Enhancing Reservoir Design by Integrating Resilience into the Modified Sequent Peak Algorithm (MSPA 2024)

Abstract

This study introduces the modified sequent peak algorithm (MSPA 2024), an advanced reservoir design framework that incorporates resilience as a key performance metric alongside traditional measures like reliability and vulnerability. By integrating resilience thresholds, MSPA 2024 addresses the complexities of water resource management under diverse hydrological conditions and demand scenarios. Comparative analyses reveal that MSPA 2024 surpasses traditional approaches, such as behavior analysis (BA) and earlier MSPA versions, particularly in maintaining higher resilience and sustainability at lower reliability levels. Although it requires greater storage capacity and experiences higher evaporation losses, MSPA 2024 proves effective in ensuring resilience under challenging conditions, making it especially suitable for long-term water management in drought-prone regions. The key findings highlight its performance across varied demand scenarios, emphasizing the importance of tailoring MSPA 2024 applications to specific hydrological contexts. While achieving 100% resilience is optimal, this study demonstrates the practicality of lower resilience thresholds (e.g., 75% and 50%), showcasing MSPA 2024’s adaptability to diverse operational needs. These results underscore MSPA 2024’s potential to enhance resilient and sustainable water systems, offering a vital tool for addressing increased water scarcity driven by climate change.

1. Introduction

Artificial reservoirs are fundamental to water resource management, serving as essential systems for meeting fluctuating demands under diverse hydrological conditions [1,2]. As climate change exacerbates variability and uncertainty in water availability, the need for advanced reservoir design methodologies has become increasingly urgent [3,4]. These methodologies are crucial for ensuring reservoirs can adapt to extreme weather events, prolonged droughts, and variable water inflows and demands. A resilient and robust reservoir design is vital for sustainable water resource management, supporting agricultural productivity, industrial operations, and domestic water supplies while maintaining socio-economic stability and environmental sustainability [5]. Addressing these challenges requires methodologies that integrate resilience into reservoir design, enabling long-term sustainability under uncertain future scenarios.

Reservoir design typically employs models based on mass balance equations, evaluated period by period under specified conditions [2,6]. These models are generally categorized into storage-based simulations and deficit-based approaches such as the sequent peak algorithm (SPA). While storage-based simulations incorporate storage-dependent processes like evaporation and seepage, they require an operating policy, even for single-reservoir systems [2]. Developing such policies during the design phase is often complex, particularly without heuristic methods such as standard operating policies [7].

Storage-based simulations commonly involve trial-and-error processes to achieve no-failure performance over historical records. Accounting for failures in these models can lead to oscillatory behavior, complicating efforts to achieve unique solutions. Such oscillations may cause unexpected fluctuations in mean and median storage levels, as documented in prior research [8]. In contrast, the SPA is procedurally simpler, directly determining the storage capacity required to meet a specified reliability over historical periods [9]. Despite its procedural simplicity, storage-based simulation remains widely used, especially for complex multi-reservoir systems [2,10].

Among reservoir design methods, the SPA is valued for its ability to determine storage capacity to achieve specific reliability levels. Introduced by Thomas and Burden, the SPA has evolved to address storage-dependent losses, such as evaporation, which were absent in its original formulation. The modified sequent peak algorithm (MSPA) represents a significant advancement, integrating hydrological processes like evaporation and rainfall into mass balance equations and refining storage estimates through nonlinear area–storage relationships [11]. However, further refinements are necessary to meet the growing demand for resilience in reservoir designs, particularly in the face of climate change and increasing hydrological variability. Incorporating resilience as a performance metric represents a notable improvement over traditional methods, enabling a more comprehensive evaluation of system sustainability [12,13,14].

Complementary to the SPA, behavior analysis (BA), also known as storage-based simulation, provides a versatile framework that integrates storage-dependent processes such as evaporation and release constraints [15]. BA facilitates the estimation of key performance metrics, including reliability, resilience, and vulnerability [16,17]. Its ability to simulate long-term reservoir dynamics makes it a valuable complement to the SPA, offering a broader understanding of reservoir behavior under varying conditions.

Evaluating reservoir performance is fundamental to effective water resource management. Metrics such as reliability, resilience, and vulnerability are essential for assessing system performance [18,19,20]. Reliability measures the probability of meeting water demands, resilience quantifies the ability to recover from failures, and vulnerability assesses the severity of unmet demands [14,17,21]. More recent advancements have introduced unified performance indices, such as figures of merit (FMs) and sustainability metrics, which integrate these measures into a single evaluation framework [12,22,23].

Each reservoir design method has inherent limitations [9]. The basic SPA focuses on achieving 100% reliability over historical records but does not explicitly address vulnerability or resilience. Modified versions incorporate reliability and vulnerability norms, adjusting for storage-dependent losses such as net evaporation (i.e., evaporation minus rainfall) over the reservoir surface [24]. In contrast, BA performs comprehensive water balance computations, accounting for inflows, outflows, net evaporation, and seepage. However, achieving specific time-based reliability with BA often requires trial and error, limiting its intrinsic control over vulnerability and resilience metrics. These limitations highlight the need for precise hydrological data and system characterization to achieve reliable and accurate results.

