Balancing operator’s risk averseness in model predictive control for real-time reservoir flood control

DP

dynamic programming

FWL

flood water level

GA

genetic algorithm

LP

linear programming

LWL

low water level

MAVE

maximum allowed value estimate

MPC

model predictive control

NHWL

normal high water level

PD-MPC

parameterised dynamic model predictive control

RWL

reservoir water level

Many multipurpose reservoirs are designed and constructed with flood control as an important objective. They play a crucial role in mitigating flood risks downstream by retaining a significant portion of inflows in a given period. Additionally, the efficient operation of existing reservoirs is becoming increasingly important since costs and growing environmental concerns in society are making it challenging to build new reservoirs (Scudder 2012). At the same time, climate change may lead to more severe and unpredictable flood events (Havens et al. 2016), which are difficult for operators to manage efficiently using conventional simulation-based methods (Watts et al. 2011).

Decisions regarding flood control operations for such reservoirs significantly influence basin flood conditions. Due to the importance of reservoir flood control, achieving optimal outflows of reservoirs has long been a focus (Giuliani et al. 2021; Jain et al. 2023). Various optimisation methodologies, including linear programming (LP), dynamic programming (DP), and their variants, have been widely applied due to their capacity to guarantee optimal solutions (Labadie 2004) for this type of problem. Another popular class of algorithms in the literature on reservoir optimisation is randomised search techniques, mainly evolutionary algorithms and the genetic algorithm (GA). These have also gained traction, particularly for addressing complex problems that are not analytically expressed and less tractable to pose as mathematical optimisation problems (Ahmad et al. 2014). Some relatively recent literature has also focused on the approaches associated with control theory, primarily model predictive control (MPC), with different optimisation algorithms employed to solve the resulting optimal control problem (Breckpot et al. 2013; Delgoda et al. 2013; Castelletti et al. 2023).

Reservoir operators use only a limited horizon of rainfall/inflow forecasts, which are often uncertain. These forecasts are typically shorter than the whole length of a flood event (Breckpot et al. 2013). Operators must make decisions based on uncertain short-term forecasts. These decisions are not only about one outflow for the current time but also outflows for some time horizon, i.e., an outflow schedule. However, operators implement only the first (sometimes a few) outflows in the outflow schedule because, in general, forecasts are updated regularly, and new forecasts are considered less uncertain. Therefore, decisions should be updated repeatedly to reflect updated forecasts and hydrological conditions. In this sense, the receding horizon MPC concept, in which only the first control input is implemented and the optimisation processes are repeated at each time step iteratively (Van Overloop 2006), coincides with the reservoir flood control.

While optimisation-based control approaches including MPC have made substantial contributions in many applications (Schwenzer et al. 2021), their practical application within reservoir flood control remains somewhat limited. This discrepancy can be mainly attributed to the disparities between research assumptions and the pragmatic necessities of real-time reservoir flood control.

The first disparity concerns objectives. Many researchers have introduced objectives for optimal reservoir operation (Ko et al. 1992; Reddy & Kumar 2006; Malekmohammadi et al. 2011; Lin & Rutten 2016; Tang et al. 2019), such as minimising outflows via spillway gates, maximising hydropower generation, and maintaining reservoir water level (RWL) for water supply. However, during a flood event, most of these objectives may not be a priority.

What operators want in practice for flood control, i.e., operators’ objectives for short-term planning, is not generally specified in the regulations and guidelines but is considered ambiguous ‘experience’ or ‘expert knowledge’. Here, we call these ‘practical objectives’. Note the distinction from mathematical objective functions, where mathematical equations, including system models and constraints, are used to calculate values of control objectives. Researchers seem to have paid insufficient attention to how to define and adopt these practical objectives in the optimisation process (Ritter et al. 2020; Teegavarapu & Simonovic 2001). A recent survey conducted on water supply companies in England and Wales by Pianosi et al. (2020) has shown that factors, i.e., objectives and constraints, in decision-making were too complex to be included in an optimisation process. Hence, operators generally hesitated to adopt optimisation tools. Moreover, even if we successfully formulate practical objectives but with a number of highly nonlinear mathematical formulas, the problem then becomes intractable by the MPC approach (Allgower et al. 2004; Berberich et al. 2022). This is often a reason why many researchers apply only a limited number of objectives or linear and/or quadratic objectives, which are tractable even with several objectives, for a reservoir flood control problem (Breckpot et al. 2013; Qi et al. 2017; Uysal et al. 2018b) and so are deemed feasible to solve.

