv) If the estimated number of days is greater than the value derived from a baseline drought scenario, the corresponding year is regarded as a severe drought year. If the estimated number is less than the baseline value, the corresponding year is regarded as a moderate drought year. If the estimated number is zero, the year is regarded as a normal year.
Figure 10.9 shows how the probabilities of drought occurrences are treated. As shown in Figure 10.9, three potential outcomes, i.e., normal (no drought), moderate, and severe drought, are postulated. The overall drought damage cost is calculated as the weighted sum of three different drought damage costs: i.e., overall drought damage cost = ^—normal ^{x} ^—normal + ^—moderate ^{x} ^—moderate + P—severe ^{x} C_severe (Ihm et al., 2019).
Drought Damage Cost
Drought damage cost can be calculated as the product of the unit price of water and the amount of water deficit. Assuming that the drought damage is proportional to the length of drought, Ihm et al. (2019) calculated the drought damage cost as presented in Equation (10.23).
where Q = the drought damage cost, JV_{t/} = the water demand per day, W_{s} = the water demand per day under the severe warning stage, WP_{w} = the unit price of water, and D_{t}i = the average number of days under the severe warning stage (Ihm et al., 2019).
Figure 10.9 Evaluation of the uncertainty factor by postulating multiple potential conditions. c: drought damage cost for each condition; p: the probability of drought for each condition (Ihm et al., 2019).
Furthermore, the stress cost, which is an additional damage defined as an amount of mental stress due to restricted water supply, can be considered as a monetary value. Then, the term WP_{w} in Equation (10.23) becomes (WP_{w} + S_{w}) where S_{w} is the unit price of mental stress due to water supply restriction.
ROA Modeling Using the Decision Tree Approach
Three options, “invest,” “delay,” and “abort,” are used in the case study. With the investment option, a project is installed immediately and operated until the end of the planning horizon. Conversely, if the delay option is chosen, the installation is postponed and reevaluated in the following time step. Finally, when the installation and delay of the project are no longer economically feasible, the abort option is suggested. With the abort option, the project is canceled so that no further investment is considered during the planning horizon. Figure 10.10 demonstrates the decision tree generated for ROA based on the three options.
With the calculated ENPV value for each year, the abort option is suggested when the OP value is zero. Otherwise, when the ENPV is greater than the NPV (i.e., the OP value is positive), the invest (delay) option is chosen if the option value (sum of the net benefit of the option) of invest is greater (lesser) than the option value of delay. When the delay option is selected, all the options are available for the next year. The optimal option is determined based on the previously mentioned selection mechanism. The outcomes of the existing feasibility study on the Conduit Project (KDI, 2016) are compared with those of the ROA model to evaluate the developed ROA model with the decision tree approach.
Results
Ihm et al. (2019) reevaluated the economic feasibility of the Conduit Project and discussed that the abort option could be the optimal choice for the lowest economic loss on the project. In reality, the Conduit Project caused a nationwide controversy due
Figure 10.10 Decision tree based on the three options: invest, delay, and abort (Ihm et al„ 2019).
to its enormous capital cost despite its advantage in alleviating water shortage immediately. Table 10.4 presents the economic feasibility results evaluated using various BCA models. As the total expenses of the Conduit Project were greater than the initial estimation of the project, all the BCA results (total benefit in Table 10.4) were negative, which indicates that the project was not economically feasible at the time of construction.
Table 10.4 Results of BCA for Boryeong Dam Conduit Project (Ihm et ah, 2019)
Models 
dcf_{kdi} (baseline)^{0} 
dcf_{unc}» 
^{R}0AUNC^{C} 
^{DCF}UNC_S^{d} 
^{R}OA_{UNC}_s^{e} 
Total benefit (million USD) 
NPV 83.4 
NPV 89.3 
ENPV 2.0 
NPV 94.3 
ENPV 79.5 
^{a}The traditional DCF model conducted by KDI (2016). ^{b} The DCF model reflecting climate uncertainty.
^{c} The ROA model reflecting both climate uncertainty and flexibility in the decisionmaking process. ^{d} The same model as (b), but the stress cost is added to drought damage. ^{e}The same model as (c), but the stress cost is added to drought damage.
Compared with the DCF_{K}dj, the NPV of the DCFunc model decreased by approximately 7% (5.9 million USD). This decrease indicates that the economic feasibility of the project was overestimated with no consideration of uncertainty in climate. Conversely, the ROAunc (which selects the abort option), was the most valuable economically (i.e., the optimal option), although the ENPV was still negative. This suggests that the invest option is not better than the abort option. The OP obtained using the abort option was 81.4 USD, (= (2.0)  (83.4)). Including the stress cost in the drought damage, in the case of DCF (by comparing the difference between the total benefits of DCFunc ^{a}nd DCFunc_s)> the impact of the additional stress cost on the economic feasibility was relatively less, as the conduit can reduce the water shortage. Conversely, there was a notable difference between the total benefits (ENPV) of ROAunc and ROAunc_s This difference indicates that the additional stress cost due to water shortage was added to the ENPV by selecting the abort option for the optimal decision (Ihm et al., 2019).
Furthermore, sensitivity analyses were implemented to find other conditions that can increase the economic worth of the current project (Ihm et al., 2019). In this study, the onefactoratatime (OFAT) method (Czitrom, 1999), which is one of the most commonly used methods, is applied for the sensitivity analysis. Theoretically, OFAT experiments change only one variable at a time while maintaining others fixed. Figure 10.11 shows the results of the sensitivity analyses of the case study.
The ROA model determined that the invest option would be better than the abort option when the probability of severe drought increases by approximately 20%. This indicates that the Conduit Project can be economically feasible if the probability of severe drought is increased by 20%. Moreover, if the total project expense decreases by approximately 27%, the invest option becomes economically feasible. Thus, the sensitivity analyses provide insights into a possible strategy for maximizing the economic feasibility of the project (Ihm et al. 2019).