This study aims to advance reservoir design methodologies by enhancing the MSPA model with explicit resilience controls, addressing a critical gap in current practices. Traditional SPA and MSPA models primarily focus on reliability and vulnerability, often overlooking resilience—a key factor for long-term water system sustainability [12,25]. The proposed MSPA 2024 model integrates resilience norms into the reservoir design process, ensuring balanced performance across all key metrics. By embedding resilience, the MSPA 2024 model prevents consecutive failure periods and ensures a minimum recovery threshold, offering a robust and sustainable approach to reservoir design. This innovation enhances system resilience and supports rapid recovery from failures, making a significant contribution to water resource management in the face of climate variability and evolving water demands.

2. Materials and Methods

2.1. Case Study and Data

This study employs streamflow data from the Cidadelhe (R.E.) (08O/02H) river gauging station, located on the Côa River within the Douro River watershed. Positioned at 40.915° N and 7.101° W (WGS84), the station has a catchment area of 1685 km² (Figure 1) and an average annual flow volume of 466.18 Mm³. The analysis assumes the presence of an artificial reservoir at the gauging station site, with no restrictions on maximum storage capacity. Streamflow data were derived from daily records spanning a 51-year period, from October 1960 to September 2011, sourced from the Sistema Nacional de Informação de Recursos Hídricos (SNIRH) database, accessed on 18 Marrch 2024 at https://snirh.apambiente.pt/. Managed by the Portuguese Environment Agency, SNIRH provides high-quality hydrological and meteorological data extensively utilized in water resources research and engineering across Portugal. The data follow the hydrological year, which in Portugal extends from October 1st to September 30th.

To align with the study’s objectives, daily streamflow data obtained from the SNIRH were aggregated into monthly volumes. This approach facilitates the evaluation of performance metrics such as time-based reliability, vulnerability, and resilience, which are typically assessed on monthly or annual timescales, particularly during the initial phases of reservoir design [15]. Aggregating daily flows into monthly volumes enables the analysis to capture patterns and trends critical for evaluating long-term reservoir performance.

The annual reservoir demand was set at 25%, 50%, and 75% of the mean annual flow at the Cidadelhe gauging station. Monthly demand patterns reflected seasonal variability, peaking during the warmer months (Figure 2). Monthly allocations ranged from 5% to 15% of the annual demand, accounting for cumulative irrigation and urban supply needs. Figure 2 illustrates the distribution of annual demand across different months on the primary vertical axis (dimensionless) and the monthly demand volumes for the 75% demand scenario on the secondary vertical axis. While this scenario formed the basis for primary results, the 25% and 50% demand scenarios are addressed in the discussion section.

In the absence of precipitation data specific to the hypothetical reservoir at Cidadelhe, monthly precipitation series were obtained from the nearest rain gauge, Castelo Melhor (07O05UG) (Figure 1 and Figure 3), which shares the same observation period as the streamflow data (October 1960 to September 2011). These precipitation records were integrated into the reservoir analysis.

Monthly reservoir evaporation was estimated using the Thornthwaite formula, which calculates potential evapotranspiration based on temperature data alone [26]. This method is particularly suitable due to its minimal data requirements and proven accuracy in approximating reservoir evaporation [27,28]. As temperature data specific to the Cidadelhe site were unavailable, climate-normal temperature data (1981–2010) from the Guarda meteorological station (10N/03), located at 40.533° N and 7.267° W (WGS84), were used (Figure 1). The estimated mean monthly evaporation rates were replicated for each hydrological year under study.

The mean monthly net evaporation, defined as the difference between mean monthly evaporation and precipitation, is depicted in Figure 3. Mean monthly evaporation estimates were based on climate-normal temperatures from the Guarda station (10N/03), while mean monthly precipitation data were sourced from records at Castelo Melhor (07O05UG) over the analysis period. Figure 3 illustrates that annual evaporation consistently exceeds precipitation, underscoring the importance of incorporating net evaporation into reservoir design to avoid underestimating storage capacity [11].

To include net evaporation in the mass balance equation, an empirical relationship between the flooded area and the stored water volume was established for the hypothetical reservoir at the Cidadelhe station (08O/02H). This relationship was developed using topographical data from a Geographic Information System (GIS) and modeled with a second-degree polynomial (Figure 4). The polynomial achieved a high coefficient of determination (R² = 0.9952), ensuring accurate representation. The analysis assumes zero dead storage volume, with the origin point (0,0) in Figure 4 corresponding to the river gauging station’s elevation.

2.2. Performance Measures and Methods for Reservoir Design

2.2.1. Introduction

Incorporating performance indices into reservoir design is critical to ensuring efficient operation throughout the reservoir’s lifespan, particularly under varying demands and hydrological conditions [16,29]. The evaluation of reservoir performance typically hinges on specific criteria, often identifying instances of inadequate operation during periods of low inflows, known as failures. Failures indicate the reservoir system’s inability to meet target demands. Performance metrics are pivotal in guiding decisions related to reservoir capacities, system configurations, operational strategies, and objectives. The subsequent subsections provide a concise overview of these performance norms to establish a foundation before exploring reservoir design methodologies.

2.2.2. Reservoir Performance Measures

Planning reservoir systems necessitates evaluating performance under diverse demand and hydrological conditions [30,31]. Performance criteria, rooted in the identification of failures during low inflows, inform critical decisions on capacity, configuration, and operational policies [18,32]. Central to performance assessment is reliability, which measures the likelihood of the system meeting its target demand. Additional metrics, such as vulnerability and resilience, further enhance the evaluation of reservoir performance. Vulnerability assesses the consequences of system failures, while resilience quantifies the system’s ability to recover after such failures [2,33]. Together, these measures form a comprehensive framework for evaluating and optimizing reservoir system performance.