In addition, the preferences imply the selecting criteria of operators among control alternatives, which are calculated from different weight sets and/or parameters. Therefore, selected weights and parameters in the multi-objective optimisation problem can be referred to as the preference (Wang et al. 2017). In many studies focusing on reservoir flood control and applying the receding horizon MPC approach, weighted multi-objective optimisation-based methods have been adopted (Wang et al. 2013; Hu et al. 2014). Hence, it is necessary to produce a Pareto set, which is a vector of non-dominant control inputs, using a weighted-sum method or apply one fixed weight representing the operators’ preference, i.e., allocating high weights to the objective to which operators pay more attention. However, producing a Pareto set at every time step is computationally expensive (Peitz & Dellnitz 2018). Additionally, we cannot be sure that the relative importance of objectives remains constant during a flood event.

To fill these research gaps, this article presents an MPC framework to generate an optimal flood control decision, which, in our opinion, would be practical and more acceptable to operators. The main idea is explicitly defining the practical objectives for reservoir flood control based on the first author’s operational experience. We assume dynamic preferences, which we then formulate using parameterised linear MPC and dynamic optimisation of weights and parameters of objectives to efficiently handle an otherwise intractable multi-objective nonlinear optimisation problem. This methodology is tested for historical flood events with different features, such as the number of inflow peaks and event lengths. Such verification is important because outflows and water level at the end of the first/second peak hugely influence the outflows of the next time steps.

The manuscript is organised as follows. In Section 2, we present the objective functions for practical flood control and propose a framework to efficiently incorporate linear and nonlinear objectives under the assumption of dynamic preference. Section 3 describes the detailed MPC problem and a numerical experiment. The contribution and limitation of this research and the result of the numerical experiment are then presented in Section 4, followed by conclusions.

Model predictive control (MPC) is an optimal control framework that uses a system model to predict and evaluate the system’s behaviour over a finite time horizon. The control input, which is the optimal outflow schedule in reservoir flood control, is calculated to optimise objectives and satisfy constraints based on the current state and predicted behaviour. The main characteristic of the MPC approach is the combination of prediction and optimisation. Only the first outflow in the schedule is implemented, and the prediction and optimisation processes are repeated at each time step. This iterative process is generally referred to as receding horizon control (Van Overloop 2006). In some applications, such an approach is also called a rolling horizon method (Wang et al. 2014).

Unlike randomised search algorithms such as GA, MPC can explicitly consider constraints. Another significant advantage of MPC is its robustness to disturbances, such as uncertain inflow forecasts and system uncertainty. This robustness stems from MPC’s ability to re-estimate predictions and recalculate optimal control inputs at each time step based on updated hydrological states, including revised inflow forecasts (Schwenzer et al. 2021).

The flood control of large reservoirs requires multiple practical considerations; one well-known conflicting objective is the need to reserve enough water to supply to contractors at the end of a flood event. In this section, we propose additional important objectives and motivate their necessity based on the receding horizon MPC concept.

First, using the full capacity of control facilities such as spillway gates is not reasonable for large reservoirs. Instead, operators tend to desire less outflow generally. Second, it is preferable to limit the frequency of operations of spillway gates to prevent wear and malfunction. Furthermore, immediate and frequent changes in outflow schedules are not preferred because they can hinder the predictability of the flood situation of the downstream area for other flood control agencies. It is worth noting that even though we are focusing on flood-relevant objectives, other objectives for reservoir operation, such as maintaining minimum flow requirements or ensuring adequate flow for fish migration and downstream habitats, can be formulated similarly.

In defining the practical objectives and the subsequent methodology, we introduce a control input vector, ⁠, at time step k, defined as where and are the total outflow and spillway outflow decided at time step k for time ⁠, respectively. H is the length of the prediction horizon. We define an augmented state vector that consists of the RWL, predicted inflow, and the outflows decided at the previous time step as ⁠, where and are the storage volume and predicted inflow variables for time k, respectively. To avoid ambiguity, we define the term ‘time step k’ as the iteration of MPC and ‘time t’ as the exact hour at which a control input is supposed to be implemented. The detailed objectives for practical reservoir flood control are defined below.

It is noteworthy that and complement each other to generate an outflow sequence with less variability. When outflows need to increase, e.g., when RWL is expected to rise significantly due to substantial inflow, leads to dropping the last outflow in the outflow schedule to minimise the total spillway discharge over the prediction horizon. This occurs because, in the discretised reservoir model, which will be detailed in Section 3.2.1, the last outflow does not affect RWLs within the same prediction horizon, as outflows and inflows at a given time, e.g., t, only influence the reservoir’s hydrological condition at the subsequent time, ⁠. can prevent this undesirable schedule formation. Conversely, when outflows need to decrease, e.g., when inflows are insufficient to maintain RWLs for water supply despite the earlier prediction of substantial inflows, may impede the reduction of the final outflow, but can enforce this reduction. This complementarity may be deemed less crucial when the prediction horizon is long, given the diminishing significance of outflow changes distant from the current time step. However, in scenarios with a short prediction horizon, such as the 6-h case in our numerical experiment in Section 3, these objectives can be essential for making the optimal outflow practically applicable.

This equation shares a similar purpose to and ⁠, in that it penalises changes in outflow schedules. However, it holds a greater significance because it also regulates the opening and closing of gates. The reason is that and ⁠, which penalise total discharge, can have the minimum zero values while both turbine and spillway gate states change in a schedule. Especially, has zero values when the change of spillway gate states is planned in a previous time step. Adding the objective shown in will ensure that changes in spillway gates are specifically avoided unless necessary. However, this objective is a penalty form of what is called a complementarity constraint (Powell et al. 2016), so linearising this objective is difficult for MPC without adding binary variables representing resulting in a mixed-integer problem (Anitescu 2000). In Section 2.4, this objective is included only in the Evaluator to get around this issue of optimisation problem complexity.

Note that the peak outflow represents the maximum outflow within a prediction horizon, whereas the peak inflow denotes the maximum inflow up to the current time step, as shown in Equation (7). This objective can be easily linearised, but we incorporate it into the Evaluator in Section 2.4 rather than the MPC formulation because it is optimised indirectly in the middle of minimising the peak outflow in in most cases.

As mentioned in Section 1, numerous studies aimed to find the best set of objective weights while assuming that the preference remains constant. This is because the relative importance of each objective typically does not change significantly when hydrological conditions remain stable. However, in the context of reservoir flood control, operating preference can often shift with changes in hydrological states. This variability is evident from the objectives outlined in Section 2.2.

Some objectives may conflict with one another, while others may complement each other, and this dynamic depends on the current state. For instance, when RWL approaches FWL or ⁠, the importance of increases, necessitating a substantial increase in outflows with less importance for other objectives. Conversely, if RWL remains below and spillway outflows are stable, and should be prioritised. When a significant increase in outflows becomes unavoidable, the highest weight should be assigned to ⁠, followed by ⁠. Consequently, we can conclude that the relative importance of each objective should vary depending on hydrological conditions.

Some parameters, like the target water level, have been considered static as well. However, the parameters such as ⁠, ⁠, and in Equation (5) directly impact the objective value of and the optimal control inputs. Hence, we can not assume that these parameters remain static when the preference is not static.

MPC is widely used in reservoir flood control, leveraging system models to predict future states and determine optimal control inputs (Breckpot et al. 2013; Delgoda et al. 2013; Castelletti et al. 2023). When we assume dynamically changing objectives’ weights and parameters, both control inputs and objectives’ weights/parameters need to be optimised simultaneously. The problem becomes complicated due to their interdependency. Without additional criteria, this co-optimisation problem would be intractable.