The most widely used performance metric is time-based reliability, R_t, which represents the probability of a reservoir meeting its demand within a specified time interval (e.g., a month or a year). It is calculated as follows [34]:

$$R_t={\displaystyle \frac{N_S}{N}};0\le R_t\le 1$$

(1)

where N_S is the number of successful time intervals in which the demand was met, and N is the total number of time intervals during the simulation.

Another measure of reservoir performance is volumetric reliability, R_v, defined as the proportion of actual water supplied to total water demanded over a given period [18]:

$$R_v=1-{\displaystyle \frac{\sum _{j\in f}(D_j-D_j^{\prime })}{\sum _{j\in N}{D}_j}}\mathrm{and}0<R_v\le 1$$

(2)

In this equation, f (Figure 5) represents the number of failure periods (equal to N−N_S), $D_j^{\prime }$ denotes the actual supply during the jth failure period, and D_j signifies the target demand for the same period. Both time-based and volumetric reliabilities are typically employed together to ensure a thorough evaluation of reservoir system performance.

Resilience,$\phi$, reflects the speed at which a system recovers after failure. It is defined as the probability of immediate recovery following failure [17,18,35]:

$$\phi ={\displaystyle \frac{f_s}{f}};0<\phi \le 1$$

(3)

where f_s is the number of sequences of continuous failure periods (Figure 5). A higher resilience indicates a faster recovery, making it preferable for reservoir systems to swiftly return to a satisfactory state.

Vulnerability, η′, quantifies the severity of failures by measuring unmet demand. It is calculated as the average of the maximum shortfalls in continuous failure periods [18]:

$${\eta }^{\prime }={\displaystyle \frac{\sum _{k=1}^{f_s}\mathrm{m}\mathrm{a}\mathrm{x}\left({sh}_k\right)}{f_s}}\mathrm{and}0\le {\eta }^{\prime }$$

(4)

where max(sh_k) denotes the maximum shortfall during each of the kth continuous failure sequence (Figure 5), and f_s has the meaning previously presented.

For standardization, vulnerability, $\eta$, is often expressed as a dimensionless ratio relative to the target demand during failure periods:

$$\eta ={\displaystyle \frac{{\eta }^{\prime }}{D_j}}$$

(5)

This ensures that the vulnerability falls within the range $0\le \eta \le 1$, akin to the reliability criteria.

There has been a shift towards developing unified performance indices to address the complexity of managing multiple performance metrics in water resource systems [7]. Moy et al. [36] explored the interplay between reliability, resilience, and vulnerability, highlighting challenges in managing these metrics individually. Loucks [23] introduced the concept of unified indices, termed “figures of merit” (FMs), to simplify decision-making. Similarly, Simonovic [37] proposed an FM that integrates reliability, resilience, and vulnerability into a single metric, offering a holistic view of system performance.

Zongxue et al. [38] presented the drought risk index (DRI), $\kappa$, which combines time-based reliability, resilience, and vulnerability into the following equation:

$$\kappa ={\epsilon }_1\left(1-R_t\right)+{\epsilon }_2\left(1-\phi \right)+{\epsilon }_3\eta \mathrm{and}0\le \kappa \le 1$$

(6)

where R_t, $\phi$, and η are as previously defined, and ε₁, ε₂, and ε₃ are weights reflecting the relative importance of each metric. By default, these weights are assigned a value of 0.333.

Sandoval-Solis et al. [12] further advanced this concept by introducing a sustainability metric, λ, for reservoir design and operation, which integrates reliability, resilience, and dimensionless vulnerability:

$$\lambda ={{(R}_t\phi \left(1-\eta \right))}^{(1/3)}\mathrm{and}0\le \lambda \le 1$$

(7)

This progression toward unified indices underscores the need for a holistic perspective in water resource management, enabling more informed and effective decision-making [39,40].

2.2.3. Reservoir Design Methods

As previously mentioned, reservoir design models based on the principle of mass balance include storage-based simulation or behavior analysis (BA) and the sequent peak algorithm (SPA). The basic SPA [41] is often regarded as the automated counterpart to Rippl’s graphical approach [42], adapted for computerized execution. However, Lele [43] introduced a simplified formulation that has gained recognition due to its enhanced compatibility with computerized execution, as noted by McMahon and Mein [15].

In its fundamental form, SPA estimates the storage capacity required to maintain uninterrupted reservoir operation, ensuring 100% reliability over an N-year period based on historical data. The equations for the basic SPA are expressed as follows [11]:

$$K_{t+1}=\max \left\{0.0,\left(K_t+D_t+{EV}_t-Q_t\right)\right\}\mathrm{f}\mathrm{o}\mathrm{r}t=1,2,\dots ,2N$$

(8)

$$S=\max \left(K_t\right)\mathrm{f}\mathrm{o}\mathrm{r}t=1,2,\dots ,2N$$

(9)

where K_t and K_t+1 represent the volumetric sequential storage deficits at the beginning of periods t and t+1, respectively; Q_t denotes the inflow volume during period t; D_t represents the demand volume during the same period; and EV_t signifies the net evaporation volume during period t, although it is assumed to be zero in the basic SPA. S denotes the estimated reservoir storage capacity, and N represents the number of time intervals in the dataset. The algorithm assumes an initially full reservoir, setting K₀ to zero. Equation (8) is typically applied twice to the data record (t = 1, 2, …, 2N).