If we have the means to evaluate the derived optimal control inputs, which we call an evaluator here, we can simplify this problem. While traditional approaches typically generate a Pareto set, a set of alternatives, and select the optimal one using a single criterion, such as the possibility for system failure (Chen et al. 2020), minimising the worst-case impact (Yu et al. 2023), or integrate simple but diverse evaluation criteria (Zhu et al. 2018; Myo Lin et al. 2020), computing the complete Pareto set becomes computationally intractable with multiple objectives (eight in our case), particularly for the online optimisation process of MPC.

At each time step, we first solve the Equation (11b) to obtain control actions for a specific weight set, then solve the Equation (11a) to optimise both the weights/parameters and their corresponding control inputs through the evaluator, ⁠. To solve this evaluation problem, we can employ heuristic optimisation techniques, such as GA. Heuristic algorithms explore the weight and parameter space to find weights and parameters with the optimal value from the evaluator at each time step of the receding horizon recalculation. Therefore, this methodology enables us to decouple the optimisation process into two separate tasks: finding optimal weights and determining control actions, which simplifies the co-design problem, as illustrated in Figure 1.

In this research, we linearise all proposed practical objectives for a linear MPC and integrate objectives presented in the previous section to formulate the evaluator, without limiting the evaluator to linear and nonlinear objectives that are straightforward to implement in mathematical optimisation. Moreover, to reflect real hydrological processes, the evaluator uses a nonlinear system simulation (i.e., nonlinear height-volume curve) to calculate water levels. This allows the PD-MPC framework, primarily optimising a linear MPC problem to find a release schedule, to select an outflow schedule that actively aligns with the operator’s preferences. The reason why all objectives are linearised for ⁠, instead of applying quadratic equations as presented in Section 2.2, is to penalise even minor exceedances and linearising an absolute form is straightforward. For detailed explanations, including formulas, please refer to Supplementary material, Appendix A for the evaluator and Section 3.3 for the objectives, weights and parameters.

In addition, we utilise GA as our heuristic optimisation algorithm. The genetic algorithm (GA) is one of the most prominent and easy-to-implement gradient-free algorithms which imitate the natural selection process (see, e.g., Katoch et al. 2021). It continuously develops the population (a set of vectors seen as potential solutions) over each iteration, employing reproduction, crossover, and mutation operators, aiming at preserving the vectors with lower objective values and iteratively recombining them.

The Daecheong reservoir has enough spillway capacity with the spillway crest level significantly lower than NHWL. The outflow capacity at NHWL is greater than 6,000 ⁠, so we do not need to consider the spillway outflow capacity according to RWL. However, the flood control storage is relatively small. For example, the 200-year frequency inflow is 10,700 ⁠, and it takes only 6.5 h to completely fill the flood control storage. In addition, there is no restricted water level for the flood season in this reservoir. In essence, the Daecheong reservoir is susceptible to flooding, necessitating rapid and well-informed flood control decisions by operators to avert extreme conditions.

Three flood events with a large amount of peak inflow and one to three peaks are selected from historical data as presented in Table 2. To the best of our knowledge, most studies have primarily concentrated on floods lasting 1–2 days (Uysal et al. 2018a; Breckpot et al. 2013; Delgoda et al. 2013). In the case of short flood events, generating reliable outflows is relatively straightforward because the trade-offs among objectives are evident. For instance, it is clear that minimising outflows leads to an increase in reservoir water level, and vice versa. However, the situation becomes more complex over longer time frames. Minimising outflows can ironically result in increased outflows if peak inflow occurs after the reservoir water level has already risen significantly. Therefore, our research examines floods lasting more than eight days, featuring one to three peaks, to provide a more practical case study.

Table 2

Study flood events

Periods .	Duration .	Peak inflow .	Feature .
From 2 July 2016 to 11 July 2016	197 h	3,655	Double peak
From 25 August 2018 to 7 September 2018	311 h	2,590	Triple peak
From 1 September 2020 to 12 September 2020	269 h	4,468	One peak

Periods .	Duration .	Peak inflow .	Feature .
From 2 July 2016 to 11 July 2016	197 h	3,655	Double peak
From 25 August 2018 to 7 September 2018	311 h	2,590	Triple peak
From 1 September 2020 to 12 September 2020	269 h	4,468	One peak

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Although these events can present significant operational challenges demonstrating our method’s practical utility, more extreme floods, such as a 200-year return period flood and a probable maximum flood (PMF), would provide more rigorous validation cases. However, the Daecheong reservoir, our case study site, was constructed over 40 years ago, and to the best of our knowledge, the hydrographs for the 200-year flood and PMF are unavailable. Moreover, the 200-year flood and PMF typically span only 1–3 days with a single peak, representing less uncertainty and total inflow volume.

The operational constraints can be formulated as follows:

1. The total outflow is more than the sum of the agreed amounts with water users and the in-stream flow, i.e., the downstream demand:

   

   (14)

   where and are the total outflows and the downstream demand at time t, respectively.
2. RWL should be between FWL and LWL:

   

   (15)

   where is the reservoir storage at time t, while LWS and FWS are the reservoir storage at LWL and FWL, respectively.
3. Outflows via the turbine and spillway gates are not able to exceed their capacity:

   

   (16)

   where and represent the turbine and spillway outflow at time t, respectively. and are the outflow capacity of turbines and spillway gates. We take into account the spillway crest level in our analysis, but spillway outflow capacity depending on RWL is disregarded as explained in Section 3.1.

The important difference in the objective formulas presented in Section 2.2 is about in Equation (19b) and in Equation (4). Note that is the first outflow in the optimal control inputs at time step k. First, changing the outflow for time k at time step k is practically impossible because it involves implementing the current outflow while calculating it. To address this, we separate this to ensure it, rather than assigning significant weight to the change in the first outflow in Equation (19d). Hereby, the outflow decision is delayed for a time step.

Since the Korea Meteorological Agency (KMA) publishes 6-h quantitative rainfall forecasts every hour, in the experiment, we employed four different prediction horizons: 6, 12, 18 and 24 h. Starting from the 6 h, the longer horizons allow us to explore the effect of horizon length on performance. The control horizon is the same as the prediction horizon. For the initial run at ⁠, we set the initial storage to the corresponding level of NHWL (EL. 76.5 m), while the initial outflows via the turbines and spillway gates are set to 150 ⁠, which is the average hourly outflow during the flood season from 2015 to 2020, and 0 ⁠, respectively.

In this experiment, the GA optimises 10 parameters, comprising of 9 weights for the objectives and 1 parameter for the highest RWL, denoted as ⁠. To reduce the running time of GA, we impose search range limits for each weight and parameter, as outlined in Table 3. In in Equation (18), and are fixed at twice and twenty times ⁠, respectively, because maintaining RWL under for dam safety is a higher priority than maintaining it between target levels, additionally, in order to reduce the computational complexity by decreasing the number of weights which need to be optimised. Again, the reason why nominators for and are larger than others is to reduce the number of dynamic weights and to reflect the fact that the objectives corresponding to and are more important than the objective for ⁠. Here, 40 and 400 are selected somewhat arbitrarily because precise values are not critical due to the PD-MPC, which finds optimal weights dynamically. Similar to the value of 20 for other multipliers, the values 40 and 400 are selected to prevent the solver from neglecting objectives when objective values are divided by a substantial value, e.g., FWS for ⁠, ⁠, and ⁠. Instead of fixing these nominators and weights, we could extend the search range from 1–20 to 1–400 or 1–500, and so on. However, the results are the same, although it takes considerably more time. This is why we fixed the nominators instead of extending the search range while setting the same value for all nominators. Therefore, the number of weights and parameters explored becomes 8 from 10. All weights share the same length of the search range, except for ⁠. This is because the weight of ⁠, i.e., minimising the total outflows via spillway gates, can be relatively small due to its similarity to ⁠, i.e., minimising the peak outflows via spillway gates, and we want to emphasise ⁠. To ensure is not neglected, the search ranges for ⁠, ⁠, and start from 1, not from 0.