The modified sequent peak algorithm (MSPA) extends the basic SPA by addressing storage-dependent losses in reservoir capacity estimation. Natural reservoir systems are subject to evaporation and rainfall, necessitating their inclusion to avoid misestimating storage requirements [9].

Direct rainfall contributes to inflow to the reservoir, while evaporation constitutes an outflow. The net evaporation, e_t, is the difference between evaporation and rainfall. A positive e_t indicates evaporation exceeds rainfall, causing a net outflow, while a negative e_t signifies net inflow. Incorporating net evaporation in volumetric terms requires knowledge of the reservoir’s surface area, which varies nonlinearly with storage [44]. This study adopts the following second-degree polynomial to describe the area–storage relationship (Figure 4):

$$A_t=-8.158\times {10}^{-6}{Z_t}^2+0.02970\times Z_t$$

(10)

where A_t represents the reservoir’s surface area at the start of period t (km²), and Z_t is the stored water volume (Mm³). Subsequently, evaporation volume is calculated as:

$${EV}_t=e_t\left(A_v\right)$$

(11)

where e_t represents net evaporation depth and A_v which is the average surface area during period t to t+1 is computed as:

$$A_v=0.5(A_t+A_{t+1})$$

(12)

Equation (11) is incorporated into the SPA using Equations (8) and (9). Initially, the basic SPA estimates sequential deficits (K_t; t = 1… 2N) and approximate storage capacity (S). Storage states (Z_t) are then derived by subtracting the sequential deficits (K_t) from the estimated capacity S. These Z_t values are used to compute evaporation volumes via Equation (11). Iterative refinement proceeds until a stable storage capacity estimate is achieved, typically within three to four iterations. The algorithm utilizes a convergence criterion (Equation (13)), which compares successive estimates of storage capacity for a same time step (S′ and S) [9]. If the absolute difference between them falls below a predefined threshold, the iteration terminates, and S is considered the final estimate of the reservoir storage capacity:

$$\mathrm{A}\mathrm{b}\mathrm{s}\left[\left(S^{\prime }-S\right)/S\right]\le 0.0001$$

(13)

One notable advantage of the MSPA is its compatibility with nonlinear reservoir area–storage relationships (e.g., Equation (10)). This adaptability arises from the independent calculation of evaporation estimates, enabling the use of more accurate area–storage relationships instead of relying on linear approximations.

The MSPA integrates reliability and vulnerability indicators to design reservoir systems that can tolerate occasional failures while maintaining desired reliability levels. Traditional reservoir design approaches aim for an uninterrupted supply over the entire record length, neglecting scenarios where partial supply during limited failure periods is acceptable.

To incorporate reliability and vulnerability indicators into SPA, the number of failure periods, f, is first determined based on the target time-based reliability, R_t, as follows [11]:

$$f=\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\left\{N-N{R}_t\right\}$$

(14)

where round denotes rounding to the nearest integer, and N represents the total number of periods. This approach enables the identification of storage capacity accounting for both time-based reliability, R_t, and vulnerability, $\eta$.

The iteration process over f cycles involves sequentially identifying failure periods and adjusting their releases by a factor $(1-\eta$) accordingly. Initially, the critical period for full supply is identified using the basic SPA. Then, in each iteration cycle, the releases are iteratively adjusted to ensure that no single period is considered more than once for reducing the release. This process guarantees that during any iteration, f independent periods have their supplies reduced [9].

While the original MSPA allows for the regulation of time-based reliability and vulnerability indices, it does not inherently control the resilience index, a critical performance metric. Additionally, if a failure period overlaps with a previous one, the algorithm shifts it one period backward to find an unused period. This often results in consecutive failure periods, leading to a minimal resilience index.

To address this limitation, this study introduces an enhanced algorithm, MSPA 2024, which includes control over a minimum resilience threshold. The algorithm ensures that consecutive failure periods are avoided, achieving a resilience level of 100%. For instance, if a failure period coincides with a previous one, the new algorithm shifts it two periods backward, ensuring that no two failure periods are consecutive. To set a minimum threshold of resilience for MSPA, with the number of failure periods, f, calculated using Equation (14), the algorithm follows a structured process with the following eight stages:

    I.

Set the minimum resilience threshold, φ_m.

   II.

Assign K_i based on φ_m:

• If φ_m ≤ 0.10, then K_i = 1.
• Else if φ_m = 1, then K_i = f +1.
• Otherwise, K_i = round $\left({\displaystyle \frac{2}{1-{\phi }_m}}\right).$

  III.

Initialize calculated resilience, φ: set φ = 0.

  IV.

Check the value of calculated resilience, φ:

• If φ ≥ φ_m, then stop and finalize the process.
• Otherwise, proceed to V.

    V.

Set D_t′ = D_t for all t = 1 to N and initialize I_r to 0 (Dt′ represents the release volume during period t, while I_r denotes the failure period counter).

   VI.

Increment I_r by 1. Execute the MSPA, incorporating net evaporation losses, to estimate the storage capacity for the current release configuration.

VII.

Check the reset condition:

• If I_r equals (f +1) (i.e., if drought yields have been reset for f periods), then:

  −

  Increment K_i by 1;

  −

  Estimate resilience φ (Equation (3));

  −

  Return to IV.

• Otherwise, proceed to VIII.

VIII.

Find the current failure period, FP:

• Identify FP as the last period of the critical period (i.e., the time when a full reservoir is depleted without refilling during the intervening period).
• If FP matches any previous iterations, adjust it as follows:

  −

  If the remainder of I_r divided by K_i, is zero, set FP = FP − 1; otherwise, set FP = FP −2;

  −

  Verify whether this adjusted period has been used previously. If not, use it as the new period for resetting the release; otherwise, repeat this stage until an unused period is found.