Table 3

The possible range for PD-MPC and fixed weights/parameters for the baseline MPC

P .	.	.	.	.	.	.	.	.	.	.
Multiplier										–
Search range	0–19	0–2	0–19	0–19	0–19	0–19	1–20
Fixed-1	3	1	3	3	20	20	15
Fixed-2	20	5	3	3	3	3	15

P .	.	.	.	.	.	.	.	.	.	.
Multiplier										–
Search range	0–19	0–2	0–19	0–19	0–19	0–19	1–20
Fixed-1	3	1	3	3	20	20	15
Fixed-2	20	5	3	3	3	3	15

Note Multipliers are introduced to normalise objective values using MAVE as the denominator of each objective, and nominators are set to prevent objective values from approaching zero in any case as well as to reduce the range of search space.

indicates the same value as in the previous column.

Fixed-1 and Fixed-2 represent MPC baselines with fixed weights and a fixed parameter set.

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Moreover, to simplify the optimisation process and work with integer values in GA, we introduce multipliers based on these MAVEs. Because MAVE represents the largest anticipated magnitude of each objective, it is often adopted to normalise objectives to maintain numerical balance across different objectives. Each weight is then calculated by multiplying a selected integer value by its corresponding multiplier. This approach ensures that all objectives contribute as a proportion of the selected values in the search range, regardless of their natural scales, while allowing GA to work with simpler integer values during the search process.

We prepare MPC baselines with fixed weights and a fixed parameter set, denoted as ‘Fixed’, which would be used in standard operation, as presented in Table 3. Our comparative analysis aims to demonstrate that operators’ decisions can be improved by dynamically optimising weights in real-time. This approach allows for adapting risk preferences depending on the system state. To ensure the feasibility of the problem, two weight/parameter sets, which emphasise minimising the changes in outflow schedules and minimising peak outflow, are selected from the best weight/parameter sets generated from various PD-MPC tests and verified through trial and error (Uysal et al. 2018a). Throughout this paper, we refer to the value before being multiplied by the multiplier as the weights unless this would cause confusion otherwise.

In Table 3, a searching range of is the storage where RWL is in EL. 78.5 m, 79.0, 79.5 m and of the baselines, i.e., (Reservoir Water Storage for Fixed weights/parameter set), which is the stored amount of water at EL.79.0 m. A computer code is developed using Python. In detail, pyomo (Hart et al. 2011), GLPK solver (Makhorin n.d) and pyGAD (Gad 2021) packages were applied to implement the numerical experiment of PD-MPC.

Parameterised dynamic model predictive control (PD-MPC) delivers reliable results across all events and prediction horizons. The results show only a few changes in outflow schedules, and all peak outflows remain below the peak inflow.

The detailed result of the numerical experiment is presented in Table 4 for uncertain inflow and Table 5 for certain inflow. As expected, the performance of MPC under certainty generally surpasses the results obtained under uncertainty. Peak outflows and RWLs from uncertain inflow exceed those from certain inflow, and the reservoir should change the outflow schedule more often. Additionally, under uncertain inflow conditions for Event 1, a comparison of the results between PD-MPC and the Fixed cases is presented in Table 6. PD-MPC outperforms the Fixed cases for all items in the table.

Table 5

The detailed result of PD-MPC under certain inflow

Prediction horizon .	Event .	Peak outflow (⁠⁠) .	Peak RWL (EL. m) .	Lowest RWL (EL. m) .	Changes between time steps .
6	1	2,332	79.89	76.50	9
12	1	2,153	79.27	76.00	4
18	1	2,290	78.60	76.00	11
24	1	1,873	77.13	75.93	4
6	2	2,051	79.84	76.19	9
12	2	1,499	79.09	75.99	8
18	2	990	77.38	76.00	11
24	2	1,870	76.92	76.00	17
6	3	2,078	79.82	76.32	8
12	3	1,051	78.59	76.00	11
18	3	1,104	78.33	75.95	9
24	3	1,374	77.91	76.00	18

Prediction horizon .	Event .	Peak outflow (⁠⁠) .	Peak RWL (EL. m) .	Lowest RWL (EL. m) .	Changes between time steps .
6	1	2,332	79.89	76.50	9
12	1	2,153	79.27	76.00	4
18	1	2,290	78.60	76.00	11
24	1	1,873	77.13	75.93	4
6	2	2,051	79.84	76.19	9
12	2	1,499	79.09	75.99	8
18	2	990	77.38	76.00	11
24	2	1,870	76.92	76.00	17
6	3	2,078	79.82	76.32	8
12	3	1,051	78.59	76.00	11
18	3	1,104	78.33	75.95	9
24	3	1,374	77.91	76.00	18

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Table 6

The comparison between PD-MPC and the Fixed cases for Event 1 under uncertain inflow

.	Prediction horizon .	Peak outflow (⁠⁠) .	Peak RWL (EL. m) .	Lowest RWL (EL. m) .	Changes between time steps .
PD-MPC	6	2,367	79.88	76.50	9
Fixed 1	6	3,710	79.10	75.97	13
Fixed 2	6	3,617	79.31	76.31	28
PD-MPC	12	2,005	79.35	76.00	9
Fixed 1	12	3,267	79.19	76.37	10
Fixed 2	12	2,759	79.35	76.34	37
PD-MPC	18	2,259	78.67	76.00	11
Fixed 1	18	2,578	79.35	76.34	12
Fixed 2	18	2,551	79.60	76.31	48
PD-MPC	24	2,128	77.13	75.98	9
Fixed 1	24	2,259	78.03	75.95	16
Fixed 2	24	2,223	79.32	76.28	33

.	Prediction horizon .	Peak outflow (⁠⁠) .	Peak RWL (EL. m) .	Lowest RWL (EL. m) .	Changes between time steps .
PD-MPC	6	2,367	79.88	76.50	9
Fixed 1	6	3,710	79.10	75.97	13
Fixed 2	6	3,617	79.31	76.31	28
PD-MPC	12	2,005	79.35	76.00	9
Fixed 1	12	3,267	79.19	76.37	10
Fixed 2	12	2,759	79.35	76.34	37
PD-MPC	18	2,259	78.67	76.00	11
Fixed 1	18	2,578	79.35	76.34	12
Fixed 2	18	2,551	79.60	76.31	48
PD-MPC	24	2,128	77.13	75.98	9
Fixed 1	24	2,259	78.03	75.95	16
Fixed 2	24	2,223	79.32	76.28	33

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In Section 1, we discussed that some researchers had focused on finding appropriate weights of objectives when defining preferences in a multi-objective setting. To demonstrate the parameters should also be regarded as important components of the preference, we conducted an experiment for Event 1, where we fixed to the storage level at EL. 79.0 m and compare it with the results of PD-MPC where is a dynamic parameter to be optimised.

In Section 2.3, we discussed the complexity of trade-offs among objectives, which led us to assume that the operator preferences among them can be dynamic.

It appears that the detailed design of practical objectives for reservoir flood control has not been thoroughly presented (Pianosi et al. 2020; Ritter et al. 2020), and it seems complicated to formulate practical objectives with only linear equations, as we described in Section 2.2. The receding optimal control of an MPC problem with numerous nonlinear objectives and constraints can become intractable to solve online (Allgower et al. 2004; Berberich et al. 2022). Given that preferences can be expressed as objective weights to represent their relative importance (Wang et al. 2017), the optimal preferences (weights) should adapt to varying hydrological conditions of the reservoir. However, it seems that the operators’ preferences have not been adequately incorporated, and constant weights/parameters have been used to optimise reservoir flood control (Van Overloop 2006; Xu et al. 2011; Breckpot et al. 2013; Qi et al. 2017; Uysal et al. 2018b; Aydin et al. 2022).

In this study, we harness the advantages of solving linear MPC problems by parameterising the nonlinear and dynamic preferences around the different operating points. This allows us to optimise weights/parameters and control inputs simultaneously. Our approach is also in the spirit of multi-objective MPC methods for linear systems (Bemporad & de la Peña 2009) where a time-varying, state-dependent decision criterion can be taken into account using parametric optimisation. First, we present objectives for practical flood control in detail. Some of them have not been extensively covered in prior literature despite their significant importance in practice, such as minimising the changes in outflow schedules. Subsequently, we categorise these objectives into linear and nonlinear ones following the parameterisation of the operator’s preference, i.e., the weights of objectives and parameters. We employ the genetic algorithm (GA) to optimise nonlinear objectives and constraints to derive the optimal weights/parameters of the linear MPC formulation at each time step. We refer to this framework as a parameterised dynamic model predictive control (PD-MPC).