• Update D_t′ = (1−η) D_t for all periods corresponding to identified failure periods.
• Return to VI.

Thus, this enhanced algorithm enables more refined control over the minimum threshold of the resilience index within the MSPA, thereby improving the overall performance assessment of the reservoir.

Behavior analysis (BA), also known as storage-based simulation, provides greater versatility compared to the sequent peak algorithm (SPA). BA effectively accounts for storage-dependent processes such as reservoir evaporation and release restrictions in a more straightforward and comprehensive manner. Furthermore, behavior analysis can accommodate a wide range of reliability definitions and is capable of handling complex reservoir system configurations, making it a more robust approach for system performance evaluation [15].

At its core, behavior analysis operates based on the water balance principle, which involves tracking storage inputs and outputs over time. This is expressed as follows [45]:

$$Z_{t+1}=Z_t+Q_t-D_t^{\prime }-e_t{A}_v-L_t$$

(15)

subject to the constraint

$$0\le Z_{t+1}\le S$$

(16)

where Z_t and, Z_t+1 represent storage volumes at the beginning of periods t and t+1, respectively. L_t denotes seepage during period t, A_v is calculated using Equation (12), and other variables are as previously described. The operations are iteratively conducted until convergence, enabling the estimation of reliability, vulnerability, and other performance metrics.

To analyze a single reservoir system with a known or assumed capacity, Equation (15) is utilized to simulate the state of the storage contents. This involves employing an uninterrupted discrete streamflow record along with known or assumed demand levels and net evaporation estimates.

In summary, behavior analysis provides a holistic approach to assessing reservoir systems by accounting for factors such as inflow, outflow, evaporation, and seepage. Through iterative simulations of storage dynamics, it facilitates the estimation of reliability and other critical performance metrics essential for efficient water resource management. However, it is important to note that while performance indices can be derived from the analysis, adjustments beyond time-based reliability are generally not feasible.

3. Results

3.1. Constraining Parameters for the Applied Reservoir Analysis Methods

In implementing the BA methodology, time-based reliability was varied from 80% to 100% in increments of 2.5%. Key reservoir characteristics were calculated based on this range. In contrast, the MSPA method imposed constraints not only on time-based reliability, but also introduced a vulnerability threshold of 30%. This threshold aligns with Fiering’s findings [46], which suggest that shortfalls exceeding approximately 25% can cause severe adverse effects on water-dependent populations.

The MSPA 2024 methodology further refined these constraints, addressing both time-based reliability and vulnerability while capping resilience at 100%. This refinement was prompted by observations of significantly low resilience in both the BA and original MSPA methods for reliabilities below 100%. By limiting resilience to this threshold, the MSPA 2024 approach reduces consecutive failure periods, thereby enhancing the system’s overall resilience and sustainability.

3.2. Comparison of the Results for the Different Reservoir Analysis Methods

The results from the BA, MSPA, and MSPA 2024 methods, applied to the hypothetical Cidadelhe reservoir with a mean annual demand of 75% of the mean annual flow (MAF) (Figure 2), are summarized in Figure 6. These results are illustrated as curves sharing a common horizontal axis that represents time-based reliability. The vertical axis varies according to the performance metric or reservoir characteristic (e.g., dimensionless storage capacity and mean annual evaporation loss, MAEL). This approach allows for a direct comparison of outcomes across the different methods at identical time-based reliability levels. Although the results are depicted as continuous curves, they were derived exclusively from specific time-based reliability values ranging from 80% to 100%, with increments of 2.5%. The obtained discrete data points were connected by straight lines to enhance visualization.

At 100% time-based reliability (indicating that demand is consistently met), all methods yield similar results, with dimensionless storage capacity and MAEL, normalized by MAF, measuring 2.8 and 0.03, respectively (Figure 6). At this reliability level, volumetric reliability, resilience, and sustainability are maximized at 100%, while vulnerability and the drought risk index are minimized to 0%. For reliabilities below 100%, the key reservoir characteristics are detailed as follows:

Dimensionless storage capacity—ratio of storage capacity to mean annual flow. Figure 6a illustrates distinct trends across methods as the reliability drops below 100%. The BA method exhibits a sharp reduction in dimensionless storage capacity, reaching a value of 0.5 at 80% reliability. In contrast, the MSPA method shows a more gradual decrease, stabilizing at 1.5 by 82.5% reliability. The MSPA 2024 method demonstrates the slowest reduction, decreasing to 2.2 by 92.5% reliability and maintaining this value as the reliability declines to 80%. Overall, while the BA method shows a significant reduction in storage capacity with decreasing reliability, the MSPA and MSPA 2024 methods display more gradual reductions, with the MSPA 2024 method consistently maintaining the highest capacity throughout the range.

Dimensionless mean annual evaporation loss (MAEL)—ratio of mean annual evaporation loss to mean annual flow. As depicted in Figure 6b, MAEL varies significantly across methods. The BA method demonstrates a sharp decrease in MAEL from 0.030 to 0.007 as reliability falls from 100% to 80%. The MSPA 2024 method exhibits minimal variation, with MAEL declining slightly from 0.030 to 0.025. The MSPA method shows a moderate reduction, decreasing from 0.030 to 0.018. These results suggest that while the BA method minimizes evaporation losses at lower reliabilities, the MSPA 2024 method, with its higher storage capacity, incurs relatively greater evaporation losses.

Volumetric reliability (RV)—ratio of actual water supplied to total water demanded (Equation (2)). Figure 6c illustrates that the BA method experiences a linear decline in RV, reaching 82% at 80% time-based reliability. The MSPA and MSPA 2024 methods exhibit more gradual declines, maintaining higher RV values of 94% at 80% reliability. This indicates a steeper reduction in RV for the BA method compared to MSPA and MSPA 2024 approaches.

Resilience—the system’s ability to recover after failure (Equation (3)). As shown in Figure 6d, the MSPA 2024 method maintains consistent resilience at 100%, even as reliability decreases to 80%. In contrast, the BA method exhibits resilience values ranging from 12% to 20%, while the MSPA method has the lowest resilience, between 2% and 8%. This underscores the superior resilience of the MSPA 2024 method compared to its counterparts.

Sustainability—an integrated metric of time-based reliability, vulnerability, and resilience (Equation (7)). Figure 6e reveals that the MSPA 2024 method maintains a higher sustainability index, ranging from 82% to 88% as the time-based reliability declines from 97.5% to 80%. In comparison, the MSPA method fluctuates between 25% and 35%, and the BA method ranges between 20% and 30%. These findings confirm the consistently higher sustainability achieved by the MSPA 2024 method relative to the other two methods. Notably, the sustainability values for the MSPA and BA methods are generally similar, with the MSPA method exhibiting slightly higher indices than the BA method.

Vulnerability—unmet demand ratio (Equation (5)). Figure 6f highlights that, for reliabilities below 100%, the BA method exhibits significantly higher vulnerability, ranging from 80% to 100%. Both the MSPA and MSPA 2024 methods maintain vulnerability around 30%, effectively limiting shortfalls under sub-100% reliability conditions.

Drought Risk Index (DRI)—a measure of drought severity (Equation (6)). As shown in Figure 6g, the DRI increases linearly with declining reliability for both the MSPA and MSPA 2024 methods. The MSPA 2024 method’s DRI rises from 10% to 17% as the reliability decreases from 97.5% to 80%, while the MSPA method’s DRI increases from 40% to 50%. The BA method exhibits the highest DRI values, fluctuating between 57% and 65%. These results emphasize the greater efficacy of the MSPA 2024 method in mitigating drought risk compared to both BA and MSPA methods.

4. Discussion

The comparative analysis of behavior analysis (BA), the modified sequent peak algorithm (MSPA), and the newly proposed MSPA 2024 method for the hypothetical Cidadelhe reservoir reveals notable performance trends across key metrics—resilience, sustainability, and vulnerability—under varying reliability levels. The MSPA 2024 method consistently outperforms the BA and MSPA methods in terms of resilience, sustainability, and reduced vulnerability to drought risks. However, these benefits come with increased storage capacity requirements, resulting in a larger reservoir and an expanded water surface area. Consequently, higher evaporation losses are observed for MSPA 2024 compared to the BA and MSPA methods.

The MSPA 2024 method exhibits superior resilience, maintaining 100% resilience even at lower reliability levels, reflecting its robust capacity to meet water demands under fluctuating supply conditions. Its sustainability indices are significantly higher, underscoring its ability to ensure long-term system performance under varying conditions. In contrast, the BA method exhibits lower sustainability and greater vulnerability at reliability levels below 100%. These results align with earlier findings, such as those by McMahon et al. [47], which reported that reservoirs modeled using BA exhibited low sustainability indices and high vulnerability under sub-100% reliability conditions. While the MSPA method shows slightly higher sustainability than BA, it has the lowest resilience among the three methods, consistent with prior research [48] reporting resilience values below 0.4 for reservoirs with demands equating to 75% of the mean annual flow under non-100% reliability conditions.

In summary, while MSPA 2024 offers significant advantages in resilience and sustainability, its higher storage requirements and the associated increase in evaporation losses require careful consideration. These trade-offs are particularly relevant when evaluating the method for specific reservoir systems. Nevertheless, MSPA 2024’s superior performance under varying reliability levels highlights its potential as a more effective approach for long-term water resource management, particularly in drought-prone regions.

Two significant issues related to the results presented in Figure 6 warrant attention. First, the analysis was conducted exclusively for a demand of 75% of the MAF, without assessing the effects of different demand scenarios. Second, the resilience threshold for MSPA 2024 was set to 100%, which may be overly restrictive. These issues prompted a re-evaluation of storage capacity, resilience, and sustainability metrics, as shown in Figure 7a–c. Similar to Figure 6, the results in Figure 7 represent discrete points obtained from time-based reliability values ranging from 80% to 100% in increments of 2.5%, connected by straight lines.

Regarding the first issue, for very high demands, such as the 75% MAF scenario—Figure 7(a3,b3,c3)—the reservoir typically operates as an over-year system [49]. However, different behaviors may arise under lower-demand scenarios. To generalize the findings, the analysis was extended to include lower- and moderate-demand scenarios at 25% and 50% of MAF, respectively, as shown in Figure 7. This extension provides a more comprehensive assessment of reservoir performance across varying demand conditions.