Through our numerical experiment, we demonstrated that PD-MPC shows robustness to the inflow uncertainty. It is important to note that the PD-MPC framework does not directly handle inflow uncertainty. Instead, we rely on the robustness of the receding horizon MPC approach to address uncertainty and produce reliable results (De Nicolao et al. 1996; Schwenzer et al. 2021). Our results reveal that PD-MPC outperformed MPC formulations with fixed weights/parameters, even when these fixed weights/parameters were specifically designed for individual objectives. We showed that the weights/parameters in MPC formulations and the weights of objectives should also vary dynamically to adapt to changing conditions. PD-MPC then effectively adapted to the changing hydrological conditions and continuously updated the weights/parameters. This adaptability allowed it to make optimal decisions in real-time, resulting in better overall performance in terms of peak outflows, reservoir water levels, and the number of changes in outflow schedules. This adaptability is the key factor in its superior performance and is also essential for generating optimal control inputs reflecting the dynamic characteristic of the operator preferences.

Nevertheless, this research has some limitations. First, the MPC formulation applied here does not consider the final states of the system. MPC with policy search algorithms (Song & Scaramuzza 2022) or a value function produced by a Reinforcement Learning (RL) model (Arroyo et al. 2022) can give a chance to find the approximation of the terminal cost such that our MPC framework can consider the whole period of a flood event. To the best of our knowledge, this approach has never been applied to a reservoir system for the purpose of flood control. The second limitation is related to uncertainty. We showed that PD-MPC can generate acceptable control inputs under uncertainty from the inherent feedback mechanism of a receding horizon implementation; however, it is advisable to explore PD-MPC with stochastic/robust MPC (Saltık et al. 2018) or Learning-based MPC (Hewing et al. 2020) to ensure robustness by considering uncertainty explicitly. The third limitation pertains to our numerical experiments, which utilised three historical flood events. Although these events present significant operational challenges that demonstrate our method’s practical utility, e.g., long event periods and two or three peaks, as well as a limitation in obtaining hydrograph of the 200-year flood and PMF, to establish broader applicability, numerical experiments under more extreme conditions are necessary. Finally, we did not consider the downstream impact of reservoir outflows as well as the upper reservoir for simplicity. This should be considered important in practical operations. The downstream impact can be considered using routing models or various heuristics suggested in many studies (Hsu & Wei 2007; Le Ngo et al. 2007; Peng et al. 2017). The entire reservoir system, including the upper reservoirs, needs to be explored, and we expect that PD-MPC could be applied to the joint optimisation problem, though this may be a topic for further study.

This study addresses the limitations of existing reservoir control approaches from a practical point of view, highlighting the current limitations of the employed optimisation approaches to take into account specific operators’ preferences, which may change over time. We assume dynamic preferences by operators in a multi-objective setting and show the dynamic characteristics of weights/parameters. We then propose a PD-MPC framework as a parameterised linear MPC with dynamic optimisation of weights/parameters via a model-based learning concept. We applied this methodology to the Daecheong reservoir, verified the dynamic-preference assumptions, and tested this framework’s effectiveness. This study lays the foundation for developing more adaptive decision-making frameworks in reservoir flood control and other related fields, being closer to the actual set of preferences in reservoir management and hence having the level of adoption by practitioners.

Despite the mentioned limitations of this study, we think it allows for making one step towards wider adoption of optimisation approaches to real-time reservoir flood control. The presented methodology is, of course, not aimed at replacing manual operation but rather gives instruments for reducing operators’ stress in critical situations and ultimately enhancing their ability to make better decisions.

We express our gratitude to Korea Water Resources Public Corporation (K-water) for their sponsorship of the first author and for sharing the data and information related to the Daecheong reservoir. The hydrological and operational data of the Daecheong reservoir is accessible to the public on K-water’s website (http://kwater.or.kr). We also thank the ICT Cooperative of Dutch Education and Research Institutions (SURF) for allowing us to utilise the Dutch national e-infrastructure with the support of the SURF Cooperative using grant no. EINF-6342.

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The authors declare there is no conflict.