The second issue pertains to the feasibility of achieving 100% resilience with MSPA 2024, which may not be realistic due to constraints such as river valley geometry or the economic feasibility of the reservoir [2]. To address this, the MSPA 2024 method was re-evaluated using lower resilience thresholds of 75% and 50%, referred to as MSPA 2024-B and MSPA 2024-C, respectively (Figure 7). At a time-based reliability level of 100%, and demands of 25% and 50% of MAF, all methods exhibit dimensionless storage capacities of 0.5 and 1.4, respectively (Figure 7(a1,a2)). These values are notably lower than the dimensionless storage capacity for a demand of 75% of MAF, which is 2.8 (Figure 7(a3)). While the MSPA 2024 variants produce similar storage capacities at high-reliability levels (e.g., reliabilities ≥ 95% for a demand of 75% of MAF), storage capacities diverge as reliability decreases. Among the MSPA 2024 variants, the original version designed for 100% resilience requires the highest storage capacity, whereas MSPA 2024-C, with a 50% resilience threshold, requires the lowest capacity for comparable reliabilities and demands.

Achieving lower resilience thresholds in reservoir design, such as 75% or 50%, impacts socio-economic outcomes, warranting further investigation. While these thresholds reduce storage requirements—Figure 7a—and associated costs, they increase the risk of water shortages during prolonged droughts, potentially affecting agricultural productivity, industrial operations, and community livelihoods [50]. Such trade-offs may be acceptable in regions with lower-demand variability or sufficient alternative water sources. However, in drought-prone areas, reduced resilience could exacerbate economic losses and social inequalities [51]. Balancing resilience with affordability and regional needs is thus critical to ensure equitable and sustainable water resource management.

Resilience thresholds play a pivotal role in decision-making for stakeholders, particularly in water-scarce regions where competing demands often arise. By quantifying a water system’s ability to recover from failures, these thresholds facilitate trade-offs between reliability, storage requirements, and socio-economic outcomes. Previous studies by Hashimoto et al. [18] emphasized resilience’s role in equitable water allocation during droughts, while Sinha et al. [50] highlighted its importance in mitigating economic disparities. Similarly, Simonovic and Arunkumar [17] demonstrated that dynamic resilience metrics optimize operational strategies. Incorporating resilience thresholds into frameworks like MSPA 2024 ensures alignment with long-term sustainability and adaptability.

Figure 7(b1,b2,b3) show that while MSPA 2024 variants effectively stabilize resilience across demand scenarios, the specified lower thresholds were not always achieved. This discrepancy arises because resilience is calculated as the ratio of two integer values: the number of continuous failure periods (f_s) to the total failure periods (f) (Equation (3)). While exact thresholds were not always met, the calculated resilience consistently meets or exceeds the defined thresholds, with MSPA 2024 approximating them closely. In contrast, BA and MSPA exhibit much lower resilience for comparable demands and reliabilities, particularly under higher-demand scenarios. These findings align with earlier studies [48,49], which concluded that reservoirs modeled with higher demands exhibit lower resilience when modeled using BA and MSPA methods.

MSPA 2024’s dynamic ability to address hydrological variations, including extreme rainfall and prolonged droughts, distinguishes it as a robust tool for climate-resilient reservoir design. Its iterative algorithm adjusts reservoir storage strategies in response to changing inflow patterns and evaporation losses, as evidenced in studies on climate-resilient reservoir designs [3]. Further, its resilience-focused thresholds mitigate the risks of system failure during hydrological extremes, aligning with recommendations for adaptive water management [17]. This adaptability ensures that MSPA 2024 maintains reliability and sustainability, even under intensified climate variability [13].

Moreover, MSPA 2024 can integrate advanced probabilistic modeling techniques to address uncertainties in demand projections and climate variability. By employing Monte Carlo simulations and stochastic optimization, the algorithm generates a spectrum of possible outcomes, enhancing robustness. Incorporating ensemble climate models, as suggested by Zhang et al. [52], ensures that climate variability is accounted for across multiple scenarios.

As shown in Figure 7(c1,c2,c3), the MSPA 2024 variants achieve higher sustainability levels under varying demand scenarios and reliability conditions compared to BA and MSPA methods. The BA and MSPA approaches do not effectively control sustainability levels, which are generally higher in lower-demand scenarios for comparable time-based reliabilities.

However, while MSPA 2024 significantly enhances resilience and sustainability, these benefits come with trade-offs. The method requires larger storage capacities, which lead to expanded water surface areas and, consequently, increased evaporation losses—a concern particularly acute in arid and semi-arid regions. These trade-offs underscore the importance of carefully balancing resilience and sustainability gains against potential inefficiencies in water conservation. Studies by Garcia et al. [24] and Srdjevic and Srdjevic [40] emphasize the considerable economic and environmental costs associated with increased evaporation losses in large reservoirs. Additionally, Simonovic and Arunkumar [17] highlight dynamic resilience metrics as a promising tool to optimize these trade-offs. Addressing these complexities requires further investigation to refine the MSPA 2024 methodology for diverse hydrological settings.

To mitigate the increased storage requirements and evaporation losses associated with MSPA 2024 while maintaining resilience, several strategies can be implemented, as follows: (i) the installation of surface-covering systems, such as floating solar panels or modular shade structures, to reduce evaporation and simultaneously enable energy co-generation [13]; (ii) the enhancement of demand management strategies, for example, by incorporating seasonal adjustments and allocation prioritization, to optimize reservoir operations and balance storage utilization with resilience [2]; (iii) the development of vegetative buffer zones around reservoirs to reduce wind speed over the water surface, thereby minimizing evaporation [24]; and (iv) the refinement of resilience thresholds to more practical levels, such as 75%, to strike a balance between system performance and reduced storage demands [17]. These strategies offer a pathway to enhance the sustainability and adaptability of MSPA 2024, particularly in responding to challenges posed by climate variability and water resource constraints.

Finally, MSPA 2024’s performance can be further enhanced by integrating predictive models such as machine learning (ML) and climate simulations. Machine learning techniques, particularly ensemble methods and neural networks, can refine forecasting accuracy by extracting intricate patterns from large datasets [53]. Similarly, climate models can simulate key environmental variables, enabling the prediction of long-term trends and helping water managers anticipate potential future scenarios [54]. Combining MSPA 2024 with predictive models creates a hybrid framework for dynamic adaptation to real-world complexities, improving resilience and response strategies [55]. These synergies highlight the transformative potential of MSPA 2024 in advancing applied forecasting and decision-making within water resource management frameworks.

5. Conclusions

This study introduces the modified sequent peak algorithm (MSPA 2024), an enhanced reservoir design methodology that incorporates resilience as a central performance metric alongside traditional measures such as reliability and vulnerability. By integrating resilience thresholds, MSPA 2024 offers a more comprehensive and robust approach to water resource management under diverse hydrological conditions and demand scenarios. Comparative analysis with traditional approaches, including behavior analysis (BA) and the earlier MSPA model, reveals that MSPA 2024 consistently outperforms both methods in terms of resilience and sustainability, particularly under lower reliability conditions.

However, these improvements come with trade-offs. MSPA 2024 demands increased storage capacity, resulting in higher evaporation losses compared to other methods. Despite these drawbacks, the method’s capacity to maintain system resilience under challenging conditions establishes it as a valuable tool for long-term water resource management, especially in drought-prone regions.

Two key issues were identified during the analysis. First, reservoir behavior under reduced demand scenarios (25% and 50% of mean annual flow) shows significant variability in system performance. This underscores the need to contextualize MSPA 2024’s application based on specific hydrological and demand patterns. Second, while achieving 100% resilience is theoretically ideal, it may not always be practical or necessary. Evaluating MSPA 2024 with lower resilience thresholds (75% and 50%) demonstrates its adaptability to varying operational requirements.

The findings of this research underscore the need for further investigation into the application of the MSPA 2024 model in different geographical, hydrological, and operational settings. Future work should focus on the following areas:

• Application across diverse hydrological systems: Expanding the analysis to include a wider variety of reservoir systems, particularly those with lower or fluctuating demand scenarios, will provide a more comprehensive understanding of the model’s versatility. This includes evaluating MSPA 2024 in smaller, seasonal reservoirs and multi-reservoir systems to assess its adaptability and performance across different scales.
• Incorporating uncertainty in demand projections and climate variability: Future research should focus on developing a hybrid framework that integrates MSPA 2024 with advanced uncertainty quantification techniques. For demand projections, methods such as Bayesian networks and Monte Carlo simulations can improve adaptability to rapidly changing demand conditions. For climate variability, coupling MSPA 2024 with high-resolution ensemble climate models can provide deeper insights into the regional impacts of extreme weather events, improving adaptability to rapidly changing environmental conditions.
• Refining resilience thresholds: While achieving 100% resilience is desirable, it may not always be feasible due to geographic, hydrological, and economic constraints. Future research should focus on identifying optimal resilience thresholds that balance resilience, storage requirements, and socio-economic impacts. Exploring the implications of lower resilience thresholds (e.g., 75% and 50%) is particularly critical, as these levels could reduce storage costs but may increase vulnerability to water shortages, especially in drought-prone regions. Such studies should evaluate trade-offs between system performance and affordability, aiming to guide practical reservoir designs tailored to specific socio-economic and environmental contexts. Furthermore, investigating how resilience thresholds influence decision-making will enhance understanding of stakeholder priorities, enabling strategies that align equitable water allocation, agricultural productivity, and environmental preservation in water-scarce regions with competing demands.
• Economic and environmental impacts: The higher storage capacities and resulting evaporation losses associated with MSPA 2024 pose significant economic and environmental challenges. While accounting for these design features may enhance resilience and sustainability, they also increase costs related to infrastructure, land use, and reservoir maintenance, and may exacerbate water loss in arid regions. Future research should focus on quantifying these trade-offs to balance resilience, cost, and environmental sustainability. Furthermore, investigating optimal resilience thresholds and assessing their socio-economic implications could inform cost-effective and context-specific reservoir designs. These efforts will enhance the applicability of MSPA 2024 across diverse hydrological and climatic scenarios.
• Integration with multi-criteria decision analysis (MCDA): Incorporating the MSPA 2024 model into a broader decision-making framework, such as multi-criteria decision analysis (MCDA), could enhance its utility in complex water management scenarios. By evaluating trade-offs between competing objectives, such as water supply, ecological preservation, and economic costs, this integration would provide decision-makers with a holistic tool for sustainable water resource management.
• Integration with predictive models: Integrating MSPA 2024 with predictive models, including machine learning algorithms and climate simulations, can significantly improve its forecasting accuracy and adaptability. These models can extract complex patterns from large datasets and simulate long-term environmental trends, enhancing MSPA 2024’s ability to address real-time uncertainties and dynamic environmental changes.

The continued refinement and application of MSPA 2024, with a focus on addressing these key research areas, will contribute significantly to resilient and sustainable water resource management systems. This is especially critical in regions facing increasing water scarcity and climate-related challenges. By balancing technical advancements with economic feasibility and environmental stewardship, MSPA 2024 has the potential to become a cornerstone methodology in future water resource planning and management.